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Renewable Energy 57 (2013) 269e282

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Renewable Energy

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Numerical modelling of the effect of turbines on currents in a tidal channel e ToryChannel, New Zealand

David R. Plewa,*, Craig L. Stevens b,1

aNational Institute of Water and Atmospheric Research, PO Box 8602, Riccarton, Christchurch 8440, New ZealandbNational Institute of Water and Atmospheric Research, Wellington, New Zealand

a r t i c l e i n f o

Article history:Received 1 June 2012Accepted 3 February 2013Available online 1 March 2013

Keywords:Tidal energyTurbinesTory channelCook straitTurbine dragHydrodynamic model

* Corresponding author.E-mail addresses: david.plew@niwa.co.nz (D.R. Ple

(C.L. Stevens).1 Private Bag 14901, Wellington, New Zealand.

0960-1481/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.renene.2013.02.001

a b s t r a c t

Numerical modelling is used to assess the effect of a turbine array on tidal currents in the Tory Channel,New Zealand. The Tory Channel is the smaller of two entrances from Cook Strait to the Queen CharlotteSound with a large island separating the narrow Tory Channel from the main entrance. The 2D depth-averaged finite element model is validated against velocities from shipboard ADCP transects froma survey during spring tide conditions, and water levels recorded at the study site. Turbine drag isintroduced to the model as a stress term in the momentum equations, and includes both the turbinethrust and the structural drag. Turbine array drag is a function of the number and size of turbines, whichcan be parameterised in a non-dimensional number. This non-dimensional turbine drag number D canbe used to represent the drag of several different turbine designs. Restrictions are placed on the size ofthe array to ensure that turbines are placed in realistic locations. In this study, turbines are restricted toareas with water depths greater than 30 m, and where spring tide currents (in the absence of turbines)are greater than 2.0 m s�1, and consequently the turbine array does not span the entire channel width orlength. The modelling shows turbines will reduce current speeds both within the turbine array, and alsothroughout much of the Tory Channel, with local increases in speed immediately adjacent the array. Cut-in and maximum or rated turbine speeds are also incorporated to compare how these factors influenceboth the power production and effect on currents. The study shows that, due to the restrictions placed onthe array location, the likely power production that can be achieved is considerably less than what ananalytical prediction suggests might be obtained from the channel. Due to the effects of turbines oncurrent speeds, optimising the area occupied by an array is likely to be an iterative procedure. The powerproduced per turbine unit could be substantially improved, with little impact on total power produced bythe array, by removing turbines from areas where power produced was low. Turbine operational limits,applied in the form of cut-in speeds below which no power is produced, and design speeds above whichload shedding occurs, affect both the magnitude and spatial distribution of power production and thusneed to be considered in array design.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Tidal currents appear to be an attractive source of “clean” energy.Tides are predictable, more so than wind, river flow or wave con-ditions, and it is possible to forecast potential power generationyears in advancewith a reasonable degree of accuracy. The first stepwhen considering the potential of a site for tidal stream powergeneration is to assess the available resource through measure-ments of current speeds [1], or through numerical modelling [2e4].

w), craig.stevens@niwa.co.nz

All rights reserved.

However, it is recognised that the extraction of energy by, and thedrag from, in-stream turbines will affect currents [5e7]. At the timeof writing, there are only a few operational, individual, in-streamturbines. However, proposed tidal power systems will use arraysof turbines that could conceivably occupy large areas. Other exam-ples of arrays of structures introduced to the marine environment,such as lines and cages used in aquaculture, affect currents overlarge areas, andwell beyond the immediate boundsof the arrays [8e10]. Similar effectsmaybe expected for tidal turbine arrays. Turbine-induced changes in currents can affect not only the extraction ofenergy, but may also have ecological implications such as changingtransport of sediment [11] and suspended material [12].

Both analytical and numerical approaches have been used toassess the effect of turbine arrays on currents and water levels, and

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282270

to estimate the optimal number of turbines as well as the powerthat can be extracted. Analytical models can provide a useful firstestimate of the likely tidal resource, but they rely on simplifiedchannel geometry and generally a 1D assumption. In the simplestform of analytical model, turbines arrays are evenly spread over thechannel. These models demonstrate that increasing the number ofturbines will initially increase the total power extracted, until thedrag from the turbines slows currents to a degree that additionalturbines results in a reduction of total power extracted. The totalarray dragmust be optimised tomaximise power output [5,7,13,14].Other analytical models have considered arrays that span part ofa channel width [6,7,13]. Common to all analytical models is thatthe array drag is evenly distributed over all or part of the channel.

In contrast, numerical hydrodynamic modelling can providemore detailed information particularly where large spatial differ-ences in current speeds occur along or across a channel. To obtainthe highest level of detail about flow modification and powerproduction, it is possible to model each individual turbine in 2D or3D numerical models [15]. This approach requires knowledge ofturbine properties and also the exact location of turbines. Thecomputational costs of such a modelling approach are high, par-ticularly if different turbine numbers, locations or designs are to becompared, and it is likely that such detailed modelling would beconducted in the later stages of assessing a tidal energy extractionproject. An intermediate approach that can be implemented morereadily is to distribute turbine drag over the area covered bya proposed array [11,12,16,17]. This level of numerical modelling,that includes turbine drag, is likely to follow assessments of thetidal energy resource based on kinetic energy flux [2,3,18,19]. Thekinetic energy flux approach does not allow for the flow reductioncaused by the turbines and so will give inaccurate predictions ofpotential power production, but may be used to identify regions inwhich more accurate assessments are desired.

There is some variation in how the effect of turbine drag isincluded in numerical models. The most common approach is toadd a sink or drag term into the momentum equation[4,11,12,15,16]. However, this term has been parameterised in dif-ferent ways. In some studies the turbine drag term is estimatedfrom the extracted power [4,11], while elsewhere the drag is cal-culated from the turbine thrust and drag terms [12]. This secondapproach is more complete in that it allows for inefficiencies inpower extraction (not all of the energy removed from the flow goesto power generation, some is lost to turbulence), and may alsoinclude the drag from the structures supporting the turbine. Whileit is convenient to assume that drag and thrust coefficients areconstant, in many cases turbines may have minimum or cut-inspeeds or maximum design speeds, beyond which turbines eithercease operating or power output is limited [20]. Consequently therelationship between drag and velocity may vary over the velocityrange in which the turbine is placed.

