222Corresponding author email<EMAIL_ADDRESS>(JM)

# Estimating the technical wind energy potential of Kansas that incorporates
the atmospheric response for policy applications

Jonathan Minz111Corresponding author email<EMAIL_ADDRESS>Biospheric
Theory and Modelling Group, Max Planck Institute of Biogeochemistry,Jena,
Germany  Axel Kleidon  Biospheric Theory and Modelling Group, Max Planck
Institute of Biogeochemistry,Jena, Germany  Nsilulu T. Mbungu  Department of
Electrical Engineering, Tshwane University of Technology, Pretoria, South
Africa  Department of Electrical Engineering, University of Sharjah, Sharjah,
United Arab Emirates  Lee M. Miller  Pacific Northwest National Laboratory,
Richland, Washington, United States of America

###### Abstract

Energy scenarios and transition pathways need estimates of technical wind
energy potentials. However, the standard policy-side approach uses observed
wind speeds, thereby neglecting the effects of kinetic energy (KE) removal by
the wind turbines that depletes the regional wind resource, lowers wind
speeds, and reduces capacity factors. The standard approach therefore
significantly overestimates the wind resource potential relative to estimates
using numerical models of the atmosphere with interactive wind farm
parameterizations. Here, we test the extent to which these effects of KE
removal can be accounted for by our KE Budget of the Atmosphere (KEBA)
approach over Kansas in the central US, a region with a high wind energy
resource. We find that KEBA reproduces the simulated estimates within 10 -
11%, which are 30 - 50% lower than estimates using the standard approach. We
also evaluate important differences in the depletion of the wind resource
between daytime and nighttime conditions, which are due to effects of
stability. Our results indicate that the KEBA approach is a simple yet
adequate approach to evaluating regional-scale wind resource potentials, and
that resource depletion effects need to be accounted for at such scales in
policy applications.

###### keywords:

Wind resource estimation, technical wind energy potentials, regional limits,
Kinetic Energy Budget of the Atmosphere, wind speeds, boundary layer,
stability

##  1 Introduction

Estimates of technical wind energy potential are important for the design of
energy transition pathways towards a future sustainable energy system [1, 2,
3, 4]. These are theoretical estimates of energy generation from hypothetical
wind turbine deployments over large regions available for wind energy
development. The standard approach to estimating them is to force a single
wind turbine’s power curve with observed or modelled wind speeds at the hub-
height of the turbine and then multiply this by the total number of wind
turbines within a deployment area [5, 6, 7, 8, 9, 10]. Higher wind speeds,
higher turbine densities, and larger turbine rated capacities are expected to
lead to a higher technical potential[11]. Attaining these technical wind
resource potentials is underpinned by the implicit assumption that the removal
of kinetic energy (KE) by the wind turbines does not affect wind speeds over
the deployment region. Single factor adjustments of 10 - 20% are used to
correct for inter-turbine interactions within individual wind farms. This
potential divided by the installed capacity of the deployment is used to
calculate a metric of efficiency known as the capacity factor. This is used in
the estimation of the economic cost of large scale wind turbine
deployments[12, 13]. Technical potentials and capacity factors are the most
significant control on the cost of wind energy [12, 13]. Thus, accurate
estimates of technical potential and capacity factors are critical for the
design of robust energy transition pathways.

However, simulations with numerical models of the atmosphere show that the
standard approach significantly overestimates technical potential and capacity
factors when wind energy is intensively used at large scales [14, 15, 16, 17,
18, 19]. This overestimation results from the standard approach implicitly
neglecting the wind speed reduction by the regional-scale extraction of KE by
the wind turbines. To understand this reduction effect on regional scales, we
need to look at how kinetic energy is generated and transported towards the
surface before it can be extracted by wind turbines.

The winds of the large-scale circulation and their associated kinetic energy
are predominantly generated in the free atmosphere by differences in potential
energy due to differential solar radiative heating [20, 21]. This KE is
transported vertically downwards into the boundary layer, the lowest layer of
the atmosphere where most of the KE is dissipated [22]. The turbines extract
some of the KE which would otherwise have been dissipated by surface friction.
Since the rate at which KE is transported into the boundary layer is limited,
it leads to a fixed KE budget being available for driving movement within the
boundary layer[18]. This means that the extraction of KE by a large number of
wind turbines leads to less KE being available for the motion of the winds. As
a result, larger rates of KE extraction from a fixed KE budget causes slower
winds and reduced capacity factors [23]. This is supported by numerical
weather model simulations, which show that mean electricity generation
potentials from onshore deployments greater than 100 km2 are limited to yields
of about 1.1 W m-2 of surface area [14, 16, 19, 24, 25, 26, 27, 17], which
contrast with standard estimates ranging from 2-6 W m-2[28, 19, 7, 6, 29, 30].
At the maximum generation potential, wind speeds are estimated to slow by 42%,
while capacity factors reduce by $\sim$ 50% relative to the standard estimate
[15, 17]. A mean of 1.1 W m-2 implies electricity generation of $\sim$ 900 -
1900 TWh yr-1 if all the available area for wind energy in a windy area like
Kansas (100,000 - 200,000 km-2) is covered with wind turbines. These
generation potentials are about a third lower than the standard expectation of
2000 - 3000 TWh yr-1 [31, 32]. Thus, the policy-side approach to technical
potential estimation needs to incorporate the effects of wind speed reductions
arising from limitations imposed by the atmospheric KE budgets.

A simple yet physical approach to deriving technical potential estimates that
includes the effects of KE removal on wind speeds is to constrain the wind
speeds and turbine yields with an explicitly defined KE budget of the
atmospheric boundary layer. In this approach, known as the Kinetic Energy
Budget of the Atmosphere (KEBA, Kleidon and Miller [18]), first the budget
available to the deployment is estimated from the sum of the vertical and
horizontal KE fluxes over the deployment. The vertical component represents
the KE input into the boundary layer from the free atmosphere while the
horizontal component represents the boundary layer wind flow. Both rates can
be estimated from wind speed observations, but also depend on boundary layer
height and surface friction. The reduction in wind speeds is estimated by
accounting for the removal of KE from the budget. The slower wind speeds are
then used to estimate turbine yields[18]. This approach has previously been
shown to compare well against numerical weather forecasting simulations of
wind turbine deployments in idealized onshore weather conditions[18] and in
real weather conditions in offshore areas in the German Bight of the North Sea
[33]. The goal here is to test this approach further in a realistic onshore
region to understand the limits of its application for it to be used in
policy-side resource estimation.

Figure 1: (a) Distribution of wind speeds averaged over a prospective
deployment area in Kansas, central US, for daytime (red) and nighttime (blue)
in the absence of wind turbines. (b) The power curve for a Vestas V112 3MW
wind turbine used in this study. (c) Deployment yields during daytime (red)
and nighttime (blue) for the 5 MW km-2 scenario from an interactive WRF
simulation (solid, "WRF"), and using the standard approach (dashed outlines).
Data taken from [15].

