By considering the momentum balance for a control volume with and without a wind farm present, derived the non-dimensional farm momentum (NDFM) equation: 1CT*λCf0β2+βγ=M, where β is the farm wind speed reduction factor that is defined as β≡UF/UF0 (where UF is the average wind speed in the nominal farm layer of height HF, and UF0 is the farm-layer-averaged speed without turbines present); the (farm-averaged) internal turbine thrust coefficient CT* is defined as CT*≡∑i=1nTi/12ρUF2nA (where Ti is the thrust of turbine i, n is the number of turbines in the farm, and A is the rotor-swept area); the array density λ is defined as λ≡nA/SF (where SF is the farm area); the natural surface friction coefficient Cf0 is defined as Cf0≡τw0/12ρUF02 (where τw0 is the bottom shear stress without turbines present); γ is the bottom friction exponent (assumed to be 2.0 in this study, following , and also as justified later in Sect. ) defined as γ≡log⁡β(τw/τw0) (where τw is the bottom shear stress with the turbines present); and M is the momentum availability factor defined by M≡MF/MF0, where MF is the net momentum flux into the farm control volume with the turbines present and MF0 the case without the turbines present. In this study we use a fixed definition of the farm-layer height HF=2.5Hhub (where Hhub is the turbine hub height) for convenience. The exact value of HF defined originally by depends on the undisturbed wind profile, to ensure that UF0 matches exactly with the undisturbed wind speed averaged over the turbine-swept area, UT0; however, as shown later by , the fixed definition of HF=2.5Hhub is a good approximation for a wide range of ABL profiles.

Note that the derivation of Eq. () given by was for an idealised case where the flow through the farm was assumed to be fully developed. However, they also discussed (in Sect. 3 of their paper) how the same form of equation could be derived for more general cases, where the net momentum transfer through the side and top surfaces of the farm control volume should also be considered part of M. See for the full expression of M.

used numerical weather prediction (NWP) simulations to calculate M for a realistic offshore wind farm site in the North Sea. They found, for most cases, an approximately linear relationship between M and β. Therefore, as proposed by , it is convenient to express M as 2M=1+ζ(1-β), where ζ is called the wind extractability factor. showed that ζ was a time-dependent parameter that varied with atmospheric conditions and inversely with farm size. More recently, proposed an analytical model of ζ, as discussed later in Sect. .

Equations () and () can be solved to calculate the farm wind speed reduction factor β for a given farm design and atmospheric condition (i.e. λ, CT*, Cf0, γ, and ζ). Using β, the farm power can be calculated using 3Cp=β3Cp*, where the (farm-averaged) turbine power coefficient is defined as Cp≡∑i=1nPi/12ρUF03nA (Pi is the power of turbine i in the farm) and the (farm-averaged) internal turbine power coefficient defined as Cp*≡∑i=1nPi/12ρUF3nA.

Generally, the internal turbine thrust coefficient CT* depends on the turbine layout . However, as suggested by and later confirmed by , an approximate upper limit of CT* (with respect to the turbine layout) can be predicted using an analogy to the classical actuator disc theory: 4CT*=16CT′(4+CT′)2, where CT′≡Ti/12ρUT,i2A is a turbine resistance coefficient that represents the turbine operating condition (assumed to be constant for all turbines in the farm), and UT,i is the streamwise velocity averaged across the rotor swept area of turbine i. Note that the two-scale momentum theory summarised in Sect. 2.1 is for general cases where the turbine thrust Ti and power Pi may vary across the farm, whereas the analytical model described here is for less-general cases where the turbine resistance coefficient CT′ is constant across the farm (such as the LES cases shown later in this paper, where CT′ is fixed at 1.94 for all turbines in the farm). showed that, for periodic arrays of turbines with a fixed CT′ value of 1.33, some specific turbine layouts could exceed this CT* value slightly, presumably due to local blockage effects .

The two-scale momentum theory described in Sect.  can be used, together with Eq. (), to predict the performance of arrays of actuator discs (or aerodynamically ideal turbines operating below rated conditions). For actuator discs Pi=αiUFTi, where αi≡UT,i/UF is the local wind speed reduction factor, and αi can be estimated as αi=α=CT*/CT′ since CT′ has been assumed to be constant for all turbines. It is useful to note that this theoretical estimation is strictly valid only for infinite regular arrays of actuator discs. The (farm-averaged) power coefficient of an actuator disc is therefore given by 5Cp=β3αCT*=β3CT*32CT′-12.

