Elsevier

Marine Structures

Volume 21, Issue 1, January 2008, Pages 23-46
Marine Structures

Including moorings in the assessment of a generic offshore wave energy converter: A frequency domain approach

https://doi.org/10.1016/j.marstruc.2007.09.004Get rights and content

Abstract

A method to include the influence of mooring cables in the frequency domain analysis of wave energy converters is presented. In brief the method consists of:

  • (i)

    A non-linear time domain solution of the mooring line in isolation and at an appropriate equilibrium condition. This is done by enforcing a sinusoidal displacement at the mooring attachment point in each translational degree of freedom. This is repeated at a number of frequencies.

  • (ii)

    The amplitude and phase of the resulting force is recorded, allowing the equivalent linear resistive and reactive contribution of the mooring line to be estimated separately. Using results at a number of frequencies, frequency dependent impedance properties of the mooring cable can be estimated.

  • (iii)

    Considering the attachment point and orientation of the mooring cables in a suitable equilibrium condition of the device, the contribution of each mooring cable is resolved to the global co-ordinates of the device and added to the frequency domain equation of motion.

The method here is applied to a generic wave energy device based on a truncated vertical cylinder of 100 tonne displacement. The results for the unmoored device are compared to the same device with moorings of varying configuration. The results indicate that moorings may have a significant impact on the performance of devices of this scale, both beneficial and detrimental. The introduction of mooring terms can upset device symmetry and introduce significant cross coupling in the overall mechanical impedance of the device. Arrangements where this can benefit as well as detriment performance are studied.

Introduction

In a family of floating ocean wave energy converters (OWECs), termed “motion dependent” by Johanning et al. [1], the power extraction is directly dependent on the motion of the floating device. A power take off machine, connected to the device provides a damping force that is used to generate useful power. Such a damping force can be linear, allowing the application of frequency domain analysis. Frequency domain motion calculations are a very convenient tool in predicting power production, particularly in the early stages of developing wave energy converters. The linear assumption allows the application of a whole host of useful statistical tools to estimate quantities such as annual power production in a certain wave climate. However, damping due to eddy formations around the body, not considered here, as well as viscous cable losses can dissipate some of the power, reducing the power available for useful production by the power take off machinery. The additional inertia and elasticity of cables can also have a reactive effect on the motion of the device. Catenary moorings are inherently non-linear structures, both due to non-linear static restoring forces as well as significant quadratic viscous damping and seabed friction. As such, they are not convenient to represent in a simple linear model and are often neglected in frequency domain assessments of wave energy converters. Furthermore, moorings are also notoriously difficult to model physically at smaller scales. Webster [2] defines no less than 12 non-dimensional values to define a given mooring system, which depend varyingly on gravity forces, viscous forces and elasticity. The scaling laws are very difficult to tie down for general applications. This may be why the role of moorings has not yet been greatly considered in wave energy production.

In the general offshore engineering industry, the influence of mooring cables is also often considered negligible when predicting the wave frequency response of large floating structures for non extreme events. Offshore standards such as Det NorskeVeritas (DNV's) [3] position mooring do dictate that the effect of mooring forces on wave induced motions should be considered for water depths less than 70 m. However this standard is developed with survivability issues in mind where a non-linear time domain calculation including mooring forces can be used in the analysis of singular extreme events. Indeed Johanning et al. [4] have highlighted some concerns for the position mooring of OWECs that are unique to this application and mean that mooring design could play an important role in the economics of wave energy. The extra mass and stiffness of the mooring cables or chains may in some cases alter, or indeed be designed to alter, the motion characteristics of the device, while the extra damping introduced by the mooring will incur some power absorption penalties. An understanding of the contributions of moorings at the scale of wave energy devices might be necessary for improving predictions of power extraction and could allow for the use of moorings as an integral part of wave energy converter designs. The development of a method to include moorings with sufficient accuracy in the frequency domain could be very useful.

Despite the severe non-linearity of extreme mooring problems, the problem can be contained somewhat if the application is restricted to the power take-off (PTO) performance of OWECs. For example, it can be restricted to wave frequency excitations at some nominal amplitude representative of waves captured for power production. This can make the linearisation assumptions more applicable.

One source of non-linearity is the changing geometry of catenary structures. The equivalent mass and spring (reactance) characteristics of the cable vary with the offset of the attachment point. This is visible in the static restoring force of catenary structures. However, for limited amplitudes about a moderately offset equilibrium condition, the static forces can very often be represented with a linear equation (see Fig. 6 for the static results of the cable used in this report). Aranha et al. [5] noted similar behaviour for dynamic excitation. “The geometric non-linearity is of little concern for this class of problems”, even for tight moorings and strong excitation where geometric non-linearity is expected to increase. Aranha goes on to validate an algebraic approximation for the dynamic tension due to harmonic mooring excitation where “the dynamic equations can be derived ignoring the geometric non-linearity, writing them directly in terms of the (equilibrium) static geometric configuration”. This may have applications in approximating the performance impact of moorings. The important thing to note, however, is that geometric linearity can only be a reasonable approximation for limited amplitudes about a suitable reference static condition of the cable. Therefore, if performance calculations are desired for a device with numerous identical mooring cables, at an equilibrium position where the moorings provide differing static restoring forces, then the cables must still be linearised separately about their appropriate equilibrium conditions. There are already some limitations then, as the performance might be different under different steady loading (wind, current, tidal height, etc.) of the device as well as slowly varying performance due to drift forces, etc. The linearisation calculation may therefore need to be repeated at a few different device excursions, to see if this effect is large.

