Within the southern portion of the North Sea, a new wind farm cluster, the Princess Elisabeth cluster (PE) (red dots in Fig. ), will be built. Although the final installed capacity is still under discussion, ranging from 3.15 to 3.5 GW, only one possible wind farm scenario is considered in this study, corresponding to a newly installed capacity of 3.5 GW with 15 MW turbines with a rotor diameter of D=236 m and a hub height of zh=146 m . While the impact of the new wind farm cluster on all neighboring farms is investigated, the Belgian–Dutch (BE–NL) wind farm cluster, located just downstream of the prevailing southwesterly wind direction, is the primary focus of this study. However, as other nearby wind farms are likely to be affected , they are also considered in the numerical weather prediction model of this study (Fig. , Appendix ). In total, the influence of the Princess Elisabeth wind farm zone on 1409 wind turbines is investigated. The turbines are modeled in the numerical simulations with the appropriate turbine manufacturer curves or approximations thereof . The locations of the existing wind turbines are extracted from , while the locations for the new Princess Elisabeth wind farm are from and . The latter layout results from a preliminary distribution of turbines over the lease area, accounting for a known exclusion zone, without delving into the specifics of micrositing based on sea surface topography.

The simulations are performed for the year 2016 as it is a representative year for the long-term wind climate at 100 m above mean sea level for the study area. This was determined based on the methodology outlined in , based on hourly ERA5 data for the period 1990–2020 (Appendix ). This methodology identifies a representative year by comparing the wind speed and wind direction distributions for a single year with the long-term distributions. Other parameters, such as atmospheric stability and wave conditions, were not explicitly considered and could be included in future work. The goal of using a representative year is so that this study can provide an appropriate basis for long-term representation of the influence of building the new Princess Elisabeth wind farm cluster close to neighboring wind farms.

The simulation results are compared with offshore measurements. For the year 2016, five measurement platforms (yellow diamonds Fig. , Table ) measured wind speed and wind directions at different altitudes and with a temporal resolution of 10 min . These measurements were taken by boom- or platform-mounted anemometers, wind vanes, or a lidar. At the location of Wandelaar (WA) both anemometers and vanes are present, while at Lichteiland-Goeree (LG) both an anemometer and a lidar are present. Table  also shows the uncertainty in the wind speed measurements, including the uncertainty in the calibration, mounting, data acquisition, and local site conditions . For wind direction the measurement uncertainty is not known. Furthermore, Table  indicates the recovery rate at each observational mast, representing the percentage of available measurements relative to the total expected measurements during 2016.

The numerical model used is the Weather Research and Forecasting (WRF) model v4.3 with three nested domains (150 × 150, 190 × 190, and 220 × 190 grid cells) with a horizontal grid spacing of 18 km for the outermost domain, 6 km for the middle domain, and 2 km for the innermost domain (Fig. ) following the recommendation of . The vertical resolution has 80 vertical levels that are stretched, resulting in more levels close to the surface, following the recommendations of . Within the first 150 m there are approximately 15 vertical levels. The model top is located at 1000 Pa. The Global Multi-resolution Terrain Elevation Data (GMTED2010) is imposed for the geographical data with a resolution of 30 arcsec. The initial and lateral boundary conditions come from the ERA5 reanalysis data set provided by the European Centre for Medium-Range Weather Forecasts , which has a 0.25° × 0.25° horizontal grid resolution, 1 h temporal resolution, and 37 pressure levels up to 1 hPa. The physical parameterizations in this study are based on previous studies for similar applications in the same geographical region (Table ) . The parameterizations are a double-moment microphysics scheme , the long- and shortwave Rapid Radiative Transfer Model for General Circulation Models (RRTMG) , the revised MM5 surface layer scheme , the Noah land surface scheme , the Kain–Fritsch cumulus scheme (only applied to the parent domain as the two innermost domains have a convective permitting resolution), and the Mellor–Yamada Nakanishi Ninno (MYNN) planetary boundary layer scheme . The simulations were performed using the atmospheric WRF model in a stand-alone configuration without explicit coupling to an ocean or wave model (e.g., ROMS, SWAN, or WAM). When wind turbines are included, the parameterization of is used with a turbulent kinetic energy generation parameter of 0.25, which currently is the default value used in WRF .