In this study, numerical modelling is used to investigate thepotential of a channel for tidal stream power generation. Themodelling is based on the Tory Channel Entrance, located near thenorthern end of the South Island, New Zealand. The Tory ChannelEntrance has attracted the interest of tidal power developersbecause of its strong currents and proximity to infrastructure.

A unique feature of the Tory Channel is that it is formed by anisland, thus there is an alternative entrance for flow into the QueenCharlotte Sound. The goals of this study are to

� predict how increasing numbers of turbines will affect hydro-dynamics at the array/channel scale with flow being divertedaround the array

� and similarly at the Sound scale, whether flow is insteaddiverted to the other side of the island forming the channel

� estimate the power that could be extracted by different num-bers of turbines, and how this power varies over tidal cycles

� determine if incorporating limits to turbine operation (cut-inand design speeds) substantially alters the effect of the turbinearray on hydrodynamics and power production

In the Methods section of this paper, we consider the drag on, andpower produced by a single turbine. Further to this, we then examinehow cut-in and maximum design speeds affect the drag and power,and how these can be incorporated into a numerical model usinga non-dimensional turbine drag parameter that is independent ofturbine size. A numerical model and the study site it is applied to arealso described. In the Results, numericalmodel predictions of theflowin the absence of turbines are compared to field data, and then themodel is used to investigate how currents and power production areaffected over a range of non-dimensional turbine drag, both with andwithout turbine operating limits. Key findings, limitations of thepresent approachand improvements are considered in theDiscussion.

2. Methods

2.1. Turbine power and drag

With regard to modelling turbines, the two parameters of in-terest are the power produced by the turbine, and the thrust (drag)produced by the turbine. The power P produced by a turbine maybe expressed by the following equation:

P ¼ 12CPrAU

3: (1)

The thrust on the turbine T is

T ¼ 12CTrAU

2: (2)

A is the cross-sectional area of the turbine (generally taken as thearea swept by the blades),U the upstreamflowvelocity, r the densityof sea water, CP a power coefficient, and CT a thrust coefficient. Pub-lished values of the power and thrust coefficient for scale modelturbines are often derived from axial force and torque measured atthe turbine hub so do not include mechanical losses in generatorefficiencies (which reduce the power produced), or drag from themain body (nacelle) and supporting structures [21]. Thus, the totaldrag FD froma turbine should includeboth thrust and structural drag:

FD ¼ 12rCTAU

2 þ 12rCDAsU2

¼ rU2

2ðCTAþ CDAsÞ:

(3)

The first term on the right side of the equation above is thethrust, while the second term is the structural drag. CD is a dragcoefficient for the structural components, and AS the projected areaof the structural components. As described previously, in somemodels the turbine drag is parameterised from the power term asF 0D ¼ P=U ¼ 1=2CPrAU2. If this method is used, the drag will beunderestimated as the structural drag is not included, and wemight reasonably expect that the thrust coefficient is greater thanthe power coefficient (i.e. CT > CP) otherwise more power would beproduced than work done against the flow. This is also supportedby data for both horizontal and vertical axis turbines [21,22].

The power and thrust coefficients are influenced by design andoperation factors includingbladepitch, tip speed ratio andyaw. Someturbine designs may allow the blade pitch angle and yaw to beadjusted, while other designs may use a fixed blade and/or yaw toreduce mechanical complexity and the likely cost of maintenance.

Fig. 1. (a) Total force, thrust and structural drag on a 18.6 m diameter turbine pro-ducing a rated power of 500 kW at 2.0 m s�1, cut-in speed of 1.0 m s�1, and CT0 ¼ 0.80;(b) turbine drag components parameterised as an equivalent bed friction assuming anelement area of 390 m2. Bed friction is shown for comparison.

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282 271

As Equation (1) shows, the power extracted is proportional to thecube of velocity. At low velocities, very little power is produced, andturbines typically have a “cut-in” or minimum flow speed Uc belowwhich the turbine does not operate. Published values or estimatesfor the cut-speed are in the range 0.5e1.5 m s�1 [20]. Turbines mayalso have amaximumor rated power output PDwhich is obtained ata design speed UD. Above this speed, blade pitch may be adjusted tolimit power output to PD and protect the generator.

Shapiro [17] suggests that the thrust on a turbine is betterapproximated by a linear drag law than the quadratic drag parame-terisation used here. A generic turbine thrust coefficient of CT ¼ 7/Uhas been proposed for wind turbines if turbine specific data is notavailable [23]. This recommendation is based on wind turbine datashowing thrust coefficients decreasing with increasing velocity.However, the relationship between thrust coefficient and velocity isturbine design specific, and also depends on whether the turbine isstall or pitch controlled. For many turbines, particularly those withpitch control, the thrust coefficient is constant at low speeds, anddecreases at higher speeds [23,24]. The proposed generic thrust co-efficient is based on a stall-controlled turbine, yet only fits data wellovera limited rangewhere the thrust coefficient begins todecrease. Itis also intentionally conservative, over predicting thrust at high andlowwind speeds [23]. In our study, reductions in thrust coefficient athigh flow speed are included by altering the thrust coefficient inorder to limit power production to a rated power output.

As an example of a turbine and likely operation, we consider themodel three-blade turbine tested by Bahaj et al. [21]. They givethrust and power coefficients for a range of tip speed ratios, bladepitch, and yaw conditions. The power coefficient CP was dependentonpitch,with highest values obtained at a pitch angle of around 20�.The thrust coefficient decreases as pitch angle increases. In thisstudy, we assume that the pitch is fixed and the tip speed ratio isconstant for velocities between the cut-in speed Uc and rated speedUD, and that these parameters are chosen to maximise power out-put. Consequently the power and thrust coefficients are constant forvelocities between Uc and UD. These constant values are denoted asCP0 and CT0 for the power and thrust coefficients respectively. BelowUc we assume that the turbine does not produce power and thepower coefficient Cp ¼ 0. If the velocity exceeds UD, we assume thatthe pitch is increased to restrict power output to the rated power PD.The power coefficient for U > UD can then be estimated as

CP ¼ 2PDrAU3: U > UD (4)

As pitch is increased, the thrust coefficient also decreases. Whileit is possible to develop a relationship between CP and CT from testdata [21], for simplicity we assume that the ratio is constant. Theeffect of this approximation is that the efficiency of the turbineremains constant; the energy lost to turbulence is constant relativeto the power produced. Bahaj et al. [21] suggest design values ofCP0 ¼ 0.45 and CT0 ¼ 0.80 (Uc � U � UD) which we use henceforth.The thrust coefficient is then parameterised as

CT ¼ 0; U < Uc: (5)

CT ¼ CT0; Uc < U < UD: (6)

CT ¼ CT0CP0

2PDrAU3; U > UD: (7)

The rotor diameter required to obtain the rated power as a func-tion of velocity can easily be determined. To achieve the same ratedpower at lower speeds requires greater turbine diameter, which re-sults in greater thrust to obtain the samepoweras the design speed is

reduced. It is clear that large turbinediameters are required to extractsignificant power, especially at low velocities. Longer blades requiremore material, and will also experience greater forces, so will bemore expensive to construct. Furthermore, water depth will placea restriction on the maximum physical size of a turbine.

