## Abstract

Theories of in-stream turbines are adapted to analyse the potential electricity generation and impact of turbine arrays deployed in Minas Passage, Bay of Fundy. Linear momentum actuator disc theory (LMADT) is combined with a theory that calculates the flux through the passage to determine both the turbine power and the impact of rows of turbine fences. For realistically small blockage ratios, the theory predicts that extracting 2000–2500 MW of turbine power will result in a reduction in the flow of less than 5 per cent. The theory also suggests that there is little reason to tune the turbines if the blockage ratio remains small. A turbine array model is derived that extends LMADT by using the velocity field from a numerical simulation of the flow through Minas Passage and modelling the turbine wakes. The model calculates the resulting speed of the flow through and around a turbine array, allowing for the sequential positioning of turbines in regions of strongest flow. The model estimates that over 2000 MW of power is possible with only a 2.5 per cent reduction in the flow. If turbines are restricted to depths less than 50 m, the potential power generation is reduced substantially, down to 300 MW. For large turbine arrays, the blockage ratios remain small and the turbines can produce maximum power with a drag coefficient equal to the Betz-limit value.

## 1. Introduction

The Bay of Fundy has the world’s highest tides, with a tidal range of over 17 m in Minas Basin. The basin is connected to the Bay of Fundy by Minas Passage, a channel that is approximately 5 km wide, 15 km long and up to 150 m deep, as shown in figure 1*a*. During flood and ebb tides, the water flux through the passage can reach 1×10^{6} m^{3} s^{−1}, with water speeds exceeding 5 m s^{−1}. These high water speeds and the huge volume of water flowing through Minas Passage make it one of the world’s most interesting sites for in-stream tidal power development. The Nova Scotia Department of Energy is spearheading a project to deploy test turbines in Minas Passage to explore the possibility of further commercial development (http://www.fundyforce.ca).

A critical aspect of tidal power development is an accurate assessment of the power resource. Initial assessments of Minas Passage, based on the kinetic energy flux through the passage, estimated a maximum mean power of 1900 MW [1]. (Throughout this paper, when we refer to power or turbine power, we are referring to the mean power, i.e. the average power over a diurnal tidal cycle.) From this maximum power, it was estimated that less than 15 per cent of the power would be available for electricity generation, giving an estimate of approximately 170 MW of generation capacity for Minas Passage and approximately 300 MW for the entire Minas Channel [2]. These estimates are considerably smaller than the 10 000 MW estimated using the potential energy of the tides in Minas Basin [3].

A more recent assessment of the power potential of Minas Passage was presented by Karsten, McMillan, Lickley and Haynes [4] (hereafter KMLH). KMLH adapted the results of Garrett & Cummins [5] to the case of a channel connecting a tidal basin to the ocean. They found that the maximum extractable mean power is given by
1.1
where *ρ* is the water density (taken to be 1025 kg m^{−3}), *a* the amplitude of the forcing tides at the entrance of the channel, *g* the acceleration due to gravity and *Q*_{0} the undisturbed peak volume flux through the channel (see also [6]). Since the power depends linearly on the volumetric flow rate, the formula does not differentiate between thin channels with strong flow and wide channels with weaker flow. It should be emphasized that for a channel connected to a basin, the potential power does not depend on the existing tidal head across the channel, which is often small, but on the potential tidal head—the tidal head when the forcing tides and basin tides are 90^{°} out of phase. From our numerical simulation, we find that for Minas Passage *a*=4.5 m and *Q*_{0}=8.4×10^{5} m^{−3} s^{−1}, giving a maximum time-mean, extractable power of *P*_{avg}=7400 MW!

