efficiency of tidal turbine farms

8132019 Efficiency of Tidal Turbine Farms

httpslidepdfcomreaderfullefficiency-of-tidal-turbine-farms 18

Realizing the potential of tidal currents and the ef 1047297ciency of turbine farms

in a channel

Ross Vennell

Ocean Physics Group Department of Marine Science University of Otago 310 Castle St Dunedin 9054 New Zealand

a r t i c l e i n f o

Article history

Received 2 December 2010

Accepted 29 March 2012

Available online 11 May 2012

Keywords

Tidal stream power

Tidal turbine

Farm ef 1047297ciency

Power from tidal channels

a b s t r a c t

Tidal turbines in strong 1047298

ows have the potential to produce signi1047297

cant power However not all of thispotential can be realized when gaps between turbines are required to allow navigation along a channel A

review of recent works is used to estimate the scale of farm required to realize a signi 1047297cant fraction of

a channelrsquos potential These works provide the 1047297rst physically coherent approach to estimating the

maximum power output from a given number of turbines in a channel The fraction of the potential

realizable from a number of turbines a farmrsquos 1047298uid dynamic ef 1047297ciency is constrained by how much of

the channelrsquos cross-section the turbines are permitted to occupy and an environmentally acceptable 1047298ow

speed reduction Farm ef 1047297ciency increases as optimally tuned turbines are added to its cross-section

while output per turbine increases in tidal straits and decreases in shallow channels Adding rows of

optimally tuned turbines also increases farm ef 1047297ciency but with a diminishing return on additional rows

The diminishing return and 1047298ow reduction are strongly in1047298uenced by how much of the cross-section can

be occupied and the dynamical balance of the undisturbed channel Estimates for two example channels

show that realizing much of the MW potential of shallow channels may well be possible with existing

turbines However unless high blockage ratios are possible it will be more dif 1047297cult to realize the pro-

portionately larger potential of tidal straits until larger turbines with a lower optimum operating velocity

are developed

2012 Elsevier Ltd All rights reserved

1 Introduction

Underwater turbines in strong tidal currents can contribute to

the need for renewable energy sources To make a signi1047297cant

contribution many turbines must be grouped into large tidal farms

to exploit the energy of high 1047298ows through narrow tidal channels

This leads to a critical question how does power production scale

as a turbine farm expands from a single turbine into a large energy

farm While adding turbines to a farm increases its generation

capacity power extraction also enhances drag on the1047298owalong thechannel This enhanced drag slows 1047298ows along the channel

reducing power production by all turbines Flow reduction

becomes more signi1047297cant as a farm grows and ultimately limits

a channelrsquos potential to produce power ([1] hereafterGC05) So it is

the combined effects of 1047298ow reduction and installed generation

capacity which determines how much power an expanding farm

produces

Part of addressing the question of how power production scales

is estimating the maximum power a particular channel can

generate ie the channelrsquos ldquopotentialrdquo (see review [2]) Most recent

estimates of potential have used the approach developed by GC05

(eg [3]) To realize that potential requires the turbines to occupy

the entire cross-section of the channel have perfect electro-

mechanical ef 1047297ciency that there are no losses to mixing behind

the turbines and that there is no drag on any structure supporting

the turbines Realistically none of these are achievable however

GC05rsquos potential provides a useful upper bound for production

The need for navigation along a channel will often restrict

turbines to only part of the channelrsquos cross-section Fig 1 Thus justas important as estimating a channelrsquos potential is to determine

how much of a channelrsquos potential can be realized from a given

number of turbines when some 1047298ow can bypass the turbines

through gaps left to permit navigation Fig 2 In addition 1047298ow

reduction may also have environmental impacts such as reduced

tidal exchange and sediment transport (eg [4]) Thus another

constraint on turbine numbers is likely to be an acceptable 1047298ow

speed reduction This requires answering the question how does

power production scale with farm size within constraints on how

much of the cross-section can be1047297lled with turbines andhow much

1047298ows within the channel can be reducedE-mail address rossvennellotagoacnz

Contents lists available at SciVerse ScienceDirect

Renewable Energy

j o u r n a l h o m e p a g e w w w e l s e v i e r c o m l o c a t e r e n e n e

0960-1481$ e see front matter 2012 Elsevier Ltd All rights reserved

doi101016jrenene201203036

Renewable Energy 47 (2012) 95e102



An older method for estimating a channelrsquos potential was to use

the Kinetic Energy Flux through the undisturbed channel and an

arbitrary loading factor of 10e15 to estimate power output from

the channel [2] However the KE 1047298ux through a channel without

turbines is unrelated to a channelrsquos potential as it does not account

for the 1047298ow reduction due to power extraction [5] Also the KE 1047298ux

does not provides a means to estimate the power available from

a given number of turbines which may occupy only part of thecross-section nor a means to estimate the 1047298ow reduction Both

these aspects are critical in assessing the tidal current resource at

a particular site the farmrsquos impacts and the economics of devel-

oping it Building on GC05 and GC07 the recent [67] (hereafter V10

and V11) works provide the 1047297rst physically coherent approach to

maximizing the poweroutput from a given numberof turbinesThe

upper bound for the power available for electricity production

given by these models can be used to answer the question about

how power production scales within constraints as well as how

best to arrange and con1047297gure the turbines

In this work a review of GC05 [8] (hereafter GC07) V10 and V11

is used to discuss the question and indicate the scale of farm

required to realize a signi1047297cant fraction of a channelrsquos potential

How much of a channelrsquos potential can be realized ie farm ef 1047297-

ciency is strongly linkedto turbine ef 1047297ciency Not only the turbinersquos

electro-mechanical energy conversion ef 1047297ciency but also the

turbinersquos 1047298uid dynamic ef 1047297ciency This work focuses on 1047298uid

