Beyond the First Generation of Wind Modeling for Resource Assessment and Siting: From Meteorology to Uncertainty Quantification

Abstract

Increasingly large turbines have led to a transition from surface-based ‘bottom–up’ wind flow modeling and meteorological understanding, to more complex modeling of wind resources, energy yields, and site assessment. More expensive turbines, larger windfarms, and maturing commercialization have meant that uncertainty quantification (UQ) of such modeling has become crucial for the wind industry. In this paper, we outline the meteorological roots of wind modeling and why it was initially possible, advancing to the more complex models needed for large wind turbines today, and the tradeoffs and implications of using such models. Statistical implications of how data are averaged and/or split in various resource assessment methodologies are also examined, and requirements for validation of classic and complex models are considered. Uncertainty quantification is outlined, and its current practice on the ‘wind’ side of the industry is discussed, including the emerging standard for such. Demonstrative examples are given for uncertainty propagation and multi-project performance versus uncertainty, with a final reminder about the distinction between UQ and risk.

1. Introduction

In the age of multi-megawatt turbines, which began what I will roughly label as the ‘second generation’ of industrial wind energy, a number of significant differences have arisen compared to the first few decades (‘first generation’) of the wind energy industry. Some of these changes have essentially come to crucially (re-)define industrial wind energy, from research to operation. This is true not only of wind energy as a whole, but also particularly within the areas related to quantitatively describing the wind, from meteorology to uncertainty quantification.

A single article about meteorology in wind energy would be too long (books such as [1,2,3] already exist), as would an article on uncertainty quantification in wind energy, about which a book could be written. The present article is thus centered on the emergence of UQ in wind energy, on the side of the primary ingredient of such energy production—the wind itself, as informed by applied meteorology and meteorological advances. This involves creating a bridge between theory and practice, including the crucial statistical aspect of application and interpretation.

An Exceptional Multi-Disciplinary Problem

For wind energy production, our main ‘ingredient’—the wind—can be quite uncertain to quantitatively characterize in the long term, let alone predict in the short term. This is different than other industries, where individual ingredients in production are well known or even quality-controlled.

The environment in which wind turbines operate, which is known as the atmospheric boundary layer (‘ABL’), is one whose quantitative description involves a daunting number of physical variables—most of which range over multiple orders of magnitude, arising over and/or characterized by different lengthscales and timescales that also span multiple orders of magnitude [4,5,6]. With numerous flow regimes and behaviors that can depend on the time of day, season, geographic location, and distance from the surface, wind in the ABL—and UQ of wind resources within it—can be quite challenging.

As detailed in Section 2, the large number of physical variables describing flow in the ABL can be reduced from (at least) 10 down to seven by combining them into dimensionless groups, and perhaps further to four or fewer near the surface; the latter situation was exploited in the first generation of industrial wind energy (with its implicit assumptions persisting in some IEC wind standards). However, non-stationarity of the flow and inhomogeneity of the flow environment complicate the picture, effectively adding several more variables to deal with, which are also outlined in the next section. Further, for large windfarms, particularly those which are offshore, the windfarm–atmosphere interaction introduces several more variables. Increases in turbine size have also meant that upper-ABL variables can no longer be neglected.

Thus, the statistics that wind engineers ‘think’ they obtain from measurements can be different than what has actually been obtained: such wind statistics are conditional on quantities which have either not been measured in a typical industrial pre-construction campaign, or accounted for in the statistical calculations. This has implications for measurement-driven modeling in wind energy, where models with a limited number of inputs are used in predicting wind statistics and associated UQ at different sites; the uncertainty can depend on quantities that have not been measured. Doing validation, UQ, and model parameter tuning is challenging here, because one essentially needs to account for the influence of numerous variables, including those which may vary from site to site. In order to cover a representative space, many simulations are needed, demanding an untenable amount of computational resources unless sufficiently simple (fast) models are used; there is a tradeoff between model complexity and ability to validate.

There is also the issue of workers, disciplines, and companies within the field of wind energy having different engineering or scientific backgrounds and aims, which bring with them different nomenclature, terminology, assumptions, tools, and industrial practices. Many wind engineers have backgrounds that are more closely related to mechanical engineering or disciplines associated with wind turbines, with only a small minority of such engineers having expertise in boundary-layer meteorology or applied statistics. To model complex systems lacking complete unified models, such as ABL flow (or perhaps wind turbines themselves), it is worth noting that uncertainty quantification may demand more expertise in the area of application, compared to the modeling itself; one way of stating this is that simply running Monte Carlo simulations of a complicated model might not provide sufficient or applicable uncertainty estimates. Additionally, the quantities of interest, and assumptions about them, may differ depending on the application; e.g., some quantities demanded by both wind resource assessment and site assessment might differ depending on their use, and wind resource analysts may have a substantially different understanding of some variables compared to loads engineers. The above factors further lead to a level of community uncertainty, due to differing understandings and treatment of wind modeling. The latter can be accommodated by clearer reporting of industrial wind calculations, and is ultimately addressed by an industrial standard for wind modeling and its uncertainty; creation and finalization of such a standard (IEC 61400-15-2) [7] has taken roughly one decade. The increase in the size of turbines and projects, with subsequently higher total costs and capitalization, as well as maturation of the commercial side of the wind industry, has also meant that the need for and value of UQ has dramatically increased in importance within the meteorologically associated areas of resource, energy yield, and site assessment.

The present work delves into describing the problem of industrial wind quantification by first describing the ABL parameter space via meteorological variables and proceeding into applied meteorology in Section 2. This is followed by statistical characterization and how it is practically applied in Section 3. Uncertainty quantification is the basis of Section 4, followed by industrial applications in Section 5 and lastly some summarizing discussion.

2. Some Wind Energy Meteorology

2.1. Describing the Parameter Space for Wind

Flow in the ABL can be considered most simply through the variables that describe it. These include the wind speed U for a given height z, a surface roughness length $z_0$, the surface-layer kinematic stress ${\tau }_0/{\rho }_0$ (where ${\rho }_0$ is the near-surface air density), the surface-air heat flux $Q_0$, the Coriolis parameter $f_C$ (inverse timescale characterizing the Earth’s rotation effect on lateral motion as a function of latitude), the air’s viscosity $\mu$, the ABL depth $h_{\mathrm{ABL}}$, the strength of the ABL-topping temperature inversion, and the strength of the large-scale horizontal pressure gradient $|{\nabla }_{h}p|$ driving the flow. Here, by ‘large-scale’ we mean tens of kilometers or more, with the driving pressure gradient expressible in meters per second as the geostrophic wind defined by $G\equiv -|\nabla p|/\left(\rho f_C\right)$. The term ‘kinematic’ indicates a quantity normalized by the air density. Thus, we use a kinematic viscosity of $\nu \equiv \mu /\rho$ with units of m² s⁻¹, and in micrometeorology the kinematic surface stress is expressed via the definition of friction velocity $u_{\ast }$ through $-u_{\ast }^2\equiv {\tau }_0/{\rho }_0={\langle uw\rangle }_0$, so the stress can be effectively replaced by $u_{\ast }$ with units of m s⁻¹; lower-case u and w are fluctuations about the mean for streamwise and vertical velocity components, angle brackets denote an average (typically 10 min), and subscript ₀ indicates surface-layer values (from a height much smaller than $h_{\mathrm{ABL}}$, typically 2–10 m above the surface). The ABL-capping inversion strength is commonly expressed as the Brunt-Väisälä frequency $N_c$ defined by the vertical gradient of virtual potential temperature $\partial {\theta }_v/\partial z$ above the ABL ($h_{\mathrm{ABL}}<z<\sim 2h_{\mathrm{ABL}}$) via $N_c\equiv \sqrt{(g/{\theta }_v)\partial {\theta }_v/\partial z}\simeq \sqrt{(g/{\theta }_v)\partial T/\partial z}$, where $g=9.8$ m s⁻¹ is the acceleration due to gravity [4,8]; here the potential temperature $\theta$ is the buoyancy variable or “meteorologist’s entropy” [9] accounting for the change in temperature due to decreasing static pressure with height, and the “virtual” aspect (subscript _v) accounts for the effect of humidity to give ${\theta }_v\equiv (1+0.61q)\theta \equiv (1+0.61q)T{(p_0/p)}^{R/c_p}$ where $R=287$ J kg⁻¹K⁻¹ is the specific gas constant of air and $c_p$ is the temperature-dependent specific heat capacity for constant pressure (with the unitless ratio $R/c_p=0.286$ often assumed to be constant). The surface heat flux $Q_0$ can be simply defined by the covariance of fluctuations in ${\theta }_v$ and vertical velocity w, with units of degrees Kelvin times meters per second, as $Q_0\equiv {\langle w{\theta }_v\rangle }_0$; in micrometeorological equations and application it is scaled by $g/{\theta }_v$ to have units of energy per time.

