Wake steering of multi-rotor wind turbines: a new wind farm control strategy

In this paper, wake steering is applied to multi-rotor turbines to determine whether it has the potential to reduce wind plant wake losses. Through application of rotor yaw to multi-rotor turbines, a new degree of freedom is introduced to wind farm control such that wakes can be expanded, channelled or redirected to improve inflow conditions for downstream turbines. Five different yaw configurations are investigated (including a baseline case) by employing large-eddy simulations (LES) to generate a detailed representation of the velocity field downwind of a multi-rotor wind turbine. Two lower fidelity models from single rotor yaw studies (curled-wake model and analytical Gaussian wake model) are extended to the multi-rotor case and their results are compared with the LES data. For each model, the wake is analysed primarily by examining wake cross sections at different downwind distances. Further quantitative analysis is carried out through characterisations of wake centroids and widths over a range of streamwise locations. Most significantly, it is shown that rotor yaw can have a considerable impact on both the distribution and magnitude of the wake velocity deficit. Two of the configurations lead to a significant increase in wake expansion and a corresponding decrease in downstream velocity deficit. The remaining two yaw arrangements demonstrate a capability to redirect and channel the turbine wake. The lower fidelity models show small deviation from the LES results for specific configurations, however, both are able to reasonably capture the wake trends over a large streamwise range.

In this paper, wake steering is applied to multi-rotor turbines to determine whether it has the potential to reduce wind plant wake losses. Through application of rotor yaw to multi-rotor turbines, a new degree of freedom is introduced to wind farm control such that wakes can be expanded, For each model, the wake is analysed primarily by examining wake cross sections at different downwind distances. Further quantitative analysis is carried out through characterisations of wake centroids and widths over a range of streamwise locations. Most significantly, it is shown that rotor yaw can have a considerable impact on both the distribution and magnitude of the wake velocity deficit. Two of the configurations lead to a significant increase in wake expansion and a corresponding decrease in downstream velocity deficit. The remaining two yaw arrangements demonstrate a capability to redirect and channel the turbine wake. The lower fidelity models show small deviation from the LES results for specific configurations, however, both are able to reasonably capture the wake trends over a large streamwise range.

K E Y W O R D S
multi-rotor turbine, turbine wakes, velocity deficit, wake steering