dynamic ef 1047297ciency thus addressing the ldquopower availablerdquo for

electricity production from a farm with a given number of turbines

Turbines extract energy from the 1047298ow passing through the area

spanned by their blades To maximize turbine ef 1047297ciency all turbines

must have the strength of the 1047298ow passing through them adjusted

or tuned typically achieved by varying blade pitch To maximize

farm output the 1047298ow speed through the turbines must be tuned for

a particular channel and fraction of the channelrsquos cross-section

taken up by the turbines as well as tuned in relation to each

other (V10 and V11) This makes maximizing the 1047298uid dynamic

ef 1047297

ciency of a farm complicatedThe four fundamental works GC05 GC07 V10 and V11 are

necessarily mathematical and complex This work aims to 1047297rstly

review the essential results from these works to make them

accessible to a wider audience and then use the results to discuss

the number of turbines required to realize a signi1047297cant fraction of

the potential of two example channels This work begins by

reviewing the physics of tidal channel potential and turbine ef 1047297-

ciency contained in the works by outlining four key concepts in

Section 2 Understanding the physics of farm ef 1047297ciency hinges on

understanding how the farmrsquos gross drag coef 1047297cient affects the

1047298ow along the channel Section 3 looks at the ef 1047297ciency of a single

row of turbines Section 4 examines the effects of adding more

rows Section 5 looks in detail at how many turbines are required to

achieve a signi1047297cant fraction of the example channelsrsquo potentials

when constrained by cross-sectional occupancy and 1047298ow reduction

2 Background physics

The physics underlying the latter sections are reviewed in the

next four sub-sections each of which presents an essential concept

The underlying aim of this section is to make the connection

between the farmrsquos drag coef 1047297cient and the power which is avail-

able for electricity production clear The farmrsquos drag coef 1047297cient is

the link between the number of turbines in the farm and the power

available for electricity production The essential idea is that thefarmrsquos drag coef 1047297cient increases as the turbines 1047297ll more of the

cross-section ie the blockage ratio increases or as more rows of

turbines are added to the farm The drag coef 1047297cient also changes as

the 1047298ow through the turbines is adjusted by tuning the pitch of

their blades Channel and farm speci1047297c tuning is critical to

maxmising the farmrsquos output V10

The underlying models are those of GC05 and GC07 adapted by

V10 and V11 In GC05rsquos model for a turbine farm in a short narrow

channel Fig 1 the farm is modelled as a drag on the 1047298ow The

model has oscillating tidal 1047298ow driven along the channel by a water

level difference between the ends of the channel This difference or

headloss is due to the differing tidal regimes in the two large water

bodies which are connected by the channel The water bodies are

assumed to be so large that any water 1047298owing through the channel

does not affect water levels within them Thus water levels at the

ends of the channel are unaffected by a power extraction within it

This is the simplest useful channel geometry One extension not

included here has a lagoon at one end of the channel and a large

ocean at the other [9] The 1047297nite reservoir of the lagoon means that

tides within it depend on the volume of water which1047298ows through

the channel As a result a lagoon can increase or decrease a tidal

channels potential depending on whether the amplitude of the

head between the ends of the channel is less than or greater than

the tidal amplitude in the ocean [10] In addition any large deep

ocean will likely have a shallow continental shelf between it and

the entrance to the channel Frictional dissipation and resonances

over the shelf may also in1047298uence the amplitudes of the tides at the

entrance to the channel driven by tides in the deep ocean [11]

The GC05 model is given in terms of volume transport here it is

presented in terms of velocity In short uniform cross-section

channels the cross-sectional average velocity does not vary signi1047297-

cantly along the channel [1213] Thus the tidal velocity everywhere

along a short channel with a rectangular cross-section depends only

ontimeand can beexpressedas u frac14 u0sinethut thorn fuTHORN where u0 is the

amplitude of the velocity and fu its phase GC05rsquos momentum

balance for a uniform cross-section channel can be written in the

form

vu

vt frac14

g z0

L sinethut THORN

C Dh thorn

C F

L

uu (1)

In Eq (1) the 1047297rst term represents the inertia of the 1047298ow the

second term the sinusoidal pressure gradient or head which forces

Fig 1 Schematic of a turbine farm in a narrow constricted channel connecting two

large water bodies Differing tidal regimes in the two large water bodies drive oscil-

lating tidal 1047298ow through the channel The example farm has 3 rows of turbines The

arrows around each turbine indicate the stronger 1047298ows passing around the turbines

and the weaker 1047298ows passing through the turbines

Fig 2 Schematic of 1047298ow through a row of identical turbines and the 1047298ow through and

around a single turbine within the row Relative sizes of the velocities are u4 u u1

u3 After V11

R Vennell Renewable Energy 47 (2012) 95e10296





adding turbines decreases the power lost to the turbines ie C F u30

decreases as the bene1047297t of growing the farm to increase C F is out-

weighed by the reduction in u0 due to the additional drag Between

these two extremes there is a maximum power which can be lost

by the 1047298ow to the turbines (GC05) as seen in the power lost curves

in Fig 3b) This maximum power lost to the turbines P maxlost is

a channelrsquos potential which is the upper bound for how much

power a channel can generate The optimal gross farm drag coef-

1047297cient C F peak required to realize the potential can be estimated

from the analytic solution to GC05rsquos model using (V10-A4) or (V11-

210) To realize a channelrsquos potential P maxlost turbines must occupy

the entire cross-section Were it possible around 25 turbines would

1047297ll the cross-section of the shallow channel and 2500 the cross-

section of the tidal strait Table 1 Many turbines sweep a circular

area thus cannot 1047297ll a rectangular channelrsquos cross-section unless

contained within a square shroud Thus the number of turbines

required to 1047297ll the cross-section N 0 may not always be of practical

use However N 0 is a useful as a reference value for the size of a farm

in later discussions on turbine numbers where they only 1047297ll part of the cross-section or are spread amongst rows

GC05 found a remarkably simple approximate expression for

estimating the potential of a channel based on the transport

amplitude through the undisturbed channel U 0UD frac14 Au0 and the

amplitude of the headloss z0 which is given by

P maxlost frac14 022r g z0U 0UD (4)

[10] gives a method to estimate both the potential and the 1047298ow

speed reduction which only requires the transport along the

undisturbed channel which is easily measured using a vessel

mounted ADCP eg [18]