The turbines require some form of structural support. We as-sume that the area of the structural supports and housings are 10%of the turbine swept area. Furthermore, the structural supports areassumed to be represented by a circular pile. Using a realisticdiameter of 2.0 m and flow velocity of 2.0 m s�1, the Reynoldsnumber of 3.6 � 106 indicates that a smooth cylinder would havea drag coefficient CD w 0.6. However, any structure in a marineenvironment will rapidly obtain surface fouling, so a more con-servative value of CD w 0.9 is assumed.

2.2. Parameterising drag in a 2D FE model

In the majority of studies of turbine arrays, the drag (and thrust)from the turbines is distributed over the area covered by the tur-bine array [4,11,12]. In high resolution models, it may be possible toapply the turbine momentum sink terms in the actual turbine lo-cations [15]. However, this level of detail may not always be prac-tical or required, particularly if exact turbine locations or design arenot known, or if the model grid elements are larger than the tur-bines. In this study, we distribute the drag over the area of the array.

Turbine drag is added to the model as a body force term in themomentum equations. In the 2D finite element model used here,the turbine drag is parameterised as a friction coefficient in a sim-ilar form to bottom friction. The turbine drag force is distributedover the area Ae of the element containing the turbine, and con-sequently is a stress term.

For a grid elementwith a single turbine, the turbine stress term is

sturb ¼12rCTAT jUj þ

12rCDAsjUj

AeU

¼ ðCTAþ CDAsÞ2Ae

rjUjU:(8)

Fig. 2. (a) Location map showing location of study site in New Zealand, (b) Queen Charlotte Sound, and the Tory Channel formed by Arapawa Island, the stars indicate points wherewater levels are compared in Fig. 5; (c) locations of points where field data and model time-series data are compared (yellow squares) overlain on bathymetry with the ADCP vesseltrack shown in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282272

More generally, if the number of turbines is expressed asa spatial density n (the number of turbines per m2), then

sturb ¼ nðCTAþ CDAsÞ

2rjUjU: (9)

The thrust coefficient is a function of velocity while the struc-tural drag coefficient can either vary with Reynolds number, orassumed to be constant as is done here.

As an illustration, the force and stress terms (in the form ofa friction coefficient) that would be obtained for a turbine witha blade diameter of 18.6 m, cut-in speed of 1.0 m s�1, rated (max-imum) speed of 2.0 m s�1 are shown in Fig. 1. A turbine of this sizewould produce 500 kW at 2.0 m s�1. Stress terms have been cal-culated assuming a single turbine in an equilateral triangular gridelement with sides of w30 m, or 390 m2. The turbine structuraldrag is constant, while the thrust is zero below the cut-in velocity,

constant between cut-in and the design velocity, then decreases asthe velocity increases further (Fig. 1b). Consequently the greatesteffect on flow is likely to occur when water velocities are betweencut-in and the design velocity.

By fixing the structural drag as a proportion of the swept area,i.e. CsAs¼ aCT0A, the additional turbine stress term can bewritten as

sturb ¼ DrjUjU; (10)

where the dimensionless turbine drag parameter D is defined as

D ¼ 12nðCTAþ CsAsÞ z1

2ðCT=CT0 þ aÞnCT0A: (11)

Larger values of D represent greater drag from turbines, whichcan result from amore dense array (n), larger turbines (A) or greaterthrust (CT0). A range of turbine sizes and densities can be modelled

Fig. 3. Contours of simulated maximum velocities at spring tide in the Tory Channelentrance in the absence of turbines. The black envelope shows the area wherevelocities > 2 m/s and depth > 30 m.

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282 273

through the choice of D. With regard to the power produced perunit channel plan area Pe,

Pe ¼ 12CPrnAU

3 ¼ DCT=CT0 þ a

CPCT0

rU3: (12)

This equation holds for all velocities as CP and CT are altered if Uis less than the cut-in speed or greater than the design velocity.

From here onward in this study, we consider two forms of tur-bine parameterisation. In the first form, we ignore cut-in andmaximum or design velocities. This means the turbine has nomaximum rated power and the blade pitch is not altered to shedload. The CP and CT are held constant. In the second scenario, theturbines are given cut-in speeds of 1.0 m s�1 and design (max-imum) velocities of 2.0 m s�1. At higher speeds the power andthrust coefficients are reduced to limit the output power, andCT ¼ CP ¼ 0 below the cut-in speed.

2.3. Study site

In this study we simulate the effect of turbines located in the ToryChannel, which is the southern arm of the Queen Charlotte Sounds,

Table 1Turbine drag parameters, and number the number of turbines represented by thatdrag parameter in the area identified in Fig. 3.

D Total arrayrated power(MW)at 2.0 m s�1

Turbine diameter and turbine rated power at 2.0 m s1

5 m (36 kW) 10 m (145 kW) 18.6 m (500 kW)

Equivalent number of turbines

0 0 0 0 00.001 5 138 35 100.002 10 276 69 200.005 25 690 173 500.01 50 1380 345 1000.02 100 2760 690 2000.04 200 5521 1380 4000.06 300 8281 2070 6000.08 400 11,042 2760 8000.10 1000 13,802 3451 1000

South Island New Zealand (Fig. 2). The Tory Channel is 16.8 km inlength, has an averagewidth of 1.1 km, and the centreline depth variesfrom 40 to 65 m. The average depth in the channel (excluding sidearms) is 39.2 m. The strong tidal currents in the Tory Channel area consequenceof theoceanographyof theCookStrait region. TheCookStrait separates the North Island and South Island of New Zealand.Together, these islands present a northesouth oriented barrier ofapproximately 1400 km to the prevailing west to southewest windsand oceanic currents [25]. At its narrowest point, the Cook Strait is24 kmwide. On the north-western side, the bathymetry is relativelyflatwithdepthsdownto100m.On the south-eastern side, theheadofthe Cook Strait Canyon descends rapidly to 1000 m, and joins the3000 m deep Hikurangi Trench. There are complex interactions be-tween the strong tidal and wind forcing and regional oceanography[25e27]. The largest tidal constituent in the region is the lunar semi-diurnal M2 tide which has a phase difference of 145� (5 h) across theCook Strait, of which 100� occurs over the 40-km narrowest section[28,29]. This leads to large currents through Cook Strait, complicatedby strong winds funnelled between the islands and density gradients[25]. There are similarly large phase differences in theN2 and P1 tides.The M2 tidal wave propagates in from either end of the Cook Strait,creating a standing wave component with a node near the narrows.This node appears as a virtual (degenerate) amphidrone near Well-ington at the south end of the North Island [29].

