Publications

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 converters (WECs) are designed to produce useful work from ocean waves. This useful work can take the form of electrical power or even pressurized water for, e.g., desalination. This report details the findings from a wave tank test focused on that production of useful work. To that end, the experimental system and test were specifically designed to validate models for power transmission throughout the WEC system. Additionally, the validity of co-design informed changes to the power take-off (PTO) were assessed and shown to provide the expected improvements in system performance.

Improved power take-off (PTO) controller design for wave energy converters is considered a critical component for reducing the cost of energy production. However, the device and control design process often remains sequential, with the space of possible final designs largely reduced before the controller has been considered. Control co-design, whereby the device and control design are considered concurrently, has resulted in improved designs in many industries, but remains rare in the wave energy community. In this paper we demonstrate the use of a new open-source code, WecOptTool, for control co-design of wave energy converters, with the aim to make the co-design approach more accessible and accelerate its adoption. Additionally, we highlight the importance of designing a wave energy converter to maximize electrical power, rather than mechanical power, and demonstrate the co-design process while modeling the PTO's components (i.e., drive-train and generator, and their dynamics). We also consider the design and optimization of causal fixed-structure controllers. The demonstration presented here considers the PTO design problem and finds the optimal PTO drive-train that maximizes annual electrical power production. The results show a 22% improvement in the optimal controller and drive-train co-design over the optimal controller for the nominal, as built, device design.

The “Pioneer WEC” project is targeted at developing a wave energy generator for the Coastal Surface Mooring (CSM) system within the Ocean Observatories Initiative (OOI) Pioneer Array. The CSM utilizes solar photovoltaic and wind generation systems, along with rechargeable batteries, to power multiple sensors on the buoy and along the mooring line. This approach provides continuous power for essential controller functions and a subset of instruments, and meets the full power demand roughly 70% of the time. Sandia has been tasked with designing a wave energy system to provide additional electrical power and bring the CSM up-time for satisfying the full-power demand to 100%. This project is a collaboration between Sandia and Woods Hole Oceanographic Institution (WHOI), along with Evergreen Innovations, Monterey Bay Aquarium Research Institute (MBARI), Eastern Carolina University (ECU), Johns Hopkins University (JHU), and the National Renewable Energy Laboratory (NREL). This report captures Phase I of an expected two phase project and presents project scoping and concept design results. phase project and presents project scoping and concept design results.

The wave energy control competition established a benchmark problem which was offered as an open challenge to the wave energy system control community. The competition had two stages: In the first stage, competitors used a standard wave energy simulation platform (WEC-Sim) to evaluate their controllers while, in the second stage, competitors were invited to test their controllers in a real-time implementation on a prototype system in a wave tank. The performance function used was based on converted energy across a range of standard sea states, but also included aspects related to economic performance, such as peak/average power, peak force, etc. This paper compares simulated and experimental results and, in particular, examines if the results obtained in a linear system simulation are borne out in reality. Overall, within the scope of the device tested, the range of sea states employed, and the performance metric used, the conclusion is that high-performance WEC controllers work well in practice, with good carry-over from simulation to experimentation. However, the availability of a good WEC mathematical model is deemed to be crucial.

This report describes testing conducted related to the development of a “hydrostatic power takeoff” (HPTO) system for a wave energy converter. Tests were conducted with an experimental electric motor rig to provide preliminary results and de-risk future testing. Efficiency mapping tests were conducted as well as hardware-in-the-loop (HIL) testing. The results of the efficiency mapping tests provide good insight into how to systematically perform efficiency mapping tests. The HIL testing indicates good overall performance of the system and provides a stepping stone towards more complete system tests in the future.

The open-source WecOptTool was developed to make wave energy converter (WEC) control co-design accessible. WecOptTool is based on the pseudo-spectral method which is capable of efficiently dealing with any linear or nonlinear constraints and nonlinear dynamics by solving the WEC optimal control problem in the time domain using a gradient based optimization algorithm. This work1 presents a control co-optimization study of the AquaHarmonics Inc. heaving point absorber WEC sized for ocean deployment to solve practical industry design problems. Components such as the specific type of generator, the hull shape, and the displaced volume are pre-determined. We co-optimize the WEC’s mass versus mooring line pretension in conjunction with the controller. The optimization is subject to the power-take-off (PTO) dynamics and the rated constraints of the components. In particular, the continuous torque rating is implemented as an explicit constraint, a novel approach for WEC optimization. The PTO dynamics are incorporated into the optimization algorithm via a combination of first principle methods (linear drivetrain model) and empirical efficiency maps (electrical generator) represented as a power loss map. This is a practical method applicable to a variety of PTO architectures and transferable to other WECs. A discussion between using an efficiency coefficient versus a power loss map and their implication for the optimization method is presented. This application of WecOptTool represents a real world WEC by combining simplified models with empirical efficiency data. The WEC, as a dynamically coupled, oscillatory system, requires consideration of the time trajectory dependent power loss for optimizing the average electrical power. This objective function, the modelling approach, and the realistic loss terms makes the common practice of artificially penalizing the reactive power needless.