Three simulations for the year 2016 are performed: a first one where no wind farms are included, which is used as a reference simulation; one simulation with only the existing wind farms in 2022 (black dots, Fig. ; see Appendix ); and one simulation with the existing wind farms in 2022 and the new Princess Elisabeth wind farm cluster (black and red dots, Fig. ). The names of these three simulations throughout the paper are referred to as “no WF”, “without PE”, and “with PE”, respectively. The simulations including wind turbines (without PE and with PE) have exactly the same setup as the no-WF setup, grid size, and employed parameterizations except for the additional wind farm parameterization.

Because atmospheric stability has significant effects on the wakes of wind turbines , it is important to investigate how the wake impacts of the new PE wind farm are influenced by different atmospheric stability regimes. The atmospheric stability in this work is defined by the Monin–Obukhov length (L), which categorizes atmospheric stability based on the ratio of turbulence production by mechanical shear to turbulence production by buoyancy associated with the surface layer . The different stability classes based on L are shown in Table . This classification is proposed by and has been utilized in various studies . The Monin–Obukhov length is output in 10 min intervals by WRF and is calculated by the surface layer scheme . A stability classification based on the bulk Richardson number between model levels at 10 and 100 m produced similar results, showing a Pearson correlation of 72 % with the Monin–Obukhov length classification. Consequently, only the Monin–Obukhov length method is applied in the following analyzes.

In this study, we quantify the total wake energy losses and decompose them into internal and external contributions. External wake energy losses are defined as the additional wake-induced energy losses caused by the new PE wind farm cluster on the existing wind farms. Internal wake energy losses, as defined here, represent the wake energy losses experienced by the existing wind farms in the absence of the PE wind farm cluster. It is important to note that this definition differs from the strict definition of internal wake energy losses, which includes only wake effects generated by turbines within the same wind farm. Due to the simulation setup employed in this study, this distinction cannot be made. Consequently, the internal wake energy losses presented here include inter-farm wake effects among the existing wind farms but exclude wake effects originating from the new PE wind farm cluster. Note that the internal wake effects of the Belgian–Dutch cluster virtually exclusively originate from wind turbines within the dense cluster, as other wind farms included in the without-PE simulation are at least 58 km away.

Three different averaging procedures are employed, where Ns is the number of turbines, Nt the number of time steps, and Pi(t) the instantaneous power output of turbines within grid cell i at time t.

Temporal average: P‾i=1Nt∑t=1NtPi(t), i.e., the average over time for turbine-containing grid cells

The absolute internal wake energy loss is defined in Eq. () as the power reduction of the reference power (no WF) associated with the existing wind farms only (without PE). The reference power P‾no WF is obtained by sampling the hub height wind speed from the no-WF WRF simulation at each turbine location and converting it to power using the turbine power curves. The total (internal + external) absolute wake energy loss is defined in Eq. () as the difference between the reference power and the mean power obtained in the presence of both the existing wind farms and the PE cluster (with PE). The absolute external wake energy loss induced by the PE wind farm cluster is defined in Eq. () as the difference between the total and internal losses: 1ΔP‾internal=P‾no WF-P‾without PE,2ΔP‾internal+external=P‾no WF-P‾with PE,3ΔP‾external=ΔP‾internal+external-ΔP‾internal=P‾without PE-P‾with PE.

From the absolute wake energy losses defined in Eqs. ()–(), the corresponding percentage wake energy losses are obtained by normalization with the reference power P‾no WF. The percentage internal wake energy loss is defined in Eq. () as the relative power reduction associated with the existing wind farms only (without PE), normalized by the reference power. The total (internal + external) percentage wake energy loss is defined in Eq. () as the relative power reduction in the presence of both the existing wind farms and the PE cluster (with PE), normalized by the reference power. The percentage external wake energy loss induced by the PE wind farm cluster is defined in Eq. () as the difference between the total and internal percentage losses: 4Lossinternal=100×ΔP‾internalP‾no WF,5Lossinternal+external=100×ΔP‾internal+externalP‾no WF,6Lossexternal=100×ΔP‾externalP‾no WF.