The Queen Charlotte Sound has two entrances to the Cook Strait,thenorthernmainarmwhich is9kmwide, and theToryChannel. Thephase difference in the M2 tide is 50� between these two openings.

2.4. Numerical model

The hydrodynamic model RICOM [30] is used to conduct depth-averaged simulations of tidal currents. This finite-element modelsolves themomentum and continuity equations on an unstructuredgrid with triangular elements, semi-implicit time-stepping, anda semi-Lagrangian advection scheme [31]. Further details and de-scriptions of the numerical schemes used in the model are descri-bed elsewhere [32,33]. The model grid used in this study coversa domain of w490 � 210 km, spanning the greater Cook Strait re-gion in order to reproduce water levels and currents within CookStrait, as these drive the flows through the Tory Channel asdescribed above.

The grid used in this study is based on that used byWalters et al.[25] but with further refinement in Queen Charlotte Sound and theTory Channel. The grid is composed of triangular elements, gen-erated using the software described by Henry and Walters [34]following the procedure described by Walters et al. [25]. Thelength of triangle edges varies from approximately 25 m in theentrance to the Tory Channel, to 4 km on the open ocean bound-aries. The grid contains 125,635 nodes and 240,430 triangular el-ements. Bathymetry is derived from several sources includingswath bathymetry and digitizing of contours and soundings fromthe Land Information New Zealand Hydrographic Charts (NZ 463,4633, 614e615, 6142, 6151e6154, 6212).

Tides were applied by specifying the water level at the openocean boundary points, and these were calculated from tidal har-monic constituents derived from a regional tidal model [29]. Sim-ulations were started with an initially flat surface with zerovelocities. Tidal amplitudes at the boundaries were ramped up overa 6 h period to reduce start-up transients, and simulations run for 6full M2 tidal periods (6 � 12.42 h) before output to allow for themodel to spin up. Validation simulations for comparison againstmeasured water level and velocity data (see Section 2.5) wereconducted using 7 tidal constituents (M2, S2, N2, K2, K1, O1 andP1), which were applied at the open boundaries. For all othersimulations, currents over a representative spring-neap cycle were

Fig. 4. The upper left panel showmodelled and observed water levels in Okukari Bay, near location E (see Fig. 2). The tidal component of the observed water level is also shown. Theupper right panel shows spectra of the water level time-series for the observed and modelled data. The lower 6 panels show comparison of depth-averaged velocities (east andnorth components) from the numerical model (black line) with ADCP measured data. The red dots show all data points recorded within 150 m of the point where model data wereoutput, while the blue circles show the average of all ADCP data within 150 m of the comparison point at a similar time. The location of each comparison is indicated by the letter inthe upper left of each panel. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282274

simulated using only the two largest semi-diurnal tidal constitu-ents, the lunar M2 and solar S2 tides.

To ensure that turbines are placed in realistic places, the fol-lowing criteria are used to identify the area in which turbines willbe placed:

� Depth > 30 m to ensure the turbines remain submerged withclearance for shipping at low tide.

� Maximum spring tide velocity in the absence ofturbines > 2.0 m s�1, so that turbines are likely to be operatingat or near capacity for at least part of the time.

The area where turbines are located is selected by first runningthe model with no turbines, and recording the maximum waterspeed over a tidal cycle at spring tide. A “representative” spring tide

is selected by using only the M2 and S2 tidal constituents andrunning the simulation over a 12.42 h period where the M2 and S2constituents are in phase, producing the maximum tidal range.Other tidal constituents will combine to modulate the tidalamplitude at spring tide and are neglected in this analysis.

The area within the Tory Channel where both the depth andvelocity criteria are met covers 1.204 � 106 m2, and is shown inFig. 3. An indication of the available power resource can be esti-mated by averaging P ¼ 1=2rU3 over the area of the turbine array,giving the peak spring tide power density per unit area of turbineswept area as 7650 W m�2.

The model scenarios run in this study cover a range of0< D< 0.50, both with and without turbine operation limits. Theselimits are cut-in speeds of 1.0 m s�1 andmaximum design speeds of2.0 m s�1. The non-dimensional turbine drag parameters

Fig. 5. (a) simulated water level at the Queen Charlotte Sound entrance, Tory Channelentrance, junction between the Tory Channel and Queen Charlotte Sound, and in theinner sound at the locations indicated in Fig. 2(b), (b) difference in water level betweenthe junction and the other three sites.

Fig. 6. Time series of (a) the difference in water level between the Tory Channel entrance anentrance (positive values are flows into the channel) over a spring tide as a function of the dthe RMS flow with turbines to the RMS flow without turbines (black solid line), and the peakthe references to colour in this figure legend, the reader is referred to the web version of t

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282 275

investigated in the model, along with the total array rated power at2.0 m s�1, and number of turbines of various sizes that are repre-sented by these drag parameters are given in Table 1.

2.5. ADCP survey

Measurements of currents in and around the entrance to theTory Channel were obtained with a ship-board acoustic Dopplercurrent profiler (ADCP) over the period 17:00 2 Mar 2006 to 8:00 3Mar 2006. During this time, the ship made 12 circuits around theentrance of the Tory Channel (Fig. 2c). The ADCP data were recor-ded at 15 s intervals and averaged into depth bins of 4 m. The po-sition of eachmeasurement was recorded by differential GPS. Thesedata were used to validate the numerical model simulating flow inthe absence of turbines. A temporary water level recorder was alsoinstalled in Okukari Bay, near location E in Fig. 2c, from 21 Feb 2006to 22 Apr 2006. Water levels were recorded at 5 min intervals, andare used as a further validation of the numerical model.