Using Buckingham–Pi analysis [10,11,12,13], one can identify the minimal number of variables needed to describe the system that we are considering, i.e., ABL flow. The Buckingham–Pi theorem states that the number of non-dimensional variables needed to describe a system is $n_{\mathrm{nd}}=n_{\mathrm{dv}}-n_{\mathrm{units}}$, where $n_{\mathrm{dv}}$ is the number of dimensional variables and $n_{\mathrm{units}}$ is the number of units involved in these variables. The list of variables $\{U,z,\nu ,u_{\ast },z_0,{\langle w{\theta }_v\rangle }_0,f_C,G,h_{\mathrm{ABL}},N_c\}$ from the previous paragraph gives $n_{\mathrm{dv}}=10$; the units describing these quantities are meters, seconds, and degrees Kelvin (since kinematic quantities lack kg), giving $n_{\mathrm{units}}=3$, so that $n_{\mathrm{nd}}=7$. Accordingly, from the meteorological and geophysical fluid dynamics literature we find seven non-dimensional ‘numbers’ or ‘groups’ describing atmospheric boundary layer flow:

• ABL Reynolds number $\mathrm{Re}\equiv U{h}_{\mathrm{ABL}}/\nu$ [3,4];
• Geostrophic drag coefficient $C_G\equiv u_{\ast }/G$ [14,15,16];
• Surface-based Rossby number ${\mathrm{Ro}}_0\equiv u_{\ast }/\left(f_C{z}_0\right)$ [16,17];
• Surface-based stability $z/L_O\equiv \kappa z(g/{\theta }_v){\langle w{\theta }_v\rangle }_0/u_{\ast }^3$ [18,19,20];
• ABL Rossby number ${\mathrm{Ro}}_h\equiv G/\left(f_C{h}_{\mathrm{ABL}}\right)$ [21];
• Kazanski–Monin parameter ${\mu }_{\mathrm{KM}}\equiv$ $u_{\ast }/\left(\kappa f_C{L}_O\right)$ [22,23]; alternately, $h_{\mathrm{ABL}}/L$ [24,25];
• Zilitinkevich number $\mathrm{Zi}\equiv N_c/f_C$ [26,27].

Here, $\kappa =0.4$ is the von Kármán constant and $L_O$ is the Obukhov length. These dimensionless quantities can be seen to basically represent ratios between two (occasionally three) timescales, lengthscales, forces, or energies; they also arise when non-dimensionalizing different truncated forms of the Navier–Stokes (force balance) or kinetic energy balance equations (e.g., [4]). Thus, they typically tend to represent the dominance of one phenomenon or flow regime for very large values, and another for very small values, or conversely indicate that one term in a balance equation can be neglected (e.g., the viscous term in the Navier–Stokes equation becomes negligible for the large Re encountered in the atmosphere). So-called similarity theories can also be constructed; these analytically describe flow behavior in certain limits [11], in terms of these variables (e.g., Monin–Obukhov similarity to treat the effects of buoyancy on the wind profile as a function of $z/L_O$). One can also find a few other parameters in the literature which are constructed with the same variables, such as $G/\left(f_C{z}_0\right)$, $u_{\ast }/\left(f_C{h}_{\mathrm{ABL}}\right)$, $u_{\ast }/\left(N_c{h}_{\mathrm{ABL}}\right)$ [28], or $u_{\ast }\sqrt{f_C{N}_c}/h_{\mathrm{ABL}}$ [5]; these are simply combinations of the parameters in the bullet-point list above (note: if the parameter $g/{\theta }_v$ is not attached to $d{\theta }_v/dz$ as in $N_c$ or ${\langle w{\theta }_v\rangle }_0$ as in $L_O$, one additional variable could appear).

Nonuniformity of the surfaces underneath an area of interest, such as hills or coastlines, will introduce additional length scales of horizontal inhomogeneity [4,29,30]. Further, geostrophic shear ($dG/dz$) arises due to large-scale horizontal temperature gradients, also known as baroclinity or baroclinic shear [31,32]. Consequently, we have yet more variables to consider, in addition to the flow variables mentioned above. Temporal non-stationarity might further add a timescale to the list of variables to consider; however, this is typically not relevant, other than perhaps wake-related load-associated areas such as meandering [33,34] or U-dependent decay [35]. Furthermore, windfarm scales including the rotor diameter, characteristic horizontal turbine spacing, and hub height [36,37,38] can increase the variable space further; this can also include the addition of wake blockage scales which combine with the earlier list of ABL variables (e.g., stability, [39,40]) to form more dimensionless groups.

2.2. First Applications of Meteorology in Wind Energy

The early days of wind energy—indeed the first generation of folks working on it—were both challenged by and benefiting from a particular aspect of the wind turbines in that era: they were generally confined to the atmospheric surface layer (‘ASL’). The ASL is defined as the bottom ∼10% of the atmospheric boundary layer [4], which can vary hourly, with daily and seasonal patterns; $h_{\mathrm{ABL}}$ can vary from less than 200 m to more than ten times that, most commonly having depths of ∼500–1000 m [41]. Thus, in the first decades of industrial wind energy, where hub heights were typically below ∼50 m with rotor tops often lower than $\sim 80$ m [42], they were confined to the ASL, or at least most of the rotor was usually within the ASL. Here, the surface dominates the flow, and the physics—i.e., the micrometeorology—are relatively well known; this formed the basis of wind resource assessment in the wind energy industry.

Generally, in the ABL the Reynolds number is very large ($\mathrm{Re}\sim {10}^6$ to ${10}^8$), so flow predictions do not depend on $\mathrm{Re}$ [4] and it may thus be neglected from the list given in the previous sub-section. In the ASL, the effects of the ABL depth may be ignored, allowing us to drop the final three dimensionless variables from the dimensionless list in Section 2.1; we are then left with {$C_G$, Ro₀, $z/L_O$}, along with a need to model the effects due to local inhomogeneities.

The first wind resource modeling with a micrometeorological basis addressed three basic elements: vertical extrapolation (VE) from measurement height to turbine hub height, horizontal extrapolation (HE) from a measurement mast to another location, and treatment of effects arising from surface inhomogeneities. Originally, to deal with surface inhomogeneities, the effects resulting from variations in surface elevation (hills) and surface roughness changes ($\Delta z_0$) were treated separately, as in the classic European Wind Atlas (EYA) methodology [16]. This is accomplished through quasi-linearized modeling, using the assumption that terrain (orographic)-related effects on the flow can be superposed on effects due to surface roughness changes; perturbations of the wind speed at a given point relative to the wind speed over a flat uniform version of the surface are modelled as

$${\displaystyle \frac{\delta U_{\mathrm{inhom}}}{U}}=(1+\delta A_{\mathrm{oro}})(1+\delta A_{\Delta z_0})\simeq 1+\delta A_{\mathrm{oro}}+\delta A_{\Delta z_0},$$

(1)

where the $\delta A$ are dimensionless wind speed perturbations due to the terrain inhomogeneities that are upwind in a given direction. The orographic perturbations, often labelled in wind resource assessment (WRA) as ‘speed-ups,’ were modelled using quasi-linearized Jackson–Hunt theory [43] applied within a cylindrical Fourier framework (the ‘IBZ’ model, [44]); implementations other than the EWA/WAsP [16,44] IBZ-model also exist, but have the same character (e.g., the so-called MS-Micro model). This modeling lacks turbulence and only applies to orography with limited steepness where slope-induced flow separation is negligible, and so, consistent with (1), it is most accurate and usable when predicting $\delta A_{\mathrm{oro}}\ll 1$. The effects of a surface roughness change—a transition from an upwind wind speed profile above the so-called internal boundary layer to the local surface-induced profile—are basically handled by interpolation in $ln\left(z\right)$, assuming neutral conditions (without buoyancy effects); multiple roughness changes are handled by a multiplicative superposition model with spatial damping [16,45]. In terms of the augmented dimensionless parameter list from Section 2.1, which would include the addition of more parameters to deal with inhomogeneities, several modeling choices were made which avoid such expansion of the parameter space. Treating hills and roughness changes separately, neglecting the effect of stability and ABL depth in such treatment, and noting that the inhomogeneity effects are local so that the influence of Coriolis is negligible, means that we are still left with only $\left\{C_G,{\mathrm{Ro}}_0,z/L_O\right\}$; the first two are known to affect the flow at larger scales, and arise in horizontal extrapolation modeling, and $z/L_O$ is part of vertical extrapolation modeling in the classic EWA methodology.

For horizontal extrapolation (HE), the geostrophic drag law (GDL)

$$G={\displaystyle \frac{u_{★}}{\kappa }}\sqrt{{\left[ln\left({\mathrm{Ro}}_0\right)-A_G\right]}^2+B_G^2}$$

(2)

was the first basis for predicting the wind at a site different than the measurement location—if one dismisses the generally untenable practice of horizontally interpolating between two measurements [16]; consequently, two of the dimensionless parameters ($\{C_G,{\mathrm{Ro}}_0\}$) are reduced to one. Originally derived to relate the geostrophic drag coefficient $C_G$ to $u_{\ast }/f{z}_0$ (that is, $u_{\ast }/G$ to ${\mathrm{Ro}}_0$) [17], its use in the EWA method arose with the observation that the wind-driving pressure gradient (expressed via G) varies over scales of tens of kilometers—at least over non-mountainous terrain far from the equator, such as in northern and western Europe, where the EWA was first made and validated. Thus, for a potential turbine site not too distant from a measurement location, one can assume the (mean) G to be the same at the two sites; using a wind profile model, one can obtain $u_{\ast }$ from winds measured at a mast location, calculate G using (2), then, at the turbine location(s) calculate ‘new’ $u_{\ast }$ and $U\left(z\right)$.