All (absolute or percentage) wake energy losses are computed from the averaged power values defined above, rather than from instantaneous (absolute or percentage) losses at individual time steps. As described earlier, these losses (internal, external, and total) are analyzed using temporal, spatiotemporal, and spatiotemporal conditional averages to further elucidate the drivers of the interactions between turbines and farms. Throughout this paper, the term wake energy losses refers to the percentage definitions given in Eqs. ()–(), unless explicitly stated as absolute wake energy losses, in which case the definitions of Eqs. ()–() are used.

In addition to wake energy loss metrics, annual energy production (AEP) is computed by integrating the power of the turbines over the full simulation period. AEP is calculated separately for the without-PE and with-PE simulations, and AEP losses are reported as a relative change between these two cases. In contrast to the wake energy loss definitions above, AEP changes are normalized by the baseline production of the existing wind farms, without PE, rather than by the no-WF reference.

The performance of the WRF model representation of the wind field is first evaluated. The simulations in this study represent idealized scenarios, as only a few of the Belgian–Dutch wind farms (Thorntonbank I, II, and III; Belwind; and Northwind; 182 of 572 turbines) were operational in 2016, while the existing turbines used in the simulations correspond to the wind farm layout as of 2022. However, the year 2016 is selected because it is identified as the most meteorologically representative. As a consequence of this mismatch between the meteorological year and the wind farm layout, a true validation of wake energy losses is not possible. As such, validation will only become feasible once post-construction field observations for the fully developed wind farm area become available . Future research may then also explore the optimal representation of the wind field by selecting the most appropriate model configuration and parameterizations.

Although optimizing the model setup through an extensive sensitivity study is beyond the scope of this work and has already been addressed elsewhere , it remains essential to assess the performance of the simulated flow fields relevant to this study. For this purpose, wind speed and wind direction from the no-WF, without-PE, and with-PE simulations are compared against observations at five different locations (Table ). The mean bias (simulated minus observed) is reported in Tables  and for both wind speed and wind direction.

The validation of the ambient wind estimation largely focuses on the no-WF case because the inclusion of wind farms in the simulation causes wakes at the observational mast locations, thus deviating from the observed values, where only a few turbines generated wakes in 2016. This is especially visible at the location of WH, whereas the location in SL is less affected by wakes, as indicated by the smaller differences between the no-WF, without-PE, and with-PE simulations. This is because SL is generally located upstream of the wind farms for the prevailing southwesterly wind direction in this region of the North Sea .

The bias in the simulation without wind farms (no WF) ranges from -0.95 to 0.66 m s⁻¹ for wind speed and -9.85 to 13.71° for wind direction. This falls within the range reported in previous studies on offshore wind fields . Examining specific observation locations, namely WH, SL, and LG, the wind speed predicted by WRF tends to underestimate observed values offshore, consistent with findings by , , and . In contrast, at WA and SW, closer to the coast, the WRF model tends to overestimate wind speed , suggesting a potential coastal effect not well captured by WRF. Due to the relatively coarse coastal resolution, WRF sees a longer fetch, which results in higher wind speeds in the model . No clear trend is evident in the no-WF biases concerning the altitude of the measurement masts considered, except that the highest observation point exhibits the lowest bias, similar to , but opposite to and .

When comparing all three simulations (no WF, without PE, and with PE), the strongest wake-related biases appear at WH, WA, and SW, where the inclusion of wind farms leads to a significant change in bias compared to the no-WF case. At LG, which is located downwind of the BE–NL wind farm cluster during the prevailing wind direction, some differences between the three simulations can also be linked to wake effects. At WH the differences between the no-WF and without-PE simulations are within the uncertainty range (95 % confidence interval, which accounts for both model variability and, when available, measurement uncertainty; see Appendix ), while the wake effect of the new PE is clearly present in the wind speed bias with PE. At SL, where wakes are not expected to play a large role, the biases are nearly identical across the three simulations. This indicates that the background flow is consistently reproduced in all simulations, providing confidence in the subsequent wake energy loss analysis.

