## Abstract

Tuning wind and tidal turbines is critical to maximizing their power output. Adopting a wind turbine tuning strategy of maximizing the output at any given time is shown to be an extremely poor strategy for large arrays of tidal turbines in channels. This ‘impatient-tuning strategy’ results in far lower power output, much higher structural loads and greater environmental impacts due to flow reduction than an existing ‘patient-tuning strategy’ which maximizes the power output averaged over the tidal cycle. This paper presents a ‘smart patient tuning strategy’, which can increase array output by up to 35% over the existing strategy. This smart strategy forgoes some power generation early in the half tidal cycle in order to allow stronger flows to develop later in the cycle. It extracts enough power from these stronger flows to produce more power from the cycle as a whole than the existing strategy. Surprisingly, the smart strategy can often extract more power without increasing maximum structural loads on the turbines, while also maintaining stronger flows along the channel. This paper also shows that, counterintuitively, for some tuning strategies imposing a cap on turbine power output to limit loads can increase a turbine’s average power output.

## 1. Introduction

Power generated from tidal turbines could make a significant contribution to the global demand for renewable energy. Many of the best sites for arrays of tidal turbines lie in the concentrated energy of flows along narrow tidal channels [1–6]. To extract a significant fraction of the potential of these sites, large arrays of tidal turbines must be installed. As for wind turbines, tuning these tidal turbines is critical to maximizing their power output. Tuning is typically achieved by adjusting the blade pitch [7,8]. For wind turbines, control systems are used to maximize the power output at any given time. Tuning tidal turbines within large arrays in channels is very different to tuning wind turbines ([9]—hereafter V10, [10,11]). This work demonstrates how poor a strategy of maximizing the power output at any given time is for tidal turbines, when compared with V10’s strategy of maximizing output averaged over a tidal cycle. This work goes beyond V10 to present an even better strategy for tuning large arrays of tidal turbines.

The large arrays of tidal turbines required to realize a significant fraction of the maximum resource available within a particular narrow channel are necessarily ones large enough that power extraction by the array reduces free stream flows throughout the channel ([12]—hereafter GC05, [13]). This flow reduction due to the interaction between the array and the energy resource is not necessarily bad for power production. In some channels, expanding an array by putting more turbines in the same cross-section across the channel can increase the power output of every turbine within the array, despite a reduction in flows within the channel [10,13,14]. Large is a relative term, one turbine in a small channel would be a large array if it fills much of the channel’s cross-section, whereas 100 turbines in the 30 km wide Cook Strait would be a small array, because their total output is tiny compared with the Strait’s 14 GW potential for power production [5,14]. Turbines only have to fill 2–5% of the channel’s cross-section to begin to interact with the resource and start to become a ‘large array’. This only requires around forty 400 m^{2} turbines in the 10 km wide Pentland Firth [14].

Garrett & Cummins [15] showed that the efficiency of a row of turbines in a channel is maximized by the same tuning that maximizes the power output of an isolated wind or tidal turbine, i.e. the same tuning as a Betz turbine. However, their approach did not allow for the reduction in free-stream flow along the channel due to power extraction. V10 was the first work to show that this interaction between power extraction and the resource requires turbines in large arrays to have different tunings; tunings which depend on the array size, how they are arranged into rows and the dynamical balance within the tidal channel (figure 1). The interaction also requires tidal turbines in large arrays to have individualized tunings, which are optimized in the presence of all the other turbines [11].

Although it may deliver more power, Garrett & Cummins’s [15] tuning strategy is the same as that used for wind turbines, i.e. tune to maximize the instantaneous power output. Tuning a wind turbine to maximize power output at each instant makes sense, because even a large wind turbine array is too small to significantly affect the weather systems which drive it. Maximizing power at each instant is also sensible for isolated single tidal turbines, or turbines in small arrays within a channel. However, V10’s strategy to tune to maximize the output averaged over a tidal cycle is much better for large arrays of tidal turbines (figure 1).

To date, there has not been a comprehensive comparison of Garrett & Cummins’s [15] instantaneous tuning strategy and V10’s tuning strategy. This paper does this comparison. It also shows that, with hindsight, the advantage of the V10 strategy results from forgoing maximal power output early in the tidal cycle, in order to gain more power later in the cycle. Thus, while Garrett & Cummins’s [15] and the wind turbine approach of tuning ‘on the fly’ to maximize output ‘now’ can be described as an ‘impatient-tuning strategy’ (I-TS), the V10 approach of tuning to maximize the tidal cycle averaged output can be described as a ‘patient-tuning strategy’ (P-TS).

V10’s P-TS assumed that the optimal turbine tuning parameter was constant throughout the tidal cycle. This paper shows that V10’s strategy can be improved on by optimizing how the tuning varies during the tidal cycle. This ‘optimal tuning strategy’ generates more power, thus will be described as a ‘smart patient-tuning strategy’ (SP-TS). The SP-TS takes advantage of the interaction of the power extraction with the resource to exploit subtle manipulations of the amplitude and phase of the tidal currents. Manipulation of the resource sounds fanciful, but is a consequence of installing an array large enough to extract a significant fraction of a channel’s potential to produce power. Manipulations of the resource have already been shown to allow large tidal turbine arrays to store energy in the inertia of the flow in order to better meet short peaks in electricity demand [16]. Here it is shown that a SP-TS which manipulates the resource can have triple benefits. These are producing more power, reducing environmental impacts which may result from flow reduction, while often reducing construction costs by lowering the maximum structural load on the turbines.