KMLH [4] applied the theory to Minas Passage and compared it with a two-dimensional, depth-averaged numerical model of the tidal flow in the Bay of Fundy. The numerical model extracted power from Minas Passage by increasing the bottom friction in the passage. The numerical results agreed well with the theory, with a maximum mean extractable power of nearly 7000 MW. The impact of extracting this much power was a 40 per cent reduction in the flow through Minas Passage. It should be noted that it is critical to limit the reduction in flow through the passage because this reduction in flow directly translates into a reduction in the tidal range in the Minas Basin. Even small changes in the tidal range could have a severe impact on the sensitive and important intertidal ecology of the basin. As well, KMLH also demonstrated that any power extraction in Minas Passage pushes the Bay of Fundy–Gulf of Maine system closer to resonance with the forcing tides, resulting in increased tidal amplitudes throughout the Gulf of Maine. When 7000 MW of power was extracted, these increases in the tides could be as large as 20 per cent. Although extracting the maximum power produces significant impacts, these impacts were reduced dramatically if the extracted power was reduced. KMLH calculated that 2500 MW of power can be extracted with a maximum of 5 per cent change in the tidal amplitude throughout the Bay of Fundy and the Gulf of Maine, with the largest changes in Minas Basin.

The ability to extract such a large amount of power without drastically reducing the flow is related to the increase in the tidal head across the passage as power is extracted. In the KMLH numerical simulations of Minas Passage, the tidal head increases from 1.4 m with no power extraction to 3.2 m at maximum power extraction. This increase in the tidal head increases the hydrostatic pressure that forces the flow through the passage, partially offsetting the retarding force of the turbines.

While the results of KMLH [4] are important, their estimates were based on turbines that essentially constituted a barrage where all the water flowing through the passage flows through the turbines. It is not clear how they apply to realistic arrays of individual turbines. In this paper, we examine the use of theoretical models to assess the power potential of a turbine array placed in Minas Passage. In §2, we apply the Betz limit to the results of a numerical simulation to give a simple assessment of the power potential of the flow through the passage. In §3, the linear momentum actuator disc theory (LMADT) is combined with the theory presented in KMLH and applied to Minas Passage. This allows us to make initial estimates of the potential power generation and impact of a series of turbine fences. In §4, we further adapt the theory to examine arrays of individual turbines using the output from our numerical simulations of the flow through Minas Passage. In this turbine array model, we use LMADT and KMLH theory but also consider how the turbine wakes and varying water speeds affect the positioning of the turbines in the array. In §5 of the paper, we conclude and discuss our results.

## 2. Betz formula applied to numerical simulations

A simple approach to estimating the power of individual turbines in a flow is the use of the Betz formula:
2.1
where *u* is the water speed and *A* the turbine cross-sectional area, taken here to be 400 m^{2}, which is similar to that of Marine Current Turbines’ SeaGen turbine. The overbar indicates that we are taking the time average over a tidal cycle. *C*_{P} is the power coefficient, which in the Betz theory has a maximum value of 16/27=0.59, indicating that only 59 per cent of the kinetic energy flux is available to generate electrical power. Actual turbines do not achieve such a high power coefficient. For example, the SeaGen turbine is rated as producing 1.2 MW at a water speed of 2.4 m s^{−1}, which corresponds to *C*_{P}=0.42.

In order to apply the Betz formula to Minas Passage, we simulated the tides and currents in the upper Bay of Fundy using the two-dimensional, finite-volume model FVCOM (see [7] for model details). The specific model grid was adapted from grids developed by Triton Consultants and David Greenberg and Jason Chaffrey at the Bedford Institute of Oceanography. The model domain covers the upper portion of the Bay of Fundy, and the model is forced by specifying the tidal amplitude and phases at the open boundary. The model has been validated through comparisons to tide gauge data and recent current measurements from Minas Passage. More details of the model and the domain can be found in KMLH [4] and [8].

We apply the Betz formula, equation (2.1), using the simulated speed at each grid point to calculate the potential mean power. The result is shown in figure 1*b*. The figure illustrates several key points. There is a large area where a turbine could generate in excess of 2 MW, and much of the passage can generate 1 MW. By comparing the top and bottom plots in figure 1, it can be seen that some of the highest potential is in deep water—water deeper than 70 m—which could provide challenges for deployment. But well over half of the energetic area lies in water between 30 and 70 m in depth. The figure emphasizes the potential size of a turbine farm in Minas Passage. The challenge is to estimate the power potential and the impact of such a turbine farm.

