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SPECIAL ISSUE PAPER 181 Vertical-axis tidal-current generators and the Pentland Firth S H Salterà and J R M Taylor School of Engineering and Electronics, University of Edinburgh, Edinburgh, UK The manuscript was received on 1 March 2006 and was accepted after revision for publication on 13 November 2006. DOI: 10.1243/09576509JPE295 Abstract: This paper extends ideas presented to the World Renewable Energy Conference [1, 2]. One idea involves the impedance of flow channels and its relevance to the maximum tidal- stream resource. Estimates of the inertial and damping terms of the impedance of the Pentland Firth suggest a much higher resource size than studies based purely on the kinetic flux, because adding extra turbines will have less effect on flow velocities than in a low impedance channel. This very large resource has pushed the design of the turbine towards the stream velocities, depth, and seabed geology of this site. A second idea is an algorithm to control the pitch of close-packed vertical-axis generators to give an evenly distributed head. Finally, there are sug- gestions for a seabed attachment aimed specifically for conditions in the Pentland Firth and intended to allow rapid installation of a self-propelled tidal-stream generator. Keywords: vertical-axis turbine, tidal stream, marine-current generator, channel impedance, low-head hydro, variable pitch, Betz momentum theory, Darrieus, troposkien, tri-link, inflatable 1 BACKGOUND perform the function of a geometrically tolerant bearing providing axial and radial location. The The earliest field work on Darrieus-type underwater blade pitch was controlled only by the choice of a turbines was reported by Fraenkel [3]. It involved torque limit which allowed blades to adjust their testing models of fixed pitch vertical-axis rotors and own pitch-angle above a chosen level of pitch confirmed the performance in water to be compar- torque so as to prevent stall. Mooring was by tension able to results obtained with fixed-pitch vertical legs which passed through the centre of pressure of axis wind turbines. the rotor to avoid inducing pitching torque. How- Pitch variation was an essential feature of the Edin- ever, tank tests showed that tension legs would burgh vertical-axis tidal-current generator which was suffer unacceptable snatch loads following any slack- first described at the 1998 European wave energy ening, even in quite small waves. conference at Patras [4] and is shown in Fig. 1. The design had several layers of short blades with vari- able pitch. The layers were separated by elliptical 2 CHANNEL IMPEDANCE AND THE PENTLAND section rings and cross-braced by streamlined wires FIRTH which gave torsional and shear rigidity. Power take- off was above the surface and out at the rim, An important question for all tidal projects is the housed in a floating torus. It used a quad ring-cam, extent to which the introduction of generating variable-displacement, high-pressure oil pump to plant will reduce the flow. This can be thought of give the correct torque to convert from a variable in terms of the ‘impedance’ of the flow, a concept input speed to the constant speed required by syn- which is familiar to electrical engineers but less so chronous generators. The cam rollers could also to mechanical, marine, and civil engineers even though there are many instances where it can be ÃCorresponding author: School of Engineering and Electronics, useful. One anthropomorphic way to think of impe- University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JL, dance is as the determination needed for a current UK. email: [email protected] to overcome obstacles placed in its path. One of the JPE295 # IMechE 2007 Proc. IMechE Vol. 221 Part A: J. Power and Energy 182 S H Salter and J R M Taylor two is L Á C f (3) Z Á Cp If this ratio is unity, the power already being lost as bed friction will be equal to that of one bank of rotors filling the entire cross-section and the current status is on the edge of the change from a low to a high impedance channel. Note that aeronautical engineers and turbine designers use half for drag Fig. 1 An artist’s impression by Carn Gibson of the and power equations but oceanographers do not, first ideas for the Edinburgh vertical axis turbine and so care is needed in collecting values of friction coefficients. A friction value of 0.017, has been measured at the extremes would be a source of constant current Menai Strait by Campbell et al. [5]. If this is used in a which has to flow and which will create whatever very simplified model of the Pentland Firth with a head (or voltage) is needed to overcome all resist- channel length L of 23 000 m, a depth Z of 70 m, ances placed in its path. This is the tendency with and a turbine performance coefficient of 0.4, the long, shallow, rough flow channels where many LCf/ZCp ratio is 14 and so a channel of these dimen- banks of turbines might be installed in series up to sions is clearly a high impedance source. If the chan- some limit at which the increased upstream head nel width was 10 000 m, the seabed dissipation at a would cause flooding. Less desirable would be the flow velocity of 3 m/s would be 53 GW. Friction other extreme of a low impedance source where the would be setting the flow velocity and many banks flow could be stopped by an obstruction or easily of turbines would reduce it only a little. Perhaps find an alternative passage. one third of 53 GW could be captured giving an aver- In electrical engineering, the impedance of a age output well above previous estimates [6]. source would be the ratio of the open-circuit voltage The Pentland Firth has an irregular coast line, pits divided by the short-circuit current. In a non-linear going to below 100 m, shallows known as the Merry device, like a zener diode, the local impedance Men of Mey which generate eddies at the surface, a could be obtained from the local slope of the cur- 308 bend and two islands – Stroma and Swona – rent/voltage curve. For turbines a resistive impe- which must suffer large drag forces. The Menai fric- dance can be calculated from the turbine power tion coefficient of 0.017 is only four times more divided by the square of flowrate through the turbine than that of a highly polished wing of a fighter air- flow window or from the square of a water head craft at zero angle of incidence. divided by the power being generated. The effects The velocities of tidal streams have been studied of side channels could be calculated, just as in elec- by Bryden et al. [7], who has shown that they are trical network theory for series and parallel resistors. more complicated, turbulent, and variable than Ratios of turbine power to bottom friction power might be suggested by a first look at Admiralty tide are of particular interest. Consider a very simple tables. Work must be done to measure direction case of steady flow in a channel of length L, width and velocity texture. There is also a need to measure W, depth Z, and bottom friction coefficient Cf, impedance values in all flow channels and their side having a current velocity U of fluid density r comple- branches before making an accurate assessment of tely filled with rotors having a performance coeffi- the tidal stream resource. cient of Cp. The power at present being wasted by bed friction is 3 MEASURING IMPEDANCE 1 The driving function of the tides depends on the Pf ¼ Á r Á W Á L Á U 3 Á C (1) 2 f skewness, non-circularity, and precession of the orbits of the moon round the earth and their com- The power from a single full rotor bank would be bined orbit round the sun. Further complications are that large masses of water attract one another 1 by detectable amounts and that any non-linearity Pr ¼ Á r Á W Á Z Á U3 Á C (2) 2 p caused by propagation in shallow water will make every component generate frequencies which are By cancelling common terms, the ratio of the the sum and difference between its frequency and Proc. IMechE Vol. 221 Part A: J. Power and Energy JPE295 # IMechE 2007 Vertical-axis tidal-current generators 183 all the others. A useful listing of the components, level and velocity components for points at each periods, and relative phases of 18 components for end of the Pentland Firth (R. Protor, 2006, personal Delaware Bay is given in reference [8]. communication). The slope of the water surface Thevenin’s theorem says that an electrical network could be calculated from the difference in head pre- of any complexity can be reduced to a single voltage dictions. The eastward (conventionally known as U) source in series with a single impedance driving the component of velocity was almost the same at each load under study. The drive voltage would be what- end. The phase relationship of head to velocity was ever voltage would be present if the load under of particular interest. There are two ways to get an study were absent. Its analogy in water flows is the estimate. The easiest and crudest, suitable for engin- head difference that would be developed if a dam eers in a hurry, is to look at the times of zero cross- were built across the channel entry. The impedance ings and assume that the records are well-behaved in series with that head would be the total series/par- sine waves.

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