| INTRODUCTION
As the wind energy sector continues to expand, it is of increasing relevance to optimise and improve turbine and plant technologies. Currently, in terms of overall power production, the most significant source of loss in a wind plant is the impact of upstream rotor wakes on downstream machines [1,2]. Wakes of turbine rotors are typically characterised by velocity deficit and high levels of turbulence, as detailed by prior studies [see the review of 3, and references therein].
Consequently, turbines operating in these wakes are not able to extract as much energy as their upstream counterparts, which tends to result in a reduced overall power output from the plant [4].
To mitigate power losses in wind farms, it has been proposed that wake steering may offer an appropriate solution.
This technique involves yawing upstream rotors such that wake flows are directed away from downstream turbines.
While this leads to a decrease in upstream turbine power, this is outweighed by an increase in power output downstream caused by lower velocity deficit [5]. The potential of this technique has been examined in a number of studies such as Jiménez et al. [6], in which the effectiveness of wake steering is demonstrated using LES, and a simple analytical model is developed to describe wake deflection. Gebraad et al. [7] conducted an alternative analysis by adapting the 'top-hat' profile proposed by Jensen [8] and optimising rotor yaw angles within a wind plant using game theoretic methods. More recently, Bastankhah and Porté-Agel [9] built on earlier findings [10], employing a Gaussian distribution to describe velocity deficit. Details of cross-stream components of wake velocity have been examined by Martínez-Tossas et al. [11], in which a 'curled-wake' model is formulated, and by Shapiro et al. [12] in which yawed turbines are viewed as lifting lines. In recent developments, several wind tunnel studies [e.g., 13], numerical simulations [e.g., 14,15] as well as field studies [e.g., 16,17] have examined wake steering for wind farms of single-rotor wind turbines, in which improvements have been seen in terms of both power output and reliability.
Another obstacle facing the industry from a more structural and logistical standpoint is that associated with the 'square-cube law', expounded primarily by Jamieson and Branney [18]. This problem essentially stems from the fact that energy extracted from a rotor scales with its area (proportional to square of diameter), but the production cost scales approximately with volume (proportional to cube of diameter). With the objective of lowering cost of energy, rotor sizes have therefore rapidly increased in recent years, though marginal gains have become steadily smaller due to higher manufacturing costs [19]. Further problems associated with large rotors, such as high shipping, assembly and maintenance costs also indicate that the practice of rotor upscaling may be reaching its limit [20].
In tackling the problems associated with rotor upscaling, a promising solution is that of multi-rotor turbines. This concept involves installing a number of smaller rotors on a single support structure, rather than just one large rotor [21]. Primarily, this alleviates the growing expense of rotor manufacture since the total material volume associated with the smaller rotor components is less, leading to lower production costs. Jamieson and Branney [18] also showed that further cost reductions are associated with installation and shipping, leading to overall savings estimated at 30% for a 20 MW turbine. Advantages are also seen from a reliability standpoint, since multi-rotor turbines are still able to operate at a reduced capacity when one generator fails, as noted by Jamieson et al. [22]. Where failure of a single rotor turbine generator would constitute a significant loss to the overall power output of a wind farm, failure of one multi-rotor generator would be less severe since it only makes up a fraction of the turbine's overall capacity.
Further to this, one of the key merits of multi-rotor designs is a lower susceptibility to wind veer, an effect caused by the presence of the Coriolis force in the atmospheric boundary layer which results in varying wind direction with height [23,24,25,26,27]. For current large turbine designs the variation in direction across the rotor is non-trivial, and efficiency losses typically result since the wind direction is not normal to the rotor at all points within the swept area [28]. Wind veer also leads to the generation of a skewed wake, as modelled by Abkar et al. [29], which can affect the efficiency of downwind turbines. In multi-rotor turbines, however, these effects are less pronounced, primarily due to smaller rotor sizes and the ability to yaw individual rotors such that they are properly adjusted to the wind direction at a given height.
Finally, multi-rotor turbines are able to deliver an improved power output when compared with single rotor equivalents. Field measurements and numerical simulations carried out by van der Laan et al. [30] showed a 1.8 ± (a) Wake expansion by coning (left) and multi-rotor yaw (right) (b) Wake channelling by coning (left) and multi-rotor yaw (right) F I G U R E 1 Introduction of wake control through coning and multi-rotor yaw. Fluid velocity is in the left to right direction. 0.2% increase in power production of the Vestas 4R-V29 demonstrator, an improvement attributed primarily to rotor interaction. When tested in wind farm arrays, van der Laan and Abkar [31] estimated an increase of 0.3-1.7% in annual energy production for a 4 × 4 (16 turbine) arrangement. Wind farm power increases are primarily seen as a result of faster wake recovery in the range of 5-8 rotor diameters, the typical streamwise turbine spacing for most wind farms.
Bastankhah and Abkar [32] found that this was the result of individual rotor wakes remaining distinct in this range, where at greater distances they merge to form a single wake. LES studies of Ghaisas et al. [33] also demonstrated lower wake losses, which was linked to a higher planform energy flux and greater flow entrainment in the wake. Further to this, Ghaisas et al. [34] showed that wake losses were further reduced with increases in rotor tip spacing. Such improvements in wake recovery lead to lower velocity deficits downstream, facilitating greater power production from downwind machines.
This paper aims to bring together advances in both wake steering and multi-rotor research to introduce a new degree of freedom to wind farm control. It is proposed that wake steering is applied to individual rotors of a multi-rotor turbine such that the wake can be expanded, channelled or redirected to reduce downstream losses. In regards to wake modulation, a yawed multi-rotor turbine can be compared with rotor coning, in which single rotor turbine blades are angled in the streamwise direction, as illustrated in Fig. 1. While coning is primarily a load alignment strategy in which cantilever blade loads are converted to tensile ones [35], it can be seen in Fig. 1 that this method also allows some control of wake expansion. However, since this design is still very much at a conceptual stage, there exist several practical issues surrounding its implementation. Details of rotor mounting may be challenging, and the extent to which benefits of a fully redesigned turbine outweigh its cost have been questioned [36]. By comparison, rotor yaw is already a mature technology, currently utilised in adjusting single rotor turbines to the incoming wind. This same functionality could therefore feasibly be extended to multi-rotor turbines to gain even greater control over wake expansion.
A number of different multi-rotor yaw configurations are modelled in this paper, evaluated qualitatively by examining wake cross sections and quantitatively by characterising wake centroid locations and wake widths. LES data forms the basis of the results, which is compared with a Gaussian analytical wake model and a curled-wake model. The formulation and application of each of these models is explained in §2. The results are presented in §3, with an exact description of the model setup and tested configurations. Wake cross sections are first presented and examined to investigate the nature of wake development in each cases. Finally, the wake centroids and widths are characterised and compared for all models to further assess the merits of each yaw configuration.