For vertical extrapolation (VE), so-called similarity profiles have long been in use; these began with the logarithmic wind profile (‘log-law’) which ignores buoyancy, and Monin-Obukhov similarity (’M-O theory’), which corrects the log-law for non-neutral conditions based on $z/L_O$. The EWA/WAsP incorporates GDL-based perturbations into Monin-Obukhov theory, leading to quite complicated expressions for stability-affected adjustment of the Weibull–A parameter (or long-term mean wind speed) at a given height relative to the surface roughness [46]. It is applied as a factor after ‘removing’ local terrain effects from observed winds via (1) just before GDL (2) use to obtain G, and just after subsequent GDL use to obtain $u_{\ast }$ at the prediction site before applying terrain effects there. The factor perturbing the logarithmic profile at observation or prediction sites can be expressed as $[1+p/ln(z/z_0)]$, meaning that the perturbation is simply also an additive correction to $ln(z/z_0)$ of the form

$$p\equiv {\displaystyle \frac{z}{z_m}}\left[a_G{\displaystyle \frac{0.6H_{\mathrm{rms}}}{f{G}^2}}ln\left({\displaystyle \frac{z_m}{z_0}}\right)-{\psi }_W\left(z_0,H_{\mathrm{off}},H_{\mathrm{rms}},G,f\right)\right]+2.5\left({\displaystyle \frac{g}{{\theta }_v}}\right){\displaystyle \frac{H_{\mathrm{off}}}{f{G}^2}}$$

(3)

where ${\psi }_W$ is the effective Monin–Obukhov stability correction function, accounting for the long-term mean effect (‘offset’) and fluctuations (‘rms’), and $z_m(z_0,G,f)$ is a scale height (ca. 65–80 m), with the offset heat flux $H_{\mathrm{off}}$ and fluctuation amplitude $H_{\mathrm{rms}}$ in °K m s⁻¹. The EWA method also perturbs the Weibull–k parameter per height for stability with analogous expressions [16,47] to predict Weibull distributions.

Alternately, in many industrial applications, VE has been performed using the power-law profile, which does not have a meteorological basis (though meteorological connections were later derived [48,49]). By calculating the power-law exponent $\alpha \equiv \partial lnU/\partial lnz\simeq ln(U_2/U_1)/ln(z_2/z_1)$ from winds measured at two heights ($z_2>z_1)$, measured winds $U_2\left(z_2\right)$ can be scaled by the factor ${(z_{\mathrm{pred}}/z_2)}^{\alpha }$ to predict wind speed at height $z_{\mathrm{pred}}$. To compare with the EWA perturbation form (3), binomial expansion provides an $\alpha$-based factor of $U_{\mathrm{pred}}/U_2\simeq [1+\alpha (z_{\mathrm{pred}}/z_2-1)$.

The EWA and similar WRA methodologies could essentially be considered as perturbation-type models; for relatively small corrections to the observed wind and/or wind at prediction site, its symmetric modeling (where the GDL is applied alternately in opposite directions, as is the GDL-perturbed stability correction if it is used) means that the effect of biases in GDL parameters is reduced, since they are taken to be the same at both sites, other than $z_0$. This can be seen by expressing the EWA method’s wind prediction based on observations:

$${\displaystyle \frac{U_{\mathrm{pred}}}{U_{\mathrm{obs}}}}={\displaystyle \frac{{(1+\delta A_{\mathrm{oro}}+\delta A_{\Delta z_0})}_{\mathrm{pred}}}{{(1+\delta A_{\mathrm{oro}}+\delta A_{\Delta z_0})}_{\mathrm{obs}}}}·{\displaystyle \frac{f_{\mathrm{VE},\mathrm{pred}}}{f_{\mathrm{VE},\mathrm{obs}}}}·{\mathrm{G}}^{-1}\left(\mathrm{G}(u_{\ast \mathrm{obs}},z_{0,\mathrm{obs}}),z_{0,\mathrm{pred}}\right),$$

(4)

where $\mathrm{G}\left(\right)$ denotes the use of (2) and ${\mathrm{G}}^{-1}\left(\right)$ denotes the use of the inverse of (2), i.e., numerical solution of $u_{\ast ,\mathrm{pred}}$ from G obtained by first use of the GDL; here, $f_{\mathrm{VE}}$ is the VE model factor, whether it is calculated using the EWA form (3) or power-law (in the latter case, $f_{\mathrm{VE},\mathrm{pred}}/f_{\mathrm{VE},\mathrm{obs}}\to {(z_{\mathrm{pred}}/z_{2,\mathrm{obs}})}^{\alpha }$). Because the $f_{\mathrm{VE}}$ can be expressed as an additive perturbation for $\alpha$-based VE (given the values of $\alpha$ and $z_{\mathrm{pred}}/z_{2,\mathrm{obs}}$ used in practice), and similarly for the EWA form (the ratio of $u_{\ast }$ disappears with the symmetric GDL double-use), then $f_{\mathrm{VE}}$ also satisfies superposition, like the terrain modeling. It can be written as $(1+\delta A_{\mathrm{VE}})$, so that when multiplying by the local surface inhomogeneity corrections, it is simply added to $(1+\delta A_{\mathrm{oro},\mathrm{pred}}+\delta A_{\Delta z_0,\mathrm{pred}}-\delta A_{\mathrm{oro},\mathrm{obs}}-\delta A_{\Delta z_0,\mathrm{obs}})$. If some of the flow perturbations become large enough, then the superposition assumption fails and the results become less certain.

2.3. Meteorology Beyond the Surface-Layer

Farther from the surface, above the ASL, where most multi-megawatt turbine rotors are located, two things generally happen to the flow: (1) the effects of surface inhomogeneities are reduced, with surface variations farther upwind having an increasing effect on the flow (albeit more weakly) and with larger length scales characterizing the wind field; (2) more influence from the capping ABL-inversion is felt. Simply put, less of the ‘bottom-up’ effects and more of the ‘top-down’ effects are impacting the wind above the ASL. The amount of modeling needed to capture local effects of terrain elevation and surface roughness changes tends to decrease above the ASL, though roughness changes such as coastlines that are farther away (∼50–100 times the z considered) can begin to affect the flow; but because the associated internal boundary layer (IBL) grows with distance downwind of a roughness change $\Delta z_0$, the blending (vertical interpolation) between upwind and downwind profiles becomes spread over a larger vertical span. Generally the surface inhomogeneity aspect of wind resource prediction becomes easier above the ASL (smaller corrections), while the VE modeling becomes more challenging.

The buoyancy effects associated with the surface are constant or decrease with height through the surface layer (the shear $dU/dz\propto u_{\ast }/L_O$ in stable conditions according to M-O theory, while $dU/dz$ decreases as ${(-z/L_O)}^{-4/3}$ in unstable/convective conditions [50]), with overall higher shear than neutral conditions due to the cumulative effect of stable stratification surpassing the mixing effect of unstable conditions [51]. However, for heights above the climatological mean ASL depth (typically 50 m or more), the picture becomes more complicated. Long-term mean shear is increased due to the ‘top–down’ entrainment of very stable air from the ABL-capping inversion, with relatively rare shallow ABLs (low $h_{\mathrm{ABL}}$) causing an outsized impact; on the other hand, $h_{\mathrm{ABL}}$ tends to be larger in unstable conditions, which have a nearly zero-shear mixed layer around half the depth. The authors of [46] extended the form of [52] to capture the combined long-term mean effect of these, including distributions of stability and $h_{\mathrm{ABL}}$, and adapted it to be consistent with the geostrophic drag law; the latter form, which was implemented in WAsP, has not gained operational release. Refs. [53,54] derived similarity theories to account for the strength of the inversion for a given ABL depth, extending wind profiles in a way that includes $N_c$ with the previously mentioned dimensionless parameters $\{{\mathrm{Ro}}_0,\mathrm{Zi}\}$; but such profile forms are not (yet) commonly used in industrial WRA.

However, the full EWA method/WAsP actually includes a simple treatment of ABL depth in its vertical extrapolation; it dampens the stability effect with height (as $e^{-6zf/u_{\ast }}$ [16,55]), and also incorporates the mean effect of $\{\partial A_G/\partial {\mu }_{\mathrm{KM}},\partial B_G/\partial {\mu }_{\mathrm{KM}}\}$ on perturbing $u_{\ast }$ for a given G [16,47]. The shear-perturbing effect of baroclinity (large-scale horizontal temperature gradients) and shear-increasing effect of the Coriolis force (${\mathrm{Ro}}_{\mathrm{ABL}}$) derived by [56] were more recently added to WAsP [32,55,57], though for typical VE use (less than 50 m distance between prediction and measurement height) these perturb the speed significantly less than the surface stability.

Wake modeling has also come to include micrometeorology as turbines and windfarms grow larger, because wake recovery—and the energy available for large wind farms from above—depends on entrainment affected by the capping inversion; wakes are also affected by stability due to surface heat fluxes. The recent wake model of [58] integrates all the aforementioned effects (ABL depth, inversion strength, and surface heat fluxes), though it is not yet in common use. Industrial EYA tends to involve simpler engineering wake models, which lack meteorological basis. Blockage is also known to be affected by stability due to the capping inversion [59] and surface fluxes [39], but models including these effects have not yet been adapted into most industry workflows. An exception is Fuga [60] (currently available via open-source pywake [61]), which incorporates surface-layer M-O similarity for both wake deficit and blockage, though it does not treat top–down effects due to $h_{\mathrm{ABL}}$ and $N_c$.

