In a recent paper, suggested a theoretical limit for the energy extraction of offshore wind farms in terms of the capacity factor. applied simplified analytic models of a wind turbine power curve, the wind resource, and the interaction between the atmospheric boundary layer and a wind farm of infinite size. Furthermore, a largely unspecified model correction for finite wind farms is applied that depends on the wind farm layout, wind rose, and neighboring wind farms. The model is validated against measured net capacity factors of 72 offshore wind farms situated in the Baltic, North, and Irish seas. Finally, national policies for planned offshore wind farms are investigated, and some of these are stated to assume capacity factors significantly in excess of the proposed theoretical limit, especially for the Netherlands. The latter led to a public hearing in the Dutch parliament where the authors were interviewed regarding their findings. Hence, the work of has had a major political impact in the Netherlands.

addresses an important issue regarding the saturation of offshore wind farms in waters with limited space leading to reduced energy yield, mainly due to wind turbine and farm wake losses. The wind industry and academia have developed a range of numerical models to calculate such losses (, , and ), which due to their complexity and computational cost, may not be accessible to policy makers. For this reason, proposed a simplified analytical model that can potentially be applied in a simple spreadsheet, based on the work of  and .

In this paper, we address several concerns regarding the validation and application of the model in . We also offer a number of clarifications, as the chosen terminology in can lead to misinterpretation of their results. Our main concerns are as follows.

The proposed limit of the capacity factor is not a theoretical limit in the same way as, for example, the Betz limit is considered to be for the power coefficient of a single turbine. Instead, the proposed limit is based on an analytical expression that is formulated using a number of heuristic model assumptions, including unknown model parameters. A different choice in these parameters can lead to distinctly different limits.

To understand our principal concerns with the work of , the main model equations are summarized and discussed in Sect. . The finite wind farm correction is addressed in Sect. , where we also compare results of an automated method with the results of the undocumented manual method of . Several simplifications are shown in Sect. , which we use to understand the main parameters of the model. Finally, we address the problems with the validation method in Sect. .

It should be noted that original authors have added several community comments during the open review process of this article. We have included their main results in the present work to show that our main concerns remain unresolved.

The main model equations of are repeated here. We start with a simplified model of the wind distribution at the location of a wind farm, which is the well-known Weibull distribution of wind speed, being a function of a shape parameter, k, and a scale parameter, λ: 1f(U)=kλUλk-1exp⁡-Uλk,U0=λΓ1+1/k.

Here, U0 is the mean wind speed and Γ is a Gamma function. Note that we use the subscript 0 to denote freestream conditions, while the subscript ∞ is used for infinite wind farm variables. employed a constant Weibull shape parameter (k=2.4) for simplicity, while the Global Wind Atlas 4.0  suggests that k varies between 2.0 and 2.6 for the 72 offshore wind farms.

Subsequently, a simplified model of a wind turbine power curve, P, as function of the wind speed, U, is defined as 2P(U)=PrUUr3U≤UrP(U)=Pr,Ur<U,Ur=2PrρCPπD23. Here, Pr is rated electric wind turbine power generation and Ur is the wind speed at which rated power is achieved. Furthermore, ρ=1.225 kg m⁻³ is the air density, CP is the constant below-rated power coefficient, and D is the rotor diameter. The wind turbine model does not have a cut-in, Ucut-in, and cut-out wind speed, Ucut-out. For the energy yield calculations, this simplification holds if Ucut-in/λ≪Ur/λ≪Ucut-out/λ. In , values of k=2.4 and CP=0.46 are assumed for all sites.

The integration of the power curve and Weibull distribution, normalized by the rated power, leads to the capacity factor of a single turbine, which can be expressed as function of x≡Ur/λ and k: 3Cf,0=f(x,k)=x-3Γ1+3kΓic1+3k,xk+exp⁡-xk, with Γic as the incomplete Gamma function. The equation for Cf,0 multiplied by a loss factor for losses unrelated to wake effects, floss, is the analytical upper limit that propose, though this is not clearly stated in their work and often misunderstood by readers of their work. Hence, the limit represents a normalized gross AEP × floss. The variable x is a parametric variable in Eq. () which gives a single line for a fixed k, which is explained in more detail in Sect. . This variable can also be defined as the ratio of Ur/U0, using the constant c0=Γ1+1/k from Eq. (). Furthermore, Eq. () is reused several times by (their Eqs. 12, 13, and 16), but it is the same as Eq. () with different x definitions.