## 3. Theoretical assessment of turbine fences

The power potential of a fence of turbines in a channel can be estimated using linear momentum actuator disc theory (LMADT). The theory uses momentum balances and Bernoulli equations to derive formulae for the flow past a turbine fence. Garrett & Cummins [9] illustrated how the Betz theory could be adapted to a finite channel, illustrating that, since the channel restricts the flow, the potential power of a turbine fence changes as it occupies a larger portion of the channel cross section. Whelan *et al*. [10] illustrated how free-surface effects and a finite Froude number could be included in the theory. Houlsby *et al*. [11] gave a detailed derivation of the models, from the Betz formulation to the full LMADT. Finally, Draper *et al*. [12] presented the LMADT theory and compared it with numerical simulations. Here, we give a brief review of the formulation of LMADT as presented in [9,12].

Figure 2, adapted from [12], gives the configuration of the flow past a partial turbine fence. The numbers along the bottom of the figure label the region of the domain: region 1 is far upstream from the turbine, region 2 is immediately upstream from the turbine, region 3 is immediately downstream from the turbine, region 4 is in the wake of the turbine, and region 5 is far downstream from the turbine, where the wake has mixed with the surrounding flow. The theory assumes the incoming flow, *u*_{1}=*u*, is uniform with depth and across the channel.

The flow through and past a turbine fence can be calculated in terms of the blockage ratio,
3.1
which is the ratio of the turbine fence cross-sectional area, *A*, to the cross-sectional area of the channel, *A*_{c}, and thus is the portion of the channel cross section occupied by turbines. In the problem formulation, we have one free variable. Choosing the value of this variable can be interpreted as tuning the turbine to extract more or less power from the flow. Mathematically, it is most convenient to choose the free variable to be
3.2
which is the ratio of the water speed in the turbine wake, *u*_{t4}, to the upstream water speed, *u*.

Following [12], we can calculate *β*_{4}=*u*_{b4}/*u* by finding the solution of the quartic equation
3.3
and thus obtain the speed of the flow outside the turbine wake, *u*_{b4}. Here, *F* is the Froude number, defined by , which we will calculate based on the undisturbed depth and flow speed. When the Froude number is set to zero, equation (3.3) reduces to a quadratic equation in *β*_{4} that is equivalent to eqn. (2.22) in [9]. After finding *β*_{4}, we can calculate the flow through the turbine, *u*_{t2}, by calculating *α*_{2}=*u*_{t2}/*u* using the formula
3.4
We require a physically acceptable solution to equations (3.3) and (3.4), that is, *β*_{4} is a real number with *β*_{4}≥1 and 0≤*α*_{2}≤1. This is not possible for all values of *B*, *F* and *α*_{4} (see [12] for more details). Fortunately, this is not an issue for the small blockage ratios considered in this paper.

Having calculated the flow through the turbine, we can calculate the turbine thrust. Here, we choose to write the thrust as a quadratic drag
3.5
where the turbine drag coefficient is given by
3.6
Thus, using LMADT, we can calculate the drag coefficients of the turbine fence knowing only *B* and *α*_{4}. Specifying the flow in the wake of the turbine is an odd way to describe the design of a turbine. Here, we choose to present our results in terms of the turbine drag coefficient, *C*_{D}. Alternatively, one could specify the reduction of flow through the turbine, *α*_{2}, or the axial induction factor, 1−*α*_{2}.

In order to combine the theory with the theory of KMLH [4], we can write the turbine thrust as if it were generated by a quadratic drag on the upstream flow acting across the entire channel cross section; that is, we write
3.7
where the effective fence drag coefficient is given by
3.8
Using this, we can calculate the total power extracted from the flow,
3.9
Finally, we can calculate the turbine power,
3.10
with the power coefficient given by
3.11
As discussed in [9], the difference between the turbine power, *P*, and the total extracted power, *P*_{ext}, is the power lost in the merging of the turbine wake and the free stream. The details of power lost in the wake are discussed in more detail in [11,12].

