The dynamics of wave energy

In the quest for green energy, engineers have increasingly viewed tides and ocean waves as sources from which they can harvest energy. But there is no one single technology that has emerged as a standard for harvesting wave energy. There are several viable methods being studied. Most of them have several functions in common. These functions include a captor which collects energy from the oscillatory motion of waves, power take-off equipment which converts mechanical energy into electricity, and an anchoring system which holds the wave energy converter (WEC) in place.

For example, consider a device called the P2 from Pelamis Wave Power in the UK. Its captor is a ‘snake-like’ construction on the surface of the water consisting of several cylindrical sections joined together by hinged joints. As waves pass down the length of the machine these sections flex relative to one another. Hydraulic cylinders at each joint resist wave motion and pump fluid into high pressure accumulators. The power take-off equipment consists of a hydraulic motor powered from fluid in the accumulator. The motor turns a generator which sends electrical power to shore. The anchoring system consists of several anchors connected to a central point, plus a yaw-restraining line.

Another WEC from Ocean Power Technologies looks like a buoy. Called the PowerBuoy, its collector consists of a buoy that moves freely up and down as waves pass by. The buoy sits on a vertical shaft anchored to the ocean floor. As it rides the waves it moves hydraulic cylinders up and down to power a hydraulic motor. The power take-off includes the hydraulic motor which drives an electrical generator transmitting power via an underwater power cable. The anchoring system consists of a foundation on the seabed.

A third example is the Oyster wave device developed by Aquamarine Power in Scotland. The collector consists of a buoyant, hinged flap attached to the seabed at depths of between 10 and 15 m. The hinged flap, which is almost entirely underwater, pitches back and forth under the influence of waves. The movement of the flap drives two hydraulic pistons which push high pressure water onshore via a subsea pipeline to drive a conventional hydro-electric turbine. The power take-off consists of the hydraulic pistons and pumping mechanics. The anchoring system consists of a foundation on the seabed.

Resolving forces

All these WECs use different approaches for harvesting wave energy. But essentially, engineers analyze the dynamics of all WECs with the same approach, applying Newton's second law: Inertial force balances the forces acting over the WEC captor, or, simply, F = mA. The typical approach is to decompose forces into those from hydrodynamic and external sources. The hydrodynamic source includes the excitation force inflicted by the incident waves, the buoyancy force which comes from the variation in submergence caused by wave oscillations, and radiation force related to the pressure over the submerged surface caused by fluid displaced from device oscillations. Moreover, forces from the PTO and mooring system could also constrain the motion of the WEC.

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In general, the hydrodynamic interaction between WECs and ocean waves is a complex process described by high-order nonlinear equations. Fortunately these equations can be simplified to linear versions under particular conditions. This is true when waves and the resulting oscillations have a low amplitude. And it turns out that the linear hydrodynamic approach is acceptable throughout the WEC's operational regime.

The first steps in modeling WEC dynamics work with parameters to the frequency domain. Thus all the physical quantities vary sinusoidallly with time according to the frequency of the incident wave. Under these conditions, the motion equations become a linear system that can be solved with widely used mathematics software such as Matlab.

Usually, direct methods of solution are faster and more generally applicable to the sorts of relationships involved in WEC hydrodynamics. These methods are basically variants of Gaussian elimination and are often expressed as matrix factorizations such as LU or Cholesky factorization.

(As a quick review, the Cholesky factorization is a way of factoring a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. A Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate transpose - that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column. An LU factorization is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix.)

To facilitate work in the frequency domain, software packages such as MatLab contain large collections of predefined functions to compute and manipulate Fourier transforms. Specifically, dimensional or non-dimensional fast Fourier transforms can efficiently decompose a time signal into the constituent sinusoids.

Engineers typically use frequency domain analysis to assess how the WEC performs and to optimize the WEC geometry. Special numerical codes help compute the linear radiation and excitation forces acting on the device. One such code is called Wamit. It was originally developed at M.I.T. and is based on linear and second-order potential theory. The velocity potential is solved by means of boundary integral equation method, also known as the panel method. Wamit can represent the forces by continuous B-splines according to classic linear water wave theory.

Still, in many practical cases WEC dynamics are strongly non-linear. For example, forces from the mooring system and PTO might arise only when the WEC yanks against them. When non-linear forces are in play, handy shortcuts, such as superposition, can't be used to help with the analysis. The only way to get a handle on the forces involved is to employ non-linear techniques and work in the time domain.

In particular, the computation of radiation force can be tough in the time domain because it involves a convolution between the velocity and an impulse response function. One way to handle such problems is through use of the state-space method. It represents the convolution integral of the radiation force as a small number of first-order linear differential equations with constant coefficients. There are several ways of deriving the coefficients of these differential equations. All methods either derive them explicitly from the impulse-response function of the system or directly from the system transfer function. The transfer function is computed through standard hydrodynamic radiation-diffraction codes.

There are two methods in wide use because they lend themselves to numerical computation. The first approximates the impulse-response function by a combination of exponential functions in the time domain. The second approximates the transfer function by a complex rational function. Both alternatives can be found amongst signal processing tools provided by packages such as MatLab. In the case of MatLab, a method called prony finds an IIR filter with a specified time domain impulse-response function, and a function called invfreqs identifies continuous-time filter parameters from a prescribed transfer function.

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Small waves

When modeling the WEC, it is practical to treat incident waves as the input to the linear system model. That means modelers need to characterize the incident waves. There are special software packages that can simulate and statistically represent random waves and random loads. For example, one such tool, called WAFO for wave analysis for fatigue and oceanography, handles such tasks as predicting fatigue life for random loads; calculating the theoretical density of rainflow cycles; simulating linear and non-linear Gaussian waves; estimating frequency spectra and directional wave spectra; and estimating joint wave height, wave steepness, and period distributions.

Once the loads are modeled, designers must build a sub-system in which the input is the incident wave and output is the wave excitation force acting on the WEC. As with the description of the radiation force, this subsystem can be built in the form of a state space description. And as with impulse radiation response, the excitation impulse response comes from the application of radiation-diffraction codes.

Finally, the hydrostatic force is added to the model. Analysts generally model this force as a non-linear function of the WEC's displacement, which depends on its geometry. Again, this function can be linearized for small motions. So the hydrostatic sub-block usually takes the form of a static linear function of the WEC displacement meaning it has no dynamics.

Similarly, analysts usually neglect any mooring dynamics as the anchoring cables have no memory effect. So the mooring can be modeled by an additional non-linear restoring force. If the WEC displacement is small, the restoring force caused by the mooring system might also be linearized.

In contrast, the PTO has strong nonlinearities because of its stroke and force limitations. It also usually has strong dynamic components due to, for instance, the inertia of a turbine, or a control strategy that includes on times and off times.

All in all, the total dynamics of a WEC consists of a set of non-linear ordinary differential equations with variable order. The order depends on the complexity of the dynamics of each sub-block. For instance, the order of each sub-block of a WEC with a single degree of freedom is typically 3 to 6 for the excitation, 3 to 5 for the radiation, 1 to 3 for the PTO, and two for the rigid body dynamics. To compute the total dynamics of a WEC, analysts invoke a collection of initial value problem solvers in a software package such as MatLab. These solvers cover stiff, moderately stiff, and non-stiff differential equations.