Using the analytical model of CT* (Eq. ), Eqs. (), (), and () can be solved to give a theoretical prediction of near-ideal wind farm performance, denoted Cp,Nishino. We describe this as near-ideal since this is close to but slightly less than the maximum possible (as shown later). If we assume γ=2.0, we can derive a single analytical expression for Cp,Nishino, i.e. 6Cp,Nishino=64CT′(4+CT′)3×-ζ+ζ2+416CT′(4+CT′)2λCf0+1(1+ζ)216CT′(4+CT′)2λCf0+13. Meanwhile, the power coefficient of an isolated turbine Cp,Betz is given by 7Cp,Betz=64CT′(4+CT′)3, which gives a maximum turbine performance of Cp,Betz=16/27 when CT′=2.0. Note that Eq. () reduces to Eq. () in two special cases: (i) when λ/Cf0=0 and (ii) when ζ is infinitely large. These two theoretical predictions of performance (Cp,Nishino and Cp,Betz) will be used to define the turbine-scale and farm-scale efficiencies later in Sect. .

Here we reanalyse the results of the wind farm LES performed by , who reported that the farm normalised power relative to the first-row power (i.e. wake efficiency) varied from 0.48 to 1.23 for the same turbine layout and wind direction. The aim of this section is therefore to investigate the physical mechanisms behind such a large change in the farm normalised power.

have introduced three different “efficiencies” (or power ratios) for wind farm performance. Firstly, the wake efficiency ηw (sometimes called “normalised power”) is defined as 8ηw≡PfarmP1, where Pfarm is the farm-averaged turbine power, and P1 is the first-row-averaged turbine power. Secondly, the “non-local” efficiency ηnl is defined as 9ηnl≡P1P∞, where P∞ is the power output of an isolated turbine under the same atmospheric conditions. This represents the power loss due to the velocity reduction in front of the farm, i.e. due to farm blockage. Finally the “farm efficiency” ηf is defined as 10ηf≡PfarmP∞≡ηwηnl. The farm efficiency ηf quantifies the overall power losses caused by placing turbines together in a farm.

As noted by , the farm LES results show a relatively strong negative correlation between ηw and ηnl (Fig. ). When the farm blockage increases (i.e. ηnl decreases), the downstream power losses decrease (i.e. ηw increases). These two effects counteract each other to a certain extent. This means that ηw is affected by not only turbine–wake interactions but also larger farm-scale flow effects causing farm blockage.

The correlation between ηw and ηnl is caused by the induced pressure gradients across the farm. To illustrate this, the pressure perturbation for cases H300-C2-G1 and H300-C8-G1 is shown in Fig. . Case H300-C2-G1 has a low degree of farm blockage (ηnl=0.857) and a relatively small induced pressure gradient. Conversely, H300-C8-G1 has a high degree of farm blockage (ηnl=0.437) and a large induced pressure gradient. Essentially, H300-C8-G1 has a larger driving force across the farm, meaning that the velocity tends to stay high across the farm despite the lower velocity at the front. This causes ηw to be higher (ηw=1.00 compared to ηw=0.501 for H300-C2-G1).

The LES results also show that the overall farm efficiency ηf is not well correlated with either the wake efficiency ηw or the non-local efficiency ηnl. Figure a shows the weak correlation between ηf and ηw. This shows that the wake efficiency or normalised power is not a good indicator of wind farm efficiency. The correlation between ηf and ηnl is also relatively weak, as shown in Fig. b. The negative farm blockage effect is mostly counteracted by the increased pressure gradient across the farm, which increases ηw.

To better understand why the same turbine layout results in a very wide range of ηw from 0.48 to 1.23, now we will investigate whether this could be explained by either (1) different “effective” turbine layouts caused by changes in local wind directions within the farm or (2) different wake recovery rates. In the following, we will again focus on the two illustrative cases, H300-C2-G1 and H300-C8-G1. The capping inversion height is the same for both, but H300-C2-G1 gives ηw=0.501, whereas H300-C8-G1 gives ηw=1.00.

First, we show that the large difference in ηw cannot be explained by different effective turbine layouts. The local flow direction for cases H300-C2-G1 and H300-C8-G1 is shown in Fig. . Both cases have an outward flow direction of approximately 5° at the sides. However, both cases have similar variations in the local flow directions across the farm. Figure  shows there is not a strong relationship between ηw and the farm-averaged absolute turbine yaw angle |Φ|‾ for all 38 cases, indicating that the large variation in ηw cannot be explained by the change in effective turbine layout.

Next, we show that the wake recovery behind each turbine is also uncorrelated with the wake efficiency ηw. The farm flow profiles are shown in Fig. a for H300-C2-G1 and in Fig. c for H300-C8-G1. The individual wake deficits look similar, but rather it is the farm-scale flows that are different. A closer view of individual wakes towards the centre of the farm is shown in Fig. b and d. Despite one case having ηw=0.501 and the other having ηw=1.00, the wakes look almost identical. This suggests that the characteristics of individual turbine wake recovery are not contributing to the difference in ηw.

