How Much Power can be Obtained from the Tides? Chris Garrett Dept. of Physics and Astronomy University of Victoria Patrick Cummins Institute of Ocean Sciences

“Conventional” tidal power (for a bay of surface area A s ) The maximum potential energy is If this is released instantaneously, the average power is This formula gives about 240 MW for surface area A s =100 km 2, a = 2m and semidiurnal frequency  There will be less power for realistic operating regimes. Examples: La Rance (France), Garolim Bay (South Korea).

There are also proposals to exploit strong tidal CURRENTS and at various locations in the Salish Sea

Uldolmok South Korea (Kang Sok Kuh)

Johnstone Strait, British Columbia Seymour Narrows 6 m/s currents Vancouver

For both basin and channel situations, we need both pressure head and flow. Power = Head X Flow Rate For a basin, the head is primary. For a channel, the flow rate is primary. But the two situations are more similar than sometimes realised.

For a basin, Garrett and Cummins (2004) considered replacing a conventional barrage with an array of fences of marine turbines. The power increases and then decreases as the number of turbines is increased (solid: linear drag; dashed: quadratic drag). For quadratic drag, the maximum power is 0.76 times the reference value of, or Maximum flow in undisturbed state

The power may also be expressed as a function of the reduced range in the bay. At maximum power, the tidal range is reduced by only 26%, so that water exchange is largely maintained (for aquaculture and pollution control). Linear drag Quadratic drag

How much power is available from a tidal channel between two large basins? The kinetic energy flux is commonly used, where A is the local cross-sectional area, and u the current. As Au is fairly uniform, this estimate is very sensitive to where it is evaluated. There is no reason to believe that this formula provides any indication of the maximum power that can be extracted.

Maximizing power is a problem in optimization Expected power Number of turbines No turbines = no power, but adding too many turbines blocks the flow and reduces the power output Model of a tidal channel

The governing equations may be simplified to The power generated at any instant is We want to maximize this.

An easy limit is for quasi-steady flow (drag dominates the acceleration) This is achieved with 2/3 of the original head loss now across the turbines and Q reduced to 58% of its natural value in the absence of turbines. the maximum average power is

In general, where  varies from 0.24 to 0.21 (with a dip to 0.196) as the regime changes from acceleration dominated, with the current lagging the head by 90 0, to friction dominated, with a phase lag of zero.  phase

Comments: The maximum power is independent of the location of the turbines along the channel, but assumes they are in “fences” across the whole channel with all the flow through the turbines. Allowance can be made for many tidal constituents. There may be corrections for long channels with non-uniform volume flux and for feedbacks which change the forcing. There is no general relation between our formula and the flux of kinetic energy, though for the special case of short channels dominated by exit separation: See Garrett & Cummins (2005) Proc. Roy Soc. A, 461, for more details. Speed at exit in natural regime, probably less than in the most constricted section.

Detailed numerical modelling by Graig Sutherland (UVic) and Mike Foreman (IOS) has been used to check assumptions and allow for splitting of the flow between Discovery Passage and Cordero Channel. Flow diversion reduces the potential of using just one channel. Johnstone Strait, British Columbia Our formula gives 1.32 GW using a = 2.11 m Q 0 = 311,000 m 3 s -1

Maximum power estimates (not additive) with tidal fences represented as additional friction in the 2D model. The current in the channel with turbines is reduced to about 57% of its natural value. For turbines in Cordero Channel or Discovery Passage, the current in the other channel increases by 14%. (Sutherland, Foreman, Garrett, J. Power & Energy, in press.) 1.34 GW 277 MW 401 MW

Limitations The power will be reduced by drag on the supporting structures and internal losses in the turbines. Flow will tend to bypass partial fences (laterally or vertically) and further head losses will occur downstream as flows with different speeds merge. (Recent analysis shows a loss of 1/3 to 2/3, depending on the fraction of the cross-section occupied by turbines.)

Conclusions A theoretical foundation for assessment of the power potential of tidal channels has been developed. Some fairly general formulae are available. Detailed numerical modelling studies seem to support the simple theory and are being used to extend it. Further progress will be aided by collaboration between oceanographers and engineers. Impacts on marine life and navigation require consideration. Tidal power can be useful, but it is not abundant.

Another use of strong, cold, tidal currents is to provide cooling water for nuclear power plants. 640 MW Candu reactor Point Lepreau World’s highest tides

The End