This paper is divided into six sections. Section 2 outlines the channel and actuator disc models used in this work, along with describing the example channels used to present results. Section 3 explains the three tuning strategies and how they are implemented within the model. Section 4 compares power output, flow speeds and loads for the three strategies in arrays and channels of different sizes. Section 5 illustrates a counterintuitive effect of limiting the power output of turbines within large-arrays. Section 6a discusses how the higher output of the new tuning strategy is a result of a higher upper limit for power production which can significantly exceed GC05’s upper limit.

## 2. The model

A one-dimensional (1D) model is used to compare tuning strategies. This simple model allows rapid computation of its numerical solution, and thus permits a wider exploration of the influence of array and channel size on the results. However, the results need to be verified with more realistic two-dimensional (2D) models, such as [6,17]. The 1D model requires two components. The first, a ‘channel-model’ which models the interaction of the enhanced drag coefficient due to power extraction and the free-stream flow along the channel, GC05. The second, an actuator disc model for the drag and power coefficients of a row of turbines in a tidal channel [15]. These two models were first properly combined in V10 to show that tuning required for large arrays of tidal turbines in an oscillating flow is very different from that given in [15] for a constant flow.

### (a) Channel model

This work uses the GC05 model for a channel with a uniform rectangular cross-section. They model the tidal turbine farm as an enhanced drag coefficient in a short narrow channel which connects two large water bodies, so large that they are unaffected by power extraction within the channel. As in V10, and following [13], the starting point for the short channel model is the depth-averaged along channel shallow water momentum balance
*u* is the tidal velocity, *η* the free surface elevation and *F* the additional drag due to power extraction by the turbine farm, *h* is the water depth and *C*_{D} the bottom drag coefficient.

In dynamically short channels, the mass flow rate due to an oscillating tidal forcing does not vary along the channel [18]. Even large channels, like the 100 km long, 25 km wide Cook Strait NZ, can be dynamically short [19]. Integrating (2.1) over a uniform rectangular channel cross section, where lateral friction can be ignored, and then integrating along a dynamically short channel gives a simplified equation ∂*u*/∂*t*=*g*(*η*|_{x=0}−*η*|_{x=L})/*L*−(*C*_{D}/*h*+*C*_{F}/*L*)|*u*|*u*. Technically, this integration along the channel results in a zero-dimensional model, though for clarity it will still be described as ‘1D’. In this equation, *C*_{F} is the drag coefficient due to the farm based on channel cross-sectional area, and the flow is forced by the water level difference between the ends of the channel. Here the periodic forcing by the tides in the connected water bodies is given by *u*′ is the tidal velocity non-dimensionalized by the peak flow in a frictionless channel. λ_{F}=*αC*_{F} is the drag coefficient due to power extraction by the farm, λ_{0}=*αC*_{D}*L*/*H* is a constant rescaled natural bottom drag coefficient and the dimensionless constant *α* is as defined in table 1. Non-dimensional time, *t*′ is measured in radians (2*π*=*one* tidal cycle), though times will be presented as a fraction of the tidal cycle. Within (2.2) the first term is the acceleration of the flow and the second is the periodic forcing due to the water-level difference between the ends of the channel. The last term is the combined drag due to natural background bottom friction and the drag due to power extraction by the farm.

The only difference between (2.2) and GC05’s model is that the farm’s drag coefficient is based on a tuning parameter *r*_{3}, given in the next section (2.3) which varies in time. Given an *r*_{3}(*t*), equation (2.2) was solved numerically for the velocity, *u*′(*t*′), using a Runge–Kutta 4th order algorithm with 360 time steps per tidal cycle. Convergence of the numerical solution was tested by solving examples with between 50 and 500 steps per tidal cycle. Above 360 the step size had minimal effect on the solution’s maximum velocity, average velocity and the average power output. The model was run repeatedly for a number of tidal cycles until the velocity was sufficiently periodic.

### (b) Turbine model

The actuator disc model of a turbine in a channel given in [15] is used to represent the drag on the tidal flow associated with power production from one row of turbines. This model applies mass continuity and Bernoulli’s equation to the flows around the turbine. The model is valid for small Froude numbers (in contrast with [20]) and the drag coefficient is expressed as a function of the four velocities around the turbine defined in figure 2. The model assumes flows passing though the turbines are fully mixed with the bypassing flow before the next row is encountered. The drag coefficient for a row can be expressed in terms of *u*_{4}, the velocity of the flow bypassing the turbine and *u*_{3}, the velocity in the wake of the turbine before significant mixing occurs with the bypassing flow.
*r*_{i}=*u*_{i}/*u* relative to the free-stream velocity *u*. From [15] the velocity of the flow bypassing the actuator disc is given by
*ϵ*=*MA*_{T}/*A*_{c} is the fraction of the channel’s cross-section blocked by the sweep of the turbine blades (i.e. the turbine blockage ratio), *M* is the number of turbines in a row spanning the channel and *A*_{c} is the cross-sectional area of the channel. The turbine blockage ratio *ϵ* is likely to be limited by the need to maintain a navigable channel. If *r*_{3}=1 then *r*_{4}=1, and the turbines produce no power. As *r*_{3} falls below 1, power is produced and *r*_{4} increases above one to maintain mass continuity in the expanded wake behind a turbines (figure 1).

