# Increasing the Power Production of Vertical-Axis Wind-Turbine Farms Using Synergistic Clustering

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## Abstract

Vertical-axis wind turbines (VAWTs) are being reconsidered as a complementary technology to the more widely used horizontal-axis wind turbines (HAWTs) due to their unique suitability for offshore deployments. In addition, field experiments have confirmed that vertical-axis wind turbines can interact synergistically to enhance the total power production when placed in close proximity. Here, we use an actuator line model in a large-eddy simulation to test novel VAWT farm configurations that exploit these synergistic interactions. We first design clusters with three turbines each that preserve the omni-directionality of vertical-axis wind turbines, and optimize the distance between the clustered turbines. We then configure farms based on clusters, rather than individual turbines. The simulations confirm that vertical-axis wind turbines have a positive influence on each other when packed in well-designed clusters: such configurations increase the power generation of a single turbine by about 10 percent. In addition, the cluster designs allow for closer turbine spacing resulting in about three times the number of turbines for a given land area compared to conventional configurations. Therefore, both the turbine and wind-farm efficiencies are improved, leading to a significant increase in the density of power production per unit land area.

## Keywords

Vertical-axis wind turbines Wind energy Wind farms Wind-farm layout## 1 Introduction

Despite the concerted effort to improve energy efficiency and decouple economic growth from energy consumption, the U.S. Energy Information Administration projects that global total energy consumption will grow by about 45% between 2015 and 2040 (U.S. Energy Information Administration 2013). Mitigating the concomitant large increase in greenhouse gas emissions necessitates exploring alternative lower-emission energy sources, particularly since the majority of the current fossil-based energy resources are finite and have other adverse side effects on the environment. Wind energy is expected to be one of the primary sources of clean, renewable energy that would allow a rapid transition away from fossil-fuel-based energy. In the USA, for example, wind power is projected to contribute around 20% of electrical energy by the year 2030 (Marquis et al. 2011). As a result, increasingly larger wind farms are being deployed, and the continued spread and expansion of these farms poses a challenge since the required land area will increase. A major goal of current research is thus to increase the wind-farm power density, i.e. how much energy can be produced per unit land area used.

In a wind farm, turbines should be far enough apart to allow wind speeds to recover, through lateral or vertical momentum entrainment, after deceleration by the upwind generator (Cortina et al. 2016). Spacing the turbines also reduces the fatigue load generated by turbulence from the upstream turbines and thus increases turbine lifetime (Chamorro and Porté-Agel 2009). The large majority of existing farms use horizontal-axis wind turbines (HAWTs); the behaviour of horizontal-axis wind turbines in large wind farms, and the required spacing between them, have been extensively studied (Wu and Porté-Agel 2012; Troldborg and Sørensen 2014). Calaf et al. (2010) investigated the vertical transport of momentum and kinetic energy in a fully-developed HAWT-array boundary layer (defined as the internal boundary layer developing above a wind farm). They showed that, for large wind farms, regeneration of the kinetic energy is mainly from downward vertical fluxes across the plane delineating the top of the farm, unlike farms with a limited number of wind-turbine rows where the streamwise advection of kinetic energy dominates. The concept of a wind-turbine-array boundary layer is particularly useful for wind farms where streamwise farm length is an order of magnitude larger than the depth of the atmospheric boundary layer (ABL) since the influence of such farms extends to the top of the ABL. Meyers and Meneveau (2010) used an actuator-disk model and large-eddy simulations (LES) to model large HAWT wind farms and investigate their interaction with the ABL. They have shown that a staggered wind farm can extract 5% more power than an aligned configuration and in a follow-up study (Meyers and Meneveau 2012), investigated the optimization of turbine spacing in fully-developed wind farms. They showed that the ratio of land cost to turbine cost in the financial optimization analysis (maximizing power per unit cost) influences the deduced optimal spacing. Meyers and Meneveau (2012) indicate that the optimal turbine spacing is higher than that currently being used in HAWT wind farms. Recently, it has also been shown that the highest achievable mean wind-farm power is strongly dependent on the alignment of the turbine arrays relative to the mean wind direction, and the optimal alignment angle is significantly smaller than that in a perfectly-staggered farm (Stevens et al. 2014). For wind-farm sites with a dominant wind direction, these findings can be implemented to improve wind-farm performance.

