The standard wind turbine power equation can be written as 2P(Ueq)=12Cp(Ueq)ρAUeq3, where P is the turbine power output, Ueq is a representative rotor-equivalent wind speed, Cp is the power coefficient, ρ is air density, and A=πR2 is the rotor-swept area with rotor radius R. In the simplest interpretation, Ueq is represented by the hub-height wind speed. In rotor-equivalent wind speed approaches, however, Ueq is instead derived from the kinetic-energy flux through the rotor-swept area, for example using Eq. ().

Equation () is therefore not introduced here as a new model, but as the basis from which rotor-integrated approaches follow. The contribution of this study is to retain the rotor-integrated interpretation while expressing the vertical inflow profile as an analytical function. This makes it possible to decompose the effects of shear, veer, and turbulence without reducing all inflow variability to a single scalar wind speed at the start of the calculation.

As shown in Fig. , the coordinate z is defined relative to the turbine hub, with z=0 at hub height and z=-R and z=R at the lower and upper rotor tips. A vertical profile v(z) therefore describes the horizontal wind speed magnitude sampled through the rotor layer. Directional veer is treated by projecting the wind vector onto the rotor-normal direction before applying the same vertical integration.

In complex terrain, wind velocity varies across the rotor-swept area due to terrain-induced acceleration, directional channeling, vertical shear, turbulence, and wake effects. In undisturbed boundary-layer flow, vertical variations are generally more important than horizontal variations across the rotor disk because vertical gradients in wind speed and direction are a dominant feature of the surface layer and lower boundary layer . This assumption does not necessarily hold inside turbine wakes or in strongly heterogeneous terrain-induced flow; wake-contaminated profiles are therefore treated separately in Sect. .

For the analytical turbulence treatment, turbulence is represented by the second-order statistics of the wind speed at each measurement height. For each height z, the 10 min wind speed distribution is described by a mean v‾(z) and standard deviation σv(z). These are not rotor-equivalent quantities; they are height-dependent statistics derived from the measured or prescribed wind speed time series. The latter is derived directly from the WindCube lidar wind speed dispersion parameter as described in .

In the analytical implementation, the probability density function modeling the distribution of instantaneous wind speed values at height z is approximated by a Gaussian distribution, 3fz(V)=12πσv(z)2exp⁡-V-v‾(z)22σv(z)2, where V denotes the instantaneous wind speed used as the integration variable. The Gaussian assumption is used here as a controlled second-order representation of turbulent wind speed variability. Rather than attempting to reproduce all higher-order or non-Gaussian features of atmospheric turbulence, it provides a reproducible and analytically tractable formulation that can be directly combined with the turbine power curve.

It has been shown by that heavier-tailed wind speed distributions, such as Student's t distribution, are commonly observed at Gotthard Pass. In the present formulation, however, turbulence is represented through a single standard-deviation measure. Within this second-order framework, the effect of heavier tails can be incorporated through a simple scaling of the wind speed standard deviation by a factor of 1.81, which effectively captures their influence on turbine power output .

The observational evaluation uses the Gotthard Wind Park in the central Swiss Alps. The wind farm is located at 2106 m a.m.s.l. on the north–south-oriented Gotthard Pass and consists of five Enercon E92 wind turbines with a rated power of 2.35 MW, a hub height of 98 m, and a rotor diameter of 92 m. Real-condition wind power output and nacelle wind speed data at 10 min resolution were taken from .

During the Gotthard transect campaign, Doppler wind lidar measurements were collected at several locations around the wind farm. The lidar measured vertical profiles of wind speed, wind direction, and wind speed variability at 10 min resolution, with measurement heights of 40–150 m in 10 m increments and 160–300 m in 20 m increments. For the Enercon E92 rotor layer, this typically provides six to nine measurement gates within the rotor-swept area, depending on the lidar location and local elevation difference relative to the turbine base.

OpenFAST is used as a numerical reference framework to test the analytical formulation under controlled inflow conditions. Because the detailed blade geometry and controller configuration of the Enercon E92 are proprietary, the simulations use the open-source WindPACT 3.0 MW reference turbine. This introduces a known limitation: differences between the Enercon E92 and WindPACT power curves can affect the comparison independently of the inflow-integration method.