Both wind and tidal turbines must be tuned to maximize their power output. In this work, this is achieved by adjusting *r*_{3}, which is a mathematically convenient tuning parameter for the actuator disc model given in [15]. In their work, the free-stream flow is unaffected by power production and they found that *r*_{3} value found by Betz [21] for an isolated turbine (figure 1). However, V10 showed that in tidal channels where power extraction affects the free-stream flow, the constant *r*_{3} which maximizes power production lies between *r*_{3} varies with time.

The last step in this section is to relate the non-dimensional farm drag coefficient in (2.2) to a number of identical rows of turbines (2.3),
*N*_{R} is the number of rows of turbines in the array.

In order to present its main idea this work uses the simplest actuator disc model for a row of turbines [15]. There are more sophisticated actuator disc type models. For example, models which allow for the array scale turbulence due to rows extending only part way across the channel [22], or allow for staggered rows of turbines [23]. Owing to limited space, the effects that the incorporation that these aspects of actuator discs might have on the results presented here will be explored in a future work.

The enhanced flow between the turbines will experience higher bottom friction, while the retarded flows behind the turbines lower bottom friction (figure 2). This variation in bottom friction is not allowed for within the channel model (2.2). Thus there is an implicit assumption that the total horizontal area occupied by all the rows of turbines and their wakes is small compared with the horizontal area of the channel. The influence that variations in bottom friction, due to the flow distortion around the turbines, has on the results presented and localized environmental effect requires further work with more realistic 2D and three-dimensional (3D) models.

### (c) Example channel

To present results from the model, it is necessary to choose channels to use as examples. For this work, a 20 km long channel loosely based on the Pentland Firth, UK, is used. The strong tidal currents through the Firth, which lies to the south of the Orkney Islands, have considerable potential for tidal current power generation [6,24,25]. The values in table 1 are based on a bottom drag coefficient of *C*_{D}=0.0025, which Adcock *et al*. [6] found best fits the observations in the Firth. Recent works have modelled the Firth and estimate the undisturbed volume transport due to the M2 tide at 1.1×10^{6} m^{3} s^{−1} and for M2 tides λ_{0}≈1.0 [25,26]. The dimensions of the example channel in table 1 have been chosen to match these values for *C*_{D}, transport and λ_{0}. The result is peak undisturbed flows of 2.5 m s^{−1}, which is reasonable, but the estimated potential of 2.3 GW is less than the 4 GW based on the more realistic 2D model of Draper *et al.* [25]. However, the idealized Firth’s λ_{0}=1.1 provides an example representative of a channel where inertia and bottom friction are of equal importance in the undisturbed channel (2.2).

To investigate how tuning strategies might vary with channels of different sizes, a whole series of example channels based on the Pentland Firth were constructed. These examples had their length, width and depth in the same proportion as the idealized Firth. They also had the phase difference of the surface tide between the ends of the channel adjusted to give the same maximum undisturbed velocity of 2.5 m s^{−1}. Table 1 shows the sizes and parameters λ_{0} and *α* for the smallest and largest of these geometrically similar examples. The smallest has dimensions 0.20 times those of the Firth, while the largest 2.97 times those of the Firth. The largest is 60 km long and has λ_{0}=0.3, but is not as large as the 100 km long Cook Strait [5,19] where λ_{0}=0.13 [5].

To present examples, a representative area swept by the turbine blades must be chosen in order to calculate channel blockage ratios for a row of turbines. Here a 400 m^{2} turbine is used, an area similar to the largest commercially operating tidal turbine [27].

## 3. Tuning strategies explained

All tuning strategies use the combined model given in the previous section, varying only in how the tuning parameter, *r*_{3}, is optimized and whether *r*_{3} varies with time. The strategies maximize the non-dimensionalized power calculated from
*r*_{1}=*r*_{3}(*r*_{4}+*r*_{3})/(*r*_{4}+2*r*_{3}−1) gives the fraction of the power lost by the flow which is available for power production (the remainder is lost to mixing within the turbines’ wakes [15,28]). The factor of 4 in (3.1) is included to ensure that the average non-dimensional power has the same value as that used V10. Below, the first strategy seeks to maximize the power now, whereas the second two strategies maximize the power (3.1) averaged over a tidal cycle,

In addition to comparing the power output for each tuning strategy, the maximum loads on the turbines due to power production were also compared. This maximum load will affect the cost to build the turbines to withstand the structural loads imposed by power production. The gross load on one turbine due to power production at any time was calculated using
*ρ* is the density of seawater and *A*_{T} the area swept by the turbine blades. For some tuning strategies *C*_{T} varies in time, so that the maximum load may not occur at the same time as the maximum velocity.