All of the above, and other previous work, have focused on wind farms consisting of horizontal-axis wind turbines (Chamorro and Porté-Agel 2010; Lu and Porté-Agel 2011; Yu-ting 2011; Meyers and Meneveau 2012). However, recently Dabiri (2011) has suggested the possibility of an order of magnitude increase in power densities for wind farms when vertical-axis wind turbines (VAWTs) are used. Due to their axis of rotation, VAWT wakes and the flow in a VAWT farm are distinctly different from their HAWT counterparts. This potential increase in power density can be achieved by configuring VAWT farms with a closer spacing to better exploit the flow patterns created by upstream turbines. Dabiri (2011) and collaborators (Kinzel et al. 2012) performed experiments on various counter-rotating configurations of 9-m tall vertical-axis wind turbines and demonstrated that, unlike the typical performance reduction of horizontal-axis wind turbines with close spacing, there is an increase in VAWT performance when adjacent turbines are arranged to interact synergistically. However, high experimental costs and time requirements prevent the extension of these field investigations to large farm scales or the assessment of a large number of configurations. The previous findings thus only pertain to a limited number of turbines where the mean kinetic energy is primarily replenished by streamwise advection and cross-stream turbulent transport, rather than by vertical transport as in large farms. Our aim here is to bridge this research gap and assess the feasibility of increasing power density in large VAWT farms using a synergistic clustering of turbines. Building on Hezaveh et al. (2016), where a LES model for vertical-axis wind turbines was extensively validated and the flow recovery in the wake of a single turbine investigated, here we simulate the interactions of multiple vertical-axis wind turbines in small clusters, and subsequently use these clusters to design large VAWT farms.

## 2 Numerical Model

*u*,

*v*,

*w*) are its streamwise, cross-stream and vertical components, respectively. This instantaneous velocity is decomposed into a mean

*U*

_{i}and a resolved perturbation

*u′*

_{i};

*x*

_{i}is the position vector with components (

*x*,

*y*,

*z*) in the streamwise, cross-stream and vertical directions respectively, \( \tilde{p}^{*} \) is a modified pressure that includes the resolved and subgrid-scale turbulent kinetic energies,

*ρ*is the air density,

*F*

_{i}is the mean pressure gradient driving the flow,

*τ*

_{ij}is the deviatoric subgrid-scale stress tensor; and \( F_{i}^{t} \) represents the aerodynamic forces of the turbine blades on the airflow. Note the omission of the Coriolis force, which is assumed to have no significant impact at such small distances (about 10 m) from the Earth’s surface. At each timestep, \( F_{i}^{t} \) is computed using the actuator-line model as detailed in Hezaveh et al. (2016). The horizontal boundary conditions are numerically periodic, but non-periodic flows can be simulated using an inlet sponge region as shown later. At the top boundary, zero vertical velocity and zero shear stress are imposed. The bottom boundary has zero vertical velocity, while the surface shear stress is imposed using an equilibrium logarithmic-law wall model with a wall roughness length

*z*

_{0}= 10

^{−6}

*z*

_{i}, where

*z*

_{i}is the depth of the computational domain used to normalize all length scales in the model (

*z*

_{i}= 25 m). The details of the wall and subgrid-scale models are provided in Bou-Zeid et al. (2005). The model details are summarized in Fig. 1: an angle of attack (

*α*, the angle between the blade chord and the flow velocity relative to the blade \( \overrightarrow {V}_{\text{rel}} \)) is first computed by knowing the location of each blade represented as a vertical line in the actuator-line model, the upstream undisturbed flow velocity (\( \overrightarrow {U}_{\infty } \)), and the rotational speed of the turbine (

*ω*). This then allows us to obtain the lift and drag force coefficients,

*C*

_{L}and

*C*

_{D}respectively, to be calculated from experimental data, blade-resolving Reynolds-averaged simulations, or tabulated airfoil data after applying a dynamic stall correction, as in Hezaveh et al. (2016).

*C*

_{L}and

*C*

_{D}are then used to compute the normal and tangential force coefficients,

*C*

_{N}and

*C*

_{T}respectively,

*c*is the blade chord length,

*θ*is the azimuthal angle of the blade at a given time, and d

*z*is the vertical length of blade occupying each LES grid cell. Finally, the forces in the LES Cartesian coordinate system are obtained as,

*C*

_{L}and

*C*

_{D}herein, actual dynamic curves for lift and drag forces determined experimentally were implemented into the numerical model. Since we aim to mimic the turbines used in the field experiments for vertical-axis wind turbines reported by Dabiri (2011) and Kinzel et al. (2012), we simulate the same DU-06-W-200 airfoil. Claessens (2006) has conducted several numerical and wind-tunnel experiments at various Reynolds numbers and tabulated the lift and drag curves for different airfoil blade types. Figure 2 depicts their measured values for the DU-06-W-200 airfoil, which include the dynamic-stall effect (directly accounted for in the wind-tunnel measurements). Based on the sign of d