23 Gaps between turbines allows 1047298ow to bypass turbines reducing the power available for power production GC07

When there are wide gaps between turbines within a row to

allow for navigation some 1047298ow bypasses the turbines altogether

and does not contribute to power production The mixing of the

retarded 1047298ow passing through the turbines with the faster 1047298ow

passing around the turbines also dissipates some of the 1047298owrsquos

energy as heat [19] and GC07 The 1047298ows near a row of turbines

are shown schematically in Fig 2 GC07 extended classic

Lanchester-Betz actuator disc theory for an isolated turbine [2021]

to a row of turbines in a narrow channel Fig 2 They found that the

rowrsquos drag coef 1047297cient C R depends only on the blockage ratio ε the

fraction of the channelrsquos cross-sectional area occupied by the

turbines and the ratio r 3 frac14 u3=u which quanti1047297

es the 1047298

ow

reduction in the wake behind the turbines ie they found C Rethε r 3THORN

Though the functional relationship is complex and given in equa-

tions (GC07-223 V10-26) and (GC07-29 V10-27) for the purpose

of this work it is only essential to understand that the drag coef 1047297-

cient of a row only varies due tothe changes in the blockage ratio or

changes in the 1047298ow reduction behind the turbines C Rethε r 3THORN

increases as either ε is increased or the 1047298ow in the wake r 3 is

decreased The fraction of the cross-section occupied by the

turbines ie the blockage ratio is simply

ε frac14 M AT

A

where M is the number of turbines in the row AT is area blocked by

the rotating by the blades of a single turbine

The average power lost by the 1047298ow is given by Eq (3) This is not

the same as the power available to the turbine for electricity

production due to mixing losses behind the turbines GC05 and

[19] Using their results the power available is the smaller work

done by the 1047298ow through the turbine Fu1 whose average is

P avail frac14 4

3pr Ar 1N RC Ru3

0 (5)

where r 1 frac14 u1=u is the ratio of the velocity though the turbines to

the velocity upstream of the row Comparing Eqs (3) and (5) shows

that dueto mixing lossesonly thefraction r 1 of the power lostby the

1047298ow tothe turbines is available forelectricity productionwhere0

r 1 1 The power available Eq (5) represents an upper bound for

electricity which could be produced from a farm in a channel

24 Turbines must be adjusted or tuned for a particular channel

and in relation to each other to maximize the power available V10

amp V11

Maximizing the powerlost by the1047298owdue tothe farm Eq (3) is

not the same as maximizing the power available for generation as

they differ bya factorof r 1 Eq (5) Fig 3b) demonstrates this where

the power lost curve peaks at C F peak while the lower power

available curvespeak at a smallerdrag coef 1047297cient C F opt indicated by

the dots To maximize the power available the 1047298ow through the

turbines must be adjusted or tuned to give the optimal farm drag

coef 1047297cient C F opt Typically tuning is done by adjusting the pitch of

the turbinersquos blades Here this tuning is done mathematically by

adjusting the value of the 1047298ow reduction r 3 which affects both C Rand r 1 GC07

GC07rsquos extension of Lanchester-Betz theory to turbines in

a channel assumed the 1047298

ow upstream of the row u in Fig 2 was

0 05 1 1504

06

08

1

F l o w r e d u c t i o n

U 0

U 0 u n d i s t u r b e d

CF C

Fpeak

a

0 05 1 150

02

04

06

08

1

P o w e r P

m a x

CF C

Fpeak

Decreasing tuning parameter r3 minusgt

b

Fig 3 Effects of increasing farmrsquos drag coef 1047297cient for two example channels in Table 1 Solid lines are for shallow channel and dashed lines for tidal strait Horizontal axis is the farm

drag coef 1047297cient relative to the drag coef 1047297cient required to realize a channelrsquos potential ie at the peak in the power lost curve a) Reduction in 1047298ow relative to 1047298ow in undisturbed

channel b) Thick lines are the power lost to the turbines and thin lines are the power available from 5 rows of turbines in the shallow channel and 40 rows in the tidal strait when

turbines occupy 20 of the channel rsquos cross-section Solid dots show peak in power available curves at optimal tuning ie the upper bound for how much power is available from

these turbines




1047297xed and found an optimal value of r 3 frac14 1=3 (equivalent to

r 1 frac14 2=3) which maximised the power available the same optimal

values as those for an isolated Lanchester-Betz turbine They also

found that at most 23 of the power lost to the turbines was

available for power production which occurred if the turbines took

up the minimum possible fraction of the cross-section However in

a channel with 1047298ows driven by head loss between its ends the 1047298ow

upstream of the row u isnot 1047297xed but decreases as the farmrsquos gross

drag coef 1047297cient C F frac14 N RC Rethε r 3THORN increases Thus C F depends on

the tuning r 3 Consequently changing the tuning changes the

strength of the 1047298ow along the entire channel via its effect on the

gross drag coef 1047297cient V10 V10 went on to show that a conse-

quence of this is that turbines need r 3 to be tuned to valuesbetween

13 and 1 to maximize the power available Thus tuning a large tidal

farm is very different from tuning a single isolated device A farmrsquos

optimal tuning r opt3 depends on a channelrsquos geometry and

dynamical balance as well as the blockage ratio ε V10 found that

by optimally tuning turbines it is possible to exceed GC07rsquos

maximum of 23 of a channelrsquos potential which is available for

power production

V11 went on toshow that not only does a row of turbinesneed to

betuned fora particularchannelit must also betuned in thepresence

of other rows tomaximizethe poweravailableThe needto tunerowsldquoin-concertrdquo has implications for modellers who must include

idealizedturbines in theirhydrodynamicmodelsto assess the power

available from a proposed site V11 and for the operators of turbine

farms as turbines come in and out of service Tuning in-concert may

require many model runs or complex interdependent adjustment of

operating turbines to 1047297nd the optimal set of turbine tunings The

need to tune turbines in-concert arises because rows of turbines in

narrow channels interact with each other via the farmrsquos gross drag

coef 1047297cient C F even if they are separated widely enough for the

disturbed 1047298ow through onerow tofullymix before encountering the

next row as C F affects 1047298ow along the entire channel

3 Farm and turbine ef 1047297ciency for a single row

31 Farm 1047298uid dynamic ef 1047297ciency

A measure of a farmrsquos ef 1047297ciency is the fraction of GC05rsquos

potential which is available for power production V10 ie

FE frac14 P avail

P maxlost

(6)