### (a) On the fly tuning: an ‘impatient-tuning strategy’

The first strategy simply seeks to maximize how much power is extracted at each moment in time. Superficially, this is the same as optimizing the power output for a constant free stream flow along a channel, as given in [15], for which the optimal tuning is *r*_{3} which maximizes power calculated from (3.1) at the next model time step. This approach allows for very small differences in the velocity at the next time step due to changes in *r*_{3}. However, despite searching at every time step for the optimal *r*_{3}, it was always found to be very close to *b*).

### (b) Constant tuning factor: a ‘patient-tuning strategy’

The second tuning strategy assumes *r*_{3} does not vary with time and searches for a single value which maximizes the power (3.1) averaged over a tidal cycle. The model was run for an initial value of *r*_{3}. The power from this model tidal cycle is used to search for a better *r*_{3} using a standard active-set constrained optimization routine to enforce 0≤*r*_{3}≤1. This V10 strategy is referred to as ‘patient, tuning’ because optimizing to maximize the power output from a tidal cycle can result in a higher average output than on the fly tuning. Figure 3*a* demonstrates this, where for this example peak power output for P-TS is 15% higher than for I-TS. The optimal value of *r*_{3} for P-TS is also higher than the *b*).

### (c) Variable tuning factor: a ‘smart patient-tuning strategy’

The third strategy optimizes how *r*_{3} varies with time in order to maximize the power output over a tidal cycle. One way to achieve this for a periodic tidal flow is to represent the tuning as a Fourier series in time
*a*_{0}, *N*, the number of harmonics included is 15 for all cases presented here. To be physically realistic 0≤*r*_{3}≤1, however, *r*_{3}=0 implies the turbine is a solid disc stopping any flow passing through. This cannot be achieved with a bladed turbine. Consequently, linear constraints are placed on the Fourier coefficients so that 0.3≤*r*_{3}(*t*)≤1 at every model time step. Had values of *r*_{3} below 0.3 been allowed it would have enhance the ability of the SP-TS to control the flow within the channel, allowing a SP-TS to generate more power than given in this paper. However, this additional control would result in unrealistically high loads on the turbines. Thus, the lower limit of *r*_{3}=0.3 was a pragmatic choice, set to be just below the optimum *r*_{3} for an isolated turbine and that for an I-TS, thus a choice in the range of what might be possible with a bladed turbine.

Standard active-set constraint techniques were used to search though 2*N*+ 1D space for the optimal coefficients. To speed up this search, an adjoint problem derived from (2.2) and (3.1) was solved to give the gradient of the average power with respect to each of the Fourier coefficients ([17], the technique is outlined in appendix A). The adjoint technique evaluates the gradient vector at a numerical cost of approximately a single model run, whereas standard techniques require 2*N*+1 model runs at each step of the optimization in order to estimate the gradient vector. For this work, the adjoint technique improved optimization times by a factor of 20. The optimization was started with a randomly chosen feasible set of coefficients. Tests of hundreds of random sets of starting coefficients for the same model case produced the same optimal *r*_{3}(*t*) when power was being produced, though they sometimes differed around times when almost no power was being produced. Thus, the optimization of the power did not appear to have local maxima.

The periodic representation of the turbine array’s drag coefficient (3.3) via (2.3) is similar to that used in [16]. In that work they optimized how an array’s drag coefficient λ_{F} varies in time to show that it was possible to exploit the flow’s inertia to provide short term energy storage. The SP-TS approach used here extends their work by representing the turbine tuning, rather than the drag coefficient, as a Fourier series. This extension allows the model to take into account limitations on the maximum drag coefficient imposed by the total number of turbines in the array and how they are arranged into rows. It also allows for energy lost by the flow to mixing behind the turbines, which is not available for power production [28]. Here SP-TS is used only to maximize power output averaged over a tidal cycle, but it could easily be extended to being optimized to meet a power demand curve to extend [16].

## 4. Comparing strategies

Comparing I-TS and P-TS in figure 3, not only does P-TS produce 15% more power than I-TS, it has two other benefits. Firstly, while both strategies reduce maximum flows along the channel, which may lead to environmental impacts (e.g. [29]), peak velocity for P-TS is 10% higher than I-TS. Thus, adopting a P-TS may have lower environmental impact (figure 3*c*). Secondly, power extraction imposes physical loads on the structure of the turbine. Peak loads for P-TS are 20% lower than for I-TS (figure 3*d*). As a result, choosing a P-TS over an I-TS allows turbines to be less robustly designed, thereby reducing construction costs. Thus, P-TS comes with triple benefits over I-TS, something which will be demonstrated to apply to all large arrays.

These benefits arise because P-TS uses a lower drag coefficient than a I-TS. Power output is proportional to the drag coefficient (2.3), thus superficially the lower drag coefficient of P-TS would be expected to produce less power. However, because power extraction interacts with the strength of the flow, a lower drag coefficient allows stronger flows within the channel. Effectively, P-TS forgoes generating some power when flows are weak, allowing greater acceleration of the flow immediately following the flow’s reversal. Thus by being patient, P-TS is then able to extract more power from enhanced flows later in the half tidal cycle (figure 3*c*).