*α*/d

*t*, the instantaneous rate of change of the angle of attack, different paths emerge in these curves in Fig. 2:

*C*

_{Lm+}versus

*C*

_{Lm−}and

*C*

_{Dm+}versus

*C*

_{Dm−}. We note that the DU-06-W-200 blade type is redesigned from the NACA 0018 airfoil and is 2% thicker, and cambered rather than symmetric (Claessens 2006). This blade type and the experimental results reported in Fig. 2 are used throughout.

## 3 Results and Discussions

### 3.1 Validation Against Field Measurements

As mentioned before, Dabiri (2011) and Kinzel et al. (2015) report on several field measurements with different configurations of 9-m tall vertical-axis wind turbines using the same blade type and rotor configuration we adopt here. They have shown that, by using counter rotating vertical-axis wind turbines and special configurations, the turbines can exploit the flow deflection from upwind adjacent turbines and there is a potential of a one order-of-magnitude increase in power density. To complement the previous validation of this ALM–LES model performed against laboratory experiments (Hezaveh et al. 2016), and to ensure that the simulations accurately represent the flow in between multiple turbines and therefore within and in the wake of turbine clusters, we compare our LES results to the field-measured data described in Kinzel et al. (2015) for two adjacent counter-rotating turbines. This is the first validation of our model against data from real-sized VAWT field measurements and, to the best of our knowledge, the first validation of any vertical-axis wind turbine ALM–LES against field data.

*D*apart (

*D*is the rotor diameter). The velocity profiles were measured at 16 points with streamwise coordinates (relative to the line joining the centre of the turbines) of

*x*= − 15

*D*, − 1.5

*D*, 2

*D*, and 8

*D*and elevations above ground of

*z*= 3, 5, 7 and 9 m. All velocity components are normalized using a measured 10-m wind speed from a meteorological tower in the vicinity of the experiments (Araya et al. 2014; Kinzel et al. 2015), which took place in Antelope Valley, north of Los Angeles, California. Further details about the measurements are provided in Appendix A—also see Dabiri (2011) for further information.

Variable | Symbol (units) | Value |
---|---|---|

Number of turbines | | 2 |

Number of blades per turbine | | 3 |

Rotor diameter |
| 1.2 |

Blade vertical length |
| 6.1 |

Blade chord length |
| 0.11 |

Airfoil section type | – | DU 06-W-200 |

Solidity | | 0.275 |

Tip-speed ratio (selected at maximum | | 2.18 |

The computational domain has *L*_{x} × *L*_{y} × *L*_{z} = 31.2 m × 15.6 m × 25 m, respectively spanned by 128 × 64 × 192 grid nodes. This resolution yields about 5 × 5 horizontal grid points spanning each turbine rotor (five in each direction). The distance between the domain inflow and turbines was set equal to the distance between the furthest upstream measurement point and the turbines in the experiment, that is 15*D.* In order to match the inflow conditions such as turbulence intensity and mean upstream wind speed profile in the LES to the observed field data, a precursor periodic simulation was run to generate the inflow. The roughness length and friction velocity of this precursor simulation were calibrated (with adopted values of 0.001 m and 0.5 m s^{−1}, respectively) to yield the experimentally-observed logarithmic velocity profile measured 15*D* upstream of the turbines. The inflow and validation simulations were conducted with neutral stability and, as detailed later, field experimental periods were selected during near-neutral conditions. Then, *y*–*z* slices of instantaneous velocity and pressure were saved at each timestep and fed to the simulation with the turbines as upwind-inflow boundary conditions.

### 3.2 Vertical-Axis Wind Turbine Cluster Design: Geometric and Shading Considerations

*β*, where

*β*= tan

^{−1}(2

*D*/2

*L*) (Fig. 5, left),

*L*being the turbine spacing (centre to centre) in a cluster. We note that this is a purely geometric consideration that does not account for the expansion of the wake. On the other hand, when the flow is approximately perpendicular to the centre-to-centre axis, the higher induced speed in-between the two turbines is not being exploited.