| LES Model
The LES code described here solves the filtered continuity and momentum equations for an incompressible turbulent fluid as where (u 1 , u 2 , u 3 ) = (u, v , w ) are the filtered/resolved velocity components, where i = 1, 2 and 3 indicate the streamwise x , spanwise y and vertical z directions, respectively. The filtered/resolved pressure is denoted by p. t is time, ρ o is the fluid density, τ i j is the subfilter stress tensor, and f i represents wind turbines' effects on the air flow. The code utilizes a second-order finite difference discretization in the vertical direction together with a pseudo-spectral method in the horizontal directions. The time integration is carried out using a second-order Adam-Bashforth method. The molecular viscous forces are neglected in the momentum equation, hence the flow is at nominally infinite Reynolds number. In the code, the subfilter stress tensor is parameterized using the Lagrangian scale-dependent dynamic model [37]. The instantaneous wall shear stress is computed based on the local application of Monin-Obukhov similarity theory [38,39]. In order to generate the inflow condition for the wake flow simulations, a precursor technique is used in which a fully-developed boundary-layer flow under neutral condition is simulated. The size of the computational domain is 1600m × 800m × 355m, and it is discretized uniformly into 160 × 160 × 72 computational grid points in the x , y and z directions, respectively. The boundary-layer flow is driven by an imposed pressure gradient. The effective surface roughness height is set to 0.005m. The turbine-induced forces are modelled using the standard non-rotational actuator-disk method described by Calaf et al. [40] using the wind velocity at the rotor plane and the disk-based thrust coefficient, C T . Values of C T and the nominal turbine thrust coefficient C T are related to the turbine thrust force T as where A is the rotor area,ū d is the time-averaged normal velocity at the rotor. The time-averaged upstream undisturbed velocity is denoted byū ∞ , and γ is the yaw angle. The thrust force is distributed uniformly over the rotor area, and the same value of C T = 4/3 is used for all simulations. From Shapiro et al. [12], C T and C T are related based on The LES framework described here has been well validated and used in earlier wind-energy research publications.
The reader may refer to Refs. [41,23,26] for a more detailed description of the LES framework and the solver. F I G U R E 2 Schematics of a yawed multi-rotor turbine