The lidar at LG was designed specifically for assessing offshore wind resources. It provides vertical wind profiles up to 215 m, enabling model evaluation at heights comparable to the level of turbine hub heights. Comparing the different WRF simulations with the lidar observations (Table ) reveals a small mean bias across all altitudes and different simulations. The bias for wind speed ranges from 0.04 m s⁻¹ at 62 m to -0.10 m s⁻¹ at 215 m in the no-WF simulation, from -0.08 m s⁻¹ at 62 m to -0.22 m s⁻¹ at 215 m in the without-PE simulation, and from -0.07 m s⁻¹ at 62 m to -0.23 m s⁻¹ at 215 m in the with-PE simulation, respectively. Wind direction biases gradually decrease with height, ranging from values around 5° at 62 m to around 1.5° at 215 m. This suggests improved directional agreement at higher altitudes. The mean wind speed and direction bias of different simulations often fall within the observational uncertainty, accounting for model variability and lidar measurement error, especially between the without-PE simulation and the with-PE simulation. Larger differences are observed when comparing to the no-WF simulation; this could be caused by wakes. To conclude, the background wind field can be assumed to be similar across simulations, justifying studying the wake effects in the suggested simulations. Overall, based on all observation comparisons, a good agreement between modeled and observed values at hub height supports using WRF to evaluate wake impacts of the newly built PE on existing wind farms.

The annual energy production (AEP) and mean yearly wind speed deficit before (Fig. a, b) and after constructing the new Princess Elisabeth wind farm cluster (Fig. c, d) are shown. The wind speed deficit at the mean hub height (96 m) of the wind farms is defined as the temporally averaged wind speed of the simulation, which could be with PE or without PE, minus the temporally averaged wind speed of the no-WF simulations. Figure e and f show the effect of constructing the new Princess Elisabeth wind farm cluster. The wind farm cluster most affected by the new PE wind farm cluster is the Belgian–Dutch (BE–NL) wind farm cluster (Fig. ), with differences in annual energy production (AEP) up to 14 % at certain grid cell locations and 5 % spatiotemporally averaged over the wind farm cluster (Table ). This effect is primarily due to the proximity of this cluster to the Princess Elisabeth wind farm and its downstream location for the main wind direction, which is southwesterly . The BE–NL cluster experiences the most significant impact, but smaller effects are also observed for wind farms further away. Table  shows that distance alone does not explain the magnitude of the AEP impact, highlighting the importance of the relative location of the wind farm with respect to the prevailing southwesterly winds. For instance, a wind farm located at a distance of 181 km exhibits a small AEP loss of 0.12 %. While the 95 % confidence interval of the simulated AEP difference excludes zero, indicating a statistically significant difference, such a small value is unlikely to represent a physical wake impact and is more probably explained by numerical variability in the WRF simulations. It is important that further research investigates these small wake impacts and assesses whether such small values hold physical or economic significance. The observed long-distance impacts underscore the importance of accounting for AEP losses in both existing and planned wind farms during future siting decisions and yield assessments. This applies not only to nearby wind farms, but also to more distant farms. This consideration becomes increasingly important as the density of offshore wind farms is projected to increase in the coming years . It should also be noted that very small increases (0.11 %–0.15 %) in AEP are seen for three wind farms after the construction of the PE wind farm. Given the low magnitude and the fact that these occur at only three locations, it is difficult to draw any strict conclusions regarding actual increases in AEP. These increases in AEP may be attributable to numerical variability, possibly related to coastal effects or model variability in WRF, and further research would be needed to determine whether they have physical (flow speedup) or economic significance.

Due to the presence of the new PE wind farm cluster, wind deficits are intensified (maximum deficit increasing from 2.7 to 2.9 m s⁻¹) and spread over a larger region (Fig. b, d, f). This effect is attributed to the increased momentum extraction of the 15 MW turbines of the PE wind farm cluster. The wind speed deficit exerts the strongest influence in the wake of the main wind direction, which is southwesterly . However, wind speed deficits are also observed southwest of the PE farms and extend toward the northwest of the future concession. These wind speed deficits contribute to changes in AEP, although the relationship between wind speed reduction and power loss is not proportional, as power production depends nonlinearly on wind speed. This nonlinearity has also been highlighted in recent studies on cluster wake-induced energy losses . Consequently, regions experiencing the strongest wind speed deficits generally correspond to areas of larger AEP losses, but even small changes in wind speed may produce disproportionately larger or smaller impacts on AEP depending on the turbine operating regime. It is also important to note that the wind speed deficits mentioned here are temporally averaged (annual) deficits, and their strength and direction are strongly dependent on wind direction and stability, as discussed in the following sections. Previous work showed that it is less dependent on the TKE coefficient .