As part of the development process, scaled testing of wave energy converter devices are necessary to prove a concept, study hydrodynamics, and validate control system approaches. Creating a low-cost, small, lightweight data acquisition system suitable for scaled testing is often a barrier for wave energy converter developers’ ability to test such devices. This paper outlines an open-source solution to these issues, which can be customized based on specific needs. This will help developers with limited resources along a path toward commercialization.

As part of the development process, scaled testing of wave energy converter devices are necessary to prove a concept, study hydrodynamics, and validate control system approaches. Creating a low-cost, small, lightweight data acquisition system suitable for scaled testing is often a barrier for wave energy converter developers’ ability to test such devices. This paper outlines an open-source solution to these issues, which can be customized based on specific needs. Furthermore, this will help developers with limited resources along a path toward commercialization.

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This paper compares four different formulations of model predictive control that attempt to maximise electrical power generated by a wave energy converter (WEC). Control laws include (1) pure maximisation of mechanical power, (2) maximisation of mechanical power with a control penalty factor, (3) maximisation of electrical power using power conversion efficiency, and (4) maximisation of electrical power using the full electro-mechanical model of a system. For this study, a wave-to-wire model is developed for a floating spherical buoy connected to a permanent magnet synchronous generator. The performance of the controllers, including the mechanical and electrical power outputs, is compared in irregular wave conditions for the unconstrained and force-constrained scenarios. The results demonstrate that the controller designed to maximise mechanical power is not suitable for practical applications and may lead to negative electrical power output due to the non-ideal power take-off efficiency. Moreover, the replacement of the power take-off dynamics by the efficiency coefficient does not guarantee the maximum electrical power production.

With a wide variety of wave energy device archetypes currently under consideration, it is a major challenge to ensure that research findings and methods are broadly applicable. In particular, the design and testing of wave energy control systems, a process which includes experimental design, empirical modeling, control design, and performance evaluation, is of interest. This goal motivated the redesign and testing of a floating dual flap wave energy converter. As summarized in this paper, the steps taken in the design, testing, and analysis of the device mirrored those previously demonstrated on a three-degree of freedom point absorber device. The method proposed does not require locking WEC degrees of freedom to develop an excitation model, and presents a more attainable system identification procedure for at-sea deployments. The results show that the methods employed work well for this dual flap device, lending additional support for the broad applicability of the design and testing methods applied here. The aim of this paper is to demonstrate that these models are particularly useful for deducing areas of device design or controller implementation that can be reasonably improved to increase device power capture.

Wave energy converters have yet to reach broad market viability. Traditionally, levelized cost of energy has been considered the ultimate stage gate through which wave energy developers must pass in order to find success (i.e., the levelized cost of wave energy must be less than that of solar and wind). However, real world energy decisions are not based solely on levelized cost of energy. In this study, we consider the energy mix in California in the year 2045, upon which the state plans to achieve zero carbon energy production. By considering temporal electricity production and consumption, we are able to perform a more informed analysis of the decision process to address this challenge. The results show that, due to high level of ocean wave energy in the winter months, wave energy provides a valuable complement to solar and wind, which have higher production in the summer. Thus, based on this complementary temporal aspect, wave energy appears cost-effective, even when the cost of installation and maintenance is twice that of solar and wind.

Wave energy converters have yet to reach broad market viability. Traditionally, levelized cost of energy has been considered the ultimate stage gate through which wave energy developers must pass in order to find success (i.e., the levelized cost of wave energy must be less than that of solar and wind). However, real world energy decisions are not based solely on levelized cost of energy. In this study, we consider the energy mix in California in the year 2045, upon which the state plans to achieve zero carbon energy production. By considering temporal electricity production and consumption, we are able to perform a more informed analysis of the decision process to address this challenge. The results show that, due to high level of ocean wave energy in the winter months, wave energy provides a valuable complement to solar and wind, which have higher production in the summer. Thus, based on this complementary temporal aspect, wave energy appears cost-effective, even when the cost of installation and maintenance is twice that of solar and wind.