*β*(Fig. 5, middle). However, the third turbine can benefit from the higher wind speed induced in-between the two upstream turbines or the two downstream rotors can benefit from the transverse flow deflection of the upstream turbine (depending on wind direction). This has the potential to result in power production from these three turbines that is greater than the power from three distant non-interacting ones (this improvement depends on

*L/D,*as shown below). By increasing the number of turbines in the cluster beyond three, the flow-related benefits decrease and the range of wind directions where the turbines shadow each other increases to

*n*(

*n*

*−*1)

*β*, where

*n*is the number of turbines in the cluster (e.g., Fig. 5, right). In Fig. 6, the variation of

*β*with

*L/D*and

*n*for various clusters is shown; physically,

*β*represents the total range of wind directions where shadowing influences the turbines. A value of

*β*=

*π*implies, for example, that one turbine is partially or fully shadowed for 50% of possible wind directions, or alternatively that two turbines are shadowed for 25% of possible wind directions. As such,

*β*is the cumulative (partial or full) shadowing of all turbines from all possible wind directions and it can therefore exceed 2

*π*. By increasing the value of

*L/D*of a cluster,

*β*is reduced, while on the other hand, increasing

*n*results in higher

*β*. For

*n*> 3, the

*β/*2

*π*value can become larger than 1, which indicates that there is no wind direction for which the turbines are not casting at least partial shadows on each other. However, one notes that, for

*L/D*> 5, the differences between the

*β*values for

*n*= 2 and

*n*= 3 are minor. Moreover, a clustering with higher

*n*has the important benefit of using a smaller land area. Therefore, the most efficient design for a cluster when there is no dominant wind direction at the site seems to be a triangle (

*n*= 3) since it has a limited

*β*, while at the same time allowing for compact clustering and synergistic interaction between the turbines. A value of

*n*= 4 almost doubles the shadowing angle

*β,*with no increase in the wind-direction range for which synergistic interactions occur. Therefore, we focus on triangular clusters hereafter.

### 3.3 Vertical-Axis Wind Turbine Cluster Design: Aerodynamic Considerations

In order to investigate the characteristics of the proposed triangular cluster design, we conduct a suite of large-eddy simulations in a computational domain containing three of the same turbines defined in Table 1. The basic domain size is *L*_{x} × *L*_{y} × *L*_{z} = 72 m × 48 m × 25 m = 60 × 40 × 20.8*D* and is spanned by *N*_{x} × *N*_{y} × *N*_{z} = 288 × 192 × 192 grid nodes. At such resolution (d*x* = d*y* = 0.25 m), the turbine diameter is covered by about 5 × 5 horizontal grid points, which is similar to the validation runs. These remain constant for the analyses used in this sub-section (except for the domain-size sensitivity analysis detailed later). The power coefficient *C*_{P} of a single isolated turbine, as modelled by the LES, is 0.36. Since the wake deficit increases with *D* and decreases with the distance between the turbines *L*, *L/D* emerges as an important dimensionless number to consider; that is, in addition to its impact on the cumulative shadowing angle *β* illustrated in Fig. 6 and discussed above. In order to investigate the optimal distance, various *L/D* ratios ranging from 2 to 8 were simulated.

*L/D*= 6, an analysis of the sensitivity of the results to the domain size was performed (such that the domain size and number of grid points increase proportionally, and thus the grid resolution is unchanged). Two parameters were investigated for sensitivity to the domain size: the cluster-averaged power coefficient

*C*

_{P}and the wake velocity deficit at 15

*D*and 20

*D*downstream of the cluster. The wake velocity deficit is averaged in time and over a

*y*–

*z*rectangle that is aligned in the

*x*-direction with the turbine cross-section projected area. As can be seen from Fig. 7, if one compares the 80 × 27, 80 × 40, and 80 × 54 runs, changes in domain width

*L*

_{y}can be significant when

*L*

_{y}becomes very small (27

*D*). For such narrow domains, the cross-flow area blocked by the turbines becomes large and prevents correct sideways deflection of the streamlines. Since our domain is periodic in

*y*, a small

*L*

_{y}allows the clusters to interact with “virtual” adjacent ones. The figure suggests that a minimal

*L*

_{y}≥40

*D*should be used since increasing the transverse domain size beyond that, to

*L*

_{y}=54

*D,*results in insignificant changes in the average

*C*

_{P}or in wake recovery.