2.4. Modeling Advancements and Their Consequences

Here we will focus on Reynolds-Averaged Navier–Stokes (RANS) and numerical weather prediction (NWP) models, which have both found increasing use in supplementing classic modeling such as the EWA methodology. Microscale RANS solvers calculate mean solutions with resolutions finer than ∼20–50 m and include (mean) turbulence modeling in their nonlinear velocity calculations; turbulence is basically lacking in the classic (quasi-)linearized flow modeling commonly used in industrial WRA. Mesoscale NWP models provide time series with horizontal resolutions at scales of kilometers, and also include more physics than traditional WRA flow modeling, but their limited resolution usually prevents direct use for predicting wind statistics; however, NWP can be blended or coupled with linearized microscale models, and as ‘reference’ datasets for long-term correction (LTC) of measured data.

We will not cover other computational fluid dynamics (CFD) methods such as large-eddy simulation (LES), as they are too demanding in terms of both computational resources (given current computer technology) and user expertise, though LES models adapted for graphical processing unit (GPU) architectures [62] are becoming tenable given current high-end computer trends. Driving RANS with mesoscale NWPs is also exceedingly demanding in terms of current resources, and remains an ongoing topic of research [63,64,65], though this too may enter industrial calculations in the coming years.

2.4.1. RANS Modeling

RANS solvers tend to lack much of the meteorology affecting the wind. This is due in part to the difficulties associated with conditions where RANS solutions are used, namely over complex terrain, which can include mountains and forests. In areas with intermittent recirculation, such as behind hills, a mean solution may be nonsensical due to multiple flow regimes and a local velocity (or speed) distribution which is not unimodal. Adding extra physics into the RANS equations and solver can make convergence to a mean solution even more difficult, and more dependent on the expertise of the person using the solver. Unlike simplified flow models such as the IBZ mentioned earlier, RANS solvers do not separately model hill-induced speedups and perturbations due to roughness changes; rather, these arise in solutions to the RANS equations due to the terrain elevation and $z_0$ variations being prescribed through lower boundary conditions, along with stability effects arising from prescribed surface temperatures or ${\langle w{\theta }_v\rangle }_0$. Thus, these different phenomena can interact together, with larger perturbations becoming possible without linear superposition or associated requirements.

In non-neutral conditions, the ability to use a RANS solver to determine the speedups independent of wind speed disappears, as the appearance of buoyancy causes a wind-speed (actually Froude-number) dependence in the nondimensionalized RANS equations. The same thing occurs if one includes the Coriolis force: again, a wind-speed dependence appears, via the Rossby number in the normalized RANS equations. To show this, one can normalize the vector RANS equation by rewriting it with normalized variables $\{t_{nd},{\mathit{x}}_{\mathit{nd}},{\mathit{U}}_{\mathit{nd}},P_{nd}\}$, assuming characteristic length and velocity scales ${\ell }_{\mathrm{cs}}$ and $U_{\mathrm{cs}}$ to substitute $\mathit{U}\to U_{\mathrm{cs}}{\mathit{U}}_{\mathit{nd}}$ with $\mathit{x}\to {\ell }_{\mathrm{cs}}{\mathit{x}}_{\mathit{nd}}$ and $t\to ({\ell }_{\mathrm{cs}}/U_{\mathrm{cs}})t_{nd}$, and also $P\to \left(\rho {U_{\mathrm{cs}}}^2\right)p_{nd}$ with Reynolds-stress tensor $\mathit{\tau }\to U_{\mathrm{cs}}^2{\mathit{\tau }}_{nd}$ [4,66]. Noting that $𝛁\to {𝛁}_{nd}/{\ell }_{\mathrm{cs}}$ in ${\mathit{x}}_{\mathit{nd}}$-space and then dividing the whole equation by $U_{\mathrm{cs}}^2/{\ell }_{\mathrm{cs}}$, we obtain the RANS equation in terms of the nondimensional variables:

$$\left({\displaystyle \frac{\partial }{\partial t_{nd}}}+{\mathit{U}}_{\mathit{nd}}·{𝛁}_{nd}\right){\mathit{U}}_{\mathit{nd}}=-{𝛁}_{nd}·{\mathit{\tau }}_{nd}-{𝛁}_{nd}{p}_{nd}-{\displaystyle \frac{{\widehat{\mathit{e}}}_z\times {\mathit{U}}_{nd}}{\mathrm{Ro}}}-{\displaystyle \frac{{\widehat{\mathit{e}}}_z}{{\mathrm{Fr}}^2}};$$

(5)

here, the Rossby number is $\mathrm{Ro}=U_{\mathrm{cs}}/\left(f_C{\ell }_{\mathrm{cs}}\right)$, and the Froude number is defined by ${\mathrm{Fr}}^2\equiv U_{\mathrm{cs}}^2/\left(g{\ell }_{\mathrm{cs}}\right)$. In (5) we have not included drag due to forests, but note that as it is typically expressed proportional to $U^2$, it remains independent of the Reynolds number, though it may introduce an additional characteristic length scale [67] to add to the list of dimensionless parameters. We see that over sufficiently long horizontal distances per wind speed (where $\mathrm{Ro}\sim 1$), e.g., in mid-latitudes with $f_C\sim {10}^{-4}$ s⁻¹ then for mean winds of 10 m s⁻¹ the Coriolis term will become significant over distances of $\sim 10$ km or more; this is relevant in large windfarms and where neighboring farms exist. For buoyancy, (5) would appear to indicate that only the vertical velocity is affected. However, we must note that a contribution to horizontal velocity arises through the (turbulent) Reynolds stress divergence ${𝛁}_{\mathit{nd}}·{\mathit{\tau }}_{nd}$ (e.g., [68]), which, following on from the rate equations for shear stress and turbulent kinetic energy, is of the order $z/L_O$ [48,49]; this is also evidenced by Monin–Obukhov similarity theory, which describes the effect of buoyancy on the wind profile in the ASL.

Related to the above is the difficulty of RANS solvers in capturing unstable (convective) conditions over hilly terrain, where large circulating eddies can arise both due to buoyancy and the terrain shapes, which tend to defy mean solution—they require a sufficiently large horizontal domain extent (∼5–10 times $h_{\mathrm{ABL}}$), require a capping inversion for sufficient representation (e.g., [4,69]), and can have a significant unsteady component. Two-equation turbulence closures, which are commonly used in commercial RANS solvers, also cannot generally handle convective conditions [70] without (ad hoc) modification; however, due to convection-enhanced mixing, whose effects reach an asymptotic limit (zero shear) for larger surface heat fluxes, one may simplify by considering only mildly unstable ABLs (i.e., a smaller range of $L_O^{-1}<0$ compared to the range of $L_O^{-1}>0$ in stable conditions [71]) but allowing for the larger ABL depths encountered in convective ABLs. Further, the RANS solution over complex terrain in stably stratified conditions has not yet been shown to be universally achievable using conventional (1- or 2-equation) turbulence closures, though some commercial software attempts to do so in essence via implicit damping or limitation of the terrain and buoyancy effects (thus only providing an approximate treatment). RANS with higher-order algebraic models have recently made this more likely (e.g., explicit algebraic Reynolds-stress models [72,73]), but such models are not yet used industrially. At any rate, employment of RANS solvers that include buoyancy and/or Coriolis forces requires multiple simulations at different speeds per wind direction, with atmospheric stability further requiring simulations for different $L_O^{-1}$ in order to attempt to represent long-term effects (as described in [71]). Subsequently, validating such RANS requires covering a larger joint variable space due to the addition of two variables (or more, if a capping inversion is imposed). This can also be seen from the perspective of adding new dimensionless variables, which include combinations of terrain metrics (also possibly RANS solver grid metrics) together with the variables listed in Section 2.1 above. A related implication, as discussed more generally in Section 4 and Section 5.1.3 below, is that RANS solvers exhibit a sensitivity to wind speed that depends on stability and vice versa, as well as a joint sensitivity to other parameters (e.g., turbulence-model constants, inflow profile prescription, grid and resolution); the behavior of RANS solvers includes nonlinearities arising from the combined effects of different variables and model parameters, which become increasingly complicated with the treatment of buoyancy and Coriolis, as well as the use of more complex turbulence models.

2.4.2. Mesoscale Modeling

In contrast with industrial RANS solvers addressing complex microscale flow, mesoscale numerical weather prediction (NWP) models such as WRF (the Weather Research and Forecasting model [74]) are built on meteorology, though they tend to have effective horizontal resolutions of several kilometers; their usable resolution ranges from roughly 2 to 8 times the horizontal grid spacing [75]. Thus, NWP models capture the physics at these (meso) scales, but not turbulence nor local terrain effects. The latter can lead to prediction bias onshore [76], with offshore bias also occurring due to several factors (e.g., [77]). Hills, roughness changes, and heat fluxes are bottom boundary conditions, as is the case for microscale RANS solution. However, NWP resolutions mean that terrain elevation variations are smoothed out, so resolved mountains and hills cannot be very steep; and NWP does not give $\Delta z_0$-induced IBLs matching microscale models or theory, but rather an analogous adjustment of the flow depending on the NWP advection scheme [78,79]. As opposed to the mean values (per specified boundary conditions) given by RANS solvers, NWP gives time series, including time-varying values of surface heat fluxes, temperature, and heat flux profiles (thus also the implied heights and strengths associated with the capping inversion). Several years of NWP data tend to cover the seven-dimensional space listed in Section 2.1, but the joint PDFs of the variables may be limited by the time series duration, as well as model resolution and subgrid turbulent flux parametrization (planetary boundary layer or ‘PBL’ scheme; e.g., [80]).