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While a great deal of research has been performed to quantify and characterize the wave energy resource, there are still open questions about how a wave energy developer should use this wave resource information to design a wave energy converter device to suit a specific environment or, alternatively, to assess potential deployment locations. It is natural to focus first on the impressive magnitudes of power available from ocean waves, and to be drawn to locations where mean power levels are highest. However, a number of additional factors such as intermittency and capacity factor may be influential in determining economic viability of a wave energy converter, and should therefore be considered at the resource level, so that these factors can influence device design decisions. This study examines a set of wave resource metrics aimed towards this end of bettering accounting for variability in wave energy converter design. The results show distinct regional trends that may factor into project siting and wave energy converter design. Although a definitive solution for the optimal size of a wave energy converter is beyond the reaches of this study, the evidence presented does support the idea that smaller devices with lower power ratings may merit closer consideration.

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The potential for control design to dramatically improve the economic viability of wave energy has generated a great deal of interest and excitement. However, for a number of reasons, the promised benefits from better control designs have yet to be widely realized by wave energy devices and wave energy remains a relatively nascent technology. This brief paper summarizes a simple, yet powerful approach to wave energy dynamics modeling, and subsequent control design based on impedance matching. Our approach leverages the same concepts that are exploited by a simple FM radio to achieve a feedback controller for wave energy devices that approaches optimal power absorption. If fully utilized, this approach can deliver immediate and consequential reductions to the cost of wave energy. Additionally, this approach provides the necessary framework for control co-design of a wave energy converter, in which an understanding of the control logic allows for synchronous design of the device control system and hardware.

Interest in wave energy converters to provide autonomous power to various ocean-bound systems, such as autonomous underwater vehicles, sensor systems, and even aquaculture farms, has grown in recent years. The Monterey Bay Aquarium Research Institute has developed and deployed a small two-body point absorber wave energy device suitable to such needs. This paper provides a description of the system to support future open-source access to the device and further the general development of similar wave energy systems. Additionally, to support future control design and system modification efforts, a set of hydrodynamic models are presented and cross-compared. To test the viability of using a linear frequency-domain admittance model for controller tuning, the linear model is compared against four WEC-Sim models of increasing complexity. The linear frequency-domain model is found to be generally adequate for capturing system dynamics, as the model agreement is good and the degree of nonlinearity introduced in the WEC-Sim models is generally less than 2.5%.

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The present work addresses the need for an efficient, versatile, accurate and open-source numerical tool to be used during the design stage of wave energy converters (WECs). The device considered here is the heaving point-absorber developed and tested by Sandia National Laboratories. The smoothed particle hydrodynamics (SPH) method, as implemented in DualSPHysics, is proposed since its meshless approach presents some important advantages when simulating floating devices. The dynamics of the power take-off system are also modelled by coupling DualSPHysics with the multi-physics library Project Chrono. A satisfactory matching between experimental and numerical results is obtained for: (i) the heave response of the device when forced via its actuator; (ii) the vertical forces acting on the fixed device under regular waves and; (iii) the heave response of the WEC under the action of both regular waves and the actuator force. This proves the ability of the numerical approach proposed to simulate accurately the fluid–structure interaction along with the WEC’s closed-loop control system. In addition, radiation models built from the experimental and WAMIT results are compared with DualSPHysics by plotting the intrinsic impedance in the frequency domain, showing that the SPH method can be also employed for system identification.

The main objective of this letter is to consolidate the knowledge about the dynamics and control of oscillating-body wave energy converters (WECs). A number of studies have shown that control systems strongly affect power absorption; however, there remains a need for a concise and integrated explanation of the theoretical and practical implications that control can have on both performance and the broader WEC design process. This short letter attempts to fill this gap by presenting a discussion on the key practical aspects concerning the dynamics and control of oscillating-body WEC. In particular, the focus is on the choice of control models and a simple causal control scheme suitable for real-time implementation. Finally, consideration is given to the effect of the power takeoff (PTO) on the maximization of electrical power, thus leading to the derivation of useful conditions for the control co-design of the PTO system.

This study presents a numerical model of a WEC array. The model will be used in subsequent work to study the ability of data assimilation to support power prediction from WEC arrays and WEC array design. In this study, we focus on design, modeling, and control of the WEC array. A case study is performed for a small remote Alaskan town. Using an efficient method for modeling the linear interactions within a homogeneous array, we produce a model and predictionless feedback controllers for the devices within the array. The model is applied to study the effects of spectral wave forecast errors on power output. The results of this analysis show that the power performance of the WEC array will be most strongly affected by errors in prediction of the spectral period, but that reductions in performance can realistically be limited to less than 10% based on typical data assimilation based spectral forecasting accuracy levels.