FEis the farmrsquos 1047298uid dynamic power ef 1047297ciency and is maximised

at the optimal tuning r opt3 V11 This is the headline ef 1047297ciency which

is theupper bound forthe fractionof a channelrsquos estimated potential

that can be turned into electricity from a given number of turbines

Fig 4a) shows FE for the two example channels as optimally

tuned turbines are added to a single row The curves for both

examples converge on 1 as turbines take up most of the cross-

section making most of the channelrsquos potential available for

power production For the shallow channel farm ef 1047297ciency initially

grows rapidly as turbines are added with reduced gains at higher

numbers indicating a diminishing return on additional turbines In

contrast for the tidal strait FE increases more rapidly at the higher

occupancies indicating an increasing return on new turbines added

to the row V11 Though a farm in the shallow channel is more

ef 1047297cient than the tidal strait for a given fraction of the cross-section

occupied in absolute terms the tidal strait has a much larger power

available and requires many more turbines to make up this given

fraction The thin lines in Fig 4a) show the reduction in 1047298ow speeds

due to adding turbines to the row Flow in the shallow channel

decreases more rapidly as turbines are added consistent with its

higher farm ef 1047297ciency

Fig 4a) shows the 1047297rst of two extreme ways to almost realise

a channelrsquos potential ie approach 100 farm ef 1047297ciency is to have

the optimally tuned turbines 1047297ll the cross-section ie blockage

ratio ε1 Turbines are normally thought of as extracting energy

from the 1047298owrsquos Kinetic Energy by reducing 1047298ows through the

turbines However paradoxically an optimally tuned tidal farm canalmost realise a channelrsquos potential without reducing the 1047298ow

through the turbines relative to the 1047298ow upstream ie the optimal

r 3 and r 11 as ε1 V10 The resolution lies in understanding the

source of the farmrsquos energy For an isolated turbine generation is

a result of reducing 1047298ows through the turbine where for a Betz

turbine the optimal tuning is r 3 frac14 1=3 corresponding to r 1 frac14 2=3

As the cross-section is 1047297lled with turbines the farmrsquos energy source

changes gradually from the 1047298owrsquos KE to the potential energy of the

1047298ow as optimal tunings increase So that at high blockage ratios the

energy source becomes the drop in water level between the

upstream and downstream sides of the farm ie the source is the

headloss across the farm It is also worth emphasizing that mixing

losses behind the turbines approach zero at high blockage ratios

which must happen if farm ef 1047297ciency is to approach 100 At highblockage ratios turbine farms approach an extreme form of a hydro-

electric dam with low head and high volume 1047298ow For the two

examples the head loss is only around 01 10 m while the peak

volume 1047298ows are 27 000 1 700 000 m3s1 In contrast a large

river dam has high head and low 1047298ow ie a 100 m head and

500 m3s1

32 Power available per turbine

Fig 4b) gives the power available per turbine as the cross-

sections are 1047297lled with turbines A single isolated turbine ε0

in the shallow channel makes 099 MW available per turbine

0 02 04 06 08 10

02

04

06

08

1

Fraction of crossminussection occupied ε

P a v a i l P m a x

o r U 0

U 0 U Da

F a r m

E f f i

c i e n c

y


0 02 04 06 08 10

05

1

15

2


P o w e r p e r t u r b i n e M Wb

Fig 4 Effect of the fraction of cross-section occupied by the turbines or blockage ratio on ef 1047297ciencies and 1047298ow speeds for a single row of optimally tuned turbines in a uniform

cross-section channel Solid curves are for shallow channel and dashed lines for tidal strait examples a) Thick lines are farm ef 1047297ciency Eq (6) the fraction of GC05rsquos potential which

is available for power production and thin lines 1047298

ow relative to undisturbed channel b) Power available per turbine in MW

R Vennell Renewable Energy 47 (2012) 95e102 99



whereas only 034 MW is available in the tidal strait due to much

weaker 1047298ows However as the cross-section is 1047297lled the power

available per turbine for the shallow channel decreases while for

the tidal strait power per turbine increases So paradoxically at

higher occupancies the tidal strait delivers up to six times more per

turbine despite having much weaker 1047298ows than the shallow

channel The curves in Fig 4b) are the result of two competing

effects Firstly occupying more of the cross-section increases

a rowrsquos drag coef 1047297cient C R increases the power available Eq (5)

Secondly increasing a rows drag coef 1047297cient reduces the 1047298ow along

the channel U 0 which acts to decrease the power available in Eq

(5) The net effect differs for the two examples For the shallow

channel the more rapid reduction in 1047298ow speed as the cross-section

is 1047297lled Fig 4a) outweighs the enhanced drag coef 1047297cient leading

to reduction in the power per turbine from 099 MW to only

036 MW at high occupancy In contrast the more gradual 1047298ow

reduction in the tidal strait leads to an increasing power available

per turbine as the cross-section is 1047297lledfrom 034MW to23MW A

result is that above 35 occupancy the tidal strait delivers more per

turbine than the shallow channel despite the much weaker 1047298ows in

the strait

The differing performance of the turbines as the cross-section is

1047297lled is linked to their differing dynamical balances At higheroccupancies in the shallow channel the power lost to the turbines is

similar to that lost to background friction (this can be inferred from

(V10-A4) where C F peak is around twice the scaled background

bottom friction coef 1047297cient C D for near steady state channels ie

large l0 channels) This signi1047297cant energy loss to bottom friction

and turbine drag is associated with a more rapid decrease in 1047298ows

a diminishing return on new turbines and the decreasing power

available per turbine in Fig 4 In contrast in the tidal strait bottom

friction is almost unimportant and relatively little energy is lost to

bottom friction This gives the strait a proportionately higher

potential and an increasing return on additional turbines as they

becomemore ef 1047297cient when occupying more of the cross-section of

the ldquoductrdquo formed by the channel The energetics of channels with

turbine farms is discussed in detail in [17]

4 Ef 1047297ciency of multi-row farms

Adding capacity to a farm by adding rows of optimally tuned

turbines increases farm ef 1047297ciency as demonstrated in Fig 5a) For

the tidal strait 15 rows makes only 40 of it potential available so as

would be expected the associated 1047298ow reduction is modest In

contrast for the shallow channel 3 rows make 70 of its potential

available with a signi1047297cant associated 1047298ow reduction

Fig 5 illustrates a second extreme way to realize most of

a channelrsquos potential having a large number of rows However for

both channels there is a diminishing return on additional rows

Farm ef 1047297ciency peaks at a very large number of rows indicating an

optimal farm size However the diminishing return on new rows is

so harsh near this peak that farm size will be restricted to a much

smaller of number rows [17] V11 showed that for two similar

examples the best strategy for growing a farm of optimally tuned

turbines is the intuitive one Fill the 1047297rst row up to the maximum

permitted by navigational constraints and then add rows up to

a maximum that can be economically justi1047297ed in the light of the

diminishing returns inherent in Fig 5a)