In this study, we will use this approach to evaluate a seemingly counter-
intuitive result (Fig. 1) reported by previous numerical simulations of
realistic large-scale wind turbine deployment scenarios in Kansas, central US,
under realistic weather conditions [15]. These simulations showed that wind
speeds are typically 40% lower during the day than at night (Fig 1a), but
overall daytime yields were about 50% higher than nighttime (Fig 1c). This is
an important result to evaluate with KEBA since the standard approach would
estimate the opposite, higher nighttime yields due to higher wind speeds. The
result can be understood when one accounts for the effect of lower boundary
layer heights and reduced mixing at night, which reduces the size of the
kinetic energy budget (Fig. 2). As a result the KE removal by the wind
turbines has a stronger effect on wind speed reductions at night [34, 35].
During the day, because the KE budget is larger, this depletion effect is
proportionally smaller. Solar insolation drives vertical convection and the
vertical growth of the boundary layer, resulting in higher downward
replenishment of KE from the free atmosphere and a larger reservoir of KE in
the boundary layer. The absence of solar-driven convection at night leads to
stratified or stable conditions that restricts vertical KE replenishment. This
leads to greater reduction in wind speeds at night compared to the the day,
and therefore lower yields, despite higher incoming, undisturbed wind speeds.
Thus, the differences in boundary layer characteristics during day and night
will affect wind resource potentials of regional deployments of wind turbines.

Figure 2: Differences between daytime (left) and nighttime (right) boundary
layer conditions. Gray downward arrows represent the downward flux of kinetic
energy from the free atmosphere into the boundary layer. The red and blue
arrows represent the horizontal kinetic energy in- and outflow (from left to
right) through the boundary layer volume bounding the regional scale wind
turbine deployment (dotted box). The free atmosphere represents the part of
the atmosphere in which large-scale motion is generated in the absence of
friction.

Our goal in this paper is twofold: (1) to evaluate the effect just described
to demonstrate the importance of a broader accounting of the atmosphere in
resource potential estimation, and (2) to quantify the role of KE removal in
shaping wind resource potentials at the regional scale. To accomplish our
first goal, we will explicitly evaluate this difference between day and night
by evaluating separate daytime and nighttime budgets in the KEBA approach.
This is accomplished by prescribing different boundary layer heights in KEBA,
which changes the size of the KE resource. Note that the effect of reduced
turbulent mixing within the boundary layer is not accounted for. To accomplish
our second goal, we use the scenarios of the previously published study for
Kansas [15]. We use the wind speeds of their control simulation without
deployment to evaluate wind resource potentials using the standard approach as
well as KEBA, and compare these to the estimates derived from the interactive
model simulations.

In the following section, we provide a brief description of the KE budget
approach, the turbine deployment scenarios, and the model parameters used. We
then present the KE budgets diagnosed from the simulations for the different
scenarios, the reductions in wind speeds with greater wind energy use, and
describe the effects on yield estimates and capacity factors. We also evaluate
the importance of accounting for different boundary layer heights for the
estimates. After a brief account of limitations, we then re-evaluate the wind
resource potential and compare it to previously-published estimates [32, 31].
We close with a discussion and conclusions.

##  2 Methods

We use the wind speeds, scenarions, and yield estimates from Miller et al.’s
WRF simulations [15]. We use these simulations as the reference in which the
effects of wind turbines on the atmosphere are fully accounted for, and refer
to the yield estimate as the "WRF" estimate. The simulations were performed
with the WRF-ARW v3.3.1 regional weather forecasting model [36] to simulate
different levels of hypothetical deployments of wind turbines over 112 $\cdot$
103 km2 in Kansas (Central US) using atmospheric conditions from May 15 to
September 30, 2001. This period is considered to be climatologically
representative for this region [37]. The deployment area is similar to that
assumed in previous resource evaluations [32, 31]. Wind turbines are
parameterised as elevated momentum sinks and sources of additional Turbulent
Kinetic Energy (TKE) [38]. The large, idealized deployments simulated a range
of installed turbine capacity densities from 0.3125 to 100 MW km-2 which were
equally distributed within the expansive wind farm area. Here we restrict the
comparison to a maximum installed capacity density of 10 MW km-2, yielding a
total installed capacity of 35 GW to 1.1 TW over the region. The turbine
characteristics and wind park scenarios, as well as the symbols used in the
following, are summarized in Tables 1 and 2.

Table 1: Turbine characteristics of a Vestas V112 3 MW turbine, as in [15]. Description | Symbol | Value | Units
---|---|---|---
Hub-height | $H_{hub}$ | 84 | m
Rotor diameter | $D$ | 112 | m
Rotor area | $A_{rotor}$ | 9852 | m2
Rated power | $P_{el,max}$ | 3.075 | MW
Cut-in wind speed | $v_{min}$ | 3 | m s-1
Rated wind speed | $v_{rated}$ | 11.5 | m -1
Cut-out wind speed | $v_{max}$ | 25 | m s-1
Power coefficient (max.) | $\eta_{max}$ | 0.42 | -
Table 2: Scenarios of large-scale deployment of wind turbines in Kansas, Central US, evaluated here. Based on [15]. Description | Symbol | Value | Units
---|---|---|---
Width | $W$ | $360\cdot 10^{3}$ | m
Length | $L$ | $312\cdot 10^{3}$ | m
Capacity density | - | $0.3125-10$ | MW km-2
Number of turbines | $N$ |  $11.7\times 10^{3}$ \- $3.7\times 10^{5}$ | -
Deployment area | $A_{farm}$ | $1.12\times 10^{11}$ | m2

The "standard" and "KEBA" yield, or electricity, estimates were then
calculated using hourly time series of wind speeds, $v_{in}$, from the WRF
Control simulation, i.e., without any wind turbines present. The "standard"
estimate replicates the standard approach used in existing policy side
evaluations. It is based on the power curve of the turbine, the number of
turbines in the scenario, and the wind speeds, $v_{in}$. The electricity yield
of the standard approach, $P_{el,std}$, is estimated from the turbine’s power
curve (Fig. 1b) by

$P_{el,std}=N\cdot\min(P_{el,max},\dfrac{\rho}{2}\cdot\eta_{max}\cdot
A_{rotor}\cdot v_{in}^{3})$ (1)

where $N$ is the number of turbines, $P_{el,max}$ is the rated capacity of the
turbine, $\rho$ is the air density (we used $\rho=1.2$ kg m-3), $\eta_{max}$
is the maximum power coefficient, and $A_{rotor}$ is the rotor-swept area, and
$v_{in}$ is the wind speed from the WRF Control simulation (Table 1). This
estimate assumes that the effects of the KE removal by the wind turbines is
fully compensated for by the inter-turbine spacing, which allows wind speeds
and capacity factors to be unaffected by presence of a wind turbine deployment
in the region.