This paper reports results from an ongoing investigation on potential ways to utilize small wave energy devices that can be transported in, and deployed from, torpedo tubes. The devices are designed to perform designated ocean measurement operations and thus need to convert enough energy to power onboard sensors, while storing any excess energy to support vehicle recharging operations. Examined in this paper is a traditional tubular oscillating water column device, and particular interest here is in designs that lead to optimization of power converted from shorter wind sea waves. A two step design procedure is investigated here, wherein a more approximate two-degree-of-freedom model is first used to identify relative dimensions (of device elements) that optimize power conversion from relative oscillations between the device elements. A more rigorous mathematical model based on the hydrodynamics of oscillating pressure distributions within solid oscillators is then used to provide the hydrodynamic coefficients, forces, and flow rates for the device. These results provide a quick but rigorous way to estimate the energy conversion performance of the device in various wave climates, while enabling more accurate design of the power takeoff and energy storage systems.

While some engineering fields have benefited from systematic design optimization studies, wave energy converters have yet to successfully incorporate such analyses into practical engineering workflows. The current iterative approach to wave energy converter design leads to sub-optimal solutions. This short paper presents an open-source MATLAB toolbox for performing design optimization studies on wave energy converters where power take-off behavior and realistic constraints can be easily included. This tool incorporates an adaptable control co-design approach, in that a constrained optimal controller is used to simulate device dynamics and populate an arbitrary objective function of the user’s choosing. A brief explanation of the tool’s structure and underlying theory is presented. To demonstrate the capabilities of the tool, verify its functionality, and begin to explore some basic wave energy converter design relationships, three conceptual case studies are presented. In particular, the importance of considering (and constraining) the magnitudes of device motion and forces in design optimization is shown.

Potential performance gains from optimal (non-causal) impedance-matching control of wave energy devices in irregular ocean waves are dependent on deterministic wave elevation prediction techniques that work well in practical applications. Although a number of devices are designed for operation in intermediate water depths, little work has been reported on deterministic wave prediction in such depths. Investigated in this paper is a deterministic wave-prediction technique based on an approximate propagation model that leads to an analytical formulation, which may be convenient to implement in practice. To improve accuracy, an approach to combine predictions based on multiple up-wave measurement points is evaluated. The overall method is tested using experimental time-series measurements recorded in the U.S. Navy MASK basin in Carderock, MD, USA. For comparison, an alternative prediction approach based on Fourier coefficients is also tested with the same data. Comparison of prediction approaches with direct measurements suggest room for improvement. Possible sources of error including tank reflections are estimated, and potential mitigation approaches are discussed.

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This report describes the testing of a model scale wave energy converter. This device, which uses two aps that pivot about a central platform when excited by waves, has a natural frequency within the range of the waves by which it is excited. The primary goal of this test was to assess the degree to which previously developed modeling, experimentation, and control design methods could be applied to a broad range of wave energy converter designs. Testing was conducted to identify a dynamic model for the impedance and excitation behavior of the device. Using these models, a series of closed loop tests were conducted using a causal impedance matching controller. This report provides a brief description of the results, as well as a summary of the device and ex- perimental design. The results show that the methods applied to this experimental device perform well and should be broadly applicable.

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A self-tuning proportional-integral control law prescribing motor torques was tested in experiment on a three degree-of-freedom wave energy converter. The control objective was to maximize electrical power. The control law relied upon an identified model of device intrinsic impedance to generate a frequency-domain estimate of the wave-induced excitation force and measurements of device velocities. The control law was tested in irregular sea-states that evolved over hours (a rapid, but realistic time-scale) and that changed instantly (an unrealistic scenario to evaluate controller response). For both cases, the controller converges to gains that closely approximate the post-calculated optimal gains for all degrees of freedom. Convergence to near-optimal gains occurred reliably over a sufficiently short time for realistic sea states. In addition, electrical power was found to be relatively insensitive to gain tuning over a broad range of gains, implying that an imperfectly tuned controller does not result in a large penalty to electrical power capture. An extension of this control law that allows for adaptation to a changing device impedance model over time is proposed for long-term deployments, as well as an approach to explicitly handle constraints within this architecture.

Through the use of advanced control techniques, wave energy converters (WECs) can achieve substantial increases in energy absorption. The motion of the WEC device is a significant contribution to the energy absorbed by the device. Reactive (complex conjugate) control maximizes the energy absorption due to the impedance matching. The issue with complex conjugate control is that, in general, the controller is noncausal, which requires prediction of the incoming waves. This article explores the potential of employing system identification techniques to build a causal transfer function that approximates the complex conjugate controller over a finite frequency band of interest. This approach is quite viable given the band-limited nature of ocean waves. The resulting controller is stable, and the average efficiency of the power captured by the causal controller in realistic ocean waves is 99%, when compared to the noncausal complex conjugate.