While LMADT calculates the power of a turbine fence, it does not determine *u*, the speed of the flow through the channel. As the calculation in KMLH [4] illustrates, calculating this flow is essential in determining the power potential of a channel and the impact that power extraction will have on the flow through the channel. Recently, Vennell [13] illustrated that the turbine fence model of Garrett & Cummins [9] could be combined with the power theory of Garrett & Cummins [5] to determine the power potential of a partial turbine fence in a given tidal flow. His analysis illustrated a couple of key points. First, turbines can be tuned to maximize the power generated for a specific arrangement of turbine fences in a given channel. Second, the maximum potential power of a channel can be realized with partial fences by adding sufficient rows of these fences to the channel. In our previous work [8], the theory described in [13] is adapted to Minas Passage. Here, we further adapt the theory to include the effects of a finite Froude number following the LMADT discussion above.

The theory in KMLH [4] calculates the volume flux through Minas Passage as a function of a non-dimensional drag coefficient,
3.12
where *C*_{DF} is the effective fence drag coefficient given by equation (3.8) and *C*_{D0} is the natural drag coefficient of the passage; that is, *C*_{D0} is the drag coefficient of momentum lost through bottom drag and nonlinear inertia. Therefore, *λ*_{T} represents the ratio of the turbine drag to the natural drag in the system. From the analysis in KMLH, we can estimate that *C*_{D0}=1.8 for Minas Passage.

Following KMLH [4], the flux through the passage with turbines is
3.13
where *Q*_{0} is the undisturbed peak volume flux through the passage, estimated from our numerical simulations to be 8.4×10^{5} m^{3} s^{−1}. The reduction in flow factor, *R*, is given by
3.14
The parameter *δ* in the formula is determined by the geometry of the system and the natural drag in the basin. It can be written in terms of the non-dimensional parameters given in KMLH as follows:
3.15
Using the Minas Passage values for these parameters from KMLH (*β*=7.6, ) gives *δ*=0.2. Therefore, given the effective fence drag coefficient *C*_{DF}, one can calculate the flux through the passage and, hence, the water speed in the passage given by
3.16

Finally, the calculation of the turbine power given by equation (3.10) must be adapted to reflect that the tidal flow varies with time; that is, water speed is
where, for simplicity, we will assume a single tidal constituent with frequency *ω*. (Note that this is a reasonable assumption for the tides in the Bay of Fundy, which are dominated by the *M*_{2} signal. However, there is a significant spring–neap variation in the tidal currents, and a more accurate model would require several constituents.) Then the mean turbine power is
3.17
where the factor of 4/3*π*≈0.42 arises from the reduction in power associated with the water speed oscillating from 0 to its maximum *u*. That is, the mean power is approximately 42 per cent of the maximum power during the diurnal tidal cycle.

To summarize, given a chosen blockage ratio *B* and wake speed ratio *α*_{4}, we can use LMADT to calculate the effective turbine-fence drag using equation (3.8) with equations (3.3) and (3.4). We can then use this drag to calculate the new volume flux through the passage using equations (3.13)–(3.15) and the water speed in the passage using equation (3.16). Finally, we calculate the turbine power using equation (3.17). Thus, we can estimate the power potential of a given turbine fence and the reduction in flow through the passage.

It should be noted that, for an isolated turbine, *B*≪1, the theory reduces to the Betz limit with a maximum power at , and *β*_{4}=1 giving *C*_{P}=0.59, *C*_{D}=1. For a complete turbine fence, *B*=1, the results of KMLH [4] are recovered if one assumes the Froude number to be zero.