TurbSim is used to generate the turbulent inflow fields required by OpenFAST. Non-turbulent simulations use deterministic, user-defined vertical wind profiles without turbulent fluctuations. For the turbulent Gotthard comparison, the user-defined von Kármán option (USRVKM) is used because it allows the measured height-dependent mean wind speed, wind direction, and wind speed standard-deviation profiles to be retained directly in the numerical inflow. USRVKM additionally requires an integral length-scale profile and component scaling, which are specified as described in Sect. .

To account for vertical variations in horizontal wind speed, the power input is integrated over horizontal rotor slices, as shown in Fig. . The width of a slice at height z is 2R2-z2, giving 4Pan=12Cp(Ueq)ρ∫-RR2R2-z2v(z)3dz. Here, v(z) is the vertical profile of horizontal wind speed after any required vector projection onto the rotor-normal direction. The power coefficient is evaluated at the rotor-equivalent wind speed corresponding to the same profile, not independently at each height.

The measured or idealized profile is represented by a Taylor polynomial of order N, 5v(z)=∑i=0Ncv,izi=cv,0+cv,1z+cv,2z2+…+cv,NzN. The notation cv,i is used to distinguish the Taylor coefficients of the mean wind speed profile from the turbulence-profile coefficients introduced in Sect. . In the present study, N=3 is used as a compromise between flexibility and overfitting. The same order is used for all measured and idealized profiles unless stated otherwise.

Substituting Eq. () into Eq. () gives an analytical expression for the rotor-integrated power, 6Pan=12Cp(Ueq)ρA∑i,j,k=0Ncv,icv,jcv,kRmKm,m=i+j+k,meven, with 7Km=m!2m(m/2)!2(m/2+1). Odd values of m do not contribute because the corresponding terms integrate to zero over the symmetric rotor interval. The full derivation is provided in Appendix . In the code implementation, the workflow first fits cv,i from the available height levels and then evaluates Eq. () exactly through the triple summation.

The analytical model requires both a continuous turbine power curve and the corresponding power coefficient. The turbine power curve is represented by 8Ppc(U)=Pr1-exp⁡-U-vinCB, where U denotes the scalar reference wind speed at which the turbine power curve is evaluated; in the rotor-integrated analytical calculations, it is set to Ueq. For vin≤U≤vout, Ppc is given by Eq. (), with Ppc=0 below cut-in and above cut-out and Ppc=Pr in the rated-power region where applicable. The fixed parameters vin, vout, and Pr are taken from the turbine technical specifications . The shape parameters B and C are fitted to the published power-curve table, which results in the fit of Fig. , which gives excellent agreement with the tabulated manufacturer power curve.

After fitting Ppc(U), the corresponding power coefficient is computed as 9Cp(U)=2Ppc(U)ρAU3. This makes the output of the power-curve-fitting step explicit: the fitted power curve Ppc(U) and the derived power coefficient Cp(U) are both passed to the analytical rotor-integration workflow. For the Gotthard application, the Enercon E92 2.35 MW power curve is used. For the OpenFAST comparison, the WindPACT 3.0 MW reference-turbine power curve is kept separate so that differences in turbine design are not confused with differences in inflow treatment. The exact specifications of both turbines can be found in Table .

For turbulent inflow, the mean wind speed profile is represented by the Taylor expansion in Eq. (). The wind speed standard-deviation profile is represented by an analogous Taylor expansion over the rotor layer, using coefficients cσv,i instead of cv,i: 10σv(z)=∑i=0Ncσv,izi, where cσv,i represents the Taylor coefficients of the wind speed standard-deviation profile.

At each height, the expected power under turbulent wind speed fluctuations is computed by integrating the fitted turbine power curve over the local Gaussian wind speed distribution given by Eq. (), 11〈P〉z=∫vinvoutPpc(V)fz(V)dV. The rotor-area-averaged turbulent-power estimate is then 12Pturb=1A∫-RR2R2-z2〈P〉zdz, where A=πR2. The factor 1/A is required because Ppc represents whole-turbine power. Since both v‾(z) and σv(z) are represented by continuous Taylor expansions, 〈P〉z is defined continuously across the rotor layer, and Eq. () does not require discretization into height bins.