In terms of performance, SP-TS improves on P-TS (figure 3). Peak power output for SP-TS is 7% higher and flow velocities are 4% higher. Despite a higher power output, adopting SP-TS over P-TS would not increase the structural loads on the turbines in the example. The optimal *r*_{3}(*t*) for SP-TS in figure 3*b* can explain its advantages. Just after the flow reverses around *t*′=1.1, *r*_{3} tends towards 1. This reduces the drag on the flow (2.3). As a result, flows accelerate more during the period immediately following the flow reversal, leading to slightly higher flows than both I-TS and P-TS (figure 3*c*). At around 0.1 tidal cycles after the flow reversal the optimal *r*_{3} declines, leading to a high drag coefficient at slack-water around *t*′=6.2. This high coefficient more than compensates for the power forgone by SP-TS earlier in the half cycle, resulting in a higher average output that P-TS. By the end of the half cycle the drag coefficient is higher than those required by I-TS, and would be even higher were not a lower limit of *r*_{3}=0.3 imposed on the solution. Though the drag coefficient for a SP-TS is higher at this time, maximum turbine loads still remain lower than for an I-TS.

### (a) Tuning a single row array

The thick lines in figure 4 show how the three tuning strategies compare as a single row of turbines expands across a Pentland Firth-like channel. While array output grows with array expansion for both the patient tuning strategies, beyond a particular size array power output for I-TS actually falls as the array expands. The decline in output is a result of I-TS using such a high drag coefficient that the power lost due to flow reduction outweighs any gain due to additional turbines. That is, in the language of [14], ‘channel-scale flow reduction’ outweighs the efficiency gain due to the ‘duct-effect’. I-TS also has a lower power output per turbine, greater flow reduction and higher structural loads (figure 4*c*–*d*). Thus, the figure confirms that for large arrays I-TS is a much poorer tuning strategy than the patient strategies.

Comparing P-TS and SP-TS, adopting a SP-TS produces up to 15% more power from a single row of turbines and has 15% stronger peak flows. A SP-TS results in similar structural loads to P-TS, except at unrealistically high blockage ratios. At high blockages figure 4 demonstrates that array output can exceed the upper limit given by [12]. The reason for this is outlined in the discussion. The triple benefits of the patient strategies over I-TS, and enhanced benefits of SP-TS over P-TS, also extend to a multi-row array with a fixed channel blockage of *ϵ*=0.3 (electronic supplementary material, Figure S1).

The non-smooth behaviour of the maximum force in figure 4*d* results from the finite number of frequencies used in the expansion of the tuning *r*_{3} (3.3) when it attempts to model a rapid increase and decrease in drag coefficient near the end of the half tidal cycle. This rapid change in drag coefficient is required to maximize power production when exceeding GC05’s upper limit for power production ([16], see §6a).

### (b) Tuning a multi-row array

Figure 5 compares the relative performance of SP-TS arrays of different sizes and layouts in a Firth-like channel. Contours in figure 5*a* show the largest example arrays produce 20% more power when tuned using a SP-TS, when compared with a P-TS. The largest arrays can also exceed GC05’s upper limit on power output by up to 15% (figure 5*b*). SP-TS tuned arrays maintain higher flows along the channel, typically with similar peak loads to P-TS for most arrays. It is only for the very largest example arrays that SP-TS has a disadvantage of higher structural loads.

The poor performance of I-TS relative to P-TS is further demonstrated in a contour plot given in electronic supplementary material, Figure S2. For the largest example arrays, the I-TS produces only 20% of the power from a P-TS tuned array, has peak flows 80% lower and has up to 40% higher structural loads than an array tuned with a P-TS.

### (c) Effect of the channel’s size

Channel sizes between the smallest and largest examples in table 1 were used to determine how the performance of ST-TS varies with channel size. Figure 5*a*,*c* demonstrates how the power output and flow speed advantages of SP-TS over P-TS grow with channel size, being up to 35% better in the largest channel example with the largest array. With enough turbines, SP-TS tuned arrays in channels larger than the Firth are able to exceed GC05’s potential by up to 20% (figure 5*b*) though as before this comes at a cost of higher loads than those experienced with a P-TS tuned arrays (figure 5*d*).

It should be noted that in a brief two page conference abstract [30] showed what are effectively two horizontal slices through figure 5 plotted against λ_{0} to represent channel size for cases of *N*=2 and *N*=4 terms in the Fourier series. No details were given on how the channels were scaled. Here we go much further, showing the wider context for their figure, provide physical interpretation, identify the mechanism which allows SP-TS to outperform the P-TS of V10 and show there are other benefits of a SP-TS.

For completeness, electronic supplementary material, Figure S3, reiterates I-TS’s poor performance across all example channel sizes compared with the P-TS, at worst producing 80% less power, for flows 80% weaker and at 40% higher loads.

## 5. Capping power output

Like wind turbines, tidal turbines will likely have their power output limited, or ‘capped’, to prevent overloading, e.g. the ‘SeaGen’ tidal turbine imposes a 1.2 MW cap on its output [27]. Capping output ensures the power train does not exceed its rated capacity, but can also be used to keep the structural loads on the turbines within the limits they have been designed for.