A more quantitative comparison in Fig.  shows that both cases have similar wake velocity deficits. Here we calculated the wake velocity deficit by defining new coordinates xi and yi local to each turbine (see Fig. b). xi is perpendicular to each rotor and yi parallel. We averaged the wake velocity deficits (relative to the undisturbed velocity u∞(z) recorded in the precursor simulation) for each turbine in the 11th row. This row was chosen because the flow profiles are characteristic of the average flow profile across the entire farm. Figure c–f show that both horizontal and vertical wake deficit profiles are approximately Gaussian, and the wake deficit profiles and wake recovery rate are both similar. H300-C8-G1 has a slightly smaller normalised wake velocity deficit compared to H300-C2-G1, but this is not sufficient to explain the large difference in ηw values.

The wake recovery across the entire farm is also similar for the two cases. We calculated the wake width by fitting a Gaussian function to the wake deficit profiles shown in Fig. . Note that the centre of the Gaussian function was not fixed to the rotor centre. The wake width σ was calculated as the geometric mean of the wake width in the horizontal σy and vertical σz directions, i.e. σ=σyσz. The wake width as a function of the downstream distance for turbine rows 3 to 16 is shown in Fig. . The first two rows were excluded, as the wake recovery was much slower. An approximately linear growth in wake width can be seen for both cases. We calculated the wake expansion coefficient k* using the equation σ/D=k*xi/D+ϵ where ϵ is the initial wake width. We then averaged the value of k* across the 3rd to 16th rows to obtain a farm-averaged k*, which is shown in Fig. . The value of k* was found to be higher than the values reported by . This is presumably because the turbulence levels are higher within a large wind farm. Most importantly, the average wake growth rate is higher for the case with the lower value of ηw. This again demonstrates that ηw is not strongly related to local wake recovery behind each turbine.

To confirm this trend further, we calculated the farm-averaged wake expansion coefficient k* and normalised wake width σ/D (at 10D downstream of each disc) for 29 of the farm LES cases from . The cases with the lowest capping inversion (150 m) were excluded, as they did not have Gaussian wake deficit profiles in the vertical direction due to the vicinity of the capping inversion base to the turbine-tip height. Figure a shows that there is no correlation between the farm efficiency ηf and the wake expansion coefficient k*. Figure b shows that low ηw values cannot be explained by a slower wake recovery. On the contrary, these LES results show that cases with a low ηw value tend to have a faster wake recovery. This trend can also be confirmed from the negative correlation between ηw and the farm-averaged turbine wake width (at 10D downstream of each disc) shown in Fig. c.

The wake efficiency ηw has been extensively used to analyse farm performance, as it is a relatively easy parameter to calculate, e.g. using supervisory control and data acquisition (SCADA). However, these LES results (for a fixed staggered turbine layout with different ABL conditions) suggest that this wake efficiency parameter ηw is not a good indicator of the turbine–wake interactions within the farm.

The two-scale momentum theory provides an alternative way of understanding wind farm performance. This theory is particularly useful when the two-scale separation assumption is valid, meaning that the farm internal parameters (CT* and γ) depend only on internal or turbine-scale conditions, whereas the external parameter (ζ) depends only on external or farm-scale conditions. This assumption allows the turbine-scale and farm-scale flows to be modelled separately; however, this assumption has not been fully evaluated in previous studies. In the following we present a first validation of the two-scale separation assumption using four new LES results as well as the previous LES results from .

Here we calculate ζLES from the momentum availability factor MLES and the farm wind speed reduction factor βLES obtained from the LES as follows: 11ζLES=MLES-11-βLES, where βLES is calculated directly from the values of UF and UF0 obtained from the LES, whereas MLES is calculated as 12aMLES=∑i=1nTi+τwSτw0S12b=∑i=1nTiτw0S+βLESγ12c=λCT,LES*Cf0βLES2+βLESγ. The bottom friction exponent γ was not recorded in the present LES results, but we can expect that this varies between 1.5 and 2.0 . Considering Eq. (), a typical value of λCT*/Cf0 is 17.5 for the farms in this study, meaning that the total force due to turbine thrust is much larger than the force due to the bottom friction (and hence, the impact of γ is very small). For example, if we suppose that β=0.75, using γ=1.5 gives MLES=10.5, whereas γ=2.0 gives MLES=10.4. Since the value of MLES is largely insensitive to the value of γ, we will use γ=2.0 in the following analysis.