We then refer to the contribution of turbulence to the power output of the wind turbine as the difference between this estimate and its zero-turbulence counterpart based on the same mean profile, 13ΔPturb=Pturb-1A∫-RR2R2-z2Ppc(v‾(z))dz. As σv(z) tends to zero, 〈P〉z tends to Ppc(v‾(z)) wherever the power curve is continuous. Consequently, ΔPturb tends to zero, including for a nonuniform mean wind speed profile. At exactly zero turbulence, 〈P〉z is evaluated directly as Ppc(v‾(z)).

The turbulence correction is added to the analytical mean-profile estimate to obtain Pan,turb=Pan(v‾)+ΔPturb. This distinction avoids ambiguity between the absolute expected power in turbulent conditions and the deviation caused by turbulence. Positive values of ΔPturb occur mainly in the convex part of the power curve, while negative values occur where the power curve is concave.

Idealized profiles are used to isolate the effects of vertical shear, veer, turbulence, and logarithmic wind speed structure. For each experiment, both the mean wind speed profile and the turbulence profile are specified explicitly. In Fig. , the three different wind profiles are shown: linear-shear, linear-veer, and logarithmic profiles are first evaluated without turbulence, i.e., with σv(z)=0 at all heights. The logarithmic wind speed profile is written as 14v(z)=u*κln⁡zh+zz0, where u* is friction velocity, κ is the von Kármán constant, zh is hub height, and z0 is aerodynamic roughness length. This profile is used as an idealized boundary-layer case and not as a fitted site-specific roughness estimate.

The linear-shear profile is defined as v(z)=v0+az, where a=0.01, 0.025, or 0.05 s⁻¹ is the shear coefficient, and z=0 is located at turbine hub height. Relative to the uniform reference profile, the analytical power deviation is 15ΔPshear=34R2a2v02Pref. The linear-veer profile is defined by a direction that changes linearly with height, giving a rotor-normal wind speed component v(z)=v0cos⁡(αmaxz/R). Here, αmax=αR=5°, 10°, or 22.5° is the maximum angular deviation at the upper and lower rotor tips. Its second-order analytical power deviation is 16ΔPveer=-38αmax2Pref. The homogeneous-turbulence experiment is evaluated using a uniform mean wind speed profile and constant turbulence intensity over the rotor layer. All of these wind profiles are compared to a uniform wind profile where v(z) is constant with height and σv(z)=0.

For a given idealized profile, the analytical power deviation is computed relative to a uniform reference profile with the same hub-height wind speed, 17ΔP=Pprofile-Pref.

OpenFAST simulations are used as a numerical reference to test whether the analytical response to prescribed vertical profiles is consistent with a higher-fidelity aeroelastic turbine model. They are not used to define the analytical formulation itself. All simulations use the WindPACT 3.0 MW reference turbine because a complete open aeroelastic model of the Enercon E92 is not available. Simulations are run for 12 min, with the first 2 min discarded as spin-up and the final 10 min used for comparison with the standard averaging interval of the lidar and turbine data.

For non-turbulent simulations, deterministic user-defined profiles are supplied without turbulent fluctuations. The idealized homogeneous-turbulence experiment uses the standard von Kármán model with prescribed constant turbulence intensity and internally determined length scales. The turbulent Gotthard simulations use the TurbSim USRVKM option, which permits the measured height-dependent profiles v‾(z) and σv(z) to be retained directly. This is useful because the analytical turbulence treatment is evaluated using the same observed vertical variance structure. USRVKM also requires an integral length-scale profile and component scaling. The length-scale profile is prescribed using the Eurocode formulation in EN 1991-1-4:2005+A1:2010, Annex B, Eq. (B.1) . For heights below 200 m, this formulation gives L(z)=Lt(z/zt)α, where Lt=300 m, zt=200 m, and α=0.67+0.05ln⁡(z0/(1m)), with z0=0.1m denoting the terrain roughness length. Here, the height entering the Eurocode formulation is measured above ground rather than relative to hub height. Below the terrain-dependent minimum height zmin, the length scale is held constant at L(zmin). This provides a standard-based atmospheric length-scale profile for the von Kármán spectrum while allowing the measured mean wind speed and standard-deviation profiles to be retained.