For an infinite wind farm with uniform turbine spacing, the model of and can be used to calculate the corresponding wake loss: 4ε∞=U∞U0=1+γδ1+γκπCT8s2+κδ2,γ=ln⁡Gf′h,δ=ln⁡hz0,s=AwfDNtot-1,f′=f⋅e4, with κ=0.4 as the von Kármán constant, G as the geostrophic wind speed, f as the Coriolis parameter based on the wind farm latitude, h as the turbine hub height, z0=10-4 m as the offshore roughness length, CT=0.75 as a constant turbine thrust coefficient, and s as the turbine spacing normalized by the rotor diameter, which depends on the wind farm area, Awf, and the total number of turbines in the wind farm, Ntot. It should be noted that the use of constant thrust and power coefficients are not realistic assumptions. For large wind farms, Ntot≫1, one can express the normalized turbine spacing in terms of installed wind farm power, Pwf, per unit area: s≈ρCPπUr3/8/Pwf/Awf. Equation () can be used to calculate the geostrophic wind speed using an iterative numerical method by setting CT=0. Furthermore, Eq. () can be used to calculate the capacity factor of an infinite wind farm Cf,∞ by the substitution of x=c0Ur/(λε∞) in Eq. (). The latter is an analytical model for the lower limit of the wind farm capacity factor. It should be noted that the infinite wind farm wake loss may never be reached for a large finite wind farm. For example, showed that a wind farm covering an area of 105 km² has not yet reached a wake loss limit. The latter was obtained from mesoscale simulations including a simplified wind farm model representing a square wind farm using a wind farm length of 338 km, and a range of uniform turbine spacing and wind climates. Furthermore, alternative infinite wind farm models exist, as, for example, the model of , who suggested modification of the model of , based on large-eddy simulations of infinite wind farms using finite wind farms with lateral periodic boundary conditions. These infinite wind farm models rely on a geostrophic draw law that is commonly expressed by two constants (A and B), which can be shown to depend on atmospheric conditions, as atmospheric stability , boundary layer height , and capping inversion .

introduced the wind farm wind factor, ϕ=Ur/(U0ε), which represents x=c0/λϕ in Eq. (). The wind farm wind factor corrects the mean wind speed or Weibull-scale parameter to account for wake losses. It is well known that wake losses are strongly dependent on atmospheric conditions, such as ambient turbulence intensity and atmospheric stability . Therefore, one can argue that the collapse of the wake losses into a single variable, ϕ, may not be possible. It should be noted that ε in ϕ is the wake loss for a finite wind farm. An additional model is required to calculate the capacity factor of a finite wind farm, Cf, from Cf,0 and Cf,∞, and this is discussed in detail in Sect. . Furthermore, also included external wind farm wake losses in their finite correction model. set Cf equal to Eq. () and solved for x using a numerical root finding method, from which ϕ can be calculated. An important realization is that for ε=1, we obtain the normalized gross wind farm AEP limit that is referred to as the theoretical limit in , which is often misunderstood by readers of this work.

Although not mentioned by , cut-in and cut-out wind speeds of 3 and 25 m s⁻¹ are employed when calculating the wind farm capacity factor (shown in Fig. 3 and listed in Table S1 by ) using a wind turbine power curve model with cut-in and cut-out wind speeds, as used by . Furthermore, a model of the thrust coefficient above rated wind speed is employed using a decay exponent 3.2, as discussed in . However, the cut-in and cut-out wind speeds and associated models of the power curve and thrust coefficient are not used when calculating the wind farm wind factor ϕ.