The "KEBA" estimate is derived from the KEBA model [18] augmented with
information about day- and nighttime boundary layer heights derived from the
WRF simulations. The budgeting of the KE fluxes of the boundary layer over the
deployment region results in a reduction factor $f_{red}$. This factor
encapsulates the effect of KE removal from the wind by the turbines on wind
speeds and turbine yields. In KEBA, the wake loss term is fixed as half of the
deployment yield after the work of Corten[39]. First, $f_{red}$ is applied to
the WRF control wind speeds ($v_{in}$) to quantify the reduction in the wind
speeds ($v_{eff}$).

$v_{eff}=f_{red}^{\frac{1}{3}}\cdot v_{in}$ (2)

Then, $v_{eff}$ is applied to the standard estimate (Equation 1) instead of
$v_{in}$ to derive the KEBA estimate of deployment yields ($P_{el,keba}$).
This results in the following expression for deployment yield (see [18] for
detailed derivation)

$P_{el,keba}=N\cdot\min(P_{el,max},f_{red}\cdot\dfrac{\rho}{2}\cdot\eta_{max}\cdot
A_{rotor}\cdot v_{in}^{3})$ (3)

The reduction factor $f_{red}$ is represented by

$f_{red}=\dfrac{H+2C_{d}\cdot L}{H+2C_{d}\cdot
L+\frac{3}{2}\cdot\frac{N}{W}\cdot\eta_{max}\cdot A_{rotor}}$ (4)

for wind speeds $v_{in}$ above the cut-in velocity $v_{min}$ and below the
rated velocity $v_{rated}$ when the turbine output is proportional to the
incoming wind speeds (2b); and

$f_{red}=1-\dfrac{3}{2}\cdot\dfrac{1}{H+2C_{d}\cdot
L}\cdot\dfrac{H}{L}\cdot\dfrac{N\cdot P_{el,max}}{J_{in,h}}$ (5)

for $v_{in}$ greater than the rated velocity $v_{rated}$, but below the cut-
out velocity, $v_{max}$. For this case, $f_{red}$ is computed only to simulate
the effect of wind speed reduction for comparison. Note that the case of
$f_{red}=1$, KEBA represents the standard approach.

In these equations for $f_{red}$, $H$ is the height of the boundary layer
(Table 3), $C_{d}$ is the aerodynamic drag coefficient of the surface (Table
3), $L$ and $W$ the length and width of the deployment (Table 2), and
$J_{in,h}$ the horizontal kinetic energy flux in the boundary layer
($\rho/2\cdot v_{in}^{3}\cdot WH$). The values for daytime and nighttime mean
boundary layer heights are provided in Table 3. They were derived by
comparison of the vertical velocity profiles of the WRF simulations with and
without the wind turbine deployment, yielding mean values of about 2000m (day)
and 900m (night) (see Supplementary Material).

The kinetic energy budgets for the different scenarios are diagnosed from the
time series of the velocity $v_{in}$ and $f_{red}$ and then averaged, with the
different terms estimated as in Kleidon and Miller [18]. The budget is defined
for the boundary layer air volume enclosing the deployment of wind turbines,
given by the dimensions $W$ and $L$ (Table 2), as well as the height of the
boundary layer $H$ (Table 3). The magnitude of the budget is set by the influx
of kinetic energy, which is determined by the horizontal
($J_{in,h}=WH\cdot\dfrac{\rho}{2}v_{in}^{3}$) and vertical
($J_{in,v}=WL\cdot\rho C_{d}v_{in}^{3}$) influxes of kinetic energy into the
volume. This energy is then either dissipated by surface friction, used for
electricity generation, dissipated by wake turbulence, or exported downwind.

Table 3: Atmospheric and environmental specifications needed for the KEBA estimate. Description | Symbol | Value | Units | Comments
---|---|---|---|---
Boundary layer height-Day | $H_{day}$ | 2000 | m | Mean, fixed
Boundary layer height-Night | $H_{night}$ | 900 | m | Mean, fixed
Drag coefficient | $C_{d}$ | 0.001 | - | Mean, fixed

##  3 Results & Discussion

We posited that the KE budget is central to capturing the reduction in wind
speeds with increased installed capacity to understand the difference between
daytime and nighttime yields as shown in Fig. 1, and quantifying the resulting
technical potential. Therefore, we start by showing the KE budgets during day
and night for the different scenarios, the regional reduction in wind speeds,
before we describe the estimated yields and capacity factors. We then perform
a sensitivity analysis to boundary layer height to evaluate the effects of the
day-night differences and compare these to the general effect of reduced wind
speeds with greater installed capacities. We end this section with a
discussion on the limitations, the resulting resource potential estimate, and
the broader implicatons.

###  3.1 Kinetic Energy budgets

The KE budget of the boundary layer volume enclosing the deployment is central
to KEBA estimates, with the magnitude of the budget defining the wind speed
reductions and limiting deployment yields. The horizontal influx accounts for
a larger share of the KE budget than the vertical input: 76% during daytime
and 60% during nighttime. The combination of lower daytime wind speeds
($v_{day,mean}$ = 6.8 $m\>s^{-1}$) and higher boundary heights (Hday = 2000m)
and higher nighttime wind speeds ($v_{night,mean}$ = 9.5 m s-1) and lower
boundary layer heights (Hnight = 900m) lead to similar influxes of kinetic
energy of about 150 GW in the mean. The 150 GW budget sets the overall
magnitude of the bars in Figure 3(a), with the distribution among the
different terms changing due to the different deployment scenarios.

Within the boundary layer volume, KEBA determines the partitioning between the
KE influx into frictional dissipation (red), wind turbine yields (dark blue)
and wake losses (light blue), and the downwind export of KE out of the
deployment volume (light red). KE extracted by wind turbines powers
electricity generation ($P_{el,tot}$), with the wakes being dissipated by the
mixing behind the turbines ($D_{wake}$). KE extraction consumes KE that would
have otherwise been dissipated at the surface by friction or exported
downwind. Thus, the increase in capacity density increases yields and wake
losses at the expense of KE in downwind export and surface friction. Since
individual turbine yields depend on wind speeds, higher nighttime mean wind
speeds lead to higher per turbine yield compared to the daytime. Consequently,
about 2% more KE is extracted by the turbines from the budget at night than
during the day (Figure 3(a)).

###  3.2 Wind speeds

The depletion of the KE budget with increased wind turbine deployment is
associated with a reduction in wind speeds. This reduction is shown in Figure
3(b), which shows how the mean wind speed over the deployment region
($v_{eff}$) reduces with the amount of KE extracted by the wind turbines (in W
m-2 of surface area). We chose to use the yield on the x-axis rather than
installed capacity, because in this way, it shows that wind speed reductions
are almost linear with the amount of KE extracted. Figure 3(b) shows these
wind speed reductions for the WRF estimate (red) and the KEBA estimate (blue),
while the standard approach (grey) assumes no change in wind speeds. The rates
of reduction can be quantified by the slope, $m$, of the linear regressions
(dashed lines). Nighttime wind speed reductions ($m_{KEBA,night}=-6.21$) are
almost twice as strong as during the day ($m_{KEBA,day}=-3.89$). These
reduction rates are similar in the WRF estimates
($m_{wrf,night}=-10.53,m_{wrf,day}=-3.15$). Note that despite the faster rate
of reduction in nighttime means, the wind speeds are nevertheless higher in
magnitude than during the daytime. Compared to the WRF simulations, KEBA
slightly overestimates daytime and underestimates nighttime wind speeds. Thus,
the difference in daytime and nighttime wind speed reductions can be directly
linked to the lower boundary layer height used in the nighttime KE budget in
KEBA.