Figure  shows the relationship between MLES and βLES obtained for three different atmospheric conditions. As can be seen from the figure, the wind extractability factor ζLES changes with the atmospheric conditions, but it is not sensitive to the turbine layout. The aligned turbine layouts result in a lower wind speed reduction (i.e. lower value of 1-βLES) because they present a lower flow resistance. However, the value of ζLES is almost identical for aligned and staggered layouts under a given atmospheric condition. A farm with a staggered layout but doubled turbine spacing (10D×10D) also follows approximately the same relationship (see Fig. b). This demonstrates that the linear relationship is valid for a wide range of β. It can also be seen that ζLES decreases with decreasing capping inversion height. This trend was predicted by the theoretical model of ζ proposed by . The values of ζLES for all atmospheric conditions tested in this study are shown in Fig. a. As shown theoretically by , for a given wind farm, there is a positive monotonic relationship between ζ and wind farm efficiency. Therefore the effects of different atmospheric conditions on the wind farm efficiency reported by can be explained by different wind extractability factors ζ.

The LES results also show that the internal turbine thrust coefficient CT,LES* is insensitive to atmospheric conditions (Fig. b). Apart from the lowest capping inversion cases (H150), the staggered turbine layout consistently gives a high CT,LES* value of about 1.0 irrespective of atmospheric stratification (Fig. b), whereas the aligned turbine layout consistently gives a lower CT,LES* value than the staggered one. This trend is expected, as showed that increased turbine–wake interactions reduce the value of CT*. Turbine–wake interactions reduce CT* because the waked turbines experience a lower incident wind speed and so produce less thrust.

These LES results in Figs.  and strongly indicate that the assumption of two-scale separation is valid for large finite wind farms, at least in the practical range of CNBLs tested in this study. This means that the impact of turbine-scale flows (i.e. turbine–wake interactions) and farm-scale flows (i.e. farm–atmosphere interaction) could be modelled separately through the modelling of CT* and ζ, as suggested originally by , to predict wind farm power in a less complicated and more physically meaningful manner.

Turbine-scale and farm-scale power losses (; see also ) are alternative metrics for wind farm performance, as illustrated in Fig. . The turbine-scale power losses are due to the internal flow interactions within the farm (i.e. turbine–wake interactions). Farm-scale power losses are due to the interaction between the ABL and the farm as a whole. have shown, using LESs of flow over a periodic array of actuator discs for 50 different layouts, that the near-ideal farm performance predicted by Eq. () is a good measure to differentiate the turbine-scale power losses from the farm-scale power losses. Note that when each turbine in a wind farm generates its wake, the flow bypassing the turbine locally accelerates due to the conservation of mass (at each turbine scale); hence, we consider that any reduction in farm-average wind speed is caused by external (farm–atmosphere) interactions. This means that the power losses accompanied by a reduction in farm-average wind speed are farm-scale power losses (caused by external interactions) and not turbine-scale power losses (caused by internal interactions).

It should also be noted that the 50 LES results of are for idealised infinitely large wind farms; hence, their findings are not directly applicable to finite-sized farms in general. However, as shown in the previous section, our new LES results indicate that the internal thrust coefficient CT* is insensitive to external conditions. This means that the upper limit of CT* (with respect to the turbine layout for a given value of CT’) should also be insensitive to external conditions, supporting our argument that the near-ideal farm performance predicted by Eq. () is a good measure for finite farms as well.

Here we propose a slight modification to the new metrics for wind farm performance introduced by , namely the “turbine-scale efficiency” ηTS and “farm-scale efficiency” ηFS

Note that the turbine-scale efficiency ηTS and farm-scale efficiency ηFS are related to the “turbine-scale loss factor” ΠT and “farm-scale loss factor” ΠF introduced by by the following expressions: ηTS≡1-ΠT and ηFS≡1-ΠF.

In this section we assess the ability of an analytical wind farm model to predict ηFS. This analytical model predicts the farm-scale flows only and not the turbine–wake interactions. Hence, here we only compare the predictions of the farm-scale efficiency ηFS and not the turbine-scale efficiency ηTS. A summary of this farm model is shown in Fig. (). Essentially, rather than using ζLES to calculate ηFS, here we predict ηFS using the expression 20ζ=1.18+2.18Cf0HFL1-τt0τw0, where L is the streamwise farm length, and τt0 is the undisturbed shear stress at a height HF in the hub-height wind direction. Using this approach, we can predict ηFS instantly using only the undisturbed atmospheric conditions. It is important to note that this model (Eq. ) currently only considers the impact of capping inversion height, and not capping inversion strength or free-atmosphere stratification. Figure a shows the distribution of percentage errors when predicting ηFS for the standard farm design with 29 atmospheric states (we excluded the lowest capping inversion cases “H150” where the capping inversion was below HF), whereas Fig. b shows the relationship between the LES results and analytical predictions for these cases. The model gives good predictions of ηFS, with a mean absolute percentage error of 5.68 %. It is likely that the impact of free-atmospheric stratification, not considered in the current model, causes some spread and contributes to this error. There is a slight bias for underpredicting ηFS but the median percentage error is below 5 %. The prediction accuracy for the smaller farm and greater turbine spacing cases are also summarised in Table , showing that this first-order model gives satisfactory predictions for these cases as well.