The measured lidar profiles are processed in the same way as the idealized profiles. For each 10 min time step, the measured mean wind speed profile v‾(z) and, where used, the measured wind speed standard-deviation profile σv(z) are fitted with Taylor polynomials on the order of N=3. The resulting coefficients are passed to the analytical rotor-integration model. Although third-order fits are used in this study, the formulation also supports lower polynomial orders when fewer measurement levels are available. A third-order fit requires at least four distinct measurement heights; with only two measurement heights, a linear fit (N=1) must be used unless additional constraints are imposed.

For the OpenFAST comparison, the same lidar profiles are transformed from the Enercon E92 rotor geometry to the WindPACT 3.0 MW rotor geometry. The height transformation is 18znew=RWP3.0RE92zold+Δzlidar-hE92+hWP3.0, where hE92 and hWP3.0 are the hub heights, and RE92 and RWP3.0 are the rotor radii of the Enercon E92 and WindPACT 3.0 MW turbines, respectively; the corresponding turbine dimensions are given in Table . Here, zold is the original lidar measurement height relative to the lidar base, Δzlidar is the elevation difference between the lidar and turbine base, and znew is the corresponding height in the WindPACT coordinate system. This form avoids introducing an unexplained fitted constant and can be adapted directly to other turbine geometries.

For each turbine and each prevailing wind direction, a nearby upwind lidar deployment is selected. Where data availability allows, 4 h windows are used, yielding 10 min profile–power pairs for comparison with the corresponding turbine production data. The OpenFAST power output is scaled by the rated-power ratio 2.35/3.0 only for comparison with the Enercon E92 measurements; this scaling is stated explicitly as a pragmatic comparison step rather than as a claim of aerodynamic equivalence between the turbines. A summary of the complete methodology pipeline is shown in Fig. .

Model predictions are evaluated against the measured 10 min power production of the five Enercon E92 turbines. Each turbine is considered separately for northerly and southerly flow, giving 10 reference cases. The primary case-wise metric is bias. Across the wake-free cases, RMSE, MAE, and the coefficient of determination are computed from the original unbinned 10 min data. Binned curves are used only for visualization and not as the basis for goodness-of-fit statistics.

Wake-contaminated profiles are treated separately. When a lidar range gate is clearly affected by a turbine wake but adjacent measurements remain representative of the undisturbed profile, the affected gates can be removed before fitting the Taylor polynomial. The analytical model is then applied to the reconstructed profile. This filtering, demonstrated in Sect. , is used only as a diagnostic test of the method's flexibility and is not included in the main wake-free model-performance statistics.

The comparison between the power expectation from OpenFAST and the analytical model is summarized in Fig. . For linear shear (Fig. a), we find that shear effects contribute positively to the power production of the turbine in comparison to the homogeneous wind profile. The power estimated by the OpenFAST model consistently increases with shear, reflecting the cubic dependence of power on wind speed (P∝v3), where higher wind speed contributes more significantly to the overall output. Above the rated wind speed, ΔP tends to 0 as the turbine saturates at its maximum capacity, and the extra wind speed is not able to positively contribute to the power production. This is particularly well illustrated by the sudden drop of the OpenFAST curve at v=12 m s⁻¹. For the analytical model, the rated power is reached more smoothly at v=14 m s⁻¹, which can be directly attributed to the smoother curvature of the power curve as reported in Sect. . With increasing shear coefficient a, we see similar increases in the magnitude of ΔP for both models, and the peak values scale as expected with ∝a2 as described in Eq. ().

Concerning the differences in the two models, we can observe two elements: the apparent overestimation of the analytical model and the linear behavior of the OpenFAST curve for low velocities. The overestimation of the power difference in the analytical model is most likely caused by the several reducing factors not taken into account in the analytical model. For example, OpenFast considers the reduced efficiency of energy transfer from the wind to the turbine blade due to non-ideal angle of attack, additional shearing forces, and vibrations that will inevitably cause additional friction and reduced power production in reality.