| Gaussian Analytical Model
The analytical model employed is an extension of the Gaussian wake model developed by Bastankhah and Porté-Agel [9], which was initially developed for single yawed rotors. In the case of a multi-rotor turbine, such as that illustrated in Fig. 2a, the Gaussian wake model is applied to each rotor, and the individual wakes are then linearly superposed as suggested by Bastankhah and Abkar [32] . For the n t h yawed rotor, the individual wake widths in the lateral and vertical directions respectively, can be found by where k n is the wake growth rate associated with each rotor, which is assumed to be equal in lateral and vertical directions. Lateral and vertical dimensions are denoted by y and z , respectively, and widths are normalised by rotor diameter, d . Streamwise distance from the turbine is denoted by x , and x 0 is the streamwise distance at which the onset of the far wake region occurs, as detailed by Bastankhah and Porté-Agel [9]. The rotor yaw angle is γ n , as displayed in Fig. 2b, defined as positive in the anti-clockwise direction when viewed from above. The maximum velocity deficit associated with each rotor, C n , may be given by where C T is the rotor thrust coefficient. Note that the reason for apparent discrepancy between Eq. 5 and the one in the original work is a different definition of C T adopted in the current study (Eq. 2). In fact, the value of C T cos 2 γ in the current study is equivalent to C T defined in Bastankhah and Porté-Agel [9]. It is assumed that thrust coefficient and rotor diameter are constant for all rotors. The maximum value, C n , may be used to describe the velocity deficit distribution as a three-dimensional Gaussian profile, given by where ∆ū n (x, y , z ) is the time-averaged velocity deficit of a single rotor, normalised by the time-averaged inflow velocity at hub height,ū h . The lateral and vertical rotor offsets from the turbine centre are denoted by y n and z n respectively, where z h is the turbine hub height and δ n is lateral wake deflection due to yaw. The total normalised velocity deficit distribution of the multi-rotor turbine is given by the linear sum of the contributions from each rotor, hence The above sum of individual rotor velocity deficit contributions facilitates generation of a full flow field which can be interrogated to examine velocity characteristics and features of the wake expansion.
Finally, the spanwise velocity distribution,v (x, y , z ), may be found by calculating the product of the streamwise velocity field,ū(x, y , z ), and the skew angle distribution, θ(x, y , z ). Streamwise velocity may be found directly as , and skew angle distribution for the n t h rotor can be computed from the Gaussian profile suggested by Bastankhah and Porté-Agel [9], where θ m is the maximum skew angle at each downwind location. The interested reader is referred to the original work for more information. The total spanwise velocity caused by all four rotors is obtained by linear superposition of each rotor contribution, akin to the one for the velocity deficit.

| Curled-wake Model
The curled-wake model uses a simplified version of the Reynolds-averaged Navier Stokes (RANS) equations for the velocity deficit of wind turbines in yaw [11]. The streamwise component of the simplified RANS equation is where V and W are the spanwise velocities from the analytical formulations caused by yaw, U is the inflow streamwise velocity and ν eff is the turbulent viscosity. Eq. 9 is a parabolic equation which is solved numerically using a forward-time centered-space method [11]. The initial condition for the wake deficit at the location of a turbine is computed from axial momentum theory based on the thrust coefficient (same as the one in Eq. 6 used in the Gaussian model when x = 0).
For the case of a multi-rotor, each turbine wake is initialized and the superposition of the wakes is done explicitly by solving Eq. 9. The numerical solution of Eq. 9 provides the wake deficit for all the turbines without the need to use a superposition method.

| RESULTS
The model setup of the simulated turbine is illustrated in Fig. 2, which shows four rotors labelled R 1-R 4 in a 2 × 2 arrangement. Each rotor is yawed by an angle, γ n , resulting in a wake deflection, δ n , and an individual rotor wake width,  Fig. 1a. Conversely, 'convergent rotors' are so called because rotors appear convergent, like that shown in Fig. 1b. In both of these cases, left rotors will have equal and opposite yaw angles to the right rotors. In other words, vertically adjacent rotors will have the same yaw. Finally, 'crossed rotors' describes a case where rotors appear crossed from above. This is a result of top rotors having equal and opposite yaw angles to bottom rotors. Hence, laterally adjacent rotors will have the same yaw. Since the wake is very similar regardless of which way the rotors are crossed, only one crossed rotor case was studied.
The magnitude of all rotor yaw angles was 30 • , in order to clearly show yaw effects such as kidney-shaped (curled) wakes which have previously been observed by Bastankhah and Porté-Agel [9] and Martínez-Tossas et al. [11]. Beyond 30 • , wake steering has been found to have diminishing effects on wake deflection [43,6], and negative effects on turbine loading [44]. All configurations involve yawing rotors in pairs, which are either vertically or horizontally adjacent. This choice was made so as to replicate the yaw mechanisms of utility scale turbines such as the Vestas 4R-V29 [30].