The model summarized in Sect.  can provide two values of the capacity factor, representing a normalized gross AEP, Cf,0, and a capacity factor including wake losses for an infinite wind farm, Cf,∞. These values provide analytical bounds of the capacity factor of a finite wind farm. introduced a model to interpolate between Cf,0 and Cf,∞ to obtain a capacity factor of a finite wind farm, Cf: 5Cf=wCf,0+1-wCf,∞,w=NfreeNtot, with w as a weight determined from the ratio of freestream turbines, Nfree, to the total number of turbines. The problem with this method is that the weight has a large impact on Cf and it is not trivial to obtain the weight for real finite wind farms that range in size and can have irregular shapes, as well as non-uniform turbine spacing. determined Nfree by assuming a square regular turbine layout, Nfree=aNtot, and initially proposed a=3, later revised to a value between 4.1 and 5.9 based on a fit using an engineering wake model applied to six offshore wind farms. This gave a corresponding average value of a=5.3. One problem with this method is that the weighting variable can become larger than one for a wind farm with fewer turbines than a2 (i.e., fewer than 29 for a=5.3). Hence, one should limit the weight to one. In a non-peer-reviewed work, proposed an alternative approach by determining the weight as 6Nfree=MrowsMturbines, where Mrows=2.5 and Mturbines is the number of wind farm edge turbines that operate in freestream conditions. It is not clear why the value of Mrows is set to 2.5. determined Mturbines manually for a given wind farm layout, wind rose, and neighboring wind farms, and their results were provided in a database . However, a scientific method was not provided, meaning that one cannot reproduce the results given in . In this work, we have made an attempt to automate the manual method by calculating Mturbines as follows, performed for each wind farm layout as follows.

A concave polygon shape is fitted to determine the edge turbine and the connections between them.

The model from is implicit, but it can be shown that a simple explicit expression for ε∞ and ϕ can be derived when the model is applied to the 72 wind farms. These simplifications are not used in the model validation of Sect. . However, it allows us to better understand the main model parameters.

The equation for the infinite wind farm wake loss (Eq. ) is implicit due to the need for solving the geostrophic drag law (Eq.  using CT=0). However, for the 72 offshore wind farms investigated by , ε∞ can be approximated with a simple explicit expression, which was not shown in their work: 7ε∞=1+c11+c2s2+c12,c1=γδ≈0.219,c2=πCTγ28κ2≈16.5.

Here, the only remaining variable is the normalized turbine spacing, s. The simple expression approximates Eq. () to within 2.3 % for all 72 wind farms, and a comparison with the model results are shown in Fig. . The reason why this simplification can be made is because used constant values for the roughness length and thrust coefficient. Furthermore, the latitudes of the all investigated wind farms are similar (between 50.7 and 58.5°) leading to Coriolis parameters in the range between 1.12×10-4 and 1.24×10-4 s⁻¹. In addition, the ratio of hub height to roughness length, h/z0, and the ratio of the geostrophic wind speed to hub height, G/h, are very similar in logarithmic space for the 72 wind farms, leading to values in the range between 13.3<δ<14.0 and 2.71<γ<3.36, with corresponding mean values δ=13.7 and γ=3.01, respectively. The mean values of δ and γ, together with the parameters that are chosen as constants by , are used to obtain c1 and c2. It should be noted that for much taller turbines, i.e., h≫100 m, the approximation may not hold.

The wind farm wind factor, ϕ, is solved numerically by , by setting Cf equal to Eq. (). However, one can show that for the present database of 72 offshore wind farms, ϕ can be approximated with the following explicit relationship using ε∞ from Eq. (): 8ϕ≈c0Urλ1NfreeNtot1-ε∞(s))+ε∞(s).