Changes in domain length *L*_{x} have little impact on the average *C*_{P} (compare the 54 × 54, 60 × 54, and 80 × 54 runs). However, the velocity deficit is clearly influenced by *L*_{x} (due to downstream boundary-condition effects). Based on the sensitivity analysis results in Fig. 7, a minimal *L*_{x} = 40*D* was deemed necessary (and sufficient) to avoid an impact of the domain length on the average *C*_{P} and the wake velocity deficit (compare the 60 × 40 to the 60 × 54, or the 80 × 40 to the 80 × 54 runs). Therefore, a domain size of *L*_{x} = 60*D* by *L*_{y} =40*D* is adopted for the single-cluster simulations hereafter. All these simulations were conducted using an imposed laminar logarithmic-profile inflow with a surface roughness length *z*_{0} = 0.001 m and a friction velocity = 0.5 m s^{−1}. To assess the influence of the turbulence levels in the inflow, a simulation was conducted using inflow planes from a precursor periodic turbulent run. As can be seen in Fig. 7, using a turbulent inflow reduces the deficit values at 15*D* and 20*D* downstream of clusters significantly, which is expected since the increase in turbulence intensity increases momentum entrainment into the wake and speeds up its recovery.

*L/D*ratios were conducted using, at first, the unique wind direction of 60° depicted in Fig. 7. Based on simulation results for different cases (see Fig. 8), it is obvious that increasing

*L/D*improves the performance of the first (upwind) turbine due to the decrease in upstream blockage effect from turbines 2 and 3. Due to the rotation direction of the first turbine (shown in Fig. 7), which deflects the flow towards the third turbine, the third turbine has a slightly higher

*C*

_{P}value compared to the second turbine. On the other hand, Fig. 8 illustrates that the performance of the second and third turbines first improves as

*L/D*increases from 2 to 3, then reaches a plateau until

*L/D*= 5, and finally decreases again. When

*L/D*> 5, these turbines are less able to utilize the higher wind speed induced from the flow deflection by the upstream rotor. The cluster-averaged

*C*

_{P}value (related to the upstream wind speed

*U*

_{∞}) thus peaks at an intermediate

*L/D*value. The three cases with the highest average

*C*

_{P}value, corresponding to

*L/D*values of 3, 4 and 5, were hence selected for further analysis.

*L/D*, but with a different incoming flow orientation. We aim to investigate the omni-directionality of the proposed VAWT clusters, as well as to find the most efficient VAWT spacing averaged over all wind directions. Figure 9a shows the average

*C*

_{P}value versus incoming wind direction; the case with

*L/D*= 5 has the highest

*C*

_{P}averaged over all turbines for all wind directions. This is confirmed in Fig. 9b that depicts the influence of

*L/D*on

*C*

_{P}averaged over all wind directions and all turbines, and normalized by the

*C*

_{P}value of a single isolated turbine. The set-up with

*L/D*= 5 has improved performance because the angle at which the vertical-axis wind turbines cast shadows on downstream turbines (

*β*) is reduced due increasing distance between turbines, as well as because wake recovery is improved when shadowing occurs at the larger recovery distance. Finally, a key observation from Fig. 9b is that the average

*C*

_{P}is about 10% higher than that for a single isolated turbine when

*L/D*= 5; this confirms our premise that the synergistic interaction between closely-spaced turbines can indeed result in a higher overall power generation when exploited adequately.

### 3.4 Farm Design: Cluster Wake and Recovery Considerations

*U*(

*x*,

*y*,

*z*)/

*U*

_{∞}(

*z*)) was averaged over the

*y*–

*z*planes encompassing the whole cluster (projected flow-normal area) at varying

*x*distances from the hub using data from the same simulations described in the previous sub-section. We also investigated various values of

*L/D*to confirm that our choice of

*L/D*= 5, made based on the power output of an isolated cluster, does not produce longer wakes than for other

*L/D*values. The results reported in Fig. 10 indicate that increasing the distance between the turbines in each triangle cluster significantly reduces the distance needed for the wind speed to recover to 75% of its upstream value

*U*

_{∞}. It is clear from the figure that the recovery distance to 75% speed is reduced from 25

*D*for

*L/D*= 3 to 15

*D*for

*L/D*= 5. The choice of the 75% recovery speed is somewhat arbitrary, and other thresholds can be selected, of course. However, the comparative analysis of the recovery distances would reach the same conclusions regarding the optimal

*L/D*to adopt, irrespective of the exact recovery threshold.