In the case of linear veer, shown in Fig. b, the observations are analogous to the shear but lead to a reduction in power output instead of an increase. This reduction arises from the diminished effective wind velocity component normal to the rotor plane, which is caused by the veer angle. The shapes of both power difference curves are similarly more abrupt for the OpenFAST model due to the less refined and realistic power curve, and the analytical model reports higher differences due to a lack of dampening factors that are present in OpenFAST. As we vary the maximum veer angle αmax, the analytical model follows the expected square dependence (ΔP∝αmax2) as described in Eq. ().

The OpenFAST simulations, however, show slightly different magnitudes, with relatively larger power losses at smaller angles (αmax=5°) and smaller losses at larger angles (αmax=22.5°). This deviation is most likely due to the more detailed modeling of airflow over the blades, but it remains of secondary importance compared to the dominant quadratic dependence on αmax in determining the overall turbine power output.

The ΔP curves of homogeneous turbulence are shown in Fig. c. The analytical response follows directly from the curvature of the turbine power curve, as previously described by : turbulence increases expected power where the curve is locally convex and reduces it where the response becomes concave as rated power is approached. Increasing TI therefore increases the magnitude of the analytical turbulence contribution. The OpenFAST curve shows the same broad sign change but differs in magnitude and near-rated power. Because this stochastic OpenFAST comparison additionally depends on a prescribed turbulence spectrum and length-scale setting that are not constrained as site-specific quantities in this study, it is used here only as a qualitative aeroelastic reference; the analytical turbulence result itself does not depend on those TurbSim settings.

The final idealized profile we discuss is the logarithmic profile shown in Fig. d, a common profile that is used to describe the standard wind shear of the earth's surface. The curvature effects previously described for the linear shear still hold in this simulation. A noticeable observation is the relatively low impact of the logarithmic profile, with the maximum deviation from the uniform inflow amounting to only about 15 and 20 kW for the analytical and OpenFAST models, respectively. The logarithmic profile resembles the linear shear profile in shape; however the increase in wind speed is more strongly limited than the decrease. So instead of a net positive effect, as might be expected from the convexity of the power curve, it results in a small negative effect. As discussed earlier in this section, OpenFAST predicts this behavior more abruptly, while the analytical model responds more smoothly. Interestingly, in this case the analytical model predicts a smaller magnitude of change. This is likely because, in the analytical formulation, the positive contributions more strongly compensate for the negative ones, whereas OpenFAST simulations have additional sources of power loss due to friction and control dynamics of the turbine, compared to the baseline OpenFAST simulation.

We continue by evaluating the models under real-world conditions. For this purpose we have selected 10 distinct wind events at each of the 5 turbines in the Gotthard Wind Park. Each case spans several hours and is classified by southerly or northerly flow (Table ), which are the prevailing wind regimes for this mountain pass. For each event, we make use of the Gotthard Transect Experiment dataset , which includes 10 Doppler wind lidar deployments along the pass, providing upwind vertical wind profiles from all 5 turbines and prevailing winds.

In Fig. , nine of the 10 cases are shown; the case of Turbine 1 with lidar location 9 under southerly flow is omitted. The wake effects from Turbine 2 contaminating the lidar wind profiles before reaching Turbine 1 require a separate case. This special case is discussed in Sect. .

The first sub-figure (Fig. a) compares the predicted and observed power output, with the turbulence module enabled for both models. As observed before in Fig. c, the turbulence improves power output until about v=10 m s⁻¹, after which power production is decreased, reducing deviation from the true power in all cases. Compared to the turbulence-free case (Fig. b), this model version achieves slightly better agreement with the measured power production, which only becomes clear when considering the bias values in Table . Here, the analytical model clearly outperforms OpenFAST, which often overestimates power once the rated power of the turbine is reached. This behavior can also be observed from the idealized wind profiles of Fig. .