NWP models are nonetheless commonly used for long-term correction, as their biases can be removed by incorporating the measurements, while long NWP model time series (covering many years) allow compensation for measurements occurring in windy or weak-wind years relative to means over expected turbine lifetimes (i.e., decades). NWP models are also gaining increasing use for HE and flow adjustment in geographical areas where mesoscale effects create significant differences between measurement and assessment locations, and offshore where windfarms may extend well beyond 10 km; in industrial WRA, mesoscale results are sometimes also ‘blended’ with classic HE calculations to constrain issues due to resolution or biases. Mesoscale model calculations for WRA require significant expertise and resources, as noted in the review of Haupt et al. [81] on numerical weather prediction for renewable energy.

Mesoscale NWP models have also been used to create so-called ‘wind atlases’ at national, regional, and global scales by using the GDL to ‘downscale’ their output [82,83]. These basically employ the predicted winds as a virtual mast over the mesoscale surface map, then apply the EWA methodology as described by (2) and (4) to make predictions on a much finer grid using a microscale surface map along with the EWA microscale modeling. Such wind atlases essentially provide Weibull parameters and mean winds, though the New European Wind Atlas (NEWA) [84] and newer versions of the Global Wind Atlas (GWA) [85] offer more statistics and some time series as well.

As with RANS solvers (especially those which include more physics such as surface-induced buoyancy), the more complicated nature of NWP models—which contain yet more physics, including time-varying buoyancy and ABL-capping inversions, Coriolis forces, baroclinity, and regional circulations—means that mesoscale models are sensitive to a larger number of atmospheric and model parameters than classic quasi-linearized WRA modeling. Furthermore, wind output from mesoscale NWA models exhibits significant sensitivity to the choices of air–surface (‘land’) parametrization, PBL scheme, and land surface (roughness) representation, along with the size(s) of calculation domains and subdomains [86].

3. Appropriate Statistical Characterization, from Theory to Practice

Much of the applied meteorology in wind energy is inextricably related to statistical characterization and subsequent interpretation, dependent on the application or even use case. One can view this as stemming from the nonlinearity of the Navier–Stokes equations and horizontally inhomogeneous boundary conditions or non-stationarity; the character of wind variations depending on the length and/or time scales over which the wind field is calculated; and also nonlinearity in typical turbine power curves. Averaging a nonlinear function of a variable does not provide the same result as evaluating the same function using the average value of its input variable, $\langle F\left(x\right)\rangle \ne F\left(\langle x\rangle \right)$, whether it is a temporal or spatial average; hourly, monthly, and yearly statistics of the wind differ from each other, as do microscale and mesoscale motions.

3.1. Rational Averaging Implicit in Classic WRA

When it comes to wind resource assessment, following from the EWA method [16], the dominant methodology since the 1990s calculates long-term wind statistics (histograms or Weibull parameters) that are representative of a turbine’s lifetime at a given site; the wind statistics are essentially representative of an “average year”. One convenient advantage of this (perhaps a necessity in the creation of the EWA) is that the geostrophic drag $C_G$ can be essentially reduced as a variable: by taking the long-term average of (2) and determining the GDL’s sensitivity to the parameters $\{A_G,B_G\}$, one may use characteristic long-term values of $\{A_G,B_G\}$—presuming that they do not still depend on other parameters after such averaging; the ${\langle C_G\rangle }_{\mathrm{long}-\mathrm{term}}$ becomes a function of climatological-mean $\langle {\mathrm{Ro}}_0\rangle$, as implemented in software such as WAsP [87] and wind atlases using the EWA methodology (e.g., [84,85,88]). Thus, it becomes possible to use the GDL for horizontal extrapolation from one site to another wherever the GDL is applicable, without considering other variables, as long as the ABL top does not create a large mean impact, to the extent that $\{A_G,B_G\}$ are representative; as noted in [21], default values of $\{A_G,B_G\}$ used in the aforementioned implementations were empirically chosen as a compromise to allow the model to be used over both land and sea. Section 2.2 also noted that the EWA method/WAsP incorporates a mean effect of ABL depth variation on the GDL, through perturbation of the surface heat flux to account for the change in $u_{\ast }$ due to surface-driven stability and ABL depth; this is done through ${\mu }_{\mathrm{KM}}$ using mean values of $\langle \partial A_G/\partial {\mu }_{\mathrm{KM}}\rangle =-\langle \partial B_G/\partial {\mu }_{\mathrm{KM}}\rangle$ [16,47].

The EWA method, which is designed to be driven by measurements with its two-way “up–down” use of both the GDL and long-term stability perturbations, essentially reduces the influence of uncertainties in its GDL and VE-model parameters because any biases in these perturbations cancel to first-order in (4) due to appearing in both the numerator and the denominator, leaving only higher-order (smaller) contributions. However, ‘one-way’ use of the GDL is more prone to systematic uncertainty (bias) in its constants, which is one reason why one cannot use pressure gradients from mesoscale modeling or measurements and obtain reliable results.

3.2. Refined Modeling and Consequent Sampling Issues

In general, the more that one divides data into separate categories, the higher the sampling uncertainty becomes; alternately, the higher the demands on sampling are for a given uncertainty tolerance (e.g., see Ch. 2 of [4] for the generic case of a stochastic signal’s sample-mean). In wind resource assessment, the practice of using a 24 × 12 “matrix”—where measurement data are separated into hour of the day per month of the year—is intended to better accommodate the different flow regimes occurring during the diurnal and seasonal cycles, improving predictions of the wind resource and energy production based on measurements. Dividing measurements into 288 (24 × 12) time categories for modeling results in a total of ∼180 samples per bin per year of data, which appears reasonable; but a histogram or Weibull fit requires ∼${10}^3$ or more samples to have a reasonable shape. Subsequently, separating per wind direction then further reduces statistical representativeness. Further, implementing such ‘chopping’ assumes that the WRA modeling can properly treat the flow regimes occurring at different times, and that these models are somehow informed by some additional information relevant to characterizing these regimes.

For example, night-time and winter periods over land will tend to be more affected by stably stratified conditions from the surface, as well as shallower ABL depths. For vertical extrapolation (VE), from these periods one needs to capture the $L_O^{-1}$ and possibly ${\mu }_{\mathrm{KM}}$ effect, but using the power-law with shear measurements below hub height does not capture the second; using M-O profiles for VE such as the EWA form (3) would require the ABL depth to be adjusted, in addition to the stability corresponding to the time/month regime. For horizontal extrapolation (HE), use of a RANS solver also requires representative $h_{\mathrm{ABL}}$ and $L_{\mathrm{O}}^{-1}$, while the geostrophic drag law (GDL) then requires its $A_G$ and $B_G$-parameters to be modified for the specific conditions. Not accounting for these aspects adds uncertainty to the modeling, including possible bias, and potentially worsens the result at some sites compared to use of yearly means without such data division. A similar example can be considered offshore: sampling of winds in the North Sea far from land. In the spring, winds with southerly components there will tend to induce stable conditions due to the water remaining cold from winter; so again, certain months in a 24 × 12 matrix would need to have the GDL and stability parameters adjusted—depending on wind direction.

To get around the problem of limited sampling, particularly offshore, more industrial use has been made of long-term re-analysis datasets such as ERA5, and multi-decade mesoscale datasets. These can provide full wind PDFs even after dividing per hour, month, and direction, though they must be adjusted to correct their bias using measurements. However, the seasonal and daily variations in these datasets tend to be smaller than measured, though borrowing from the weather community, one may use a combination of datasets along with techniques such as logistic regression or linear variance calibration [81].

3.3. Averaging Issues Arising with Time Series Use or Comparisons

If one tries to use common WRA methodologies such as the EWA to predict time series, a problem connected with that described in Section 3.2 can arise. That is, different values of the various modeling parameters, which have been defined and/or calibrated for long-term mean usage, must be modified to ensure that they are appropriate for the conditions in each timestep. Further, the assumption of independent effects (allowing superposition) may become untenable with time series; in reality, for times with non-neutral conditions, the effects resulting from hills and roughness changes may be different than the flow-perturbation modeling (which assumes the long-term mean gives neutral conditions). In some cases this does not incur much uncertainty, such as GDL use in simple offshore settings, but for inhomogeneous projects, particularly over land, the accuracy may be reduced and uncertainty increased.

Another issue arises with mesoscale time series, when they are compared with measured time series per timestamp or used in an attempt to directly supplement measurements. Known as a ‘phase lag’ problem, this is a phenomenon in which mesoscale predictions can have a horizontal and temporal offset (‘lag’) compared to reality, making comparisons difficult. Different methods exist to address this, such as filtering out periods where the time series have a correlation coefficient which falls below a specific threshold [89], but these are not yet standard practice.