As shown in [8], when the turbines occupy a large blockage ratio, *B*>0.5, the maximum turbine power of a single turbine fence is achieved with a very large turbine drag coefficient, *C*_{D}>20, resulting in very high power coefficients, *C*_{P}>5. That is, in theory, one can obtain in excess of 500 per cent of the kinetic energy flux for the Minas Passage system. This reflects the fact that a large fence of turbines is essentially a tidal barrage and, therefore, its potential power is related to the potential energy of the system, not the kinetic energy. Consequently, the turbines should be designed as barrage turbines with a high drag coefficient.

Here, we are concerned with realistic arrays of in-stream turbines deployed on the sea bed. The average depth of Minas Passage is about 50 m. If we assume that the turbines have a height of 20 m, then we have that *B*<0.4. Therefore, we restrict our analysis to relatively small values of *B*. However, we expect that many rows of turbine fences with small blockage ratios will be deployed. Following [13], we can model *N*_{R} rows of turbine fences by simply aggregating their effect. That is, we replace the formula for the effective fence drag coefficient given by equation (3.8) with
3.18
and the mean turbine power equation (3.17) becomes
3.19

In order to illustrate the results of the theory, we choose four blockage ratios—0.4, 0.2, 0.1 and 0.05—which approximately correspond to fences of 310, 155, 77 and 38 turbines with cross-sectional area 400 m^{2} across Minas Passage. In figure 3, we plot the mean turbine power versus *C*_{D} (*a*) and the resulting reduction in flow through the passage (*b*). For these plots, the total turbine area is set to twice the total cross-sectional area of the passage. That is, the results are for five rows at a blockage ratio of 0.4, 10 rows at 0.2, 20 rows at 0.1 and 40 rows at 0.05. In total, they correspond to arrays of approximately 1550 turbines.

The first conclusion we can draw is that such arrays of turbines can generate 1500–2500 MW of power with only a 3–5% reduction in the flow through the passage. That is, the change from a full barrage of turbines to rows of partial fences has not drastically changed the conclusions of KMLH [4]. The next conclusion we can infer is that a higher blockage ratio results in greater power. The greater power occurs for two reasons. First, at equal values of *C*_{D}, a higher blockage ratio results in a higher flow through the turbine—a larger value of *α*_{2}—and, thus, the power is greater. Second, the drag can be increased to higher values when the blockage ratio is higher. While the maximum power occurs when *C*_{D}=1 for small blockage ratio, corresponding to the Betz limit, the maximum power occurs at *C*_{D}=4 for *B*=0.4. For small values of drag, all the fences give similar results.

In figure 3, we see that the reduction in flow increases with increasing power, as one would expect. We can also conclude that tuning the turbines to reach maximum power can be costly in terms of their impact. For example, with *B*=0.4, 2500 MW of power results in less than a 4 per cent reduction in flow, while increasing this to 3300 MW of power results in an 8.5 per cent reduction in flow. Generating the extra 800 MW has a very large impact on the flow. One concludes that it is not efficient to increase the turbine drag above *C*_{D}=1. It is worth noting that, when *C*_{D}=1, the effective fence drag coefficient *C*_{DF} varies from 0.95 for *B*=0.05 to 1.3 for *B*=0.4. Thus, in all these cases, *C*_{DF}<*C*_{D0}, the effective drag of some 1500 turbines is less than the natural drag in the passage!

In figure 4, we plot results where we hold the turbine drag constant, *C*_{D}=1, and vary the number of rows of turbines. We see that the power increases almost linearly for small blockage ratios, with each turbine producing approximately 1 MW. Increasing the blockage ratio to 0.4 increases the power by almost 50 per cent, with each turbine producing over 1.5 MW. The power per turbine does decrease, albeit slowly, as the increased number of turbines begins to decrease the speed through the passage.

In figure 4, we also plot the turbine power versus the reduction in flow. The four curves here are remarkably similar, implying that the different blockage ratios and number of fences do not drastically change the relationship between the power generated and the impact on the flow. For example, for a 5 per cent reduction in flow, the power ranges from 2600 to 3000 MW. For small amounts of power (less than 1500 MW), we can generate approximately 750 MW for each percentage point of flow reduction, in agreement with KMLH [4]. This power rate is reduced as the number of turbines increases, down to only about 500 MW when the power exceeds 3500 MW. Note that, in these figures, we have limited the power to only 4000 MW. One can continue to increase the number of rows until the power reaches the theoretical maximum of near 8000 MW, but this would occur at an unrealistically high number of rows and an unacceptably high reduction in the flow through the passage.