5 Realizing the potential a trade off

The decision on turbine numbers is an economic trade off

between the income from the power made available by adding

turbines to a farm against their cost and any environmental

impacts Once the maximum permissible cross-sectional occu-

pancy has been achieved then the trade off is in the face of

a diminishing return on additional turbines Fig 5a) How this trade

off plays out depends strongly on the maximum fraction of the

channelrsquos cross-section which can be occupied in order to maintain

navigation along the channel It also strongly depends on the

dynamical balance of the undisturbed channel be it that of

a bottom friction dominated shallow channel or that of an inertia

dominated tidal strait Fig 5

This diminishing return is inherent in the upper row of plots in

Fig 6 where for both a low cross-sectional occupancy ε frac14 01 and

what may be an unrealistically high occupancy ε frac14 05 power

available increases more slowly as rows are added to an optimally

tuned farm While the ultimate choice on number of turbines

comes down to an economic and environmental cost bene1047297t

analysis which is beyond the scope of this work the plots however

do contain essential information to underpin this analysis For the

shallow channel at the high occupancy 1 row of 12 turbines makes

80 of its 9 MW potential available Fig 6a) In contrast the same

number of turbines spread across 5 rows at the lower occupancy

makes only 65 of the potential available for power productionFig 6c) shows that the power available per turbine for both occu-

pancies decreases as rows are added Fig 6e) illustrates that for

both occupancies a farm making 80 of the shallow channelrsquos

potential available will reduce 1047298ows by 30 If such a signi1047297cant

1047298ow reduction is not acceptable then at the lower occupancy

installing around 3 turbines reduces 1047298ows by only 10 while

making around 25 of its potential available

The curves for the tidal strait in Fig 6b) d) and f) are similar to

those of the shallow channel but the number of turbines involved

is much larger Almost 80 of the straitrsquos 6 GW potential can be

realized at the higher occupancy but this requires an astonishing

0 5 10 150

02

04

06

08

1

Number of Rows NR

a

Farm Efficiency

0 5 10 1504

06

08

1

Number of Rows NR

b

Flow reduction

Fig 5 Effect of adding rows to an optimally tuned farm where turbines occupy 20 of the channel rsquos cross-section ie a blockage ratio of ε frac14 02 Solid curves are for the shallow

channel and dashed lines for the tidal strait a) Farm ef 1047297

ciency Eq (6) b) Velocity relative to undisturbed velocity




16000 turbines spread over 13 rows At the lower occupancy

a more affordable 5000 turbines would make 30 of the strait rsquos

potential available while reducing 1047298ows by 7

6 Discussion

The principle aim of this paper was to show how many turbines

are required to realize a given fraction of a channel rsquos potential in

order to improve understanding of how power production scales

with farm size The models in V10 and V11 which build on those of

GC05 and GC07 provide the 1047297rst physically coherent way of con-

necting the number of turbines in a farm with the maximum power

available for electricity production The models do not require the

use of arbitrary loading or safety factors applied to account for

unknown effects The models are however simple They use an

extended form of classic turbine theory and they only apply to the

cross-sectional average velocities thusdo not account for the effects

of any variation of 1047298ow across the channel or with depth They also

do not allow for any effect of the turbulence shed by a row on the

ef 1047297

ciency of downstream rows They do not allow for minimum

turbine operating 1047298ows or for loss due to their electro-mechanical

ef 1047297ciency or transmission and conversion losses Thus there is

muchworktobedonetore1047297ne and extend the modelsHowever the

models do illuminate signi1047297cant aspects of developing large tidal

turbine farms Most of these are a result of power extraction

reducing the 1047298ows along the entire narrow channel via the farmrsquos

gross dragcoef 1047297cient C F They include thenecessity totune turbines

for the particular channel and in relation to each other which will

result in signi1047297cant computational cost to those modelling turbine

farms and will require on going optimization of turbine tuning in

large operational farms as turbines come in and out of service They

also include how farms have increasing or diminishing returns on

turbines added to thecross-section dependingon the importance of

bottom friction and how there is always a diminishing return on

additional rows aspects which are fundamental to understanding

the economics of farm development

The turbine numbers given by the simple models should be

viewed as estimates of the magnitude of the numbers required

rather than precise estimates These magnitudes are useful in

understanding how realistic it is to realize a signi1047297

cant fraction of

0 10 20 30 400

02

04

06

08

1

1

3

5

101

3

P o w e r P

m a x

Number of Turbines

aShallow Channel

0 5 10 150

02

04

06

08

1

1

5

10

20

50

1

5

10

P o w e r P

m a x

Thousands of Turbines

bTidal Strait

0 10 20 30 400

02

04

06

08

1

12

P o w

e r P e r T u r b i n e M W

Number of Turbines

c

0 5 100

02

04

06

08

1

12

P o w



d

06 07 08 09 10

02

04

06

08

1

5

10

15

2515

25

P o w e r P

m a x

Flow Reduction

e

06 07 08 09 10

02

04

06

08

1

10002000

5000

10000

15000

2000

5000

10000

15000

P o w e r P

m a x

Flow Reduction

f

Fig 6 Effect of the number of optimally tuned turbines on power available and 1047298ow reduction for the two example channels based on a 400 m 2 blade area Solid curves are for

ε frac14 01 and dashed lines for ε frac14 05 a) and b) Farm ef 1047297ciency versus number of turbines labelled dots give number of rows in the farm c) and d) Power available per turbine in

MW e) and f) Farm ef 1047297ciency verses 1047298ow reduction Labelled dots give the number of turbines in farm




a channelrsquos potential For the shallow channel around eight 400 m2

turbines corresponding to three rows with ε frac14 01 makes half of

its potential available and 13 corresponding to one row with

ε frac14 05 makes 80 available Table 1 The given values for the

power available per turbine are also useful in understanding

economic feasibility For blockage ratios of ε frac14 01 and 05 each

turbine makes around 06 MW available The detailed economics

and environmental costs are beyond the scope of this work

however making an average of 06 MWavailable over the tidal cycle

when existing turbines are producing around 1 MW at steady1047298ows

around2 m s1 maybe a reasonable return Thus it appears possible

to make a signi1047297cant fraction of the shallow channelrsquos potential

available using 5e10 turbines with a blade area similar to that of

the largest currently operating tidal turbine spread amongst 1e3

rows which occupy less than 50 of the cross-section

In contrast to make 50 of the tidal straitrsquos 6 GW potential

available takes 10000 turbines at 10 blockage This high number

is a result of the weaker 1047298ows in the tidal strait which yield only

027 MW per turbine However unlike the tidal shallow channel

the output per turbine rises as more of the strait rsquos cross-section is

occupied So that at 50 blockage only 5000 turbines are required

to make 50 of its potentialavailable at a much higher 061MW per

turbine similar to that for the shallow channel Tidal straits haveboth a much larger potential in absolute terms and in proportion to