Figure 3: (a) Daytime (left) and nighttime (right) KE budgets, with total
yields (dark blue), wake loss (light blue), frictional dissipation (red) and
the downwind export (light red). (b) Estimates of wind speeds over the
deployment region against the KE extracted by the wind turbines for the
"standard" (grey), "WRF" (red) and "KEBA" (blue) estimates. (c) Wind turbine
yields as a function of installed capacity density using a logarithmic scale
for the "standard" (grey), "WRF" (red) and "KEBA" (blue) estimates. (d)
Capacity factors against the rate of KE extraction for the "standard" (grey),
"WRF" (red) and "KEBA" (blue) estimates. Dashed lines denote linear fits, with
$m$ values providing the slopes obtained from the linear regression.

###  3.3 Deployment yields

Figure 3(c) shows the variation in the wind turbine yields with increasing
installed capacity density. Since KEBA models yields as a function of the
reduced wind speeds ($v_{eff}$) rather than the prescribed Control wind speeds
($v_{in}$), its estimates (blue) are lower than the standard estimates (gray).
KEBA estimates lower additional increments in yields with the increase in
installed capacity during both, day and night. Thus, the diminishing
increments in yields with added turbines can be attributed directly to the
reduced wind speeds shown in Figure 3(b). While KEBA estimates of nighttime
yields are higher than day, WRF estimates of yield (red) are lower at night
than during the day. KEBA captures the trends in yields increments but does
not estimate the lower-than-daytime yields at night. It underestimates WRF’s
mean daytime estimates by 8 - 15% while overestimating nighttime yields by 20
to 75%. The standard estimate overestimates yields by up to 180% during
daytime and up to 600% at night compared to the WRF estimates. The bias in
KEBA estimates of yield compared to WRF can be attributed to higher nightime
KEBA wind speed estimates.

###  3.4 Capacity Factors

The lower increments in yields with increased installed capacity indicates
that more turbines within the deployment region lowers the mean efficiency of
individual turbines. This can be shown by directly looking at the capacity
factors, as displayed in Figure 3(d). Both, KEBA (blue) and WRF (red)
estimates show that increasing KE extraction leads to lower capacity factors.
The standard estimate (grey), however, assumes no change because no reduction
in wind speeds is considered. Again, as in the case for wind speeds, the
capacity factors reduce almost linearly with increasing KE extraction. The
slopes of the linear regression show that turbine efficiencies reduce almost
twice as fast during the night ($m_{KEBA,night}=-.32$) than during the day
($m_{KEBA,day}=-.22$), which is similar to the WRF estimates
($m_{wrf,night}=-.48$) and ($m_{wrf,day}=-.20$). While KEBA, again,
underestimates the strength of the reduction at night, the close match of KEBA
estimates with the WRF estimates highlights that the removal of KE from the
boundary layer is the main effect that results in reduced turbine efficiencies
and wind turbine yields. KEBA is able to capture a large part of this trend
because of the separate definition of day and night KE budgets as opposed to a
single KE budget for the whole day.

###  3.5 Role of diurnal variations in boundary layer height

To evaluate how important the variation in boundary layer height is for
estimating yields between day and night, we performed an additional estimate
with KEBA in which the boundary layer height is fixed to the mean value of
$H=1268$m (as in [15]). This comparison is shown in Figure 4. Although the
KEBA estimate with a single mean boundary layer height represents a
substantial improvement over the standard estimate, it shows a greater
discrepancy to the WRF estimate. Nighttime yields are overestimated by 20 to
107% while daytime yields are underestimated by 12 to 31%. The addition of
diurnal variations in boundary layer height improves the estimates relative to
WRF estimate, reducing the daytime bias to 10 to 17% and nighttime bias to 20
to 60%. The improvement is more pronounced for the nighttime conditions.

Defining different day and nighttime budgets separately is thus an improvement
over neglecting this variation. It captures more of the underlying mechanism
because the daytime solar insolation drives convective motion and higher mean
boundary layer heights. The absence of these motions at night lead to much
lower boundary layer heights. The difference in the amount of mixing between
day and night differentially affects the wind speeds and deployment yields
during day and night [34, 40]. With all other variables in the KEBA model
being fixed, a fixed boundary layer height in KEBA results in a 58% lower
daytime and 30% higher nighttime KE budget compared to a variable boundary
layer height. Although the bias is not entirely compensated for by including
the varying boundary layer heights in the KEBA estimates, this information
clearly reduces the bias in the direction of the WRF estimate. However, the
effect of these diurnal variations at the daily 24 hour scale is relatively
muted. This is because the higher day and lower nighttime generations largely
compensate for each other implying that it is mainly the role of KE removal
that needs to be incorporated in the policy focused estimation of technical
potentials.

Figure 4: Daytime (left) and nighttime (right) total yields estimated by WRF
(red $\bigcirc$), KEBA with (blue $\bigcirc$) and without (blue $\Box$ )
diurnal variations in boundary layer height, and the standard approach (grey
$\bigcirc$).

###  3.6 Limitations

Although KEBA captures day and night trends produced by WRF better than the
standard approach, it is unable to reproduce the lower-than-day, nighttime
wind turbine yields from WRF. This is likely because KEBA assumes a well-mixed
boundary layer volume that is characterized by one effective wind speed,
$v_{eff}$. This assumption is valid during the day when the convective
boundary layer is well mixed. At night, however, stable conditions prevent
vertical mixing because the insolation-driven convective motions are absent.
The intensity of mixing within the boundary layer is thus an additional
control on the rate at which the KE deficit behind wind turbines is
replenished within the boundary layer. The less-mixed nighttime boundary layer
slows the replenishment rate, leading to a steeper decline in wind speeds,
capacity factors and wind turbine yields [34, 40].

This interpretation is supported by observations of velocity deficits, or
wakes, behind operating offshore wind farms that persist longer when the
vertical mixing is lower (55km) than when it is higher (35km)[41, 42]. Longer
wakes during less mixed conditions imply lower downward replenishment than
better mixed conditions, leading to slower recovery of wind speeds.

Reported simulated day- and nighttime mean wind speed reductions of 10 and 30%
[34] from Kansas are similar to the estimates of Miller et al. of 17% and 43%.
WRF estimates for wind turbine yields during day ( 42% lower than standard)
and night ( 73%) are consistent with other simulations of idealized deployment
yields over a full diurnal cycle which found that reductions were twice as
high at night (57%) than daytime (28%)[40]. Thus, it is likely that the
differences between WRF and KEBA could be reduced by accounting for stability
effects, which could be taken up as a part of future work.