Recovery is an important criterion for designing a wind farm, which further confirms the selection of *L/D* = 5. In a wind farm, it is important that downstream turbines are placed at distances were the available flow has recovered to significant levels of its upstream undisturbed speed (e.g. to over 75%, although higher levels are advantageous) so that the power-generation capacity of these turbines is not under-utilized. Furthermore, as shown in Fig. 10, by increasing the distance between the turbines in the clusters, the effect of incoming wind direction on the recovery distance is reduced. The *L/D* = 3 recovery is sensitive to the change in incoming wind direction *ζ;* the recovery distance to 75% of upstream flow speed occurs anywhere between 18*D* and 28*D* as the wind direction changes. The recovery for *L/D* = 5 on the other hand is much less sensitive to wind direction and thus yields more omni-directional farms. The results also indicate that when designing farms based on clusters with *L/D* = 5, the velocity recovery for a separation of 10*D* between the clusters is about 70–75%, while a separation of 20*D* allows a recovery to over 80% of upstream flow speed. Both of these separations are tested in the full-farm simulations below. While other separations can be examined, the outcome of the testing of our hypothesis regarding the potential benefits from synergistic interaction between vertical-axis wind turbines remains the same.

### 3.5 Farm Design: Performance Assessment

*N*

_{x}×

*N*

_{y}×

*N*

_{z}= 320 × 160 × 336 nodes and

*L*

_{x}×

*L*

_{y}×

*L*

_{z}= 96 m × 48 m × 32 m. The resolution yields 4 × 4 horizontal grid nodes per rotor diameter, which is comparable to the validation tests presented earlier. The vertical height of the simulation domain is selected based on a sensitivity analysis performed for the 10

*D*staggered-spacing wind farm. Domains with vertical heights of 32, 45 and 54 m were chosen and the total power coefficients (we use two definitions,

*C*

_{P}and

*C*

_{ P}

^{ *}, that are described below) of these three domains are compared. Due to the small blockage ratio of the wind turbines (projected area of turbines normal to the flow over the

*y*–

*z*area of the computational domain), which is < 3.75% in our runs, the variation of the total wind-farm-averaged power coefficients with domain height < 1% (Table 2). Therefore, the 32-m high domain was deemed sufficient and chosen accordingly; this agrees with Sarlak et al. (2016) concerning appropriate blockage ratios for wind-energy studies.

Vertical height of the computational domain versus farm-averaged power coefficient for a wind farm with 10*D* staggered configuration

Vertical domain height | |||
---|---|---|---|

32 m | 45 m | 54 m | |

| 0.45 | 0.46 | 0.46 |

| 0.52 | 0.53 | 0.53 |

Six of the simulated cases are prototypical wind farms with staggered or regular array configurations and with separation distances of 5*D*, 10*D* or 20*D*; these configurations result in 128, 32 and 8 turbines in the computation domain, respectively. Four additional experiments were conducted using staggered clusters (with *L/D* = 5 separation within each cluster) with wind direction of 0° or 60° and with inter-cluster distances of 20*D* or 10*D* for each direction, corresponding to 24 and 96 turbines in the domain, respectively. It should be mentioned that since the LES model uses periodic boundary conditions in both *x* and *y* directions, these simulations correspond to infinite wind farms with an infinite number of turbines. Therefore, the number of turbines in the computational domain does not influence the results when normalized per turbine. Finally, one experiment is conducted using 20*D* spacing and aligned clusters with a 60° wind direction relative to the cluster. To visualize the differences in the flow patterns in these designs, the average streamwise velocity component for a few selected configurations is shown in Fig. 12. The lower values in the staggered configurations reflects the higher power extraction resulting from a larger number of turbines, and therefore metrics that allow consistent comparison of these configurations are needed. Finding such metrics is not straightforward as discussed in Meyers and Meneveau (2010) and Goit and Meyers (2015).

*C*

_{P}value that uses as a reference speed the average streamwise velocity component in the whole wind-farm volume containing the vertical-axis wind turbines (i.e. over a volume spanning a full

*x*–

*y*plane and the

*z*domain from the bottom to the top of the blades). The comparison of this

*C*

_{P}value for different layouts is shown in Fig. 13. As anticipated the staggered cases have higher

*C*

_{P}values compared to the aligned ones, for both the clustered and regular designs. Of more interest and relevance is that the clustered designs consistently produce higher power than the prototypical design for any spacing. As indicated previously, the

*C*

_{P}value for an isolated turbine is 0.36 and the cluster-staggered designs with an inter-cluster spacing of 20

*D*surpass this value over the whole wind farm for both wind directions. This is due to the gain in average

*C*

_{P}that clusters allow, and the large inter-cluster spacing that minimizes the effects of being in the wake of the upstream cluster. By reducing the distance between clusters to 10

*D*, the average

*C*

_{P}value decreases but remains significantly higher than for the corresponding regular wind farms.