their size due to relatively low energy losses to bottom friction

This is seen in their having six times the potential per turbinewhen

turbines 1047297ll the cross-section in a single row Table 1 However tidal

straits typically have weaker cross-sectionally averaged 1047298ows than

shallow tidal channels Thus at low blockage ratios a very large

number of rows are required to make a signi1047297cant fraction avail-

able If higher blockage ratios are permitted then the enhanced

ef 1047297ciency of turbines in tidal straits seen in Fig 4 boosts the output

per turbine and reduces the total number of turbines required [17]

uses the channelrsquos energy balance to explain why and when tidal

straits have an increasing return on turbines added to the cross-

section and why shallow channels have a diminishing return

7 Conclusions

Critical questions in developing a farm are what fraction of

a channelrsquos potential is available for power production from a given

number of turbines as a farm scales up from a single turbine into

a large farm The V10 thorn V11 works provide the 1047297rst physically

coherent way to connect the number of turbines in a farm to both

the maximum power available for electricity production and to the

degree of 1047298ow reduction which is a consequence of power

extraction Though simple they illuminate several important

aspects of developing large farms which have implications for both

the economics and environmental impact of farms Such as the

necessity of tuning turbines in large farms for the particular

channel how much of the cross-section they occupy and the

number of rows as well as how to best arrange and con1047297gure the

turbines For example to maximize farm ef 1047297ciency the 1047297rst row

turbines should be 1047297lled up to the maximum permitted by navi-

gational constraints before adding new rows Also while farm

ef 1047297ciency always increases as optimally tuned turbines are added

once the cross-sectional occupancy limit is attained the power

available per turbine decreases as rows are added Thus increasing

power production by increasing the farmrsquos installed capacity faces

a diminishing return on additional rows This development strategy

may alter when more realistic models are developed which allow

for variation of 1047298ow across the channel and with depth However

the decision on turbine numbers will remain an economic trade off

between the power produced and installation maintenance and

environmental costs How this trade off plays out depends strongly

on the maximum fraction of the channelrsquos cross-section which can

be occupied and the dynamical balance of the undisturbed channel

The examples demonstrate that it may be possible with existing

technology to realize much of the MW potential of shallow tidal

channels The GW potential of tidal straits is both larger in absolute

terms and also proportionately larger than that of shallow chan-

nels due to relatively low energy losses to bottom friction At low

cross-sectional occupancies the typically lower 1047298ows of the strait

result in both a low return per turbine and a very large number of

turbines being required to realize a signi1047297cant fraction of their

proportionately higher potential Thus unless a large fraction of the

straitrsquos cross-section can be occupied to take advantage of a higher

output per turbine it will be dif 1047297cult to realize a substantial frac-

tion of the GW potential of tidal straits until larger turbines are

developed which are able operate economically in low 1047298ows

References

[1] Garrett C Cummins P The power potential of tidal currents in channels

Proceedings of the Royal Society A 20054612563e

72[2] Blunden LS Bahaj AS Tidal energy resource assessment for tidal stream

generators Proceedings of the Institution of Mechanical Engineers Part A Journal of Power and Energy 2007221(10)137e46 doi10124309576509JPE332

[3] Blanch1047297eld J Garrett C Rowe A Wild P Tidal stream power resourceassessment for Masset Sound Haida Gwaii Proceedings of the Institution of Mechanical Engineers Part A Journal of Power and Energy 2008222(5)485e92

[4] Neill SP Litt EJ Couch SJ Davies AG The impact of tidal stream turbines onlarge-scale sediment dynamics Renewable Energy 200934(12)2803e12doi101016jrenene200906015

[5] Garrett C Cummins P Limits to tidal current power Renewable Energy 2008332485e90

[6] Garrett C Cummins P Tuning turbines in a tidal channel Journal of FluidMechanics 2010663253e67 doi101017S0022112010003502

[7] Vennell R Tuning tidal turbines in concert to maximise farm ef 1047297ciency Journal of Fluid Mechanics 2011671587e604

[8] Garrett C Cummins P The ef 1047297

ciency of a turbine in a tidal channel Journal of Fluid Mechanics 2007588243e51[9] Blanch1047297eld J Garrett C Wild P Rowe A The extractable power from a channel

linking a bay to the open ocean Proceedings of the Institution of MechanicalEngineers Part A Journal of Power and Energy 2008222(3)289e97

[10] Vennell R Estimating the power potential of tidal currents and the impact of power extraction on 1047298ow speeds Renewable Energy 2011363558e65 doi101016jrenene201105011

[11] Arbic B Garrett C A coupled oscillator model of shelf and ocean tidesContinental Shelf Research 201030(6)564e74

[12] Vennell R Oscillating barotropic currents along short channels Journal of Physical Oceanography 199828(8)1561e9

[13] Vennell R Observations of the phase of tidal currents along a strait Journal of Physical Oceanography 199828(8)1570e7

[14] Byden IG Grinsted T Melville GT Assessing the potential of a simple tidalchannel to deliver useful energy Journal of Applied Ocean Research 200426198e204 doi101016japor200504001

[15] Vennell R ADCP measurements of tidal phase and amplitude in Cook StraitNew Zealand Continental Shelf Research 199414353e64

[16] Douglas C Harrison G Chick J Life cycle assessment of the Seagen marinecurrent turbine Proceedings of the Institution of Mechanical Engineers Part M

Journal of Engineering for the Maritime Environment 2008222(1)1e12 doi10124314750902JEME94

[17] Vennell R The energetics of large tidal turbine arrays Renewable Energyin press

[18] Vennell R ADCP measurements of momentum balance and dynamic topog-raphy in a constricted tidal channel Journal of Physical Oceanography 200636(2)177e88

[19] Corten G Heat generation by a wind turbine Vol In 14th IEA symposium onthe aerodynamics of wind turbines 2000 p 7 ECN report ECN-RX-01-001

[20] Lanchester FW A contribution to the theory of propulsion and the screwpropeller Transactions of the Institution of Naval Architects 1915LVII98e116