Despite this limitation, KEBA represents a significant improvement over the
standard approach, especially with greater installed capacities over the
region. Its nighttime yield estimates are within a factor of 2 of the WRF
estimate. The standard approach overestimates WRF yields by up to 6 times. Our
results highlight the critical role of boundary layer information, in terms of
height and mixing/stability, in determining KE budgets that shape the extent
to which wind speeds, turbine efficiencies and deployment yields are affected
by the removal of KE. Thus, the KEBA estimate appears to be a suitable tool to
evaluate Kansas’s technical wind energy potential.

###  3.7 Reevaluating Kansas’s technical potential

To illustrate the relevance of these KE removal effects, we compare our
estimates to the existing technical potentials for Kansas [32, 31]. This
comparison is summarised in Table 4. Previous studies estimate potentials of
3101 TWh yr-1 and 1877 TWh yr-1 for capacity densities of 5 and 3 MW km-2 over
1.9 $\cdot$ 105 km2 and 1.6 $\cdot$ 105 km2, respectively, which include a
fixed, 15% loss in array efficiency. This results in capacity factors of 37%
and 45%. Expressed in terms of yields, these estimates imply 1.86 W m-2 and
1.36 W m-2 of generated electricity per unit surface area. Multiplied by the
deployment areas, these yield technical potentials of 3101 TWh/a and 1877
TWh/a for Kansas in these previous studies.

Table 4: Comparison of previously published estimates of the technical wind energy potential of Kansas by [32, 31] with the estimates from this study. For the comparison, we used the scenarios with installed capacity densities of 2.5 and 5 MW km-2, which are close to these previous estimates. A 15% array efficiency reduction was applied to the Standard estimate. Lopez et al. (2012) |  | Standard | KEBA | WRF
---|---|---|---|---
Deployment area (km2) | 190 000 |  |  |
Capacity density (MW km-2) | 5 | 5 | 5 | 5
Capacity factor (%) | 37 | 41 | 21 | 19
Yield (W m-2) | 1.86 | 2.03 | 1.05 | 0.95
Technical potential (TWh yr-1) | 3101 | 3379 | 1748 | 1581
Difference (%) |  | +9.0 | -43.6 | -49.0
Brown et al. (2016) |  | Standard | KEBA | WRF
Deployment area (km2) | 157 890 |  |  |
Capacity density (MW km-2) | 3 | 2.5 | 2.5 | 2.5
Capacity factor (%) | 45 | 41 | 31 | 27
Yield (W m-2) | 1.36 | 1.02 | 0.75 | 0.68
Technical potential (TWh yr-1) | 1877 | 1410 | 1037 | 941
Difference (%) |  | -24.9 | -44.8 | -49.9

We first compare our standard estimate to these resource estimates, using our
scenarios with installed capacity densities of 5 and 2.5 MW km-2. For these
scenarios, the yield is on average 2.39 W m-2 and 1.19 W m-2, with a capacity
factor of 48%. We reduce these estimates by the same 15% loss, which reduces
the yields to 2.03 W m-2 and 1.02 W m-2 with a 41% capacity factor. Multiplied
by the deployment areas, these yield technical potentials of 3379 TWh/a and
1410 TWh/a, which are within $\pm 25$% of the published estimates.

KEBA estimates lower yields of 1.05 W m-2 and 0.75 W m-2 for the two
scenarios, with capacity factors reduced to 21% and 31%, respectively. These
reductions compare well with the WRF estimates of 0.95 and 0.68 W m-2 and
capacity factors of 19% and 27% from Miller et al.. Multiplied by the
deployment areas, these yield technical potentials of 1748 TWh/a and 1037
TWh/a using KEBA, and 1581 TWh/a and 941 TWh/a using WRF. These estimates for
the technical potentials are lower by 40-50% due to the reductions in wind
speeds.

Wind speed reductions are thus likely to play a substantial role in lowering
regional-scale technical resource potentials than those that use prescribed
wind speeds. Note that the potential is nevertheless 3 to 5 times the total
energy consumed by the state in 2018 ([43]).

###  3.8 Implications for technical wind energy potential estimation

The reduced technical potentials derived using KEBA are consistent with
previous climate (GCM) and weather modelling (WRF) estimates. This is shown in
Fig 5 in which the variation of technical potential in Kansas is plotted
against the capacity density deployed. KEBA estimates are represented in blue,
standard estimates in black and numerical estimates are shown in red. Broadly,
the black colour represents the exclusion of KE removal effects while blue and
red represent partial (KEBA) and complete inclusion (WRF and GCM),
respectively. The stars represent results from this study while the black
square [32] and circle [31] represent previously published estimates of
Kansas’s potential. The black dotted lines represent the variation in
potential linked to the standard estimates from this study (black stars [15]).
The red stars [15] and pentagons [17] represent previous WRF-based estimates
of Kansas. The dotted red line shows the peak average potential from large
deployments in Central USA as estimated by Adams and Keith [14]. Similarly,
the red circles [19] and squares [16] represent the trends over global land
(26% of global area) derived using GCMs. The blue band highlights the range of
peak average global potentials estimated previously [23, 19, 16, 24, 26, 26,
25]. The red circle with a blue outline represents an observations - based
study of generation from currently operating onshore wind farms in the Central
US[44]. Numerical estimates in Kansas show that beyond 1.5 MW km-2 of deployed
capacity in Kansas leads to 50% lower potential (red) compared to the standard
estimates. Even though potential increases with increasing capacity
deployment, it is not linear as implied by the standard estimates. This sub-
linear increase with installed capacity shows that capacity factors reduce
with the increasing deployment. All the numerically simulated estimates
display similar variation in potential with capacity and culminate near an
average peak of 1.1 W m-2. This variation is also consistent with global
estimates over land, albeit higher, because Kansas is winder than most
places[15]. In line with these estimates, Miller and Keith [44] showed that
the actual yield from an average estimated onshore US capacity density of 2.7
MW km-2 is around 0.90 W m-2. The agreement between estimates from independent
numerical modelling studies and relevant observational data analysis points to
the robustness of this trend. The variation in KEBA estimates (blue) is
consistent with these trends. Although the match between KEBA and the
numerical estimates is not exact, it highlights the significance of the
effects of KE removal effects on technical potential.

Figure 5: Technical potentials per unit surface area plotted against the
capacity density and number of turbines (x-axis). Black symbols represent
standard estimates (No KE removal), while red symbols represent meteorological
estimates (With KE removal). Blue symbols represent the KEBA estimates from
this study. The blue band represents the range of average peak global
potentials [23, 19, 16, 24, 26, 26, 25]. The red dotted line represents the
peak average potential of Kansas [14] while the dotted lines show the assumed
capacity factors without accounting for the removal of KE. Existing estimates
of the Kansas resource potential are shown in black filled symbols[31, 32].The
red circle with the blue outline shows an observation-based estimate [44]

The comparison in Fig 5 shows that the removal of KE is the predominant
physical influence which shapes the technical potential of Kansas. The close
agreement between KEBA, which only accounts for KE removal effects, and the
WRF trends, which includes effects arising from both KE removal and stability,
highlights the the role of the KE removal as the predominant influence on
technical potentials. The remaining difference indicates the secondary role of
stability or the degree of mixing in the boundary layer.