Another important result is that the staggered configurations, even at small separation distances, consistently perform better than the aligned ones. The streamwise separation in the staggered 10*D* case, for example, is the same as the separation in the 20*D* aligned case, and yet the staggered 10*D* layout yields a higher *C*_{P}. One physical reason for this improved performance is that in the staggered cluster farms, in addition to the synergistic interactions within each cluster, the clusters themselves probably interact favourably. One can observe, for example, in Fig. 12c, d that two adjacent clusters produce flow acceleration in between clusters, allowing the next staggered row to benefit from this flow deviation. This is exactly similar to the acceleration within a cluster but now occurs in between clusters, suggesting a fractal attribute to these synergistic interactions (although with only two fractal generations here).

*D*spacing) or cases with more efficient farm layouts, the required pressure gradient imposed in the simulations to yield a steady-state mean flow will also be higher. In the LES, at each timestep, the drag exerted on the ABL by the whole wind farm and the ground surface is computed and the needed mean streamwise pressure gradient to balance this drag is imposed. This gradient eventually reaches a steady state as the mean flow equilibrates. As a result, the different cases have unique pressure gradients over the wind farm, and the supplied power input (estimated as the product of the pressure gradient force and the streamwise velocity magnitude) into the domain is not consistent across all cases. As found in Meyers and Meneveau (2010), Goit and Meyers (2015), several approaches can be used to overcome this potential inconsistency. Since our simulations omit the Coriolis force, the best approach is to normalize the power extracted by the power supplied in the wind-farm volume to the simulations. With no Coriolis force and under steady-state conditions, the pressure gradient force has to balance the total (turbine + ground surface) drag. One can thus characterize these two equal and opposite forces with a squared friction velocity related to the total domain drag

*u*

_{τH}(defined similarly to Calaf et al. 2010; Goit and Meyers 2015). The total power input is thus proportional to

*U*

_{T}

*u*

_{ τH}

^{ 2}, where

*U*

_{T}is the streamwise velocity component averaged over the wind-farm volume (average of the domain containing the blades as defined before). In large wind farms, this is the rate of mean kinetic energy input into the domain that can be extracted by the turbines. The velocity upwind of a specific row (used to define

*C*

_{P}before) is an outcome of this input rather than the main source of energy as in very small farms. Thus, the kinetic energy that can be extracted in large wind farms scales with the pressure drop and with the farm-volume-averaged streamwise velocity component

*U*

_{T}, and since these farms influence the atmospheric pressure field as well as flow inside them significantly, they influence the power available to them. Therefore, in order to be able to compare the various configurations without this pressure-drop discrepancy, the following

*C*

_{ P}

^{*}relation normalized per unit power input, is introduced,

*P*

_{T}is the average power per turbine,

*u*

_{τH}is the root square of the total mean drag (on ground + turbines), which scales with the total pressure drop Δ

*P*

_{drop}across the farm, and

*A*is the rotor area of a single turbine. The comparison of this new performance metric for the various configurations is presented in Fig. 14. Since

*u*

_{τH}is roughly about 10% of the wind speed,

*C*

_{ P}

^{*}is about 100

*C*

_{P}and should not be interpreted in the same way as the classic power coefficient. Even after normalizing the total power generated in these layouts by the power input for each case, the cluster cases maintain the highest

*C*

_{ P}

^{*}value, implying that these cases are able to extract more energy from the applied pressure gradient in the field compared to regular wind farms. The relative differences in the performance of wind farms are expected to be closer to the differences depicted in Fig. 13 for smaller farms (

*C*

_{P}is strictly applicable only for a single row), and closer to those in Fig. 14 for larger farms.

*C*

_{P}). Therefore, power density itself cannot be used as a metric for optimizing farm layout. In order to have a more realistic and practical metric, the total capital cost per unit power generation

*T*

_{total}is computed. Since the power generation for each farm is proportional to the sum of the

*C*

_{ P}

^{*}values of all the individual turbines in a given lot area of fixed size, we use this sum, denoted as

*C*

_{Σ P}

^{*}, for normalization instead of the actual power. The capital costs consist mainly of the cost of the land and the turbines. The following normalized energy cost function can thus be computed (similar to Meyers and Meneveau 2012)

*Γ*

_{A}is the wind turbine density per unit area and

*A*

_{L}is the total lot land area.