In Fig. d, the cross-correlation between the analytical and OpenFAST models is shown. A systematic pattern can be observed here. Firstly, at low wind speeds, OpenFAST generates power earlier because it lacks a defined cut-in wind speed – a known unrealistic feature of the software. This should however not affect the results in a measurable way as the selected cases do not include the trivial case where v<vin and P=0 kW. Secondly, in the intermediate region, the analytical model produces more power due to the stronger curvature of the power curve (Fig. b). Finally, at higher wind speeds, OpenFAST diverges again to greater power predictions, reflecting its unrealistic post-rated behavior. Notably, the inclusion of turbulence only has a significant impact near rated power as the power curve flattens afterwards.

To connect these findings of the power prediction models to the hub-height reference methods, the third sub-figure (Fig. c) shows predictions based on nacelle-height wind speed alone. The trivial case is the turbine nacelle measurement itself (black), which unsurprisingly correlates well with true power since it is effectively calibrated against it. Still, small deviations indicate the importance of vertical variations in the inflow. More striking is the performance of the lidar-based nacelle-height estimations (red), which effectively serve as the hub-height-based extrapolation method. When considering the variety of the wind profiles measured by the lidar on Gotthard Pass, it becomes immediately evident that there is no single surface roughness that can be assigned to this site to perform such an extrapolation. We therefore remain at the best possible case for the hub-height reference by taking the lidar range gate closest to the elevation of the nacelle. Relative to the hub-height-based reference approach, the analytical model reports several improvements with the inclusion of the vertical wind profiles.

As shown in Table , the reduction in absolute wind power bias relative to the hub-height-based reference approach ranges from 7.8 % to 46.4 % across the evaluated cases, with a mean reduction of 27.2±12.3%. For each case, this reduction is calculated by subtracting the magnitude of the analytical model's bias from the magnitude of the reference approach's bias, dividing by the latter, and multiplying by 100. Positive values therefore indicate a reduction in absolute bias. The calculation uses the first and last columns of the bias section of Table , excluding the wake-affected T1-L9-Southerly case. The reported mean and standard deviation describe the case-wise percentage reductions. Across the evaluated wake-free cases, the analytical model also reduces RMSE by 8.8 % and MAE by 12.2 % relative to the hub-height-based reference approach. These improvements demonstrate more accurate power estimates for the investigated events; the limited selection of events does not establish a corresponding improvement in annual energy production estimates. The OpenFAST predictions, however, exhibit higher aggregate RMSE and MAE than the hub-height-based reference approach. Differences in turbine representation likely contribute to this discrepancy: the proprietary aerodynamic specifications of the Enercon E92 prevent its direct implementation in the present OpenFAST simulations. Instead, the open-source WindPACT 3.0 MW reference turbine was used, and its power output was scaled by the rated-power ratio. This scaling does not account for differences in power-curve shape or aerodynamic response, thereby limiting the comparability of the results.

For the turbulent Gotthard OpenFAST cases, USRVKM is retained specifically because it preserves the measured height-dependent σv(z) profile. As discussed in Sect. , the associated example length-scale settings are not interpreted physically. A representative rerun with the IEC Kaimal NTM for the highest-turbulence case produced essentially the same case-wise bias and did not improve the scatter. The turbulent OpenFAST results are therefore treated as a secondary numerical consistency check rather than as a site-specific turbulence-spectrum validation.

Overall, the results show that while nacelle-height extrapolation can provide reasonable fits, the analytical model generalizes significantly better for complex-terrain site assessments. Including turbulence further sharpens its predictive skill, particularly at high wind speeds, while retaining negligible computational cost compared to OpenFAST.

In Fig. a, the original lidar wind measurements from lidar location 9 for southerly wind conditions, relative to Turbine 1, are shown in orange, together with a subset of points selected outside the wake-contaminated region. From the analysis of wake direction and extent, this case was identified as the only instance where either the lidar or the turbine is affected by a wake. Specifically, for Turbine 1 and lidar location 9 under southerly wind conditions, the wake from Turbine 2 intersects the lidar beam but consistently bypasses Turbine 1. This becomes evident when considering the turbine position angle of Turbine 2: extending a line downstream on the lee side of the turbine while accounting for the swept area of the rotor shows that the wake passes over lidar location 9 but does not intersect Turbine 1. As a result, the lidar profile misrepresents the true inflow conditions of the turbine.