Including just the KE removal effect leads to a significant improvement in
estimates over the standard estimates. Although diurnal variations in
stability lead to variations in wind speeds reductions, yields and capacity
factors during day and night, the effect of these on the estimated technical
potential is marginal over the whole time period. Including additional
boundary layer information into KEBA improves the agreement during day and
night time but is unable to completely capture stability effects.

Nevertheless, KEBA still provides a straightforward physical framework through
which the role of different physical influences on the resource potential, in
this case diurnal variations, can be better understood and quantified.

Figure 6: (a.) Variation in WRF(red $\bigcirc$) KEBA (blue $\square$) and
standard (gray stippled lines) estimates of technical potential,(b.) capacity
factors, and (c.) % change in the Levelised Cost of Energy (LCOE) relative to
the standard LCOE estimate plotted as a function of capacity densities
(bottom) and number of turbines deployed (top).

The reduced technical potentials and capacity factors significantly affect the
economic potential of wind energy. This is commonly considered by evaluating
the economic cost of wind energy using the Levelised Cost of Energy (LCOE)
[13, 12]. We use the estimates from above and plot these in terms of a
relative increase in the LCOE in Fig. 6.

In the standard approach, based on its assumption of constant wind speeds,
standard capacity factors remain constant while technical potential increases
linearly. A doubling of capacity leads to a doubling of the potential. To
estimate LCOE based on standard estimates, we assume that cost of wind energy
is only a function of the number of turbines. Then, the LCOE becomes an
inverse function of the capacity (see supplementary materials for details).
Thus, there is no change in the standard LCOE as capacity factors remain
unchanged (gray stippled line).

In the case of WRF (red circles) and KEBA (blue squares), however, technical
potentials increase sub-linearly (Fig 6a) and the capacity factors reduce (Fig
6b). Each doubling of turbines from the lowest scenario to the 2.5 MW km-2
scenario leads to an average of 70 - 75% stepwise increments in potential
coupled with an average of 11 - 14% stepwise reduction in capacity factors.
Each doubling in capacity beyond this leads to average stepwise increment of
27 - 31% in potential coupled with average reductions in capacity factors of
35 - 40%. Since we assumed that LCOE is only inversely related to capacity
factor, reductions in them lead to increases in LCOE, relative to the standard
LCOE estimate. Thus, KEBA and WRF lead to estimates of LCOE that are on
average 80 - 120% higher than the standard estimates at an installed capacity
density of 5 MW km-2.

This increase in LCOE due to KE removal remains unaccounted for in policy
evaluation because the effect of KE removal on technical potentials and
capacity factors is implicitly neglected in the standard approach[11]. It
should be noted that although we present a simplified illustration, even a
detailed LCOE calculation is likely to show similar trends given that LCOE
values are highly sensitive to variations in capacity factors [12]. Thus, the
reduced potentials arising from KE removal have a significant impact on LCOE
which need to be explicitly evaluated in policy evaluations.

##  4 Conclusion

We conclude that the KE removal effect is the predominant physical influence
that shapes technical wind resource potentials at the regional scale. Although
day- and nighttime boundary layer heights and stability conditions affect the
technical potential, it is the removal of KE from the wind that, primarily,
shapes the reduction in wind speeds and capacity factors. It leads to reduced
potentials compared to the standard approach that have a significant impact on
the economic potential of wind energy at larger scales.

These impacts need to be assessed in policy evaluations of wind energy and the
energy transition. For this KEBA is a viable alternative to the standard
approach because it is simple to implement [18] and accounts for the effect of
the key atmospheric control on technical potentials. This is not to negate the
use of more physically comprehensive, numerical methods like WRF and GCMs in
policy analyses but to enable energy scenario modellers without a background
in meteorology to be able to incorporate the key physics without significantly
increasing their models’ computational complexity. The heavy computational
requirements associated with physically accurate descriptions of the
atmospheric circulations have been reported to inhibit their widespread
incorporation into policy side evaluations [45].

Lastly, despite these detrimental effects at larger deployment scales, KEBA’s
estimates agree with previous research that has shown that wind energy is an
abundant and renewable resource that can be harvested to meet a significant
part of the future energy demand through efficient, large scale deployment of
wind turbines [19, 17].

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## Supplementary Information

###  4.1 Determining boundary layer heights for initialising KEBA

The KEBA model estimates park yield and mean wind speed reduction through the
application of conservation of energy ([18]). The kinetic energy (KE)
generated in the atmospheric boundary layer is balanced by that consumed by
the wind turbines within the wind park and its wake, dissipated at the
surface, and that which powers the remnant wind. The KE conservation is
applied to a hypothetical boundary layer volume which encompasses the wind
turbine deployment and is mathematically represented as
$J_{in,v}+J_{in,h}=P_{el,keba}+P_{wake}+D_{surface}+J_{out,h}$. The left hand
side of this equation describes the horizontal and vertical flux of KE in to
the boundary layer volume while the right hand side describes how this is
partitioned within the volume. The vertical and horizontal KE fluxes into the
volume can be expanded in to $J_{in,v}=WL\cdot\rho C_{d}\cdot v_{in}^{3}$ and
$J_{in,h}=WH.\dfrac{\rho}{2}\cdot v_{in}^{3}$.

Figure S1: Day and night time wind vertical wind speed profiles estimated by
[15] which show that mean day-time boundary layer height is 2000m whereas that
at night is 900m

These expressions show that the KE budget available to the wind turbine
deployment, is dependent on its geometry (cross-wind width $W$ and downwind
length $L$) and the height of the atmospheric boundary layer ($H$). In our
analysis, the geometry of the deployment is fixed, therefore the only control
on the KE budget is the boundary layer height. Changes in boundary layer
height affect the horizontal input of KE flux ($J_{in,h}$). In line with the
general definition of the atmospheric boundary layer as the layer which
responds quickly to changes in surface forcing ([22]), the boundary layer from
a KE perspective can be also defined as a layer, the kinetic energy content of
which responds to changes in surface forcing i.e presence of large wind
turbine deployments. Then the boundary layer height can be understood as the
maximum height, up-till which the effects of kinetic energy removal by the
turbines can be observed. Since the extraction of KE reduces mean wind speeds,
changes in mean wind speeds induced by the turbines can be used estimate this
height.