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Sandia National Laboratories and the Department of Energy (DOE) have completed on a multi-year program to examine the effects of control theory on increasing power produced by resonant wave energy conversion (WEC) devices. The tank tests have been conducted at the Naval Surface Warfare Center Carderock Division (NSWCCD) Maneuvering and Sea Keeping Basin (MASK) in West Bethesda, MD. This report outlines the "MASK₃" wave tank test within the Advanced WEC Dynamics and Controls (AWDC) project. This test represents the final test in the AWDC project. The focus of the MASK₃ test was to consider coordinated ₃-degree-of-freedom (₃DOF) control of a WEC in a realistic ocean environment. A key aspect of this test was the inclusion of a "self-tunine mechanism which uses an optimization algorithm to update controller gains based on a changing sea state. The successful implementation of the self-tuning mechanism is the last crucial step required for such a controller to be implemented in real ocean environments.

This report serves as a comprehensive summary of the work completed by the "Advanced WEC Dynamics and Controls projecr during the period of 2013-2019. This project was first envisioned to simply consider the question of designing a controller for wave energy converters (WECs), without a complete recognition of the broader considerations that such a task must necessarily examine. This document describes both the evolution of the project scope and the key findings produced. The basic goal of the project has been to deliver tractable methodologies and work flows that WEC designers can use to improve the performance of their machines. Engineering solutions, which may offer 80% of the impact, but require 20% of the effort compared to a perfect result (which may be many years of development down the road) were preferred. With this doctrine, the work of the project often involved translating existing methods that have been successfully developed and applied for other fields, into the application area of wave energy.

This paper presents a nonlinear geometric buoy design for Wave Energy Converters (WECs). A nonlinear dynamic model is presented for an hour glass (HG) configured WEC. The HG buoy operates in heave motion or as a single Degree-of-Freedom (DOF). The unique formulation of the interaction between the buoy and the waves produces a nonlinear stiffening effect that provides the actual energy storage or reactive power during operation. A Complex Conjugate Control (C3) with a practical Proportional-Derivative (PD) controller is employed to optimize power absorption for off-resonance conditions and applied to a linear right circular cylinder (RCC) WEC. For a single frequency the PDC3 RCC buoy is compared with the HG buoy design. A Bretschneider spectrum of wave excitation input conditions are reviewed and evaluated for the HG buoy. Numerical simulations demonstrate power and energy capture for the HG geometric buoy design which incorporates and capitalizes on the nonlinear geometry to provide reactive power for the single DOF WEC. By exploiting the nonlinear physics in the HG design simplified operational performance is observed when compared to an optimized linear cylindrical WEC. The HG steepness angle α with respect to the wave is varied and initially optimized for improved energy capture.

Wave Energy Converter (WEC) technologies transform power from the waves to the electrical grid. WEC system components are investigated that support the performance, stability, and efficiency as part of a WEC array. To this end, Aquaharmonics Inc took home the 1.5 million grand prize in the 2016 U.S. Department of Energy Wave Energy Prize, an 18-month design-build-test competition to increase the energy capture potential of wave energy devices. Aquaharmonics intends to develop, build, and perform open ocean testing on a 1: 7 scale device. Preliminary wave tank testing on the mechanical system of the 1: 20 scale device has yielded a data-set of operational conditions and performance. In this paper, the Hamiltonian surface shaping and power flow control (HSSPFC) method is used in conjunction with scaled wave tank test data to explore the design space for the electrical transmission of energy to the shore-side power grid. Of primary interest is the energy storage system (ESS) that will electrically link the WEC to the shore. Initial analysis results contained in this paper provide a trade-off in storage device performance and design selection.

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This report details the background, design, and initial results for wave energy converter design optimization tool. This tool is intended to provide researchers and developers with a means of optimizing existing wave energy converter designs by including realistic dynamics and control algorithms early in the design cycle.

An increasing number of experiments are being conducted to study the design and performance of wave energy converters. Often in these tests, a real-time realization of prospective control algorithms is applied in order to assess and optimize energy absorption as well as other factors. This paper details the design and execution of an experiment for evaluating the capability of a model-scale WEC to execute basic control algorithms. Model-scale hardware, system, and experimental design are considered, with a focus on providing an experimental setup capable of meeting the dynamic requirements of a control system. To more efficiently execute such tests, a dry bench testing method is proposed and utilized to allow for controller tuning and to give an initial assessment of controller performance; this is followed by wave tank testing. The trends from the dry bench test and wave tank test results show good agreement with theory and confirm the ability of a relatively simple feedback controller to substantially improve energy absorption. Additionally, the dry bench testing approach is shown to be an effective and efficient means of designing and testing both controllers and actuator systems for wave energy converters.