The model was run over the period 24 Feb to 10 Mar 2006 using7 tidal constituents (M2, S2, N2, K2, K1, O1 and P1) but wind andbarometric forcing was not included. Velocity predictions from themodel were compared at the locations plotted in Fig. 2c. To com-pare with the model data, the ADCP measurements were firstdepth-averaged, and then all measurements obtained within 150m

d the junction with Queen Charlotte Sound, and (b) flow rate through the Tory Channelimensionless turbine drag parameter D (values given in the legend); and (c) the ratio ofpower produced by the array (dashed red line) as functions of D. (For interpretation ofhis article.)

Fig. 7. Change in current speed averaged over a spring tide for different turbine array densities (a) D ¼ 0.005, (b) D ¼ 0.020, (c) D ¼ 0.060, and (d) D ¼ 0.10.

Fig. 8. Comparison of predicted power production from an array with D ¼ 0.01,equivalent to 100 turbines rated at 500 kW at 2.0 m s�1, at spring tide when turbinedrag is included or not included. The graph also shows the effect of including a cut-inspeed of 1.0 m s�1 and design speed of 2.0 m s�1 (op. limits).

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282276

of a comparison point were extracted. Data points within the 150mradius recorded within 0.1 h were time-averaged. The 150 m dis-tance allows for slight variations in the ship track between circuitsand also provides some time-averaging. Differences between tracksof each circuit are partly due to the difficulty of operating a vessel instrong currents and a confined area with other shipping present.

3. Results

3.1. Model validation

A comparison between observed and simulated water levelsover the period 24 Feb to 10 March 2006 is shown in the top leftpanel of Fig. 4. The simulation was forced with only 7 tidal con-stituents, and not with wind or pressure. Consequently the modeldoes not reproduce the effects of weather patterns. This is clear inspectra (top right panel of Fig. 4) where the observed data showhigher amplitude at subtidal frequencies. A tidal decompositionwas performed on the observed water level time series to extractthe tidal signal. The residual water level (after subtracting the tide)tracks closely with the inverse of barometric pressure, with higherfrequency oscillations that may be due to large scale turbulence inthe channel, resonance of long waves, and noise resulting fromsurface waves. The standard deviation of the difference betweenthe extracted tidal water level and the simulated data is 5.2 cm,with an R2 of 0.94. The length of the observed water level record

Fig. 9. (a) Total power produced by the array over a spring tide as a function of turbinedrag parameter D (see legend), (b) power produced normalised by the total turbineswept area.

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282 277

meant that some of the tidal constituents applied in the modelcould not be resolved. For example the frequencies of the P1 and K1constituents, which both have small amplitudes, are too close forthem to be separated in a least-squares harmonic decomposition[35]. Some of the differences between the simulated time-seriesand the extracted tidal time series can therefore be attributed todifferent tidal constituents being extracted from the observed datacompared to those applied in the model. The time-series from thesimulation also contains fluctuations at frequencies higher than thetidal components, similar inmagnitude to the observed data, whichare attributed to turbulence and long-wave reflections within thechannel (top right panel Fig. 4).

Comparisons between modelled and observed velocities showthat the model generally agrees well with observation (Fig. 4). Datafrom a selection of the sites shown in Fig. 2c are plotted in Fig. 4simply to save space, but are representative of all sites. Thegraphs show each depth-averaged measurement from within150 m (red dots) and the average of these measurements at a sim-ilar time (blue circles). Standard errors (rms differences betweenmeasured and predicted velocities) are calculated for the averagedobservation data (blue circles), and has a mean value of 0.22 m s�1

averaged across both velocity components at all 11 sites. Site Arepresents flows in the entrance to the channel, site F representflows near the sides further within the channel, and site I repre-sents flows outside the channel.

Fig. 10. Comparison of power produced over a spring tide by eve

Some of the variation between observed and modelled veloc-ities may be due to strong lateral gradients in velocity, such thatmeasurements from up to 150 m away don’t accurately representvelocities at the comparison point. Other differences may be due toeddies not fully captured in the model. For example at A there arestrong fluctuations in the east-velocity component between 2 and6 h which may be due to eddies advected through the channel. Themodel also does not include meteorological forcing. While thesurvey was conducted during locally calm conditions, remnant ef-fects or more distant conditions may have affected flows throughthe Cook Strait.

A consequence of the difference in phase of the dominant M2constituent between the Queen Charlotte and Tory Channel en-trances (Fig. 5a) is that over much of the tide flows either side ofArapawa Island are in opposite directions. This is illustrated bycomparing the difference inwater level between the two entrancesand the junction (Fig. 5b). The water levels at the junction and theQueen Charlotte Sound entrance are nearly in phase such thatdifferences in water level, which drive the tidal currents, are small.However, the water levels at the Queen Charlotte Sound entranceare generally lower than at the junction when the water levels atthe Tory Channel entrance are higher and vice versa. Thus, flow intothe Tory Channel occurs while flow is out of the Queen CharlotteSound.

3.2. Effect of turbine drag on water levels and currents

Introducing turbines in the model has only a minor influence onthe difference inwater level through the Tory Channel (Fig. 6a). Theturbines have an increasing large influence on current speeds,particularly within the array area, as D is increased. Turbine draghas a relatively minor effect on flow for low D, becoming significant(greater than 2% reduction in root-mean-squared flow over a springtide) only for D > 0.01. Flow rates during spring tide into theentrance of the channel are shown in Fig. 6b, with positive values asinflows. Root-mean-squared (RMS) flows over this period areplotted as a function of D in Fig. 6c. At spring tide, as D increases to0.10, the peak flow is reduced by 27% and RMS flow over the 12.42 htidal cycle is reduced to 81% of the no-turbine situation. There arealso small phase shifts with the flow advancing as D increases(Fig. 6b).

Increasing the number of turbines has the effect of slowingcurrents within the turbine array, while causing some increase inmean speed (averaged over a 12.42 h tidal cycle) around the outsideof the turbine area (Fig. 7). Reductions in mean speed extend well

nly distributed arrays with (a) D ¼ 0.005 and (b) D ¼ 0.06.

Fig. 11. Comparison of power distribution as power per unit turbine area averaged over 12.42 h at spring tide for (a) D ¼ 0.002 and (c) D ¼ 0.02 with no cut-in and design speeds, (b)D ¼ 0.002 and (d) D ¼ 0.02 with cut-in and design speeds.

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282278

beyond the area of the turbines. There are reductions in currentspeeds along the length of the Tory channel, as well as outside andparticularly to the south of the channel. There are also smallchanges in mean current speed in Queen Charlotte Sound to theNorth of Arapawa Island, indicating that turbines in the ToryChannel can have some effect on hydrodynamics throughout theQueen Charlotte Sound region.