Here, a power cap was imposed on I-TS and P-TS models by examining the power output at the next time step. If this exceeded the cap, then a search was made for the *r*_{3} which matches the power cap at the next time step. For both of these strategies this requires an on-the-fly tuning on the same curve as an I-TS, for which the peak is very close to *r*_{3} on this curve which matches the power cap, there are two possible solutions which give the required output (figure 1). The one with the higher value of *r*_{3} is strongly preferred as it results in lower structural loads on the turbines. To ensure this, the search for the *r*_{3} to match the power cap was made between the peak at

Figure 6 gives time series for an array of capped output turbines using the same example array as that used in figure 3. An I-TS slows the flow so much that its output never exceeds the cap, while *r*_{3} for P-TS and SP-TP shows an increase at around *t*=0.35 to ensure the cap is not exceeded (figure 6*b*). Turbine structural loads decline when the cap is imposed (figure 6*d*). This is because power is proportional to velocity cubed, whereas load is proportional to velocity squared. The result is that a given power output results in lower structural loads at higher velocities. In the example, a P-TS still produces 10% more power, with 10% stronger flows and 35% lower peak loads than an I-TS, but the advantage over I-TS is slightly smaller than in the uncapped example in figure 3. Similarly, the advantage of a SP-TS over a P-TS is also reduced by imposing a power cap.

Figure 7 exhibits counterintuitive behaviour as the level of the power cap is changed. Starting on the right of figure 7*a*, the cap is so high that it does not limit output. Moving to the left causes power output for I-TS to suddenly jump up as the cap is first imposed. Thus, counterintuitively, limiting output can increase the average output of the turbines in a large array if they are tuned using an I-TS. The thin grey line in figure 6 provides an explanation for this. Imposing a 20% lower power cap on an I-TS increases average output by limiting power extraction at peak flows while increasing output for an extended period later in the half cycle. The overall effect of a modest power cap is to delay power production, allowing stronger flows to develop later in the half cycle, from which more power can be extracted than the power lost by imposing the cap. Thus, capping can lead to a net increase in average power output from large arrays tuned with an I-TS, giving I-TS some of the advantages of the patient tuning strategies. At very low power caps in figure 7*a* the three strategies produce similar power and there is little advantage in a P-TS or a SP-TS.

P-TS also subtly exhibits the counterintuitive behaviour of I-TS in figure 7*a*, which can be seen as slightly higher power output as the cap is initially imposed in the expanded scale of figure 7*b*. Even with the expanded scale, SP-TS does not appear to exhibit this behaviour. The inference from this is that both I-TS and P-TS are sub-optimal tuning strategies in terms of maximizing the total power output from the array. As a result, imposing a cap can exploit some of this unrealized power by forgoing some power at peak flows to gain more power later in the cycle, i.e. making them both more patient.

Figure 7*c*,*d* again show how the capped patient strategies maintain much higher peak flows along the channel and have much lower peak loads than I-TS, with SP-TS being the best in the example. The peak loads show uneven curves due to slight differences in the timing of the peak at the cusps seen in figure 6 not being properly resolved by the finite time step of the numerical model.

## 6. Discussion

### (a) The ability to exceed GC05’s limit

The upper limit for power production for an array tuned using a P-TS is given by GC05’s potential, as both use a drag coefficient which does not vary with time, V10. Figure 4 confirms this as

SP-TS improves on P-TS by being even more patient. SP-TS almost turns off power production for a short period just after the flow reverses, a time where little power can be produced (figure 3). This lowering of power production allows stronger flows to develop later in the tidal cycle. SP-TS takes advantage of these stronger flows by intelligently ramping up power production to maximize output later in the cycle to achieve a higher output averaged over a tidal cycle. Vennell & Adcock [16] intuitively reasoned that the extreme case of this delayed gratification approach is one where the potential of a very large channel is maximized by only extracting power at the very end of the half cycle. This, in principle, could increase a channel’s upper limit for power production by up to 2.5 times that given by GC05.

Figure 8 reproduces one given in [30] with more terms in the Fourier series representation of λ_{F}(*t*). The Fourier coefficients of λ_{F}(*t*) were optimized to find the channel’s potential by maximizing the average power that the flow can lose to the array, *r*_{1} in (3.1). Figure 8 demonstrates Vennell & Adcock’s [16] 2.5 times higher upper limit for the largest channel (λ_{0}=0), with an upper limit which converges with GC05’s potential as the channel becomes smaller (_{0}=0), the optimal λ_{F}(*t*) was found to approximate a delta function, extracting most of the flow’s kinetic energy near the end of each half cycle. An extraction strategy which confirms that intuitively reasoned in [16]. However, this increase in the upper limit for power extraction predicted by the 1D channel model needs to be tested with more realistic 2D and 3D models (e.g. [31]).

In mid-sized channels like the Firth (λ_{0}=1.1) friction dissipates some energy before the end of the half cycle, thus the optimal strategy is not to wait until the end of the half cycle to extract all of the energy. For these channels, the more moderate optimal approach is one which to extracts energy over most of the half cycle (figure 3*b*). In addition, the extreme case in [16] requires an infinite drag coefficient to stop the flow right at the end of the half cycle. When an array only blocks part of the channel cross-section, the maximum drag coefficient it can exert is limited by (2.3) evaluated at *r*_{3}=0, i.e. *C*_{T}=1, well short of that required to instantaneously stop the flow at the end of the half cycle. This limits how much power an array which only blocks part of the cross-section can extract at any instant. Thus, these arrays are forced to adopt the more moderate approach of extracting power over much of the half cycle in order to maximize the average output as in figure 3*b*.