In summary, the adaptation of LMADT presented here allows for several conclusions. First, 2000–2500 MW of theoretical turbine power can be realized for less than a 5 per cent reduction in flow through the passage. Second, if the blockage ratio is increased, it will result in greater power per turbine, even if we do not tune the turbines to have a higher drag. Third, if the turbines are tuned to increase the turbine power, it comes at the cost of a much higher reduction in flow. Finally, the relationship between turbine power and reduction in flow does not depend strongly on the blockage ratio.

## 4. Turbine array analysis for Minas Passage

The above analysis presents interesting conclusions in terms of the potential power and impact of a series of turbine fences. However, it does not address where the turbines should be placed in the passage or whether there is sufficient space in the passage to locate the thousands of turbines. In spacing the turbines, we must consider the size of the turbine wakes and the spatial variations in potential power illustrated in figure 1. In this section, we take these factors into account to produce a model that uses water speeds from numerical simulations to predict the power and impact of realistic turbine arrays.

We start by interpolating the depth-averaged speed from our numerical simulation onto a regular 20×20 m grid in a coordinate system with *x* running along the passage and *y* running across the channel, with *y*=0 marking the centre of the channel. The result is shown in figure 5*a*. A turbine is assumed to have an area of 400 m^{2}, corresponding, approximately, to a diameter *T*_{d}=20 m. The turbine can be placed in any of the grid boxes on the 461×375 grid, resulting in well over 100 000 possible turbine locations. For the calculations presented here, the turbines are deployed on the bottom. As well, we tune the turbine array by specifying the turbine drag.

For each turbine array, we calculate the turbine power and new flow through the passage as follows. First, the channel is divided into cross-channel strips that are 10 turbine diameters in width. For each strip, we apply LMADT to calculate the turbine flow factor, *α*_{2}, the effective fence drag coefficient, *C*_{DF}, and the power coefficient, *C*_{P}. Then, to apply the theory of KMLH [4] we calculate the total turbine drag coefficient of the channel, *λ*_{T}, as a weighted sum of the effective drag coefficients—the coefficients are weighted by the cross-sectional area of each strip, which varies since the width of the channel varies. We use this to calculate the reduction in the flux through the passage, *R*(*λ*_{T}), using equation (3.14).

Next, we approximate the wake of each turbine. The details of a turbine wake can be difficult to determine and will depend on the specific design of the turbine. We know that there will be a reduction in the water speed in the wake, and the turbulent flow will mix with the surrounding fluid as we move farther downstream from the turbine [14]. Therefore, we assume that the wake decays exponentially from the turbine with length scale 10 times the turbine diameter. Finally, since the wake cross-sectional area must be larger than the turbine area, we also assume that the wake has a cross-channel length scale equal to the turbine diameter. From these considerations the water speed in the wake, *u*_{w}(*x*,*y*), of a turbine located at (*x*_{0},*y*_{0}) is
4.1
where *u* is the undisturbed speed. Note that the factor (1+*α*_{4})/2 accounts for the fact that, in tidal flow, the point (*x*,*y*) is upstream from the turbines half the time and in the turbine wake the other half the time. Admittedly, this is a crude representation of a turbine wake, but it gives a rough idea of how the turbines must be spaced. At the turbine location itself, the flow is reduced by the factor *α*_{2}.

After calculating the wake speed and the flow through the turbines, we recalculate the mean velocity on each cross section so that it has the new flux as stipulated by the reduction factor, *R*(*λ*_{T}). This step will increase the speed of the flow around the turbines; that is, it is similar to the calculation of *β*_{4} in LMADT. But now, the calculation also includes the effect of the turbine wakes. In the end, the result is the calculation of a new speed that accounts for the impact of LMADT, the reduction in flow through the passage and the turbine wakes. An example of the resulting flow field around 1000 turbines is shown in figure 5*b*. The reduction in flow in the turbine wakes is clearly visible.