This report describes the set up, execution, and some initial results from a series of wave tank tests of a model-scale wave energy converter (WEC) completed in May 2018 at the Navy's Maneuvering and Sea Keeping (MASK) basin. The purpose of these tests was to investigate the implementation and performance of a series of closed-loop WEC power take-off (PTO) controllers, intended to increase energy absorption/generation.

The idea of acausality for control of a wave energy converter (WEC) is a concept that has been popular since the birth of modern wave energy research in the 1970s. This concept has led to considerable research into wave prediction and feedforward WEC control algorithms. However, the findings in this report mostly negate the need for wave prediction to improve WEC energy absorption, and favor instead feedback driven control strategies. Feedback control is shown to provide performance that rivals a prediction-based controller, which has been unrealistically assumed to have perfect prediction.

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Through the use of advanced control techniques, wave energy converters have significantly improved energy absorption. The motion of the WEC device is a significant contribution to the energy absorbed by the device. Reactive control (complex conjugate control) maximizes the energy absorption due to the impedance matching. The issue with complex conjugate control is that the controller is non-causal, which requires prediction into the oncoming waves to the device. This paper explores the potential of using system identification (SID) techniques to build a causal transfer function that approximates the complex conjugate controller over a specific frequency band of interest. The resulting controller is stable, and the average efficiency of the power captured by the causal controller is 99%, when compared to the non-causal complex conjugate.

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This paper discusses the optimal output feedback control problem of linear time-invariant systems with additional restrictions on the structure of the optimal feedback control gain. These restrictions include setting individual elements of the optimal gain matrix to zero and making the sum of certain rows of the gain matrix equal to desired values. The paper proposes a method that modifies the standard quadratic cost function to include soft constraints ensuring the satisfaction of these restrictions on the structure of the optimal gain. Necessary conditions for optimality with these soft constraints are derived, and an algorithm to solve the resulting optimal output feedback control problem is given. Finally, a power systems example is presented to illustrate the usefulness of proposed approach.

This paper presents a solution to the optimal control problem of a three degrees-of-freedom (3DOF) wave energy converter (WEC). The three modes are the heave, pitch, and surge. The dynamic model is characterized by a coupling between the pitch and surge modes, while the heave is decoupled. The heave, however, excites the pitch motion through nonlinear parametric excitation in the pitch mode. This paper uses Fourier series (FS) as basis functions to approximate the states and the control. A simplified model is first used where the parametric excitation term is neglected and a closed-form solution for the optimal control is developed. For the parametrically excited case, a sequential quadratic programming approach is implemented to solve for the optimal control numerically. Numerical results show that the harvested energy from three modes is greater than three times the harvested energy from the heave mode alone. Moreover, the harvested energy using a control that accounts for the parametric excitation is significantly higher than the energy harvested when neglecting this nonlinear parametric excitation term.

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The idea of acausality for control of a wave energy converter (WEC) is a concept that has been popular since the birth of modern wave energy research in the 1970s. This concept has led to considerable research into wave prediction and feedforward WEC control algorithms. However, the findings in this report mostly negate the need for wave prediction to improve WEC energy absorption, and favor instead feedback driven control strategies. Feedback control is shown to provide performance that rivals a prediction-based controller, which has been unrealistically assumed to have perfect prediction. It is well known in classical control engineering that perfect knowledge of past and future events will always lead to higher performing systems. However, it is also well known that the underlying system must be well-designed; control cannot fix a bad design. Additionally, one must consider the practical application of a control design, which relies on measurements and actuation systems. There are major implications to cost and reliability when relying on remote sensors requiring real-time data-streaming (e.g., remote wave buoys). This report shows that for a well-designed WEC, in which closed loop dynamics is considered since early stages of design, a suboptimal controller using no prediction can achieve more than 90% of the theoretical maximum. A predictionless feedback resonating (FBR) controller performs within 0.1% percent of a controller with perfect future knowledge (something which is not practically attainable). Given the major challenges with accurate and robust wave prediction, this result provides a major argument and incentive for utilizing feedback for WEC control. Implementation of these feedback strategies is readily attainable, while the strategy requiring perfect wave prediction will demand an unknown number of additional years to research and develop, all in the service of a marginal 1% benefit.