The error bars of both the orange and green profiles, as well as the shaded blue region, represent the standard deviation of the wind speed during the 10 min time window, in which complete wind profiles are measured every 5 s. After wake filtering, the resulting profile (blue) behaves much more consistently as if no wake were present, which is also observed by the reduced turbulence intensity, also interpolated by the Taylor expansion.

The effect on power predictions due to the filtering of the wake is shown in Fig. b. The unfiltered case, with analytical and OpenFAST predictions in blue and orange, respectively, substantially underestimates the measured turbine power. After applying the wake filter, the analytical results align much more closely with the actual turbine production. Quantitatively, the RMSE decreases from 515.8 to 242.9 kW, which is a reduction of 53 %, while the bias is reduced by 58 % to -215.3 kW. The only trade-off is a slight decrease in correlation, from 0.55 to 0.50, indicating that the effectiveness of the wake correction varies between individual profiles. This is to be expected as the procedure simply removes a standard range of wind measurements where the wake is expected based on the height of the swept area of Turbine 2 with respect to the elevation of the lidar. In some cases, particularly when the wind field contains a vertical component, wake remnants may still contaminate the higher points, or the correction may overcompensate, as visible in one outlier above the 1:1 line.

Despite these limitations, the approach works remarkably well and highlights a key advantage of the analytical model: it can be applied flexibly to almost any kind of input data. Whether only two measurement levels are available or a densely resolved lidar profile, the Taylor expansion method adapts naturally. This stands in stark contrast to the OpenFAST method, which cannot directly accommodate such filtering. Importantly, the method is also turbine-independent: even before a specific turbine type is selected, a single lidar deployment can provide accurate wind energy assessments for any size turbine. Furthermore, wake-filtered profiles can be particularly valuable in cases such as wind farm expansions, where wake contamination is unavoidable, yet filtering offers a practical way to recover representative inflow profiles such as in off-shore applications where one can expect the unaffected wind profile to be similar within a few kilometers from the lidar site.

In addition to the core results, several supplementary tests were conducted to evaluate the robustness of the analytical model. First, the shaft tilt was tested. In the main numerical simulations, the shaft tilt is set to 0° to remain consistent with the analytical formulation. A more realistic value of 5°, as reported in the Enercon E92 technical specifications, introduces a small bias of approximately 8 kW for the 2.35 MW turbine. However, since the wind profile is based on a lidar measurement located roughly 100 m from the turbine, any slight decrease in horizontal inflow or inclusion of vertical wind components caused by shaft tilt would not necessarily improve the accuracy of the comparison. The simpler 0° configuration is therefore retained, as it is more directly comparable to the idealized wind profile experiments.

Secondly, the sensitivity to air density was investigated using OpenFAST simulations across a range of values representative of wind parks at different elevations. The analytical model remains well behaved across the full range, with no need to adjust the shape of the Cp(v) curve. This confirms that the formulation can be transferred to sites at different elevations, provided that the air-density dependence of available wind power is treated consistently.

Thirdly, we tested the impact of using alternative turbine power curves. When replacing the Enercon E92 curve with the WindPACT 3.0 MW reference-turbine curve, the agreement between the analytical model and OpenFAST becomes closer, although this improvement is primarily due to a degradation of the analytical model rather than an enhancement of OpenFAST. This illustrates that the framework can be applied to arbitrary power curves, as the analytical approximation in Eq. () is flexible and well behaved.

Finally, we assessed the impact of temporal averaging on estimates of v3‾ by aggregating the 1 s lidar wind speed data over a range of averaging intervals: 1, 2, 5, 10, 20, and 30 min and 1 h. Changing the averaging window slightly modifies the resulting wind profiles, which in turn affects the analytically derived power production. These differences remain relatively small, on the order of 4 % (on the order of 3 kW for the 3 MW turbine at TI=15%). While this effect is not explored further, it is noted as a minor source of uncertainty. In line with standard wind energy practice, 10 min averages are adopted throughout this study.

Together, these tests demonstrate that the analytical model is robust to variations in shaft tilt, air density, turbine power curve, and temporal averaging. The separate choice of turbulence spectrum used to generate the secondary turbulent OpenFAST comparison is discussed in Sect.  and is not used to draw conclusions about Gotthard turbulence physics.