*T*

_{land}is the cost of land per unit area and

*T*

_{turbine}is the cost of a single turbine. Using different land-cost to turbine-cost ratios, and the cost for a typical individual turbine similar to the one simulated (≈ $US 10,000) (Dabiri 2011) in Eq. 11, the normalized energy costs were computed and plotted in Fig. 15. Using this comparison metric also indicates that the triangular-cluster staggered layout has the lowest capital cost per projected unit power generated, and is therefore the optimum design among those investigated.

A similar analysis has been made using total *C*_{P} and the results also indicate that wind farms with cluster designs are the most optimal amongst those investigated here. Again, we reiterate that the comparison with *C*_{P} is more relevant for very small farms, while if one uses *C* _{ P} ^{*} , the results are more representative of large farms.

## 4 Conclusions

We have presented a novel concept for optimizing the layout of large vertical-axis wind-turbine farms, taking advantage of synergistic interactions between closely-spaced turbines that were previously shown to yield higher power for a limited number of turbines. Using an actuator-line-model representation of the turbines, embedded in a large-eddy simulation, the modelled wake generated by two counter-rotating turbines is first successfully validated against observations from field experiments. To take advantage of the high wind speed created by the flow deflection of vertical-axis wind turbines when placed in close vicinity, we propose a triangular cluster design consisting of three vertical-axis wind turbines, which form the basis for larger wind farms. The triangular design is the one that best exploits flow acceleration, with a minimal increase in wake shadowing.

The influence of inter-turbine spacing relative to their diameter, *L/D*, was then investigated to optimize a single cluster in terms of the total generated power, the omni-directionality of its performance, and the needed downstream wake-recovery distance. Changing the turbine spacing, the cases with *L/D* values of 3, 4 and 5 were shown to generate the highest cluster-averaged power. Further tests were then performed with these three spacings only, and the case with *L/D* = 5 emerged as the one with the highest cluster-averaged power over all wind directions: the generated power for this case is about 10% higher than that produced by three isolated turbines. Furthermore, *L/D *= 5 results in the lowest variation of the generated power with wind direction, and the downstream wake-recovery distance is the shortest (since the cluster is more “porous”). Therefore, this cluster design confirmed the use of synergistic vertical-axis wind-turbine interactions to increase power production, and would generate a higher power density (power generated per unit land used) due to the proximity of the rotors. It was hence adopted for configuring large VAWT wind farms.

Farms that use this advanced cluster design, and a sufficient distance for wake recovery between clusters of 10*D* and 20*D*, were then compared to prototypical aligned and staggered configurations for infinitely-large wind farms, with different turbine horizontal spacings of 5*D*, 10*D* and 20*D*. For the very large wind farms chosen, the results show that the average wind-farm power coefficient, using two distinct normalizations, is much higher for the staggered-triangle clusters than for the wind farms with regular configurations. Using these average power coefficient results and a simple capital cost function for the whole wind farm, while varying the land-to-turbine cost ratio, we also showed that the wind-farm design with staggered-triangle clusters is the optimal design (amongst the ones considered here) in terms of cost per unit power produced.

These results strongly indicate that VAWT farms can and should be configured using different approaches than those used for horizontal-axis rotors (although the potential benefits of HAWT clustering could also be investigated). A significant increase in power and decrease in capital costs can be achieved using the ability of vertical-axis wind turbines to positively boost the power production of nearby turbines if properly configured. A further important aspect of the results is that, in addition to turbine interactions within a cluster, the clusters themselves interact synergistically, further boosting power production. It should also be mentioned that one of our criteria in optimizing the clusters and farms was omni-directionality. We sought to propose configurations with performances that are not strongly dependent on wind direction since this is also a major advantage of individual vertical-axis wind turbines. However, if this criterion is relaxed, for example in places where there is a dominant wind direction, the optimal cluster designs can be very different and can use this synergistic interaction between clusters as well, with potentially higher power densities.

Finally, one factor that plays an important role in modulating power output from a large wind farm is atmospheric stability. It has been shown that the diurnal cycle and a range of ABL stability strongly influence the performance of large HAWT wind farms (Lu and Porté-Agel 2011; Abkar et al. 2016), and the same is expected for the VAWT farms used herein. However, we focused on the basic neutral case only, and ALM–LES model investigations of the effect of atmospheric stability on VAWT wind-farm operation are left for future studies.

## Notes

### Acknowledgements

This work was supported by the Siebel Energy Challenge and the Andlinger Centre for Energy and the Environment of Princeton University. The simulations were performed on the supercomputing clusters of the National Centre for Atmospheric Research through project P36861020 and UPRI0007, and of Princeton University.

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