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In this study, we employ a numerical model to compare the performance of a number of wave energy converter control strategies. The controllers selected for evaluation span a wide range in their requirements for implementation. Each control strategy is evaluated using a single numerical model with a set of sea states to represent a deployment site off the coast of Newport, OR. A number of metrics, ranging from power absorption to kinematics, are employed to provide a comparison of each control strategy's performance that accounts for both relative benefits and costs. The results show a wide range of performances from the different controllers and highlight the need for a holistic design approach which considers control design as a parallel component within the larger process WEC design.

For a three-degree-of-freedom wave energy converter (heave, pitch, and surge), the equations of motion could be coupled depending on the buoy shape. This paper presents a multiresonant feedback control, in a general framework, for this type of a wave energy converter that is modeled by linear time invariant dynamic systems. The proposed control strategy finds the optimal control in the sense that it computes the control based on the complex conjugate criteria. This control strategy is relatively easy to implement since it is a feedback control in the time domain that requires only measurements of the buoy motion. Numerical tests are presented for two different buoy shapes: a sphere and a cylinder. Regular, Bretschnieder, and Ochi-Hubble waves are tested. Simulation results show that the proposed controller harvests energy in the pitch-surge-heave modes that is about three times the energy that can be harvested using a heave-only device. This multiresonant control can also be used to shift the energy harvesting between the coupled modes, which can be exploited to eliminate one of the actuators while maintaining about the same level of energy harvesting.

For a heave-pitch-surge three-degrees-of-freedom wave energy converter, the heave mode is usually decoupled from the pitch-surge modes for small motions. The pitch-surge modes are usually coupled and are parametrically excited by the heave mode, depending on the buoy geometry. In this paper, a Model Predictive Control is applied to the parametric excited pitch-surge motion, while the heave motion is optimized independently. The optimality conditions are derived, and a gradient-based numerical optimization algorithm is used to search for the optimal control. Numerical tests are conducted for regular and Bretschneider waves. The results demonstrate that the proposed control can be implemented to harvest more than three times the energy that can be harvested using a heave-only wave energy converter. The energy harvested using a parametrically excited model is higher than that is harvested when using a linear model.

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Empirically based modeling is an essential aspect of design for a wave energy converter. Empirically based models are used in structural, mechanical and control design processes, as well as for performance prediction. Both the design of experiments and methods used in system identification have a strong impact on the quality of the resulting model. This study considers the system identification and model validation process based on data collected from a wave tank test of a model-scale wave energy converter. Experimental design and data processing techniques based on general system identification procedures are discussed and compared with the practices often followed for wave tank testing. The general system identification processes are shown to have a number of advantages, including an increased signal-to-noise ratio, reduced experimental time and higher frequency resolution. The experimental wave tank data is used to produce multiple models using different formulations to represent the dynamics of the wave energy converter. These models are validated and their performance is compared against one another. While most models of wave energy converters use a formulation with surface elevation as an input, this study shows that a model using a hull pressure measurement to incorporate the wave excitation phenomenon has better accuracy.

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This report gives a brief discussion and examples on the topic of state estimation for wave energy converters (WECs). These methods are intended for use to enable real-time closed loop control of WECs.

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A study was performed to optimize the geometry of a point absorber style wave energy converter (WEC). An axisymmetric single-body device, moving in heave only, was considered. Design geometries, generated using a parametric definition, were optimized using genetic algorithms. Each geometry was analyzed using a boundary element model (BEM) tool to obtain corresponding frequency domain models. Based on these models, a pseudo-spectral method was applied to develop a control methodology for each geometry. The performance of each design was assessed using a Bretschneider sea state. The objective of optimization is to maximize harvested energy. In this preliminary investigation, a constraint is imposed on the the geometry to guarantee a linear dynamic model would be valid for all geometries generated by the optimization tool. Numerical results are presented for axisymmetric buoy shapes.

A linear dynamic model for a wave energy converter (WEC) has been developed based on the results of experimental wave tank testing. Based on this model, a model predictive control (MPC) strategy has been designed and implemented. To assess the performance of this control strategy, a deployment environment off the coast of Newport, OR has been selected and the controller has been used to simulate the WEC response in a set of irregular sea states. To better understand the influence of model accuracy on control performance, an uncertainty analysis has been performed by varying the parameters of the model used for the design of the controller (i.e. the control model), while keeping the WEC dynamic model employed in these simulations (i.e. the plant model) unaltered. The results of this study indicate a relative low sensitivity of the MPC control strategy to uncertainties in the controller model for the specific case studied here.