The mean turbine power is then calculated for each turbine using the formula 4.2 where the constant factor in equation (4.2) has changed because the speed is now the mean speed and not the maximum speed. It should be noted that this power will vary for every turbine since every location has a different speed. The total power is simply a sum of the powers of all turbines.

The final aspect of assessing turbine arrays is determining how to choose the location of the turbines. We are interested in arrays that maximize power. However, finding the optimal location of thousands of turbines given the 100 000 possible locations is an extremely difficult, if not impossible, problem. Instead, we take a simple sequential approach. We place the first turbine at the location with the highest speed. We then use the model to calculate the new speed. We use this new speed to place the second turbine, and so on. The result is a near-optimal placement of the turbines. Using this process, the turbines tend to align themselves into fences, as shown in figure 5*b*.

The focus of the results here is not the particular location of the turbines, since the actual positions of turbines will require detailed knowledge of the turbines, the tidal currents, the sea bottom, the connecting cables, etc. Instead we examine only one aspect of the turbine positioning, restricting the depth of water where turbines can be deployed. Currently, most developers are focusing on relatively shallow water less than 50 m deep. For turbine arrays of the size we are discussing here, such depth restrictions will play an important role. For all the results, turbines are restricted to water deeper than 30 m, a reasonable minimum depth for 20 m diameter turbines. Figure 5 shows a turbine array where the turbines are restricted to depths between 30 m and 70 m.

The results of the tidal array model are shown in figure 6, where three curves of the turbine power are plotted versus the number of turbines. The first curve has no restriction on the maximum water depth where turbines can be placed. For the other two curves, turbines are restricted to water depths of less than 70 m and 50 m, respectively. Considering the unrestricted-depth curve first, we see that the initial turbines produce a high rate of power. As the number of turbines increases, the power produced per turbine drops from more than 1.5 MW to less than 0.5 MW, since the initial turbines are placed in locations of strong flow while later turbines are being placed in locations of lower flow. However, it is still possible to achieve 2000 MW of power with a large array of nearly 2500 turbines. When there are 2000 turbines, the blockage ratio of the 200 strips has an average of 0.05 and a maximum of 0.1. If the power curve is compared with the results of the previous section, in particular figure 4, we see that the power produced is very similar to the curves for the blockage ratios 0.05 and 0.1. The biggest difference is how the power per turbine changes as more turbines are added.

When the turbines are restricted to water depths less than 70 m, we see a reduction in power. This reduction is not significant for the first 1000 turbines, but becomes large for more than 2000 turbines. In fact, deploying more than 2000 turbines results in only a small increase in power. At this point, any additional turbine has to be placed in the wake of a previously placed turbine. Thus, not only is the turbine in a location of weak flow, but it also reduces the power of the other turbines that now lie in the new turbine’s wake. Quite simply, we have run out of space to place more turbines.

If the turbines are restricted to water depths less than 50 m, we see a drastic reduction in the turbine power. This power reduction occurs immediately, as there are few high-energy locations with a depth less than 50 m. The power increases very slowly after only 500 turbines and actually decreases after 1500 turbines, where the drag on the flow is still increasing but the turbines are so closely packed that they are all operating in another turbine’s wake. It is clear from the last graph that being restricted to a maximum depth of 50 m will severely restrict the power potential of bottom-mounted turbine arrays in Minas Passage.

In figure 6, we plot the turbine power versus the reduction in flow. For all three cases, the curve is a remarkably straight line, with 800 MW of power for every 1 percentage point reduction in flow. This is a slightly higher value than was seen in figure 4. For no depth restriction, 2000 MW of power extraction causes only a 2.5 per cent reduction in flow. Once again, this reinforces the conclusion that a large quantity of turbine power is possible with little reduction in flow through the channel.