An additional averaging issue, which also involves comparison, is the width of directional sectors used when comparing observations to wake models [90]. Basically, the sector width must be sufficiently larger than the directional uncertainty (standard deviation of direction) in order to compare statistics. A more general version of this might be considered for comparisons conditional on other quantities and their bin widths.

4. Uncertainty Quantification

Models for different effects on the wind all propagate—and potentially amplify—uncertainties in their inputs, while also adding their own additional uncertainty independent of the inputs. For measurement-driven modeling, which is standard in bankable energy-yield assessment, this begins with wind measurement uncertainty.

The Guide to the expression of Uncertainty in Measurements (‘GUM’) [91] can be seen as a basis reference for much UQ work in the field of wind energy, including propagation of measurement uncertainty; its initial (concurrent 2008) supplement [92] gives guidance for propagation of distributions via Monte Carlo methods, and its second (2011) supplement [93] was extended to multiple quantities. The more recent (2023) GUM ‘introduction’ [94] outlines how GUM and all of its supplements can be used; notable is the 2020 supplement on models [95] which includes addressing poorly understood effects, shared (correlated) effects, and basic model uncertainty.

GUM essentially provides a description of first-order uncertainty estimation via propagation of measured uncertainty; this uses a Taylor series to provide a linear (approximate) response of a model $M\left(x\right)$ around the expected value $x_0\equiv \langle x\rangle$ of its input x. This can be expressed as $y=M\left(x\right)\approx M\left(x_0\right)+(x-x_0)(\partial M/\partial x){|}_{x_0}$ where $\partial M/\partial x$ is the model sensitivity in the neighborhood of $x_0$; note that this can change if $x_0$ is different. This leads to

$$\langle y\rangle \equiv \langle M\left(x\right)\rangle \simeq M\left(x_0\right)\mathrm{and}{ϵ}_y\simeq {\left|{\displaystyle \frac{\partial M}{\partial x}}\right|}_{x_0}{ϵ}_x$$

(6)

where the $\{{ϵ}_x,{ϵ}_y\}$ are the uncertainties, i.e., standard deviations of x and y (see uncertainty textbooks such as [96] for more details). When uncertainties are small compared to the actual values of measurand (i.e., ${ϵ}_x\lesssim 0.1x_0$) and the sensitivity does not change too quickly with x compared to ${ϵ}_x$ (so $|{\partial }^{2}M/\partial x^2|\lesssim 0.1\langle y\rangle /{ϵ}_x^2$), then (6) is a useful approximation, not requiring higher-order terms. Following GUM, the actual value of a measured quantity x follows either a normal distribution (‘Type A’) or a unit distribution (‘Type B’), with different uncertainty types that can be combined by scaling type B variances (${ϵ}_i^2={\sigma }_i^2$) with a coverage factor of 1/3 and summing the squares to find the total squared uncertainty [91]. Such summation presumes independent uncertainty components have the same units, or that each uncertainty is nondimensionalized by its respective mean. Reference (6) can also be generalized to include a vector $\mathit{x}$ representing multiple inputs and $\mathit{y}$ for multiple outputs, incorporating the covariances between all of these [96].

For highly nonlinear processes and models, the low-order GUM technique begins to fail due to sharp changes in model sensitivity. A way around this is to use Monte Carlo ‘simulations’, whereby the model calculation is repeated many times, but with its input(s) taken from a random (joint) distribution that reflects the uncertainty in the input(s). For example, Figure 1 shows the power output from a Monte Carlo simulation where ${10}^4$ randomly sampled speeds are put into a simple power curve, where the expected speed is $U_0=0.9U_{\mathrm{rated}}$ and its uncertainty is ${\sigma }_u=0.06U_0$ (6%) and the error is normally distributed; the GUM result following (6) is also shown.

One can see from Figure 1 that GUM fails around rated speed due to the sharpness of the power curve in that area. In cases/applications such as this example, where a bi-modal distribution results, one must consider not only using a Monte Carlo or other method, but possibly also consider output uncertainty metrics such as $P_{10}$ or others. The demonstration above used ${10}^4$ points, but still one can see considerable noise in the PDF; in this case the Monte Carlo curve becomes smooth for ∼${10}^6$ points. This reminds us of the sampling demand when using Monte Carlo simulations, meaning that it becomes computationally difficult to perform these calculations for more than a few dimensions; other more advanced methods are available, as outlined and used in [6], for example.

Uncertainty in the Complex ABL System

The uncertainty and validation framework of Huard & Mailhot [97], which is visualized in Figure 2 as adapted by Murcia [98], further indicates that UQ and validation have a conditional (Bayesian) aspect.

In Figure 2 one sees that for validation, outputs (vector $\mathbf{y}$) of a complex model $\mathcal{M}(\mathbf{x},\mathit{\theta })$ with multiple inputs (vector $\mathbf{x}$) and model parameters (vector $\mathit{\theta }$), require validation conditional on model error ${ϵ}_{\mathcal{M}}$ as well as on $\mathit{\theta }$, $\mathbf{x}$, ${\mathit{ϵ}}_{\mathbf{x}}$, and ${\mathit{ϵ}}_{\mathbf{y}}$. The Figure also indicates that optimal tuning of parameters is also conditional on $\{\mathbf{x},{\mathit{ϵ}}_{\mathbf{x}},{\mathit{ϵ}}_{\mathbf{y}}\}$. Forbes [99] showed how the classic GUM method is an approximate solution to the Bayesian approach. Refs. [100,101,102] showed how and why GUM was updated to accommodate and be consistent with Bayesian Monte Carlo methods.

However, given the discussion in Section An Exceptional Multi-Disciplinary Problem and list of inputs such as that shown in Section 2.1 for advanced ABL models, we cannot easily validate or tune models unless we have a large number of sites and measurements spanning the multi-dimensional input space $\mathbf{x}$—i.e., it is necessary to actually know the relevant variables which our results are conditioned upon, including those describing inhomogeneity, as well as the sensitivity of the model to all these. An exception to this is the EWA method, whereby within its operational design envelope, the number of parameters becomes manageable, particularly at non-complex sites; this is one reason why the EWA method and associated software has been used for bankable WRA over several decades. Mesoscale (numerical weather prediction) models and RANS (CFD) solvers can treat the effects of more phenomena and more variables, but paradoxically then require a larger validation space $\mathbf{x}$; they are currently too computationally demanding to be run millions of times in order to cover the dimensional space involved. Other exceptions to this are engineering models such as those used for calculating wakes within a wind farm.

The Validation and Verification (V&V) framework of [103] offers one way to address the computational complexity issue, including the so-called ‘PIRT’ (Phenomenon Identification and Ranking Table) which uses industry-wide expert knowledge to identify (typical) model weaknesses and reasonable operational envelopes. This framework originally arose to deal with complicated CFD calculations for heat transfer in other industries [104], but the latter did not deal with such extensive input spaces and infinite degree-of-freedom system as we have in the ABL; thus, it has informed some emerging practices in industrial wind calculation, but is not used for wind CFD yet. In essence the first Comparative Resource and Energy Yield Assessment Procedures (CREYAPs) campaigns [105,106] utilized something like the PIRT suggested by [103], identifying problem areas in the WRA-EYA process, and with the CREYAP-related summary work of [107] evaluating the relative importance of different steps in the process.

Offshore UQ could be considered a ‘simpler’ task than onshore, with narrower ranges of secondary variables involved (e.g., ABL depth and surface stability) and greatly reduced effects of inhomogeneity, particularly as floating lidar measurements have become standard practice and thus eliminate the need for vertical extrapolation. However, offshore projects tend to be larger, with more complex wake effects and potentially significant blockage; they also have taller turbines, which interact more with finite-ABL effects. Further, offshore windfarms can be affected by other large windfarms at distances of several kilometers or more, requiring mesoscale modeling or modification of typical engineering wake models [108,109]. The offshore CREYAP exercises [110,111,112] gave an indication of primary uncertainty components and some of their magnitudes in resource assessment offshore, though they did not explicitly address some of the issues mentioned above. Beyond the CREYAP experiments, following the Sandia V&V work [103], recently [113] gave a systematic framework for offshore V&V, which outlines future experiment concepts and essentially (implicitly) facilitates better UQ offshore. The advent of floating offshore wind turbines further necessitates the need for UQ of wave parameters and associated modeling; reporting of such uncertainties is an emerging industrial practice, with recent progress towards performance, design, and reliability [114,115,116].

5. Industrial Application

The way in which measurements and modeling of the wind field impact wind energy involves a web of processes and methodologies. To provide context, Figure 3 gives an example of a simple ‘digital model’ depiction (one-way coupling without automation [117]) of the complicated interconnected context within which meteorology fits into wind energy—and, subsequently, how uncertainty on the ‘wind side’ can propagate through model chains in different parts of the industry.

As seen at the top of Figure 3, wind measurements and meteorological model data can enter into both long-term pre-construction calculations, which are the focus of this article, as well as short-term operational modeling. Depending on how much integration and connection of models one wishes to pursue, observed and modelled wind data—and its uncertainty—can propagate well beyond wind resource and energy-yield assessment and siting, to regional (e.g., grid) planning in the long term; and windfarm operation/control, lifetime management, grid/transmission re-configuration, and energy trading in the shorter term. Regarding the interconnectedness shown in Figure 3 with many models coupled together, we note that there can also be uncertainty in the coupling itself; however, this is beyond the scope of the current article.