SP-TS generally has peak structural loads similar to those of a P-TS, thus implementing a SP-TS would typically not increase construction costs. However, for very high blockage ratios figures 4 and 5 show a rapid increase in peak loads at unrealistically high blockage ratios. These high loads occur for arrays using a SP-TS which are close to, or are exceeding the GC05 limit. As noted above, achieving this requires a high power output right at the end of the half tidal cycle, which results in very high peak loads on the turbines towards the end of the cycle. Thus, the cost of an array exceeding GC05’s limit is the need to build more robust turbines. However, for realistic blockage ratios there is no additional cost, while understanding there is higher limit is useful in explaining why power output by a SP-TS exceeds that of a P-TS even a lower blockage ratios.

### (b) Other aspects

The focus of this work has been to present the idea that there is a better tuning strategy for tidal turbines than that given in V10. This better strategy combines reduced power production early in the half tidal cycle, with higher power production late in the tidal cycle a strategy which can extract more power from the cycle as a whole. This focus and limited space have meant there has not been the space to explore the inclusion of aspects of more realistic turbine operating characteristics, which may have a secondary effects on array power production. For example, minimum and maximum turbine operating velocities (i.e. cut-in and cut-out velocities [27]). The SP-TS is one which delays the powering up of the turbine at the beginning of the half tidal cycle when flows are weak, but extracts power right to the end of the half cycle. Thus, SP-TS operates as a power optimized cut-in velocity at the start of the cycle, but not at the end of the cycle. How a cut-in and cut-out velocities might affect the three tuning strategies will be explored in a future work.

Another aspect is drag on the structures that support the turbines, e.g. towers and mooring structures. Support structure drag distorts power production curves [13], thus affecting the optimal tuning for P-TS and SP-TS. While support structure drag is likely to be much less than the drag due to power extraction (3.2), it will have a secondary influence on optimal tuning strategies and maximum structural loads, while it also means maximum array output falls short of a channel’s potential [13]. This also needs to be explored in a future work.

It is important to note that to exploit a SP-TS requires turbines which have the mechanical ability to vary blade pitch as needed. This requirement will make turbines more expensive to produce and maintain. Cheaper turbines with a fixed blade pitch customized for the full size array and the particular channel [11] may be limited to using a P-TS. An alternative turbine design might be to have a fixed blade pitch with a single array optimized *r*_{3}, but turn power production on and off electrically to achieve a ‘partial SP-TS’, an option which needs further work to quantify its benefits.

Only single frequency tidal forcing is addressed here, such as that due to the typically dominant M2 tide. Inclusion of other frequencies, such as S2, would incorporate a spring-neap cycle within the model. There may be further gains in power production by optimizing a SP-TS over the entire neap-spring cycle. However, SP-TS is useful over time scales which are similar to, or smaller than, the bottom frictional time scale. For the Pentland Firth the bottom frictional time scale is only hours [16]. Thus optimizing a SP-TS over many tidal cycles may not yield significant additional benefits. In many tidal channels, there is also an asymmetry between the ebb and the flow tides. It might be possible to exploit this asymmetry by using a SP-TS in channels with a long bottom frictional time scale. The channel model here does not include ebb–flood asymmetry, thus any secondary benefits from exploiting ebb–flood asymmetry will need to be explored in a future work.

At higher blockage ratios a row of tidal turbines act like a high flow low head hydro-electric dam or tidal barrage. Optimization of flow through a tidal barrage over the tidal cycle, e.g. [32], is akin to the time variable tuning SP-TS of a high blockage row of turbines. It would be useful to explore the comparison with a tidal barrage, which would require adding a lagoon to one end of the channel model (2.2) as in [3].

This work only briefly comments on the potential environmental affects that reducing peak flows along the channel may have on transport along the channel. These might be reducing the exchange of nutrients, sediments and pollutants, along with increasing residence times within embayments. There are other potential effects which may also have environmental impacts. For example, the distortion of the tidal cycle by a SP-TS extends the period of time flows are accelerating, while reduce the time flows are decelerating. This may affect sediment transport along the channel as bed load. Also, rapid changes in power production may create waves which may enhance erosion of the foreshore. These aspects need further work with 2D and 3D models to better compare any environmental effects due to the three tuning strategies.

## 7. Conclusion

The delayed gratification of the patient tuning strategies P-TS and SP-TS have triple benefits in large arrays of tidal turbines; higher power output, stronger peak flows and lower structural loads, when compared with the strategy of maximizing power output ‘now’ with an ‘on the fly’ I-TS. SP-TS improves on P-TS by being more patient. It exploits the ability to vary the drag coefficient in time, (2.3) and (3.3), to reduce drag on the flow early in the half cycle to gain more power from stronger flows later in the half cycle. SP-TS is a tuning strategy which maximizes the average power output within limitations imposed on the maximum size of the array’s drag coefficient, (2.3) and (2.5). These limitations result from the finite number of turbines, how much of the channel’s cross-sectional area they occupy, *ϵ*, and the lower limit imposed on *r*_{3} (figure 3).