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Optimal control theory is applied to compute control for a single-degree-of-freedom heave wave energy converter. The goal is to maximize the energy extraction per cycle. Both constrained and unconstrained optimal control problems are presented. Both periodic and non-periodic excitation forces are considered. In contrast to prior work, it is shown that for this non-autonomous system, the optimal control, in general, includes both singular arc and bang-bang modes. Conditions that determine the switching times to/from the singular arc are derived. Simulation results show that the proposed optimal control solution matches the solution obtained using the complex conjugate control. A generic linear dynamic model is used in the simulations. In conclusion, the main advantage of the proposed control is that it finds the optimal control without the need for wave prediction; it only requires the knowledge of the excitation force and its derivatives at the current time.

A model-scale wave tank test was conducted in the interest of improving control systems design of wave energy converters (WECs). The success of most control strategies is based directly upon the availability of a reduced-order model with the ability to capture the dynamics of the system with sufficient accuracy. For this reason, the test described in this report, which is the first in a series of planned tests on WEC controls, focused on system identification (system ID) and model validation.

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The addition of a compressible degree of freedom (CDOF) to a wave energy converter (WEC)-which results in a compressible WEC-has been shown to significantly increase power absorption compared to a rigid WEC of the same shape and mass for a variety of architectures. This study demonstrates that a compressible point absorber, with a passive power-take-off (PTO) and optimized damping, can also achieve equal or better performance levels than an optimally controlled rigid point absorber (with the same shape and mass) using reactive power from the PTO. Wave energy is converted to mechanical energy in both cases using a linear damper PTO, with the PTO coefficient optimized for each resonance frequency and compressible volume. The large compressible volume required to tune the compressible point absorber to the desired frequency is a practical limitation that needs to be addressed with further research, especially for low frequencies. While realistic, these auxiliary units would increase the CapEx and OpEx costs, potentially reducing the aforementioned benefits gained by CDOF. However, alternative approaches can be developed to implement CDOF without the large compressible volume requirements, including the development of flexible surface panels tuned with mechanical springs.

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The operation of Wave Energy Converter (WEC) devices can pose many challenging problems to the Water Power Community. A key research question is how to significantly improve the performance of these WEC devices through improving the control system design. This report summarizes an effort to analyze and improve the performance of WEC through the design and implementation of control systems. Controllers were selected to span the WEC control design space with the aim of building a more comprehensive understanding of different controller capabilities and requirements. To design and evaluate these control strategies, a model scale test-bed WEC was designed for both numerical and experimental testing (see Section 1.1). Seven control strategies have been developed and applied on a numerical model of the selected WEC. This model is capable of performing at a range of levels, spanning from a fully-linear realization to varying levels of nonlinearity. The details of this model and its ongoing development are described in Section 1.2.

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The addition of a compressible degree of freedom (CDOF) has been shown to significantly increase the power absorption compared to a traditional rigid WEC of the same shape and mass for a variety of architectures. The present study demonstrates that a compressible point absorber, with a passive power-take-off (PTO) and optimized damping, can also achieve at the same performance levels or better than an optimally controlled rigid point absorber using reactive power from the PTO. Eliminating the need for a reactive PTO would substantially reduce costs by reducing PTO design complexity. In addition, it would negate the documented problems of reactive PTO efficiencies on absorbed power. Improvements to performance were quantified in the present study by comparing a compressible point absorber to a conventional rigid one with the same shape and mass. Wave energy is converted to mechanical energy in both cases using a linear damper PTO, with the PTO coefficient optimized for each resonance frequency and compressible volume. The large compressible volumes required to tune the compressible point absorber to the desired frequency are a practical limitation that needs to be addressed with further research; especially for low frequencies. If fact, all compressible volumes exceed the submerged volume of the point absorber by significant amounts; requiring auxiliary compressible volume storage units that are connected to the air chamber in the submerged portion of the point absorber. While realistic, these auxiliary units would increase the Cap Ex and Op Ex costs, potentially reducing the aforementioned benefits gained by CDOF. However, alternative approaches can be developed to implement CDOF without the large compressible volume requirements, including the development of flexible surface panels tuned with mechanical springs.

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A new multi-year effort has been launched by the Department of Energy to validate the extent to which control strategies can increase the power produced by resonant wave energy conversion (WEC) devices. This paper describes the design of a WEC device to be employed by this program in the development and assessment of WEC control strategies. The operational principle of the device was selected to provide a test-bed for control strategies, in which a specific control strategies effectiveness and the parameters on which its effectiveness depends can be empirically determined. Numerical design studies were employed to determine the device geometry, so as to maximize testing opportunities in the Maneuvering and Seakeeping (MASK) Basin at the Naval Surface Warfare Centers David Taylor Model Basin. Details on the physical model including specific components and model fabrication methodologies are presented. Finally the quantities to be measured and the mechanisms of measurement are listed.