A comparison of ϕ using the implicit method of and our explicit expression of Eq. () is shown in Fig. . The maximum difference with the implicit calculation method of is 6 %. This shows that the main parameters to calculate ϕ are the finite wind farm correction factor Nfree/Ntot and the normalized turbine spacing s, which is further motivated from the fact that Ur/λ is often of the order of 1.

validated their model with measured capacity factors from 72 offshore wind farms. It is important to note that these measured capacity factors include losses from wake effects and any other losses, such as grid losses, curtailment, and turbine availability. The main validation was performed by plotting the net measured capacity factor as a function of the wind farm wind factor ϕ=Ur/(U0ε), where Ur is the rated wind speed (Eq. ), U0 is the mean wind speed (Eq. ), and ε is the modeled wake loss for a finite wind farm. Figure  depicts four results for ϕ, where the model implementation of (MinimalisticPredictionModel) in PyWake v2.6.18  is employed, and with different models of the finite wind farm correction in terms of Nfree. The first model results in Fig.  (red dots) employ the ad hoc finite correction factor of and replicate their original results (Table S1 of ), as shown in Appendix . We disregard the results of four wind farms, Baltic 1, Baltic 2, Princess Amalia, and Luchterduinen, because their measured capacity factors were only available in pairs (Baltic 1 – Baltic 2 and Princess Amalia – Luchterduinen), not individually. Figure  also includes the normalized analytical model gross AEP (Eq.  using x as a parametric variable) and the corresponding analytical model result × a loss factor of 0.9. The model only determines the location of x in Fig. . However, since the model is heavily dependent on the model correction for finite wind farms, Nfree/Ntot, this x location can vary between ε=1 (no wake losses, Nfree=Ntot) and ε=ε∞ (infinite wind farm wake losses, Nfree=0). The latter is depicted as a model range in Fig. , which shows that the model of the finite wind farm correction has a large influence on ϕ. The left and right error bar symbols reflect no wake losses and infinite wind farm wake losses. This means that a data point moves to the right and can even cross the normalized analytic gross AEP for Nfree→0. The fact that a data point can cross the normalized gross AEP is a serious issue with the validation presented by . If one would make sure to calibrate the model to get the correct asymptotic behavior where a data point would lie on top of the normalized gross AEP for ε=1, then most of the data points would move to the left, significantly deteriorating the validation against the line representing a loss factor of 0.9. Figure  depicts three more results for ϕ. The green squares represent results from our automated script for calculating Nfree, as discussed in Sect. , which results in a larger spread of the x values in Fig. . Results of the updated automated script from Community Comment 4  are largely the same as our results of the automated script. Finally, we use the analytic Nfree model of (black triangles), which results in lower ϕ values and an even larger spread in the model results.

The four model results of Fig.  are also depicted in Fig.  and are plotted against the measured net capacity factor. These results are fitted with a linear relationship, which shows how the r2 value decreases when going from the manual Nfree method of (Fig. a) to our automated method (Fig. b). Furthermore, the updated Nfree method of Community Comment 4  (Fig. c) does not replicate the high r2 value of 0.87 quoted in their original work (and shown in Fig. a). The r2 values are the lowest for the Nfree method of (Fig. d). Here, we remind the reader that never intended to use their method to include effects of the wind rose and neighboring wind farms, and hence, it is expected to obtain a large spread.

While the model can provide analytical bounds of the wind farm capacity factor representing no wake losses and infinite wind farm wake losses using a simple model, one should be careful when applying the model to finite wind farms. The large sensitivity of Nfree, as shown in Figs.  and , makes it impossible to draw strong conclusions about a limit for finite wind farms.

In this work, we have discussed a number of concerns regarding the paper of . We have shown that the proposed limit of the wind farm capacity factor from is not a theoretical limit but should be considered a limit obtained from a simple analytical model of normalized gross AEP multiplied by a loss factor. The application of the model to the 72 offshore wind farms and choice of model parameters by reveal two main model parameters, namely, the wind turbine spacing and the finite wind farm correction. The latter is a very sensitive model parameter that dominates the model results and validation with net measured capacity factors. Furthermore, the finite wind farm correction applied in (as briefly discussed in a non-peer-reviewed work of ) is an ad hoc manual method that is not well described, and we were not able to reproduce the results with an automated method. The original authors were also unable to reproduce their manual results with an updated version of the automated method, as published in their Community Comments . Given the sensitivity of the model to finite wind farm correction, it is impossible to use the model to assess national policies regarding the capacity factors of planned offshore wind farms. Finally, the Dutch national policy is even more difficult to assess due to the range of capacity factors and wind farm densities mentioned in the references provided by .