It is the higher channel potential made possible by a time variable drag coefficient which allows a SP-TS to produce more power than a P-TS, a potential up to 2.5 times that given by GC05, [16] and figure 8. What was not expected was that, despite this higher production, SP-TS also maintains stronger flows along the channel and structural loads are similar to those for a P-TS, except for the very largest arrays, figures 5 and 9. Thus, for both environmental impacts and turbine construction costs, adopting a SP-TS is often as good or better than a P-TS.

The counterintuitive result, that imposing a moderate power cap can increase the average power output of turbines using I-TS and P-TS (figure 7), occurs because a moderate power cap makes these strategies behave more patiently. The cap limits peak output, but this is more than compensated for by extracting more energy from stronger flows later in the half cycle (figure 6). The inference from this is that both I-TS and V10’s P-TS are sub-optimal tuning strategies. As a result, by imposing a modest cap they can exploit some of this unrealized power to behave more like the ‘optimally-patient’ SP-TS.

To exploit the significant global tidal current energy resource it is essential to build large arrays of tidal turbines. These large arrays must necessarily extract a significant fraction of a particular channel’s potential to produce power. At this scale, power extraction will reduce tidal currents throughout the short narrow channels which hold much of the global resource. This paper shows that this interaction between power extraction and the strength and phase of the tidal currents can be exploited using a SP-TS to extract more energy, while surprisingly maintaining higher flows within the channel and often without increasing structural loads. In the smallest channels, a few turbines would be a large enough array to exploit patient tuning strategies, though the gains are small (figure 9). The gains in channels the size of the Pentland Firth are significant, but arrays will need to have 100 s of turbines to benefit from the patient tuning strategies. While arrays this large may be some way off in the future, tidal turbine arrays must reach this scale to produce the 100 s MW required to make a worthwhile contribution to global energy demand. At this scale it is possible to intelligently choose when to extract power to meet short peaks in demand as in [16] or to produce more power as outlined in this paper.

The aim of this work was to present the idea that there is a better tuning strategy for tidal turbines than that given in V10, one where the combination of reduced power production early in the half tidal cycle with higher power production late in the tidal cycle can extract more power from the cycle as a whole. To better quantify the power output and environmental benefits of this SP-TS requires extensive work using more realistic 2D and 3D models; however, the optimization of a time variable drag coefficient will be much more computational challenging within these more realistic models. In addition, these models need to incorporate more realistic turbine behaviours, such as a minimum flow speed required to produce power and the effects of drag on the turbines support structures.

## Data accessibility

This work is based on data generated by the model described above.

## Competing interests

I declare I have no competing interests.

## Funding

This work was supported by a New Zealand Marsden Fund Grant number 12-UO-101.

## Acknowledgements

Careful editing by Malcolm Smeaton, Daryl Coup and Cerys Bailey was very much appreciated.

## Appendix A. Fourier model adjoint to speed optimization

To search for the optimal Fourier coefficients in (3.3) requires estimating a vector of the gradients of the average power (3.1) with respect to each of the 2*N*+1 coefficients. The brute force approach is to run the model 2*N*+1 times at each step in the optimization for small changes in each of the *a*_{n} s. The adjoint approach can compute the gradient vector with a computational effort similar to that of a single model run. Developing an adjoint starts with noting that *a*_{n}s via *r*_{1}(*r*_{3}) and *C*_{T}(*r*_{3}), and indirectly via the velocity which also depends on the farm drag coefficient λ_{F} and hence the *a*_{n}. Here we follow [17] to develop a ‘discrete’ adjoint. The full derivative of the average power can be written as
*a*_{n}’s have on the velocity, *du*/*da*_{n}, are more difficult to evaluate. However, the velocity must satisfy (2.2). Viewing (2.2) as a constraint on (A 1) of the form *F*(*u*(*a*_{n}),*a*_{n})=0, whose full derivative must be zero, yields
*Λ*,
*u*_{m} at the *m*=1…*M* model time steps. Using trapezoidal integration to calculate the average power for the periodic solution simply gives *P*_{m} is the power evaluated using *u*_{m} in (3.1). Using centred finite differences for the time derivative within *F* gives rows of the *M*−1 by *M*−1 matrix inside the square brackets on the L.H.S. of (A 4) of the form
_{m} is a function or analytic derivative evaluated using *r*_{3} at the *m*’th time step and *Λ* whose size is a *M*−1 by 1.

The other remain terms required for the gradient vector in (A 5) are the *M*−1 by 2*N*+1 matrix
*N*+1 long row vector

The adjoint gradient vector was compared with that found numerically using a change in each *a*_{n} of 0.001. Hundreds of tests with randomly chosen feasible sets of *a*_{n} gave a linear regression between numerical and adjoint gradients which had a slope within 10^{−5} of unity, an intercept smaller than 10^{−5} and a variance of the difference between the numerical and adjoint gradient of less than 10^{−7}.

The adjoint only gives the gradients with respect to the Fourier coefficients being optimized, it is not affected by imposing a power cap. The optimization algorithm determines whether it is feasible to increase power production by moving the values of the coefficients in the direction given by the adjoint without violating any imposed constraints.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3515193.

- Received January 20, 2016.
- Accepted September 20, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.