3.3. Effect of turbine drag on power production

To demonstrate the importance of including turbine drag whenestimating power production, a simulation with D ¼ 0.01, which isthe equivalent of 100 turbines producing 500 kW turbines at2.0 m s�1, was run with the turbine thrust and structure dragneglected. Fig. 8 illustrates that if turbine drag is not included, thenpeak and average power production will be overestimated. Alsoplotted in Fig. 8 are the power produced by the same array with andwith-out drag, but including a cut-in speed of 1.0 m s�1 anda design speed of 2.0 m s�1. At speeds below 1.0 m s�1, no power isproduced and the thrust coefficient set to zero. At velocities greaterthan 2.0 m s�1, the power and thrust coefficients are reduced tolimit the output power. Including the cut-in and design speedsreduces the total array power throughout the tidal cycle. This is dueto both load-shedding at high velocities where power production isrestricted by altering the turbine blade pitch, and ceasing power

production at low velocities. For this array density, more power isproduced on the flood tide than the ebb tide.

The total power produced by the array increases as D increases(representing an increase in the number of turbines) up to aroundD ¼ 0.06 before decreasing (Fig. 6a). At values of D up to 0.02 thereis more power produced on the flood tide compared to the neap,while at higher values of D the total power is similar on flood andebb tides (Fig. 9a). This asymmetry is more pronounced if the po-wer is normalised per unit area of turbine (Fig. 9b). This normal-isation is done through

PA

¼ ParrayNA

¼ ParrayCT0ð1þ aÞ2DAarray

(13)

where Parray is the total power produced by the array, Aarray thechannel area occupied by the array, A the turbine swept area asbefore, and N the number of turbines in the array. The powerproduced per turbine can be calculated by multiplying P/A by thearea of the turbine. This normalisation also shows howeach turbineproduces less power as the number of turbines is increased.

As well as varying over time, the power production varies spa-tially over the array. In Fig. 10, the mean power produced (overa spring tide) per unit channel area are plotted for D ¼ 0.005 and0.06; with a power coefficient of CP ¼ 0.45. These dimensionlessturbine drag parameters are representative of arrays of 50 and 600

Fig. 12. Comparison of (a) total power produced by turbine array and (b) power produced per unit area of turbine for an array with D ¼ 0.01 within the full area where depth > 30 mand spring tide velocities (without turbines) exceed 2.0 m s�1, and over a reduced area where mean power production over a spring tide > 250 W m�2 turbine area. The solid linesshow power production where there are no limits on turbine operation, and dashed lines show where a cut-in speed of 1.0 m s�1 and power limiting at 2.0 m s�1 have been applied.

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282 279

turbines designed to produce of 500 kW at 2.0 m s�1. Power pro-duction is greatest in the entrance to the channel, and generallydecreases to the west moving into the channel. As D increases(more turbines or larger turbines), the spatial difference in powerproduction becomes greater with a slight reduction in power pro-duced in the western portion of the array, and more produced inthe eastern part, particularly at the channel entrance.

Note that Fig.10 shows power produced per unit channel area. Asthenumberof turbines is increased, thepowerper turbinedrops, andthis is seen more clearly in Fig. 11a, c where the mean power pro-duction is shown per unit turbine area using Equation (13). It be-comes more apparent that as turbine numbers increase, turbines inthe western portion of the array produce very little power (Fig. 11c).

3.4. Effect of turbine cut-in and design speeds

Incorporating the effects of cut-in and load shedding at highcurrent speeds has further effects on the spatial distribution ofpower production (Fig. 11b, d). For low values of D (low numbers ofturbines or smaller turbines), the main effects are a reduction inmean power in the areas where power production is high due topower production being restricted when velocities are higher thanthe design speed. There are also small reductions in power pro-duced towards the western edge of the array in regions wherepower productionwas already lowdue to no power being producedwhen velocities are below the cut-in speed (Fig. 11a, b).

For higher values of D (more or larger turbines Fig. 11c, d), thereare small changes in high power production regions as velocitiesseldom exceed the design speed. However, even less power isproduced in the western portion of the array as the effect of turbinedrag reduces velocities such that they are seldom above cut-inspeed in this portion of the array.

If we consider the 500 kW turbine design described in Section3.3, 34% of a 20 turbine array (D ¼ 0.002, Fig. 11b) average at least50% of their maximum rated capacity, while only 10% of a 200turbine arraywill average 250 kW. Only 107 of the 200 turbines willproduce any power at all over a spring tidal cycle because velocitieshave been reduced below the 1.0 m s�1 cut-in speed.

3.5. Improving efficiency by reducing array size

As described above, for larger D values (high numbers of tur-bines), a portion of the array produces little or no power. Wetherefore select a third criterion for locating turbines (in addition tothe un-affected spring tide maximum velocity > 2.0 m s�1 anddepth > 30 m) that power production averaged over a tidal periodat spring tide > 250 W per m2 of turbine area. For the 18.6 mdiameter turbine described in Section 3.3, this equates to 180 kWaveraged over a spring tide. After applying this third criteria, thearray areas for non-dimensional turbine drag of D ¼ 0.001, 0.010and 0.060 are 1112 � 103 m2, 734.7 � 103 m2 and 172.9 � 103 m2

respectively. As an indication of turbine numbers, this equates to 9,61 and 86 of the 18.6 m diameter turbines (500 kW at 2.0 m s�1).

Fig. 12a shows the effect of reducing the array size to areaswhere mean power per turbine unit area >250 W per m�2 forD ¼ 0.01 at spring tide (12.42 h). There is a small reduction in thetotal power produced if no cut-in and maximum speeds are con-sidered (Fig. 12a). Averaged over the 12.42 tidal period, the fullarray averages 9.39MWwhile the reduced array averages 8.27MW.However, the power produced per turbine is substantiallyincreased. Expressed as power produced per unit area of turbine(Fig. 12b), average power is increased from 669 W m�2 from theoriginal or full array to 986 W m�2 for the reduced array. In con-trast, if the turbine cut-in and design maximum speeds areimposed, then reducing the array size as described causes a slightincrease in total array power (averaged over spring tide: 7.46 MWfull array, 7.59 MW reduced array). The increase in power per tur-bine unit area is even more significant (averaged over spring tide:276 W m�2 full array, 907 W m�2 reduced array).

3.6. Power production over a spring neap cycle

Power values averaged over a spring tide are biased by thehigher velocities at spring tide. A more realistic assessment of theaverage power output is obtained by extending the simulations tocover a spring neap cycle. The variation in power production overa spring-neap cycle is investigated by running simulations for

Fig. 13. Exceedance probability distributions of (a) total array power and (b) power per unit area of turbine over a spring-neap cycle as a function of turbine array drag parameter Dboth with (dashed lines) and without (solid lines) applying operating limits of a cut-off speed of 1.0 m s�1 and power limiting at 2.0 m s�1.