Superimposed onto the geometric non-linearity are non-linear damping effects due to viscous damping and friction between the cable and the seabed. The repetitive laying down and lifting off of chain from the seabed is also a significant source of energy loss because the chains kinetic energy is usually lost as it impacts on the seabed. All of these forms of damping are also non-linear. The best estimate of damping can be made by finding an equivalent linear damping coefficient such that the energy dissipated in one sinusoidal cycle is equivalent to that dissipated in the non-linear case. This approach is outlined by Liu and Bergdahl [6] and is applied in the frequency domain analysis of cables and was shown to compare well to time domain simulations. However, this equivalent damping coefficient will vary not just with respect to frequency but may also vary with respect to the excitation amplitude. This can invalidate the linear equation of motion.

In dynamic problems, moorings generally do not behave like rigid structures and have modal responses of their own, so the impedance depends greatly on the frequency of harmonic excitation. Johanning et al. [4] observed experimentally that quasi-static modelling methods are no longer valid once frequencies approach or exceed the cable's natural frequency or once elasticity becomes important. At low frequencies one can expect the mooring to behave like a spring, while as the frequency is increased, inertial and drag forces begin to dominate. Aranha [5] separates the problem into 2 parts, one where the mooring accommodates the imposed displacement by a change in geometry and a second, where it is accommodated by the elasticity of cable itself. The degree to which each feature dominates depends on not just the frequency, but the amplitude of motion and the static pretension of the cable. For example, if the motion induces dynamic tension amplitudes that approach the static pre-tension, then non-linear effects will become more pronounced. Frequency dependency is equally important but can be incorporated in the linear representation, provided that the relevant physical phenomena are included when calculating the frequency dependent impedance. However, the amplitude dependency has implications for the validity of including moorings as a linear impedance.

This was also studied by Webster [2] by analysing a non-dimensionalised simple catenary cable, looking at the variation of equivalent linear damping with parameters such as pretension, amplitude and excitation period. Webster notes that at small pretensions for a set period, the equivalent damping coefficient was constant with respect to non-dimensional amplitudes (amplitude/depth) from 0.02 to 0.04. However, as the pretension is increased and the dynamic tension becomes large relative to the static tension, the damping co-efficient tends to decrease for larger amplitudes, which invalidates the assumed linear relationship. Under normal power production, wave energy converters should have sufficient mooring compliance such that the induced dynamic tension is small relative to the pretension.

These features point to the following limitations in including mooring in a frequency domain performance analysis:

  • 1.

    The mooring linearisation will only be valid at a certain reference static configuration of the device and mooring system. The linearisation would need to be repeated in full to see the effects of different steady load conditions (wind, wave and current directionality, high, low tide levels) on performance.

  • 2.

    Linearisation is a valid assumption only for motion amplitudes that are small relative to the water depth and which induce load amplitudes that are small relative to the equilibrium pretension. This should generally be the case for performance wave amplitudes with partially preloaded moorings in intermediate water depths.

  • 3.

    Equivalent linear impedance can only be valid for a certain wave-induced motion amplitude. Using the same linear impedance for different amplitudes can introduce errors.

Section snippets

The generic wave energy converter

A generic OWEC is contrived for the purpose of this study. For ease of analysis it is based on a truncated vertical cylinder, for which an analytical hydrodynamic solution is readily available. Due to symmetry, the device is assessed in three degrees of freedom only: surge, heave and pitch (see Fig. 1). An idealised linear PTO damping of the device is assumed in the surge, heave and pitch degrees of freedom and the dampers are somehow connected to earth. The practical difficulties in creating

Mooring cables

Five mooring configurations will be considered for this device. These are sketched in Fig. 3. All 5 configurations are comprised of only 2 different types of cables. A mooring cable co-ordinate system is also defined in Fig. 4. It can be referred to the global co-ordinate system with, r, the position vector of its origin and an orientation angle, α, in the horizontal plane. The vertical axis is always parallel to the global heave axis. Note that the vertical direction in the cable co-ordinate

Selecting a suitable static configuration

Nominal and sensible equilibrium positions should be selected, that would commonly occur in operation. The condition of the moorings in the equilibrium condition can be found from a static analysis. In this study, a solution using the catenary equation is applied, based on the approach by Oppenheim and Wilson [14]. The static equilibrium condition for both cables was chosen such that they have a pretension with a horizontal component of approximately 10 kN. The results of the static analysis are

Sample results

The 5 mooring configurations (a–e) are devised to demonstrate the effects of cable type and attachment location. These are illustrated in Fig. 3 and the numerical values defining them are now given in Table 4. Configurations A and B are single point mooring configurations designed to illustrate the effect of weather-vaning with respect to incident wave direction. Configurations C and D are multipoint moorings comparing Cables 1 and 2 to see what effect adding a surface buoy will have on the

Single point mooring systems and pitch/surge performance

The results for configurations A and B are effectively for the same single point mooring but with opposite incident wave directions. Such an arrangement may be typical of moorings intended to self-align to prevailing wind or current. The results indicate a very large effect for weather-vaning. Configuration A is more likely to occur, with total environmental loads more likely to be closely aligned with the wave direction. This condition causes considerable performance losses in all degrees of

Conclusions

  • Evidence from literature study suggests that the use of an equivalent linear impedance for moorings is valid for the consideration of their influence on the power production of wave energy converters, provided it is understood that:

    • 1.

      The mooring linearisation will only be valid at a certain reference static configuration of the device and mooring system.

    • 2.

      Performance motion amplitudes referred to mooring attachment points are small relative to the water depth and induce mooring tensions that are

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