The mean wind speeds over the region of interest, Kansas in this case, were
extracted by [15] from their WRF simulations. Mean wind speeds were estimated
per vertical model level over Kansas for the time period of simulation. Such
mean wind speed estimates were computed for all WRF simulations i.e those
without wind parks (CTRL) and those with ($0.3125-100MW_{i}\cdot km^{-2}$).
These mean wind speeds from the different models when plotted against model
height (m) highlight the vertical variation of mean wind speeds or the
vertical wind speed profiles. These are plotted for day and night separately
in Figure S1. In both the plots, the vertical wind speed profiles for the CTRL
simulation represent the background circulation in the absence of any wind
turbines and hence represent the undisturbed circulation. Vertical wind speed
profiles for other simulations deviate from the CTRL trend because turbines
extract KE from the wind speeds, thus slowing them down. The larger the number
of turbines within the wind park, the greater is the deviation from the CTRL
or the undisturbed trend. The mean day and night boundary layer heights for
initialising the KEBA model are then those at which the vertical profiles
derived from simulations with wind parks realign themselves with the
undisturbed trend. Using this approach Miller et al 2015, estimated the day-
time boundary layer height to be 2000m and the night-time boundary layer
height to be 900m.

###  4.2 Wind speed reductions

Here we tabulate (Table S1) the mean wind speed data simulated by [15] without
any wind parks or control (CTRL), and with the impact of wind parks with a
range of turbine densities ($0.3125-10MW_{i}\>km^{-2}$) split by day and
night. Along with it we also provide the mean wind speed reductions estimated
by KEBA with different day and night mean boundary layer heights.

Capacity Density Standard Day WRF Day KEBA Day Standard Night WRF Night KEBA
Night $MW_{i}\>km^{-2}$ $m\>s^{-1}$ $m\>s^{-1}$ $m\>s^{-1}$ $m\>s^{-1}$
$m\>s^{-1}$ $m\>s^{-1}$ 0.3125 6.85 6.96 6.85 9.54 8.45 9.22 0.625 6.85 6.86
6.67 9.54 7.86 8.91 1.25 6.85 6.57 6.40 9.54 7.00 8.35 2.5 6.85 6.27 5.96 9.54
6.29 7.49 5.0 6.85 5.65 5.37 9.54 5.39 6.43 10.0 6.85 4.86 4.67 9.54 4.40 5.36

Table S1: This table contains shows wind speed predictions by WRF and KEBA
split by day and night. The column titled "standard" represents the CTRL wind
speeds i.e with out the impact of reduced wind speeds. Since the standard
approach predicts no change to mean wind speeds despite removal of kinetic
energy. Thus day and night winds speeds remain constant and same as the CTRL
wind speeds.

###  4.3 Park yield and capacity factors

This section contains tables containing information about park yields and
capacity factors estimated by [15] (WRF) and by us using the standard approach
and the 2 different implementations of KEBA i.e with a single boundary layer
height (KEBA single) and another with 2 different average heights (KEBA
variable) for day (2000m) and night(900m) for all the capacity density
scenarios simulated considered in (0.3125 - 10 MW km-2). Tables S2 and S3
contain the data split between day and night time, respectively, whereas Table
S4 contains the undifferentiated data.

Capacity Density Number of Turbines WRF Standard KEBA (fixed) KEBA (variable)
WRF Capacity Factor Standard Capacity Factor KEBA (fixed) Capacity Factor KEBA
(variable) Capacity Factor $MW_{i}\>km^{-2}$ - $W_{e}\>m^{-2}$ $W_{e}\>m^{-2}$
$W_{e}\>m^{-2}$ $W_{e}\>m^{-2}$ - - - - 0.3125 11700 0.06 0.06 0.06 0.06 0.20
0.19 0.19 0.18 0.625 23400 0.12 0.12 0.10 0.11 0.19 0.19 0.16 0.17 1.25 46800
0.22 0.24 0.18 0.19 0.17 0.19 0.14 0.15 2.5 93600 0.38 0.49 0.28 0.32 0.15
0.19 0.11 0.13 5.0 187200 0.56 0.97 0.39 0.47 0.11 0.19 0.08 0.09 10.0 374400
0.69 1.94 0.48 0.60 0.07 0.19 0.05 0.06

Table S2: day-time: This table shows all the capacity density scenarios
modelled, associated number of turbines, park yields (WRF) modelled by [15].
It also shows the park yields estimated in this study using the standard
approach, KEBA with a single boundary layer height (KEBA single) and KEBA with
different average bay and night-time boundary layer heights (KEBA variable).
The computed capacity factors, represented as fractions, from all the
approaches are also included.

Capacity Density Number of Turbines WRF Standard KEBA (fixed) KEBA (variable)
WRF Capacity Factor Standard Capacity Factor KEBA (fixed) Capacity Factor KEBA
(variable) Capacity Factor $MW_{i}\>km^{-2}$ - $W_{e}\>m^{-2}$ $W_{e}\>m^{-2}$
$W_{e}\>m^{-2}$ $W_{e}\>m^{-2}$ - - - - 0.3125 11700 0.07 0.09 0.08 0.08 0.22
0.28 0.26 0.26 0.625 23400 0.12 0.18 0.16 0.16 0.20 0.28 0.26 0.25 1.25 46800
0.19 0.35 0.29 0.27 0.16 0.28 0.22 0.22 2.5 93600 0.30 0.71 0.48 0.43 0.12
0.28 0.20 0.17 5.0 187200 0.39 1.42 0.68 0.58 0.08 0.28 0.14 0.12

Table S3:  night-time: This table shows all the capacity density scenarios
modelled, associated number of turbines, park yields (WRF) modelled by [15].
It also shows the park yields estimated in this study using the standard
approach, KEBA with a single boundary layer height (KEBA single) and KEBA with
different average bay and night-time boundary layer heights (KEBA variable).
The computed capacity factors, represented as fractions, from all the
approaches are also included.

Capacity Density Number of Turbines WRF Standard KEBA (fixed) KEBA (variable)
WRF Capacity Factor Standard Capacity Factor KEBA (fixed) Capacity Factor KEBA
(variable) Capacity Factor $MW_{i}\>km^{-2}$ - $W_{e}\>m^{-2}$ $W_{e}\>m^{-2}$
$W_{e}\>m^{-2}$ $W_{e}\>m^{-2}$ - - - - 0.3125 11700 0.13 0.15 0.14 0.14 0.42
0.48 0.45 0.45 0.625 23400 0.24 0.30 0.26 0.26 0.39 0.48 0.42 0.42 1.25 46800
0.41 0.60 0.46 0.47 0.33 0.48 0.37 0.37 5.0 187200 0.95 2.39 1.05 1.05 0.19
0.48 0.21 0.21 10.0 374400 1.10 4.78 1.30 1.28 0.11 0.48 0.13 0.13

Table S4: Undifferentiated: This table shows all the capacity density
scenarios modelled, associated number of turbines, park yields (WRF) modelled
by [15]. It also shows the park yields estimated in this study using the
standard approach, KEBA with a single boundary layer height (KEBA single) and
KEBA with different average bay and night-time boundary layer heights (KEBA
variable). The computed capacity factors from all the approaches is also
included.

###  4.4 Comparison with published numerical weather model-based estimates of
technical wind energy potential

In Fig 5, we have compared KEBA estimates of technical potential from our
analysis with those performed independently by others over comparable regional
and global scales using different numerical modelling approaches. For
comparison in Kansas and Central USA we used the studies performed by Adams
and Keith, Miller et al. and Volker et al. [14, 15, 17]. From Volker et al.,
we only use their estimates for their largest deployment scenario ( 105
$km^{2}$) in Central US. This was the most pertinent case for our analysis.