[21] Betz A Das Maximum der theoretisch moumlglichen Ausnutzung des Windesdurch Windmotoren Gesamte Turbinenwesen 192017307e9




An older method for estimating a channelrsquos potential was to use

the Kinetic Energy Flux through the undisturbed channel and an

arbitrary loading factor of 10e15 to estimate power output from

the channel [2] However the KE 1047298ux through a channel without

turbines is unrelated to a channelrsquos potential as it does not account

for the 1047298ow reduction due to power extraction [5] Also the KE 1047298ux

does not provides a means to estimate the power available from

a given number of turbines which may occupy only part of thecross-section nor a means to estimate the 1047298ow reduction Both

these aspects are critical in assessing the tidal current resource at

a particular site the farmrsquos impacts and the economics of devel-

oping it Building on GC05 and GC07 the recent [67] (hereafter V10

and V11) works provide the 1047297rst physically coherent approach to

maximizing the poweroutput from a given numberof turbinesThe

upper bound for the power available for electricity production

given by these models can be used to answer the question about

how power production scales within constraints as well as how

best to arrange and con1047297gure the turbines

In this work a review of GC05 [8] (hereafter GC07) V10 and V11

is used to discuss the question and indicate the scale of farm

required to realize a signi1047297cant fraction of a channelrsquos potential

How much of a channelrsquos potential can be realized ie farm ef 1047297-

ciency is strongly linkedto turbine ef 1047297ciency Not only the turbinersquos

electro-mechanical energy conversion ef 1047297ciency but also the

turbinersquos 1047298uid dynamic ef 1047297ciency This work focuses on 1047298uid

dynamic ef 1047297ciency thus addressing the ldquopower availablerdquo for

electricity production from a farm with a given number of turbines

Turbines extract energy from the 1047298ow passing through the area

spanned by their blades To maximize turbine ef 1047297ciency all turbines

must have the strength of the 1047298ow passing through them adjusted

or tuned typically achieved by varying blade pitch To maximize

farm output the 1047298ow speed through the turbines must be tuned for

a particular channel and fraction of the channelrsquos cross-section

taken up by the turbines as well as tuned in relation to each

other (V10 and V11) This makes maximizing the 1047298uid dynamic

ef 1047297

ciency of a farm complicatedThe four fundamental works GC05 GC07 V10 and V11 are

necessarily mathematical and complex This work aims to 1047297rstly

review the essential results from these works to make them

accessible to a wider audience and then use the results to discuss

the number of turbines required to realize a signi1047297cant fraction of

the potential of two example channels This work begins by

reviewing the physics of tidal channel potential and turbine ef 1047297-

ciency contained in the works by outlining four key concepts in

Section 2 Understanding the physics of farm ef 1047297ciency hinges on

understanding how the farmrsquos gross drag coef 1047297cient affects the

1047298ow along the channel Section 3 looks at the ef 1047297ciency of a single

row of turbines Section 4 examines the effects of adding more

rows Section 5 looks in detail at how many turbines are required to

achieve a signi1047297cant fraction of the example channelsrsquo potentials

when constrained by cross-sectional occupancy and 1047298ow reduction

2 Background physics

The physics underlying the latter sections are reviewed in the

next four sub-sections each of which presents an essential concept

The underlying aim of this section is to make the connection

between the farmrsquos drag coef 1047297cient and the power which is avail-

able for electricity production clear The farmrsquos drag coef 1047297cient is

the link between the number of turbines in the farm and the power

available for electricity production The essential idea is that thefarmrsquos drag coef 1047297cient increases as the turbines 1047297ll more of the

cross-section ie the blockage ratio increases or as more rows of

turbines are added to the farm The drag coef 1047297cient also changes as

the 1047298ow through the turbines is adjusted by tuning the pitch of

their blades Channel and farm speci1047297c tuning is critical to

maxmising the farmrsquos output V10

The underlying models are those of GC05 and GC07 adapted by

V10 and V11 In GC05rsquos model for a turbine farm in a short narrow

channel Fig 1 the farm is modelled as a drag on the 1047298ow The

model has oscillating tidal 1047298ow driven along the channel by a water

level difference between the ends of the channel This difference or

headloss is due to the differing tidal regimes in the two large water

bodies which are connected by the channel The water bodies are

assumed to be so large that any water 1047298owing through the channel

does not affect water levels within them Thus water levels at the

ends of the channel are unaffected by a power extraction within it

This is the simplest useful channel geometry One extension not

included here has a lagoon at one end of the channel and a large

ocean at the other [9] The 1047297nite reservoir of the lagoon means that

tides within it depend on the volume of water which1047298ows through

the channel As a result a lagoon can increase or decrease a tidal

channels potential depending on whether the amplitude of the

head between the ends of the channel is less than or greater than

the tidal amplitude in the ocean [10] In addition any large deep

ocean will likely have a shallow continental shelf between it and

the entrance to the channel Frictional dissipation and resonances

over the shelf may also in1047298uence the amplitudes of the tides at the

entrance to the channel driven by tides in the deep ocean [11]

The GC05 model is given in terms of volume transport here it is

presented in terms of velocity In short uniform cross-section

channels the cross-sectional average velocity does not vary signi1047297-

cantly along the channel [1213] Thus the tidal velocity everywhere

along a short channel with a rectangular cross-section depends only

ontimeand can beexpressedas u frac14 u0sinethut thorn fuTHORN where u0 is the

amplitude of the velocity and fu its phase GC05rsquos momentum

balance for a uniform cross-section channel can be written in the

form

vu

vt frac14

g z0

L sinethut THORN

C Dh thorn

C F

L

uu (1)

In Eq (1) the 1047297rst term represents the inertia of the 1047298ow the

second term the sinusoidal pressure gradient or head which forces

Fig 1 Schematic of a turbine farm in a narrow constricted channel connecting two

large water bodies Differing tidal regimes in the two large water bodies drive oscil-

lating tidal 1047298ow through the channel The example farm has 3 rows of turbines The

arrows around each turbine indicate the stronger 1047298ows passing around the turbines

and the weaker 1047298ows passing through the turbines

Fig 2 Schematic of 1047298ow through a row of identical turbines and the 1047298ow through and

around a single turbine within the row Relative sizes of the velocities are u4 u u1

u3 After V11















































es the 1047298

ow










ε frac14 M AT

A








P avail frac14 4

3pr Ar 1N RC Ru3

0 (5)









amp V11














0 05 1 1504

06

08

1


U 0


CF C

Fpeak

a

0 05 1 150

02

04

06

08

1

P o w e r P

m a x

CF C

Fpeak


b





these turbines























power production


















FE frac14 P avail

P maxlost

(6)













