| Wake Velocity Deficit Distributions
The first stage of analysis was to qualitatively inspect the wakes generated by each yaw configuration. Hence, cross predictions are used as a reference with which lower fidelity curled-wake and Gaussian models can be compared.
The three models agree well in general prediction of wake behaviour, though there are some notable differences in deflection magnitudes and the magnitude of velocity deficit that will be qualitatively discussed in the following. in opposite lateral directions, thereby increasing wake expansion rates. In these configurations rotor wakes also remain distinct over a larger streamwise distance, whereas rotor wakes interact and overlap faster in the baseline, equal yaw, and especially in convergent cases. Differences between configurations appear largest at short downwind distances, both in terms of magnitude and shape; velocity deficit levels further downstream are more similar between cases and wake boundaries are not so sharply defined.
Another pattern common to all configurations is a higher velocity deficit and deflection in top rotors compared with bottom rotors. This can be seen for all arrangements, but perhaps most clearly at x = 4D for divergent rotors in Fig. 6, where the wake forms a butterfly-like shape. A possible explanation for this may be offered by higher thrust forces exerted on top rotors, compounded by lower turbulence levels at greater heights. Since velocity increases with height due to the simulated atmospheric boundary layer, there will be a greater thrust force developed on top rotors, which will in turn lead to larger velocity deficits by conservation of momentum. Deflection has also been shown to be influenced by thrust force, as well as by turbulence intensity. Jiménez et al. [6] found that a higher thrust leads to a greater deflection and Bastankhah and Porté-Agel [9] suggested that lower turbulence intensities lead to larger wake deflection. Both results corresponded to yaw of single rotor turbines, however, it appears that similar relationships are present in multi-rotor arrangements.
Examining the baseline case (Fig. 4), the most notable feature is how the velocity deficit region begins as a square array of individual rotor wakes at x = 4D , before merging to a more circular shaped single wake as they move downstream to x = 10D . Such behaviour verifies previous findings by van der Laan et al. [30] and Bastankhah and Abkar [32] in which similar wake transitions were observed. In this arrangement, spanwise velocities are small and appear to have no distinct pattern. The results of the curled-wake model are not in agreement with the LES data at short downwind distances, while the agreement improves further downstream. It seems that the curled-wake model over-predicts flow mixing such that rotor wakes already form a single wake at x = 4D , which is not in agreement with the LES data. The Gaussian model predictions agree better with the LES data in this case, with only small disparities in terms of velocity deficit.
For equally yawed rotors, Fig. 5 shows how all rotor wakes are deflected in one direction. Velocity deficit is lower than the baseline and the wake appears to span wider across the domain. Moreover, formation of a kidney-shaped (curled) wake cross-section can be identified as the wake moves downstream, a phenomenon observed in similar studies of yawed single rotor turbines [9,45]. This feature is most likely the result of the counter-rotating vortex pair (CVP) that is also clearly present, again a typical characteristic of yawed single rotor turbines. It would be expected that there would be CVPs associated with each rotor, however, in the LES it appears that some vortices merge or cancel out, leaving only one vortex pair. Lower fidelity models appear less able to capture this merging, where CVPs remain distinct in the curled-wake model, and are not present in the Gaussian model.
The results of the equal yaw configuration also highlight the utility of visually inspecting wake cross sections in addition to mathematical characterisations of wake properties. In comparison to single rotor turbines, multi-rotor turbines are capable of producing quite different wakes depending on rotor yaw arrangement. In this case, there is a significant difference between the overall lateral width across the domain and the width at hub height. While such complex wake distributions can offer significant advantages in reducing wind plant losses, it does mean that mathematical characterisations are less effective in fully describing the nature of the velocity deficit distribution.
Hence, clear representation and close inspection of the flow field is a useful complement to quantitative analysis when developing an understanding of yawed multi-rotor wakes.
In the divergent rotor configuration, rotor wakes are deflected laterally outwards, as shown in Fig. 6. This leads to the formation of a butterfly-shaped wake at x = 4D which transforms to more of a 'V'-shape at x = 10D . The velocity deficit is much lower than the baseline at all streamwise distances, indicating potential for reduced wake losses. The opposite effect can be seen for convergent rotors in Fig. 7, where rotor wakes are directed towards the lateral centre (y = 0). This leads to formation of a narrow wake, which widens at the base due to the ground effect. individual rotor CVPs are observed in the curled-wake model but not in the LES data.
In these two configurations, deflection appears to be over-predicted by the Gaussian model, leading to two distinct wakes in the divergent case, and a high velocity deficit single wake in the convergent case. This can be explained by the fact that in these two configurations yawed rotors on each side of the turbine induce lateral velocities in opposite directions. As a result, their wake deflection, particularly in the convergent case, is expected to be less than the one of an isolated rotor. However, the Gaussian model is not able to capture this interaction between rotor wakes. In these configurations, the curled-wake model is able to provide more realistic predictions as it solves governing flow equations for all rotors at the same time.
Finally, in the crossed rotor arrangement (Fig. 8 Overall, it is clear that the divergent and crossed rotor configurations produce the greatest wake expansion and lowest velocity deficit levels across all streamwise distances. This may suggest that these configurations will be the most effective in minimising wake losses. However, turbine wake deflection caused by equally yawed rotors and the wake channelling displayed in the convergent arrangement may also find their use in wind plant control and optimisation, depending on the arrangement of downwind turbines. For example, in a staggered wind farm layout where neither expansion nor steering are likely to be useful, channelling can guide wakes between downstream turbines. The