As shown in Table , the internal wake energy losses (i.e., wake losses experienced by the existing wind farms in the absence of the PE wind farm cluster) are larger than the external wake energy losses (i.e., the additional wake losses induced by the PE wind farm cluster). Note that internal wake energy losses do not refer to the strict definition of internal wake losses but to the energy losses caused in the absence of the PE wind farm cluster (Sect. ). For all the simulated wind farms, the internal wake energy losses are greater than 20.99 %, and they reach as high as 42.01 % for the BE–NL wind farm cluster. These losses are spatiotemporally averaged wake energy losses. WRF simulations of planned wind farm deployments along the east coast of the United States also predict internal wake losses of a similar magnitude, despite differences in wind farm configuration . It is known that the WRF model with the wind farm parameterization of Fitch can overestimate internal wake power losses because turbine wakes are distributed over grid cells much larger than individual wakes, and turbulence generation is also underrepresented . As such, the results shown in Table  may be an upper estimate of internal wake energy losses. Future comparisons with operational SCADA could assess this in more detail.

Relative to the internal wake losses, the external wake losses are generally much smaller (Table ). For several distant wind farms, such as Scroby Sands, East Anglia ONE, Kentish Flats, Egmond aan Zee, Gunfleet Sands, and Prinses Amalia, the 95 % confidence intervals of the external wake losses include zero. This suggests that the estimated external wake losses for these distant farms are not statistically distinguishable from zero and may reflect numerical variability or random fluctuations rather than a true physical wake effect. In contrast, only closely located farms such as the BE–NL cluster, Galloper and Greater Gabbard, London Array, Thanet, and Luchterduinen show statistically significant external wake losses.

Comparing the external wake energy losses (Table ) with the AEP losses (Table ) shows that both tables exhibit similar spatial trends. The BE–NL wind farm cluster, Galloper and Greater Gabbard, London Array, Thanet, and Luchterduinen show the largest impacts, while more distant farms show much smaller effects. Differences between the two tables arise because the metrics are defined using different normalizations. The AEP losses are computed as relative changes between the without-PE and with-PE simulations and normalized by AEP without PE, whereas the external wake losses are defined as ratios of temporally averaged power normalized by the no-WF simulation (Eqs. –). This difference in normalization, together with the implicit weighting of high-production periods in the AEP calculation, explains why some farms show statistically significant AEP impacts, while their external wake losses are not statistically significant.

To gain more insight into the atmospheric conditions influencing external wake energy losses (defined in Sect. ) a more in-depth study of the BE–NL wind farm cluster (Fig.  – K) is conducted. This wind farm cluster is investigated because it experiences the largest wake losses. In all following figures we characterize the atmospheric conditions (wind direction, atmospheric stability class, and wind speed) based on a reference location (spatially averaged grid cell location of the considered BE–NL wind farm cluster) of the no-WF WRF simulation. The height at which the wind speed and direction are taken is 96 m altitude, which is the weighted averaged hub height over all wind turbines considered in this work. The error bars in the following figures represent a 95 % confidence interval calculated by bootstrapping with 1000 resamples.

In Fig. a, we show the annually averaged external wake energy losses of the BE–NL wind farm cluster caused by the PE wind farm cluster. We also show how this external wake energy effect depends on wind direction (Fig. b), stability (Fig. c), and wind speed (Fig. d). We note that each of these figures (Fig. b–d) only looks at a single variable in isolation and does not control for the other variables simultaneously, which can be important due to the inherent correlations between wind direction, wind speed, and stability. This means that the patterns in Fig. b–d reflect combined effects rather than the influence of a single variable. Temperature inversion height may also contribute to the observed stability dependence of wake losses and could be investigated in future work.

The external wake energy losses are spatially concentrated at the top left corner of the BE–NL wind farm cluster, resulting in external energy wake losses up to 10 % at these grid cell locations (Fig. a). These high losses are linked to the prevailing southwesterly wind direction of approximately 225°, leading to the highest losses in this part of the cluster, as the new PE wind farm is upstream of the BE–NL cluster for this wind direction (Figs. b and ). For wind directions outside the range where BE–NL is directly in the wake of PE, the new PE wind farm cluster induces small external wake energy losses (Fig. b). In some directions, negative wake energy losses are observed, which imply a limited power gain after the PE wind farm cluster was constructed. This is discussed later in the paper.