500 m3s1





0 02 04 06 08 10

02

04

06

08

1


P a v a i l P m a x

o r U 0

U 0 U Da

F a r m

E f f i

c i e n c

y


0 02 04 06 08 10

05

1

15

2






























the strait






































































0 5 10 150

02

04

06

08

1

Number of Rows NR

a

Farm Efficiency

0 5 10 1504

06

08

1

Number of Rows NR

b

Flow reduction










6 Discussion














ef 1047297






















cant fraction of

0 10 20 30 400

02

04

06

08

1

1

3

5

101

3

P o w e r P

m a x

Number of Turbines

aShallow Channel

0 5 10 150

02

04

06

08

1

1

5

10

20

50

1

5

10

P o w e r P

m a x


bTidal Strait

0 10 20 30 400

02

04

06

08

1

12

P o w


Number of Turbines

c

0 5 100

02

04

06

08

1

12

P o w



d

06 07 08 09 10

02

04

06

08

1

5

10

15

2515

25

P o w e r P

m a x

Flow Reduction

e

06 07 08 09 10

02

04

06

08

1

10002000

5000

10000

15000

2000

5000

10000

15000

P o w e r P

m a x

Flow Reduction

f










































7 Conclusions










































References








































































es the 1047298

ow










ε frac14 M AT

A








P avail frac14 4

3pr Ar 1N RC Ru3

0 (5)









amp V11














0 05 1 1504

06

08

1


U 0


CF C

Fpeak

a

0 05 1 150

02

04

06

08

1

P o w e r P

m a x

CF C

Fpeak


b





these turbines























power production


















FE frac14 P avail

P maxlost

(6)













































500 m3s1





0 02 04 06 08 10

02

04

06

08

1


P a v a i l P m a x

o r U 0

U 0 U Da

F a r m

E f f i

c i e n c

y


0 02 04 06 08 10

05

1

15

2






























the strait






































































0 5 10 150

02

04

06

08

1

Number of Rows NR

a

Farm Efficiency

0 5 10 1504

06

08

1

Number of Rows NR

b

Flow reduction










6 Discussion














ef 1047297






















cant fraction of

0 10 20 30 400

02

04

06

08

1

1

3

5

101

3

P o w e r P

m a x

Number of Turbines

aShallow Channel

0 5 10 150

02

04

06

08

1

1

5

10

20

50

1

5

10

P o w e r P

m a x


bTidal Strait

0 10 20 30 400

02

04

06

08

1

12

P o w


Number of Turbines

c

0 5 100

02

04

06

08

1

12

P o w



d

06 07 08 09 10

02

04

06

08

1

5

10

15

2515

25

P o w e r P

m a x

Flow Reduction

e

06 07 08 09 10

02

04

06

08

1

10002000

5000

10000

15000

2000

5000

10000

15000

P o w e r P

m a x

Flow Reduction

f










































7 Conclusions










































References
















































power production


















FE frac14 P avail

P maxlost

(6)













































500 m3s1





0 02 04 06 08 10

02

04

06

08

1


P a v a i l P m a x

o r U 0

U 0 U Da

F a r m

E f f i

c i e n c

y


0 02 04 06 08 10

05

1

15

2






























the strait






































































0 5 10 150

02

04

06

08

1

Number of Rows NR

a

Farm Efficiency

0 5 10 1504

06

08

1

Number of Rows NR

b

Flow reduction










6 Discussion














ef 1047297






















cant fraction of

0 10 20 30 400

02

04

06

08

1

1

3

5

101

3

P o w e r P

m a x

Number of Turbines

aShallow Channel

0 5 10 150

02

04

06

08

1

1

5

10

20

50

1

5

10

P o w e r P

m a x


bTidal Strait

0 10 20 30 400

02

04

06

08

1

12

P o w


Number of Turbines

c

0 5 100

02

04

06

08

1

12

P o w



d

06 07 08 09 10

02

04

06

08

1

5

10

15

2515

25

P o w e r P

m a x

Flow Reduction

e

06 07 08 09 10

02

04

06

08

1

10002000

5000

10000

15000

2000

5000

10000

15000

P o w e r P

m a x

Flow Reduction

f










































7 Conclusions










































References

















































the strait






































































0 5 10 150

02

04

06

08

1

Number of Rows NR

a

Farm Efficiency

0 5 10 1504

06

08

1

Number of Rows NR

b

Flow reduction










6 Discussion














ef 1047297






















cant fraction of

0 10 20 30 400

02

04

06

08

1

1

3

5

101

3

P o w e r P

m a x

Number of Turbines

aShallow Channel

0 5 10 150

02

04

06

08

1

1

5

10

20

50

1

5

10

P o w e r P

m a x


bTidal Strait

0 10 20 30 400

02

04

06

08

1

12

P o w


Number of Turbines

c

0 5 100

02

04

06

08

1

12

P o w



d

06 07 08 09 10

02

04

06

08

1

5

10

15

2515

25

P o w e r P

m a x

Flow Reduction

e

06 07 08 09 10

02

04

06

08

1

10002000

5000

10000

15000

2000

5000

10000

15000

P o w e r P

m a x

Flow Reduction

f










































7 Conclusions










































References
































6 Discussion














ef 1047297






















cant fraction of

0 10 20 30 400

02

04

06

08

1

1

3

5

101

3

P o w e r P

m a x

Number of Turbines

aShallow Channel

0 5 10 150

02

04

06

08

1

1

5

10

20

50

1

5

10

P o w e r P

m a x


bTidal Strait

0 10 20 30 400

02

04

06

08

1

12

P o w


Number of Turbines

c

0 5 100

02

04

06

08

1

12

P o w



d

06 07 08 09 10

02

04

06

08

1

5

10

15

2515

25

P o w e r P

m a x

Flow Reduction

e

06 07 08 09 10

02

04

06

08

1

10002000

5000

10000

15000

2000

5000

10000

15000

P o w e r P

m a x

Flow Reduction

f










































7 Conclusions










































References
































































7 Conclusions










































References



























efficiency of tidal turbine farms

Documents