∆ū/ū h [%]
F I G U R E 8 Crossed rotor wake cross sections at four downwind distances. Contours show normalised velocity deficit, ∆ū/ū h , and vectors indicate spanwise velocity. Black circles illustrate rotor swept area, and a white dot denotes the wake centroid.
behaviour of each yaw arrangement is analysed further in the following section, in which wake widths and deflections are mathematically characterised.

| Centroid and Wake Width
Following qualitative examination of wake cross sections, a quantitative analysis was carried out to characterise wake centroids and widths. The centroid was calculated using an arithmetic mean of velocity deficit values within a given streamwise plane. The integration domain considered was the same as that shown in Figs. 4-8, which is large enough to ensure that velocity deficit becomes zero at the boundary. In this respect the analysis is analogous to a centre of mass calculation, with the lateral centroid location from the turbine centre given by A similar calculation may be performed to find the vertical centroid location, though this typically remains close to hub height. The lateral width of the total multi-rotor wake can be represented by its standard deviation, σ y , given by if γ 1 = γ 2 = γ 3 = γ 4 (equal/zero yaws), if γ 1 = γ 2 = −γ 3 = −γ 4 (crossed rotors), where δ 1 , y 1 , and σ y 1 represent the deflection, lateral offset and lateral wake width of R 1, respectively.
The wake centroid locations of all configurations are plotted over a streamwise distance range 3 − 12D in Fig. 9.
LES results are plotted with solid lines and dashed lines denote Gaussian and curled-wake solutions in Figs. 9a-9b, respectively. The LES data suggests that wakes remains close to the lateral centre (y = 0) in all cases apart from the equal yaw arrangement, in which there is significant centroid deviation from this location. Small overall wake deflections are due to most configurations generating two sets of rotor wakes with equal and opposite deflections. However, while most cases remain close to y = 0, none are centred exactly at this location with most exhibiting a positive lateral offset.
The wake of the crossed rotors also shows some negative centroid movement after x = 6D . Fig. 9a shows how these patterns are predicted well by the Gaussian model which models the equal yaw case accurately, though does not capture the positive offset for other configurations. In all but the equal yaw case, the centroid location is approximated to y = 0.
Similarly, the curled-wake model predicts centroid location at or close to lateral centre for most yaw configurations, however the centroid variation under equal yaw conditions is under-predicted as shown in Fig. 9b.