5.1. Wind Resources

This section will primarily focus on long-term wind resource assessment (WRA), which drives energy yield assessment (EYA), in the so-called pre-construction (planning) context; this contrasts with short-term forecasting and the operational context. Operational uncertainties are not addressed in the current work, and are typically considered to be energy-based as opposed to being windspeed-based. However, the continuing advancement of windfarm control systems (e.g., those informed by nacelle-mounted lidar) may both reduce operational uncertainty and have a wind component, though these are still likely to be effectively independent of pre-construction wind predictions in terms of UQ.

Given the number of different methodologies across the industry, the increased use of ensembles of model types and data, and the number of (often unmeasured) variables in the ABL as well as its complexity, the industry has progressed phenomenologically—as mentioned in the previous section regarding the CREYAP exercises—towards simple UQ based on engineers’ experience. Also guided by GUM, the IEC 61400-15 working group (on the order of 100 members) separated WRA and EYA uncertainty into components and subcomponents, quantifying each of them through a consensus process based on expertise, aggregation of their industrial UQ practices and workflows, uncertainty modeling directly derived from WRA processes, and knowledge of (proprietary) production error statistics and associated pre-construction uncertainty estimates. An aim of this basic UQ is to not just establish a standard practice, but also facilitate reporting of uncertainties in order to reduce the variability in pre-construction uncertainty estimates from one engineer or company to the next. As discussed below in Section 5.5, an associated goal is to support reporting of UQ in WRA and EYA in order to separate uncertainty from (subjective) assignment of risk.

For wind-related (meteorological) uncertainties, the IEC working group established five categories, which are shown in

Table 1. Uncertainty components and subcomponents which depend on the wind, from the forthcoming IEC 61400-15-2 standard.

Component (bold) or subcomponent (italic)

Measurement Uncertainty

Wind speed measurement

Wind direction measurement/rose

Other atmospheric parameters

Data integrity and documentation

Historical Wind Resource (LTC)

Representativeness of long-term reference period

Reference data consistency

Long-term correction method

On-site gap-filling/synthesis

Representativeness of measured data

Wind distribution fit

Horizontal Extrapolation and flow modeling

Model inputs

Model “stress” (deviation from operational envelope)

Model appropriateness

Vertical (power-law) Extrapolation

Model representativeness †

Excess uncertainty propagated by VE-model

Project Evaluation Period Variability

Interannual variability (IAV) of wind speed

Climate change

(IAV of plant performance) ‡

Plant Performance

Turbine interaction/wake and blockage effects

(Non-wind elements) ‡

^† Through analytical derivation from the power-law, VE model ‘stress’ is effectively an $\alpha$-dependence within the model representativeness subcomponent. ^‡ Other non-wind plant performance uncertainty subcategories include turbine performance (power-curve), environmental (e.g., degradation), curtailment/operational strategy, availability, and electrical performance.

. A sixth component (plant performance) is calculated in terms of energy production uncertainty, but has a wind-related subcomponent due to wakes and blockage.

The uncertainties in Table 1 are all calculated as normalized values, i.e., as a fraction of expected wind speed (expressible as a percentage), so that they may be combined to provide a final total uncertainty. Each subcomponent is also assumed to characterize Gaussian-distributed variability. With the exception of power-law VE, whose uncertainty was derived analytically, the other uncertainty components and subcomponents are assigned based on user-specified conditions present in a pre-construction project; the values per condition are based on project experience and expertise across roughly 100 wind engineers, other IEC 61400 standards, and extant wind industry UQ workflows, and were re-calibrated based on cross-testing by the IEC working group. The various wind-related components ${\tilde{\sigma }}_i$ are described below.

5.1.1. Wind Uncertainty Components

• Measurement

Building from and consistent with the IEC 61400-50 series [118,119,120,121], IEC 61400-12 series [122,123,124,125,126], and MEASNET [127], measurement uncertainty subcomponents include contributions from station-level (documentation), monitoring level (data acquisition system and sensor combination), and sensor level (mounting, sensor classification/rating) for mast-based data. The subcategories of wind speed and direction measurement both have these contributions, with the assumptions that aberrant data and outliers have been removed, and that boom vibrations are damped/filtered away (neglected). Remote sensing device uncertainties are also treated. The measurement UQ model sums the squares of subcomponent variances, including sensor calibration, sensor classification, sensor mounting, flow distortion, and data acquisition. Formulae for these stem from the standards mentioned above.

• Long-Term Correction

A number of different methods and datasets are used for LTC of measured data, with ensembles of datasets and correction methods sometimes employed. Thus, for UQ of LTC, the adjustment is generalized, and to some extent is proportional to the long-term correction itself. Subcomponents include interannual variability adjusted for the length of the reference dataset, reference data consistency, method consistency, gap-fillng, measured data representativeness, and wind distribution fit (if applicable); an uncertainty reduction due to ensemble usage is also considered.

• Flow Modeling and Horizontal Extrapolation

Because nonlinear models have been increasingly used in places with complex terrain (i.e., RANS solvers for forests or steep hills) and over areas where mesoscale effects are not captured by the EWA methodology (e.g., mesoscale models for parts of the central USA), industrial wind modellers in the IEC 61400-15 working group generalize HE and flow modeling together. That is, a flow model whose domain covers both the measurement and windfarm site(s) in effect also does horizontal extrapolation, in addition to treating the effects of orography, roughness changes, and sometimes stability (for mesoscale models and some RANS).

For horizontal extrapolation, the so-called ‘Clerc-model’ [128] was made for EWA-type methodologies; it takes the IBZ flow model, roughness-change model, and the GDL together. It quantifies uncertainty based on the amount of flow correction (‘speedup’), along with the distance of horizontal extrapolation; it has been calibrated and used by several companies for more than a decade. However, in the work of [129] one sees that the HE uncertainty can depend upon height beyond the amount of speedup due to terrain complexity (basically $z/z_{0,\mathrm{eff}}$ from [30]). For this reason, and due to other complicating factors that are not included in the Clerc [128] model, the 61400-15 HE subgroup created a model to quantify HE uncertainty that incorporates additional factors. These include accuracy of the terrain map coordinates, elevation and roughness data (including forest info), accuracy of input stability conditions required for a site, degree of site complexity (slopes, forestry, stability), limitations of model physics (flow separation/turbulence, stability, thermally driven flow), suitability of computational resolution and domain size, and demonstrated accuracy of previous application of the model under similar conditions; for each factor, a value is assigned based on whether it is considered to be of low, moderate, or high uncertainty. The HE UQ model also adjusts for cross-prediction errors measured across multiple masts, and includes extrapolation distance as well as amount of correction (speedup and turning). Arising from evaluation of many pre- and post-construction sites, it essentially acts as a ‘pre-validated’ HE UQ model, and was calibrated and checked against additional sites from the working group. It allows both EWA/WAsP-like calculations as well as RANS solvers.

• Vertical Extrapolation

The VE uncertainty was derived analytically for shear-exponent extrapolation, and calibrated with roughly 100 sites, as reported in [130]. This applies to the application of shear-extrapolation per every 10-min period, which has become a nearly standard way of using the power-law [131]. The VE UQ propagates measurement uncertainty, along with amplifying it depending on the shear exponent and prediction and measurement heights, with adjustment for terrain complexity and low-shear (e.g. offshore) conditions.

An UQ form for the EWA extrapolation is not included, as the only existing form of [132] lacks $z_0$ dependence which may lead to systematic errors; a more complicated form including roughness has been derived, but needs re-validation due to improvements that have been made since its creation (updates that include, e.g., use of re-analysis data to improve the stability [55], complicating the WAsP VE).

• Year-to-Year Project Variability

The yearly project evaluation period availability uncertainty is calculated in order to be able to provide estimates of $P_{90}$ or $P_{75}$ over different time horizons, say 10 or 20 years. This includes interannual variability and expected wind modification due to climate change; the latter is typically less than 1% and smaller than the IAV component [133], with the change depending on geography and a likely weakening of resources in the northern hemisphere [134].

For a given project evaluation period in years ($N_{op}$), the variability contributes a magnitude of $({\sigma }_{U1yr}/U_{\mathrm{long}-\mathrm{term}})/N_{op}^{1/2}$ to the total wind resource uncertainty.

• Wakes and Blockage

Under the plant performance uncertainty component, there is one wind-based subcomponent, namely wakes and blockage/windfarm–atmosphere interaction effects. This is expressed as a percentage of wind farm energy yield; it includes subcomponents of internal, external, and future wakes, as well as internal, external, and future blockage effects. It is essentially proportional to the estimated losses, and also includes uncertainty reduction as a result of having more validation cases.