Finally, we examine the impact of changing the drag of the turbines in the array. Figure 3 suggests that the power generated by a turbine array could be increased by increasing the drag coefficient with a greater reduction in flow through the passage. Using the turbine array model, we varied the drag coefficient over a range of values from 0.7 to 1.9 for the 2000-turbine array. The maximum power was obtained at *C*_{D}=1.2, but was only 10 MW higher than the power at *C*_{D}=1. The power drops rapidly if *C*_{D} is increased beyond 1.2, and the reduction in flow increases substantially. We conclude that in this array the turbines are most efficient when running near the Betz limit and that there is little opportunity to tune the turbines to generate more power. However, there could be reason to reduce the turbine drag to reduce the turbine impact on the flow.

## 5. Conclusions

Our goal in this paper was to see whether the results of KMLH [4] would change significantly if we examined models of turbine arrays. In §3, we presented an adaptation of the theory presented by [13] that combined the LMADT of [12] with the theory of KMLH. In §4, we extended this model to a turbine array model, which took into account the variations in flow in the passage, turbine wakes and the positioning of turbines. The overall conclusion is that the results of KMLH are supported. In both models, over 2000 MW of turbine power could be generated by an array of 2000 turbines with less than a 5 per cent reduction in the flow. For smaller arrays, 700–800 MW of power is possible for every 1 percentage point reduction in the flow.

Several other conclusions come out of this analysis. For a channel the size of Minas Passage, we expect the blockage ratios will be very low, generally less than 0.1. So the importance of the LMADT theory is reduced. In the turbine array model, most of the turbines are very near the Betz limit. As well, this means there is little reason to tune the turbines to have a higher drag coefficient. By a similar argument, the change in the flow through the passage is so small, even with 2000 turbines, that it does not need to be accounted for in the power assessment of small arrays. The turbine array model results indicate that the most important aspect of assessing an array is locating the turbines in regions of high flow and not in each other’s wakes. The model results also illustrated that, if turbines are restricted to water depth less than 50 m, the size and potential power of the array will be severely restricted. Increasing the restricted depth to 70 m greatly increases the potential size and power of the array. But to achieve 2000 MW of power, either the turbines will need to be deployed in deep water or turbines that extract power from higher in the water column will also need to be deployed.

The turbine power values presented here are theoretical, corresponding closely to the Betz power coefficient of 0.59. These power numbers can easily be made more realistic by using the power curve of a given turbine. For the energetic flow in Minas Passage, the most important aspect of the power curve is the speed at which the turbine regulates power. If this is low, the turbine will not be able to take advantage of the strongest flows and the power levels will decrease. Similarly, the power calculations do not consider the asymmetry in the water speed between the flood and ebb tides or the possible changes in the direction of the flow. And, we have not accounted for the spring–neap cycle of the tides. All of these factors will change the exact power numbers calculated. Depending on the turbine design, they could result in either increases or decreases in power. We have not addressed many practical questions, such as whether the sea bottom at a given location is suitable for turbine installation. These details could strongly affect the design of the turbine and the choice of location to deploy the turbines. As well, the logistics and costs of installing, operating and maintaining such a large turbine array need to be accurately estimated to determine if such arrays are economic.

Finally, our analysis has indicated that the impact of the turbines is small in terms of reduction in the flow through Minas Passage. Although this is an important measure of the impact, it is only the first step in assessing the impact of a turbine array. The impact of even small changes in Minas Passage flow on the intertidal zones of Minas Basin still need to be examined and quantified. As well, large arrays will have a direct impact on the marine life that passes through Minas Passage. Ongoing research efforts to monitor the interaction of marine life with turbines, especially turbine arrays, is critical to the continuing development of sustainable tidal power.

## Acknowledgements

Thanks to the Offshore Energy and Environmental Research Association, the Atlantic Computational Excellence Network, and the Natural Sciences and Engineering Research Council of Canada for their financial support.

## Footnotes

One contribution of 14 to a Theme Issue ‘New research in tidal current energy’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.