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282280

354.9 h (14.79 days) because of the dominance of the M2 and S2tidal constituents.

Probability distributions of array power and power per turbineunit area over a spring-neap tidal cycle are shown for array den-sities of D ¼ 0.001, 0.010 and 0.060 in Fig. 13. As might be expected,the total power produced increases with D, although the effects ofcut-in and power-limiting are only obvious at high and low ex-ceedance probabilities. The power per turbine unit area is similarfor all arrays, although the more sparse D ¼ 0.001 array produceshigher peak power. The average power outputs over the spring-neap cycle are 1.1 MW, 5.9 MW and 7.54 MW with no cut-in anddesign speeds; or 0.88 MW, 5.3 MW and 7.1 MW with cut-in anddesign speeds applied for D ¼ 0.001, 0.01 and 0.06 respectively. InSection 3.5, it was noted that when turbines had cut-off and designspeeds, reducing the array size increased the total array power. Wealso observed that the peak power produced by the reduced higherdensity array (D ¼ 0.06) is greater when the turbines have loadshedding at design speeds (29 MW) than without load shedding(27 MW). This is due to the drag reduction as load shedding occurs(Fig. 1a) resulting in higher velocities on average throughout thearray.

The spring-neap cycle has been simulated using only 2 tidalconstituents. With the inclusion of more tidal constituents, therewill be larger variations in spring-neap cycles, and we expect thatour method has produced power estimates that are biased towardslower speeds. With the cubic relationship between flow speed andpower output (below design speed), average and peak power maybe higher if more tidal constituents are included and simulationsconducted over longer periods.

4. Discussion

4.1. Feasibility of tidal power schemes in the Tory Channel entrance

New Zealand presently has a relatively high level of renewableproduction (w70%) through lake/river hydro power, geothermaland wind infrastructure. However, with increasing societal resist-ance towind and hydro development, new renewable resources are

likely to be required if national targets around greenhouse gasemissions are to be met. One of the major hurdles to overcome forany development of the tidal resource is to transport the powerproduced from the generation site to the point of use. There arepotential local power users with the island community on ArapawaIsland as well the local Marlborough Regional District (populationw40,000, i.e. 1% of the national population) and its associated in-dustries (aquaculture, viticulture and aviation). Furthermore, thearray location is 15 km from a transmission station for the nationalHVDC (high voltage direct current) network that connects theSouth Island hydroelectric dams with the more populous NorthIsland via a submarine cable across Cook Strait. The North Islandterminus of the HVDC cable, 27 km to the east, is at the southernextent of the 140 MW Westwind wind turbine array that becameoperational in 2009. Both the terrestrial and submarine cableroutes are rugged and will require sufficient investment in cableprotection for them to operate reliably.

Other challenges to any array installation include navigation andecological impacts. The channel is heavily used by ferries. There arehowever periods when the ferries do not operate, as well as a po-tential alternate route around the other side of Arapawa Island.Furthermore the ferry operators have at various times considereda terminus that would bypass the channel altogether. From anecosystem perspective, marine mammals are common in the re-gion. However, the most endangered of which, Hectors/Maui dol-phins are not found in the Tory region. The region is already highlymodified in terms of structural substrate with a recent history ofboth shellfish and fish-cage aquaculture activity.

As mentioned in the introduction, a number of developers haveconsidered options for the Tory Channel Entrance. The lead pro-posal has envisaged somewhere between 25 and 50 one MW ratedturbines. Assuming the same design speeds (2.0 m s�1), thrust andpower coefficients (0.80, 0.45) as used in this study, this equates toa dimensionless turbine drag in the range ofD¼ 0.005 to 0.010. Thisstudy suggests that the larger array of 50 turbines is below thenumber require to extract the maximum possible power(D w 0.06), yet it is unlikely that all of the turbines will operate atfull capacity. The maximum total power for the 50 turbine array

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282 281

expected on an average spring tide is likely to be around 21 MW, orless than 50% of rated capacity, and average around 7e8 MW.Wholesale electricity spot prices in New Zealand typically rangebetween 20 and 200 $NZ per MW h. Assuming the average isaround $80 MW h, the array would likely generate $4-$6M NZ perannum. It seems unlikely that it would be economic to constructand maintain a 25-50 turbine array without tariff incentives ora reduction in operational costs.

By New Zealand standards, the tidal currents in the ToryChannel Entrance may be considered to be fast. Yet due to thereduction in velocities caused by the turbines, any successful large-scale development will require turbines that are optimised toproduce power at low velocities, withstand high current speeds, yetstill be produced andmaintained cheaply. This remains the primarychallenge facing the designers of marine current turbines.

4.2. 3D effects

Implicit in the use of a 2D-depth averaged model is that currentspeeds at the turbines can be approximated with the depth-averaged value. The current speed at the turbine may differ fromthe depth-averaged value depending on the depth of the turbine. Ifthe turbine is in the boundary layer, there may also be a strongshear gradient across the turbine. Stevens et al. [1] suggest turbinedesigns that place the turbine above the boundary layer may bedesirable, but note that even in regions with strong tidal currentsthere can be considerable vertical variability in profiles of velocityand turbulence that do not resemble a simple boundary layer flow.In addition, large arrays of turbines may act like submerged can-opies and produce a strong shear gradients in both the vertical andhorizontal, as well as promoting large-scale turbulent flow struc-tures [36]. Further work is required to either parameterise thesethree-dimensional effects into depth-averaged models as has beensuggested for other forms of artificial canopies [37], because of theirrelative speed and convenience, or into more complex three-dimensional models [4] in a manner that more accurately ac-counts for their position in the water column. The dimensionlessturbine drag parameter D introduced in this study allows variousturbine arrays to bemodelled without requiring specific knowledgeof the size and capacity of the turbines. While the current form of Dis intended for use in 2D-depth averaged simulations, it could bemodified to represent a force per unit volume for use in 3D models.