| CONCLUSIONS
In this paper wake steering was combined with the concept of a multi-rotor turbine to extend wind farm control capabilities. Large-eddy simulations, curled-wake and Gaussian modelling approaches were used to test the effectiveness of applying wake steering methods to multi-rotor turbines. A range of five different rotor yaw configurations (including a reference case) was investigated by closely examining the wakes at various downstream locations. A qualitative method was employed first, in which wake cross sections were examined in terms of their distribution and magnitude of velocity deficit. The key finding from this analysis was that the divergent and crossed rotor configurations were able to produce a significantly larger wake expansion than the baseline, and also generated much lower velocity deficits. The other two yaw arrangements were able to channel and redirect the wake, which may also be useful functions for reducing wind farm losses. Higher wake deflection was found to correspond to higher thrust forces and lower turbulence intensities, confirming previous findings from single rotor studies. Spanwise velocity was also given some attention since this can have a significant effect on how the wake develops as it moves downstream. Counter-rotating vortex pairs were identified at each rotor which, in many cases, cancelled or summed to form larger vortices.
A mathematical characterisation of wake widths and deflections was subsequently performed, which largely confirmed the findings of the cross-sectional wake analysis. The centroid calculations appear to agree well with what can be seen by inspection of the velocity field and acceptably predict the overall turbine wake movement. Similarly, the wake width variation along the streamwise range was in good agreement with what was shown in the wake cross sections.
It should be acknowledged, however, that these characterisations do not give a full picture of how the wake develops. It is therefore important to inspect wakes visually as well as characterising widths and deflections mathematically.
The two lower fidelity models showed an acceptable agreement with the LES data, however, some discrepancies were observed. The Gaussian model over-predicts deflection in some yaw arrangements, and the curled-wake model needs some tuning to correct the expansion rate. Nonetheless, the overall agreement and low computational expense of these models indicates they are worthy of refinement. Further work could be carried out in evaluating the effects of multi-rotor yaw techniques on overall plant power production to confirm that the strategy is both practically and financially viable.

Declaration of Competing Interest
No competing interests are present.

Data Availability
Data may be available by contacting the corresponding author.

A | GAUSSIAN ANALYTICAL MODEL DERIVATION FOR WAKE CENTROID AND WIDTH
The objective of this appendix is to clarify the derivation of two key wakes parameters, namely the wake centroid and wake width, based on the Gaussian multi-rotor wake model. First, the integrals of Eqs. 10 and 11 are analytically computed, and subsequently simplified using assumptions of turbine geometry and wake symmetry. This facilitates direct calculation of the centroid location and wake width and circumvents the need to generate a velocity field.

A.1 | Wake centroid prediction
The centroid of the wake generated by the multi-rotor turbine may be given by Using the Gaussian profile for velocity deficit given in Eq. 6, the n t h integral in the numerator of Eq. 14 may be evaluated 2σ 2 zn d y d z = 2πC n σ yn σ zn (y n + δ n ).
Similarly, evaluating the n t h integral in the denominator of Eq. 14 gives The remaining terms in the numerator and denominator of Eq. 14 are evaluated in a similar way, which allows the centroid to be written as y c = 4 n=1 C n σ yn σ zn (y n + δ n ) 4 n=1 C n σ yn σ zn .
Next, we attempt to simplify the above equation for the studied yaw configurations. Given the geometrical symmetry of a four-rotor turbine, the equation may be simplified by saying y 1 = y 4 = −y 2 = −y 3 . Moreover, rotors with the same yaw angle magnitude and thrust coefficient typically have similar values of maximum velocity deficit, C n , and wake widths, σ yn and σ zn . Therefore Eq. 17 may be simplified to

A.2 | Wake width prediction
The total wake width of the multi-rotor turbine can be represented by its standard deviation, σ y . Here, for simplicity the variance, σ 2 y , is written. As before, this may be represented as the sum of contributions from each rotor The denominator is the same as in the case of the centroid (Eq. 16). Evaluating the n t h integral in the numerator of Eq.
If yaw magnitudes and wake widths are again assumed to be approximately equal then the 2πC n σ yn σ zn term may be cancelled from the top and bottom of the fraction in Eq. 20, allowing the variance to be written as (y 2 n + 2δ n y n + σ 2 yn + δ 2 n − 2y c (y n + δ n )).
Note that the wake width is not dependent on the sign of deflection in the crossed rotor case, indicating the wake is the similar regardless of which way rotors are crossed. However, this is not the case for divergent and convergent rotors. In these configurations, the wake width is dependent on whether the deflection of the first rotor, δ 1 , is positive or negative.
If positive, then the wake diverges, whereas if deflection is negative the wake will converge. It is also acknowledged that neither the centroid nor the wake width in the lateral direction is dependent on the vertical spacing or hub height of the turbine, as expected.