All three of these studies use a version of WRF to model the wind turbine
yields and parametrize the wind turbines as momentum sinks. This means that
they account for the fact that turbines extract momentum and kinetic energy
from the wind thereby lowering wind speeds.While Miller et al. and Adams and
Keithuse a variation of the Fitch scheme [38], Volker et al. use the extended
wake parameterization or EWP scheme [46]. The main difference between the
Fitch scheme, its variation and the EWP is the while the latter does not
include an explicit term to account for the Turbulent Kinetic Energy (TKE)
generated by the turbine, the former 2 do. The different schemes lead to
differences in the amount of mixing generated withing the boundary layer due
to the turbine action. The Fitch scheme estimates more and the EWP relatively
less, even though their estimates of wind speeds largely agree with each
other[46].

It is important to appreciate these differences because Archer et al.[47]
highlighted two bugs in implementation of the Fitch scheme in WRF versions
prior to v4.2 that affect the Miller et al. study [48].It was shown that the
additional term in the Fitch scheme adds excessive TKE and a coding bug
prevents the TKE from being advected properly. Although preliminary analyses
have shown that the two errors actually compensate for each other giving rise
to TKE estimates that agree with observations [47, 49], it would be useful to
briefly evaluate any potential impacts on our results and conclusions.

First, according to a review by Fischereit et al. the conclusions of neither
of the 3 studies based used in this study,[14, 15, 46] are affected by the
identified bug. Secondly, were these studies affected by the bug or the impact
significant one would have expected a more prominent deviation between the
Fitch based studies and the EWP based study. This is because the EWP schemes
does not use the explicit TKE addition term with which the bug was related.
Instead, it is observed that the different studies exhibit a similar trend of
technical potential with installed capacity that culminates to a peak average
of 1.1 W m-2.

Further, the WRF trends in Kansas and Central US are consistent with previous
studies that estimate global potentials. Relevant estimates of global land
from Jacobson and Archer and Miller and Kleidon are shown in Fig 6. These
estimates and trends have been derived using global circulation models (GCMs).
These are also unaffected by the errors in the Fitch scheme. These trends show
the same variation in potentials as the WRF trends i.e. sub linear increase in
potential beyond 1.5 MW km${}^{-}{2}$ and culmination to a peak global average
range of 0.2 - 0.6 W m-2 [23, 16, 19, 26, 27, 24]. The agreement between all
the independent trends and regional and global scale highlights that the
impact of the errors in the Fitch scheme are unlikely to affect the insights
and conclusions generated from this study.

###  4.5 Technical Potential, Capacity Factors and Levelized Cost of Energy
(LCOE)

Fig 6b shows that as the number of turbines deployed over the hypothetical
wind farm area increases, the removal of kinetic energy (KE) reduces the
capacity factor. This means that with the increasing deployed capacity , each
turbine produces less energy than what it would have, had it been operating in
isolation. The reduction in per turbine efficiency increases with increasing
turbines. When the KE removal is neglected, the capacity factor remains
unchanged (dotted gray line). While the addition of turbines generally
increases the technical potential , the step-wise increments in generation
reduce as the turbine numbers increase6a. The lower increments are driven by
the reductions in capacity factors 6b. The effect of this variation in
capacity factors can be used to investigate their economic impacts using a
standard economic cost metric known as the levelized cost of energy or LCOE
[13]. LCOE is represented by the following formula [13]:

$LCOE_{wind}=\dfrac{\sum_{t=1}^{n}(I_{t}+O\&M_{t}-PTC_{t}-D_{t}+T_{t}+R_{t})\cdot\dfrac{1}{(1+i)^{t}}}{CF\cdot\sum_{t=0}^{t=n-1}P_{t}}$
(6)

In this equation, $I_{t}$ and $O\&M_{t}$ refer to the capital and operations
cost while $PTC_{t}$, $D_{T}$, $T_{t}$ and $R_{t}$ represent the credits,
levies, taxes and royalties, respectively. The term $\dfrac{1}{(1+i)^{t}}$ is
the present value factor which is used to account for the time value of money
with a discount factor, $i$, over the lifespan of a wind farm. $t$ represents
a year within the operational period of a wind farm. $CF$ is the capacity
factor, which in this calculation would be different for different scenarios
for WRF and KEBA but same for the standard approach. Since we are interested
in simply illustrating only the economic impact of reductions in capacity
factors due to KE removal, we can simplify equation 6 such that LCOE is only a
function of capacity factor. For this, we ignore tax related terms and assume
that all costs and installed capacity terms ($P_{t}$) are sunk and installed
at the once at the beginning of the operational life. The time factor also
remains constant for all scenarios. It should be noted that this calculation
is meant only to illustrate that capacity factor reduction arising from KE
removal result in non-trivial increases in LCOE which highlights their
inclusion into the policy design. In reality, turbine installation will occur
over many years and so will the cost investments. A real LCOE calculation
would need specific and quality controlled inputs about the timing and values
of costs, levies and discount rates. With the simplification, equation 6 would
take the following form.

$LCOE_{scenario}=\dfrac{1}{CF_{scenario}}\times constant$ (7)

In 7, the LCOE for each capacity density scenario is inversely related to the
capacity factor. As the cost and installed capacity terms are same for the
standard and the WRF and KEBA approaches, the percent change relative to the
standard approach for each scenario can be calculated. These values for each
of the installed capacity density scenarios are plotted for both WRF and KEBA
estimates. For example, for the 2.5 MW km-2 case the standard approach assumes
a 0.48 capacity factor while KEBA and WRF estimates 0.31 and 0.27. Then, to
estimate the percent change relative to the standard approach the following
approach is used:

$\%\,change\,in\,LCOE_{2.5}=\dfrac{LCOE_{2.5,\,KEBA/WRF}-LCOE_{2.5\,,standard}}{LCOE_{2.5\,,standard}}$
(8)

These values are tabulated in the table below .

Capacity Density Number of Turbines WRF Capacity Factor Standard Capacity
Factor KEBA (variable) Capacity Factor WRF LCOE Change KEBA (variable) LCOE
Change $MW_{i}\>km^{-2}$ - - - - % % 0.3125 11700 0.42 0.48 0.45 14 6 0.625
23400 0.39 0.48 0.42 23 14 1.25 46800 0.33 0.48 0.37 45 30 2.5 93600 0.27 0.48
0.31 77 54 5.0 187200 0.19 0.48 0.21 150 130

Table S5:  Tabulation of capacity factors estimated by KEBA, WRF and the
standard approach along with the estimated change in LCOE (%) due to KE
removal relative to the standard approach.

The change in LCOE is only calculated for the installed capacity range from
0.3125 to 10 MW km-2 because this is the range that is typically assumed in
wind energy policy scenarios. They show that as the capacity factor reduces
the economic cost of wind energy goes up because each of the turbines performs
less efficiently.