4.3. Comparison with analytical predictions

The numerical simulations indicate that the likely maximumpower extracted from the Tory Channel is around 33 MW at springtide (ignoring the effects of cut-in speeds and design speeds). Thisis considerably less than the analytical model of Vennell [7] whichpredicts a peak extractable power of about 105 MW. This analyticalapproach provides an upper bound on the extractable energy. Themajor reason for the numerical model predicting lower maximumpower is that the locations of turbines have been restricted to areaswith depths greater than 30 m, and where spring tide velocities inthe absence of turbines>2.0 m s�1. One of the consequences of thisrestriction is that the array does not span the channel width. It hasbeen shown theoretically that turbine arrays that span only part ofthe channel width are considerably less effective at extracting en-ergy than those that span the full width [6,7,13], although Divettet al. [15] found that more power could be produced by an arrayoffset to one side of a channel than in the centre. It is likelytherefore that the width of the turbine array is responsible formuch of the difference in extracted power.

The numerical model clearly demonstrates that power outputwill vary spatially over the array. Few natural tidal channels will be

of constant width and depth, and velocities are likely to vary sig-nificantly along and across the channel. As power is proportional tovelocity cubed (for a constant power coefficient), the power dis-tribution will be even more uneven. Introducing the turbines fur-ther modifies currents, causing reductions in current speedsthroughout the channel, but particularly within the turbine array.There are also areas of increased velocity around the perimeter ofthe array. Most significantly for positioning turbines, as the numberof turbines in an array increases, velocities within the arraydecrease and parts of the array may produce little or no power.

4.4. Implications for designing turbine arrays

Operational limits such as cut-in and design maximum speedsshould be considered when sizing arrays. In the scenarios modelledhere, we used a criterion for siting turbines that the undisturbedspring tide velocity would be above 2.0 m s�1. Yet in many of sce-narios, turbine drag reduced velocities below the cut-in speed of1.0 m s�1, showing current speeds were reduced by over 50%. Thestructural drag from these non-producing turbines is a further lossof energy from the tidal stream that diminishes the power that canbe extracted. By removing turbines from these regions, it waspossible to increase both the power produced per turbine, but alsothe total array power production. In the spring-neap cycle simu-lations, poor performing parts of the original turbine array wereremoved by applying a further restriction that turbines would onlybe located where themean power output per unit turbine area overa spring tide exceeded 250Wm�2. Although this was an arbitrarilychosen limit, it demonstrates that avoiding placing turbines wherethey are likely to be below the cut-in speed much of the time, andconsequently reducing the amount of structural drag caused by thearray, power production per turbine can be substantially improved.Reducing the turbine array in this fashion will, in turn, alter flowpatterns. Indeed, even after applying this limit, there were stillareas where velocities did not exceed the cut-in speed for theD¼ 0.001 and 0.010 arrays. This suggests that an iterative approachmay be required to yield an optimal array. As a further optimisationtechnique, turbines of different sizes or capacity could be used toconstruct an array. This could be modelled using the methodologyhere by allowing the turbine drag parameter to vary spatially overthe array.

Achieving maximum power output requires “tuning” the arraydrag [14,23,24]. This can be achieved through manipulating thesize, thrust and structural drag coefficients, and number (thereforearray density) of turbines. In the approach used in this study, theseparameters are combined into a single dimensionless turbine dragparameter. Thus, the optimal drag parameter (or distribution ofdrag parameters if spatial variations in drag are allowed for) can befound independently of the turbine size.

The ratio of power and thrust coefficients was assumed constantin our model. Provided that this assumption is valid, the optimalarray drag (to produce maximum power) is not dependent on thepower coefficient. Only the magnitude of the power output willchange. If this ratio changes with flow speed, it is likely that theturbine array drag to produce maximum power differs from thatcalculated using a fixed ratio. It would be relatively easy to allowthe power coefficient to vary independently from the thrust coef-ficient, or make Cp a function of velocity if this is known for theturbine design of interest. A further implication of fixing the ratio ofCP/CT is that the energy lost to turbulence remains constant relativeto the power produced by the turbine. However, CT and Cp don’tvary independently but are related in a non-linear manner, and theratio of CP/CT does change as a result of tuning the turbines and fordifferent turbine arrangements. In addition, the related ratio of thefraction of power lost by the flow which can be converted to useful

D.R. Plew, C.L. Stevens / Renewable Energy 57 (2013) 269e282282

power (due to mixing losses) is also dependent on the turbineschannel blockage ratio and channel dynamics [23,24]. The keysimplification used here is to assume that the ratio of convertiblepower is the same as for a single turbine far from any other turbine,i.e. performance of all turbines follow the manufacturer’s specifi-cations for Cp and CT. This is reasonable given the advantages ofreducing the effect of power extraction to a single non-dimensionalnumber as outlined here, which allows rapid assessment of a site.More precise estimates can then be developed allowing for changesin these ratios at sites identified as worthy of further evaluation.

The approach used here of a distributed array drag can beconsidered as an intermediate step to modelling a turbine array.More detailed predictions of both power production and localisedeffects on flow could be obtained by modelling the individual tur-bines. In a finite volume or finite difference model, this would mostlikely be achieved by applying the turbine drag to only the elementor elements containing the turbines. However, that level of mod-elling is best reserved for detailed assessments of arrays once theturbine design parameters, sizes and proposed locations areknown. The advantage of the approach used in this study is that anassessment of power production and effects on flow can be madewith a relatively simple turbine parameterisation independent ofthe size, and therefore number, of turbines.

Power production will vary both in time and space. The tem-poral variation occurs both over a tidal cycle, but also over longerperiods such as spring-neap cycles. Non-tidal influences on currentspeeds include wind and pressure forcing, as well as baroclinitywhich has an influence in the strongly tidal Cook Strait to which theTory Channel joins [1,25]. Observations in the Tory Channel col-lected to date indicate that it is well mixed (unpublished data), yetthe larger scale baroclinic effects in the Cook Strait are likely toinfluence the boundary conditions of the Tory Channel.

Practical, physical and economic factors also mean that theoptimal number of turbines is likely to be substantially less thanwhat might be predicted from theory. These factors include re-strictions on turbine placement including sufficient water depth,minimum current speeds and the physical space required to locateturbines [38]. Economically, a high power production per turbine isfavourable. Vennell [39] showed analytically that the economics ofadding turbines to an existing row of turbines are better thanadding new rows of turbines. The decision as towhat is optimal willultimately be the energy production cost, weighing the energyproduced against the cost of construction, operation and main-tenance. It is likely the optimal number of turbines in economicterms is lower than that which can extract themaximum amount ofpower.

Acknowledgements

This study was funded by the New Zealand Ministry for Scienceand Innovation. Velocity data from the Tory Channel Entrance werecollected by Phil Sutton and Murray Smith (NIWA) as part ofa project commissioned by Land Information New Zealand (LINZ).

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