Figure c shows the external wake energy losses as a function of atmospheric stability. In this figure, the highest losses are present for near-neutral conditions. This arises because the figure does not control for wind conditions, and aggregating over all atmospheric situations allows competing effects to cancel or amplify, which obscures the underlying stability dependence. To elucidate how the wake effects depend on stability more clearly, we define a restricted wind direction bin of 193 to 317°. This wind direction bin only includes wind directions for which the BE–NL wind farm cluster is in the wake of the PE wind farm cluster. These results are shown in Fig.  and show generally decreasing external wake energy losses from a very stable to a very unstable atmosphere. The lowest external wake energy losses are present for the most frequently occurring very unstable atmospheric conditions. It has been reported in earlier studies that, for the North Sea, (very) unstable atmospheric conditions are commonly seen . Stable conditions in the North Sea (BE–NL wind farm cluster) are often associated with southwesterly wind directions, when warmer air is transported over the colder sea, particularly during spring and summer. However, this pattern is site-dependent, as stability is strongly influenced by the temperature contrast with the nearest coastline. For other wind directions, particularly those bringing colder air over a relatively warmer sea, especially in winter, (very) unstable conditions tend to dominate .

External wake energy losses are expressed as percentages (Eq. ). At lower wind speeds, turbines produce little power, so even small external wake energy losses appear as relatively large percentage losses (Fig. d), even though the absolute wake effect may not be stronger. At the same time, turbine thrust coefficients are higher at low wind speeds, potentially leading to longer wake recovery distances. In contrast, when examining absolute losses (Fig. ), the largest deficits occur for wind speeds between 10 and 15 m s⁻¹ (mostly Region II). In this range, turbines operate close to rated power while still generating high thrust, so wakes cause large power losses. At higher wind speeds, the thrust coefficients decrease, and turbines typically remain at rated power even in the presence of wakes.

Interestingly, negative external wake energy losses (-1 %), i.e., positive interference from the construction of the new PE wind farm cluster, are observed for winds coming from the north to northeast (0–60°) or south (180–210°) (Fig. b). The negative external wake energy losses for wind directions from the south can be explained by a speedup at the lateral locations of the PE wind farm clusters (Fig. ). It appears that the PE wind farm cluster acts as a windbreak, causing global blockage and accelerating the flow around sides of the wind farm . A comparable lateral acceleration caused by global blockage is reported for certain North Sea wind farm clusters . While similar effects have been studied for windbreaks, where they mainly cause vertical speedups , lateral flow accelerations influencing farm-to-farm wakes have received comparatively limited attention in the literature . observed this farm effect based on measured wind turbine data from two offshore wind farms. This speedup effect in the lateral direction has been observed multiple times for individual wind turbines . In the coming years, it would be beneficial to enhance our understanding of this acceleration phenomenon around wind farms through idealized simulations; however, this is outside the scope of this work. Understanding these speedups is important, as they may have a significant impact on external wake energy losses, necessitating consideration in the planning of high-density offshore regions.

Why negative external wake energy losses are observed for wind directions from the north to northeast is less clear. As mentioned previously for the AEP losses, these may result from numerical variability or other uncertainties in the simulation. The underlying mechanisms of these minor effects, which might be physical or not, should be investigated in the future.

The yearly averaged internal wake energy losses are shown in Fig. a. At specific locations within the BE–NL wind farm cluster (grid level), the internal wake energy losses reach as high as 53 %, which are similar to values in previous studies . These high internal wake energy losses are associated with very dense wind farm layouts. As mentioned previously, WRF with the Fitch wind farm parameterization is known to overestimate the internal wake losses , and the results here thus show the upper limit.

Figure b shows the spatiotemporal internal wake energy losses as a function of wind direction. The high internal wake energy losses are associated with wind directions for which many turbines are aligned in the flow direction. Unlike the external wake energy losses (Fig. b), which are dominated by wind directions for which the wind farms are waked by the PE wind farm cluster, the internal wake energy losses are strongest for the least frequent southeasterly wind direction (occurring only 4 % of the time). This wind direction corresponds to conditions under which the largest number of turbines are influenced by upstream turbines within the same wind farm cluster.