5.1.2. Combination of Uncertainty Components

The combination of wind uncertainty components is most generally calculated via

$${\tilde{\sigma }}_{\mathrm{total}}=\sqrt{{\sum }_i\left[{\tilde{\sigma }}_i^2+{\sum }_{j\ne i}\left({\rho }_{ij}{\tilde{\sigma }}_i{\tilde{\sigma }}_j\right)\right]}$$

(7)

where the tilde indicates normalized quantities (components as percent/100), and ${\rho }_{ij}$ are the correlation coefficients between pairs of uncertainty components ${\sigma }_i$ and ${\sigma }_j$; this is in line with GUM [91]. The IEC 61400-15-2 group consensus is to assume all ${\rho }_{ij}=0$ for simple calculations; however, it is known that some of them can be nonzero (e.g., between HE and VE). The use of multiple masts (measurement locations) is also addressed by the summation for each uncertainty component

$${\tilde{\sigma }}_i^2=\sum _m\left[{\left(w_m{\tilde{\sigma }}_{i,m}\right)}^2+\sum _{n>m}\left(c_{mn}{w}_m{w}_n{\tilde{\sigma }}_{i,m}{\tilde{\sigma }}_{i,n}\right)\right]$$

(8)

where ${\tilde{\sigma }}_{i,m}$ is the $i^{\mathrm{th}}$ normalized wind uncertainty component (as a fraction of wind speed) for mast number m, $w_m$ is the fraction of plant energy represented per mast m, and $c_{mn}$ are the cross-mast correlation coefficients. Thus, for $c_{mn}<1$, different masts add more information and thus act to reduce the uncertainty for components that are decorrelated across a mast pair. Such decorrelations may arise for LTC, measurements, VE, and possibly others.

5.1.3. From Wind to Energy

Energy production, particularly in wind farms, tends to be a complicated nonlinear function of measured wind speed statistics; one can most ‘simply’ begin by considering annual energy production (AEP) as a function of inflow wind speed, denoted as $E\left(U\right)$. However, we need to convert dimensionless (normalized) wind speed uncertainties ${\tilde{\sigma }}_i$ into dimensionless energy uncertainties, in order to combine them with energy-based uncertainties and find the overall uncertainty in pre-construction yield (or e.g., $P_{75}$). In essence we need to find $\partial E/\partial U$ in order to do so. Near the calculated expected value of AEP $\langle E\rangle$, one may simply approximate this derivative by re-calculating the AEP for values of input speed above and below what has been used in the original calculation. This can be achieved directly by simply inflating and deflating the input wind data by scaling them with $(1\pm {\tilde{\sigma }}_{U,\mathrm{tot}})$ and finite differencing. A slightly more sophisticated method arises by doing Monte Carlo ‘simulation’ of the AEP calculation, using a Gaussian distribution of multipliers centered around 1 and with a width of ${\tilde{\sigma }}_{U,\mathrm{tot}}$; this is equivalent to using the nominal (say $P_{50}$) model input speeds and perturbing them by a random amount taken from a Gaussian distribution with dimensioned standard deviation ${\sigma }_{U,\mathrm{tot}}$ (without the tilde, in m s⁻¹). Such Monte Carlo usage allows for an asymmetric output uncertainty distribution, and might be seen as a ‘cheap’ way to practically supersede the issue of dealing with many variable dimensions in the modeling—but only if the propagation of measurement uncertainty is the dominant contribution; we note that this is just an estimate, and if model uncertainties unrelated to the wind measurement uncertainty are significant, this approximation must be adapted to include such additional uncertainties.

5.2. Forecasting

Yan et al. [135] reviewed uncertainties in short-term forecasting, i.e., prediction of power at horizons from less than an hour to a day or more into the future; these are used within the context of energy trading and adjusting production based on demand and cost. As [135] outlined, short-term operational forecasts are made more often by machine-learning methods than by meteorological models, though short-term meteorological ensembles can offer some degree of uncertainty estimation in the through their ensemble spread [136] rather than through detailed sensitivity and model uncertainty analysis.

5.3. Wind Atlases and Assessments Without Measurements

Intended for wind resource ‘prospecting’, datasets such as the Global Wind Atlas [85] and New European Wind Atlas [86] were not designed for bankable resource assessment, but rather for scoping out potential areas of development. However, it is likely that a (currently) small fraction of sites can be served solely by wind atlas or mesoscale data, albeit with larger uncertainty than would be afforded by conventional measurement-driven WRA following best practices. Currently such atlases do not contain uncertainty, but this is part of ongoing research.

5.4. Siting, Design, and Standards

Second-order moments, such as the variance of turbulent fluctuations, require longer averaging times (more samples) compared to simple first moments (mean values) to achieve the same sampling uncertainty [137]. Thus, uncertainties related to turbulence intensity tend to be higher than wind speed; this is also because turbulence is more variable and sensitive to local terrain and stability effects. [6] found the quantities with the greatest effect on turbine fatigue loads to be mean wind speed, turbulence intensity, shear exponent, and turbulence length scale, in that order. These are also key parameters for siting, along with the site-specific extreme (expected 50-year maximum) wind speed $V_{50}$. Generally RANS does not (yet) bankably predict turbulence intensity in addition to wind speed, and mesoscale or re-analysis data do not directly apply, in contrast with their mean wind speed and direction output. Further, neither VE nor HE of turbulence has a standard practice due to the above-stated reasons, and, subsequently, UQ of turbulence is difficult. The MEASNET guideline [138] addresses site parameters, as does the 61400-15-1 [139], giving a framework for reporting these parameters in relation to uncertainty characterization.

5.5. Distinction from Risk

The ultimate aim with UQ is to estimate uncertainty distribution widths around predicted mean or median quantities such as wind speed or AEP, based on distributions of all the inputs to WRA and EYA modeling. The general use of input and component uncertainties can depend on how variables are used, as well as how they are determined. As der Kiureghian and Ditlevsen wrote in their seminal work on quantifying different types of uncertainty [140], “the nature of uncertainties and how one deals with them depends on the context and application.” For example, roughness length ($z_0$) has a log-normal uncertainty distribution and multiplicative uncertainty of nearly $\times /÷$3, which is seemingly quite large. But this variable always occurs as $ln\left(z_0\right)$, which tends to be normally distributed and with corresponding uncertainty (standard deviation) from measurements of about 5% or less; however, $ln\left(z_0\right)$ assigned by wind engineers based on their experience can be ∼10–50% or more [141].

The UQ prescription by the IEC work group for the emerging 61400-15-2 standard, which includes a worksheet for calculations, mostly involves assignment of situation- or condition-based subcomponent uncertainty values, which are combined assuming independence of components as mentioned above in Section 5.1.2. However, along with the directly propagated uncertainties in the VE component, these were created to reflect underlying distributions without adding perceived risk as a bias, or inflation of standard deviations. In previous years, uncertainty estimation did not necessarily reflect this practice, as conservative pre-construction uncertainty estimates were sometimes used, with different consultants and companies having different assumptions and methods. An example of pre-construction uncertainty versus operational performance is shown in Figure 4 for more than 100 projects.

The data were obtained from two large companies, with two UQ practices, from a decade ago (2005–2014). These practices vary across the industry, possibly with greater spread than shown in Figure 4. One can see some evidence of conservatism for projects with uncertainties higher than about 7%, while the conditional standard deviation for those lower than 7% (using overlapping bins of width 1.5% every 0.8%) roughly matches the 1:1 line. For the projects with higher perceived uncertainty, the AEP error did not really follow pre-construction uncertainty estimates; nor did the mean bias. Overall, the projects underperformed by about 1.6%, as shown by the thick dashed line in Figure 4. It is worth noting that [143] showed that mean AEP overprediction has decreased over time, and that the numbers shown in the plot would likely be different if calculated today for the same projects and by the same companies. Other firms would likely provide different results, but with a sufficient standard practice, the expectation is that the dotted lines would overlap the green lines in Figure 4, while the thick dashed line (overprediction bias) would be at 0.

6. Summary

In this paper we have taken a summarizing tour from the micrometeorology applied during the first generation of the wind energy industry, to uncertainty quantification in the next generation. The exceptional problem of wind resource assessment—prediction in a typically turbulent environment that is affected by many more variables than commonly measured in industrial practice—is outlined, starting with the nondimensional space that describes flow in the atmospheric boundary layer. An explanation of fundamental WRA methodology is given, including the basic equations explaining its model chain. We see how the classic measurement-driven EWA methodology greatly simplified the variable space that needs to be covered, which was sufficient for the first few decades of wind turbines residing mostly inside the atmospheric surface layer. Consequences of needing to model the resources for taller turbines, and of using more complex models such as CFD and NWP to do so, are outlined. Modeling more physical phenomena requires knowledge of more atmospheric quantities and makes validation more difficult, even if it increases accuracy. The benefits and implications of long-term averages and data splitting are examined in relation to the concept of sufficient sampling. We see that older climatological-mean formulations were designed to produce long-term statistics, while use of more complex modeling requires some expertise and potentially extra consideration in order to provide representative statistics which reap the benefits of added model physics. This leads to uncertainty quantification, which is outlined in brief following the multi-industry standard GUM [91]. The limitations of first-order UQ are examined, and the commonly used Monte Carlo method is introduced in the context of WRA and EYA. Current wind industry practices and findings in UQ related to resource and energy prediction are noted, along with the context of these and wind prediction within the industry overall, and the emerging IEC standard is outlined as well. Examples and reminders of the difference between risk and uncertainty are considered, motivating the need for and emergence of a standard.

Future needs and work involves standardized reporting of the WRA and EYA methods used in wind projects, along with the UQ calculations associated with these. Further development of uncertainty (sub-)component models, based more directly on the WRA/EYA models used and able to propagate input uncertainties, is also required. Sharing of data across industry and academia is a lingering need; this will allow progress to be made in this area that will ultimately reduce project finance costs. Ongoing study of the financial value of uncertainty estimates related to different parts of the combined wind energy electrical market system is also beneficial.