Figure c shows the internal wake energy losses as a function of atmospheric stability. The internal wake energy losses generally decrease from stable to unstable conditions, with an apparent increase for very unstable conditions. Similar to the external wake energy losses, this may seem counterintuitive, but it arises because this figure does not control for wind speed or wind direction and instead aggregates all atmospheric conditions. After additionally conditioning on wind speeds between 10 and 15 m s⁻¹, a clear pattern of decreasing internal wake energy losses with increasing atmospheric instability emerges (Fig. ), as expected. Since very unstable conditions occur most frequently, a large number of low-wind-speed events are included, which increases the relative percentage losses and therefore results in higher apparent internal wake energy losses. This is consistent with the findings of , who showed that wind speed deficits do not translate linearly into power or energy losses.

For the internal wake energy losses as a function of velocity we observe the same behavior as for the external wake losses. When expressed as percentages, they are larger at lower wind speeds because the denominator is small, so even modest absolute deficits translate into large relative losses (Fig. d). In contrast, when considering absolute losses, the largest deficits occur between 10 and 15 m s⁻¹, when turbines operate close to rated power with high thrust coefficients, and wakes cause substantial power losses (Fig. ). At higher wind speeds, turbines remain at rated power, which limits the relative impact of internal wake losses.

To better understand the interaction between external and internal wake effects, we investigate how the presence of wakes from the upstream PE wind farm cluster modifies the development of internal wakes within the downstream BE–NL wind farm cluster. To this end, we analyze five successive turbine-containing grid cells for two representative wind directions, one for which the BE–NL cluster is not affected by PE wakes and one for which it is. In Fig. , we demonstrate that the structure of the internal wake of the selected wind turbines is affected by the new upstream PE wind farm cluster. If the turbines are not in the wake of the upstream PE wind farm cluster the internal wake is unaffected by the presence of the new PE wind farm cluster (Fig. a–e). However, the internal wakes of the selected wind turbines of the downwind farm that are in the wake of the new PE wind farm cluster undergo nonlinear effects (Fig. f–j). The power normalized by the first-row turbines is higher for the turbines when they are waked by the PE wind farm cluster (Fig. h). This increase in power could be caused by a higher added TKE compared to the simulation without the new PE wind farm cluster (Fig. j), which enhances the mixing of the wake and thus dissipates the wake faster. Additionally, the velocity deficit is smaller for the waked turbines, which additionally results in higher power after the front-row wind turbine (Fig. i). Together, the new PE wind farm cluster lowers the overall power of the turbines (Fig. g), but it also reduces the internal wake energy losses (Fig. h). These phenomena again emphasized the caution that should be paid in the future for farm–farm interactions, as they can induce nonlinear effects within a distant wind farm.

Due to the temporal variability in atmospheric conditions, the yearly averaged wake energy losses are not informative about which atmospheric conditions are more critical to induce larger internal or external wake energy losses. As mentioned earlier the atmospheric conditions (wind direction, atmospheric stability class, and wind speed) are based on a reference location (spatially averaged grid cell location of the considered BE–NL wind farm cluster) of the no-WF WRF simulation at a height of 96 m. There is a different behavior, at both hourly and monthly timescales, between the external and internal wake energy losses (Fig. b and d). The hourly variation is stronger for external wake energy losses compared to internal wake energy losses, and both vary seasonally. Larger external wake energy losses are observed in summer during the late afternoon, while the internal wake energy losses' variability is mainly determined by the season. The higher external wake energy losses in summer are explained by the lower wind speeds that coincide with the steep ramp of the power curve and have higher turbine thrust coefficients (Fig. e) and the higher occurrence of stable atmospheric conditions (Fig. g), when warm air is advected over the colder water . Additionally, these higher external wake losses coincide with increased fraction of time during which the wind farm is waked by the upstream PE wind farm cluster (Fig. h). In contrast, lower wake energy losses in winter occur for higher wind speed and more unstable atmospheric conditions (Fig. e and g), when cold air moves across the warmer water . It should be noted that these results are based on 1 year of data, and therefore any generalization of the diurnal and seasonal variation should be made with caution. In addition, further investigation to better understand the occurrence of negative external wake losses during the evening hours in spring is required in the future but is outside the scope of this paper.