Beneficial role of diurnal smoothing for grid integration of wind power

Smoothing of generation variability, i.e. reduction of variance in the aggregate generation is crucial for grid integration of large-scale wind power plants. Prior studies of smoothing have focused on geographical smoothing, based on distance. In contrast, we propose a novel concept ‘diurnal smoothing’ that depends on spatial variations in the timing of seasonal-mean diurnal cycle peak. Considering the case of India, which experiences a strong diurnal cycle of wind-speed, we show how spatial heterogeneity in the wind diurnal cycle can be exploited to smooth wind power variability over and above geographical smoothing. For any given separation distance between sites, the hourly wind speed correlation is highly variable. Difference in timing of the diurnal cycle peak is an important factor for explaining this variability and we define smoothing from differently timed seasonal-mean diurnal cycle as ‘diurnal smoothing’. We show that apart from separation distance, the diurnal cycle is crucial for correlation among sites separated by 200 km or more with strong diurnal cycles (amplitude more than approximately 0.5 m s−1). Thus, diurnal smoothing is a vital factor in the aggregation of large wind power plants, and grid integration is benefited by considering (in addition to distance) new wind plant sites with largely separated diurnal cycles, especially those differing by roughly 12 h. Such diurnal smoothing is relevant for regions across the world with strong wind speed diurnal cycles. Ultimately grid integration depends on variations in total wind and solar generation and demand. Hence, their combined effects must be studied.


Introduction
Decarbonisation of the electricity sector is essential to achieve net-zero emissions and carbon or climate neutrality [1]. Clean energy installations (wind and solar power plants) are expected to grow rapidly across the world. With increasing renewables, the management of variability in a renewable-rich electricity grid becomes crucial. The integration of wind energy is challenged by variability on different time scales (interannual, seasonal, diurnal, and minute to minute) [2][3][4][5][6][7]. Smoothing of wind generation or reduction of variance in the aggregate generation is required for reducing variability and fluctuations in the electric grid. Locating new wind plants in such a manner that smoothing can be achieved is vital for grid integration.
Prior studies of smoothing have focused on distance-based geographical smoothing. In contrast, we propose a concept, 'diurnal smoothing' , that depends on spatial heterogeneity in the seasonalmean of the wind diurnal cycle phase or peak. Although diurnal cycles of wind speed have been observed earlier, the impact of timing of wind diurnal cycle peak on their correlation and the resulting smoothing impacts remain unexplored. Here, we study correlations of hourly wind speed time-series and assess the impacts of differences in time of wind diurnal cycle peak (or diurnal 'phase angle' difference). Globally, wind speed diurnal cycles are highly variable in space due to complex physical effects [8], often involving generation of internal waves in the atmosphere due to horizontal contrasts in heating [8,9], and giving rise to spatial heterogeneity in the timing of wind diurnal cycle peak. The horizontal contrasts in heating arise from many factors such as differential warming of the surface, radiative cooling and sensible heat flux from the surface to the atmospheric boundary layer [8]. As such, a strong diurnal cycle in wind-speed is mainly prevalent over tropical land, with the diurnal cycle being much weaker over the ocean [8]. Internal waves in the atmosphere are also generated by orography [10], which can influence the phases of diurnal cycles inferred from observations [8]. As a result of these effects, the diurnal cycle does not simply propagate from east to west. In areas with strong wind diurnal cycle, different regions irrespective of their separation distance and orientation between the sites (east/west or north/ south), can have the maximum wind speed occurring at different times of the day. This paper, for the first time, quantifies the implications of this variation in diurnal cycle of wind speed on renewable energy integration and smoothing.
It is well documented that geographical smoothing from aggregation of electricity generated from wind plants dispersed over a large region reduces high-frequency oscillations in total generation [11][12][13][14][15][16][17][18]. The concept of 'diurnal smoothing' discussed here differs substantially from geographical smoothing. Literature show that, in the USA, although the smoothing of wind generation timeseries can be achieved by spatial aggregation, clear diurnal patterns were still evident in the aggregated wind generation, owing to similar diurnal wind speed patterns or phases for all systems being integrated [14]. For regions with strong wind speed diurnal cycles, aggregation of generation from wind plants located in different wind regimes, having their wind diurnal cycle peak at different times of the day, can result in generation smoothing irrespective of their separation distance.
Previous studies have indicated that in various regions (Texas, Switzerland, India, Australia, Canada, and the USA), 'geographical smoothing' is not enough to explain the smoothing observed there [15,[19][20][21]. Identifying other factors that explain the systematic correlation between wind power plants and, thereby the generation smoothing achievable, is a research gap. This study examines an important systematic factor, that of 'diurnal smoothing' . St. Martin et al [15] highlight the need for further work in order to better understand the large scatter in the relationship between correlation and distance. It is evident from Horst et al [21], that distance is not enough to explain the variation in wind speed correlation over India. Together these studies clearly demonstrate the need for identifying the other factors affecting the systematic correlation of wind-speed between sites, besides geographic distance.
In this paper, considering the case of India, which experiences a strong diurnal cycle of wind-speed, we show how spatial heterogeneity in diurnal cycle over India can be exploited to smooth wind power variability over and above geographic smoothing. It is shown that the daily ('diurnal') wind speed pattern is an important factor in explaining variation in wind speed correlation coefficient in India. Such analysis has not been undertaken before. We define the smoothing achieved by exploiting differences in the diurnal cycle as 'diurnal smoothing' .
We also note that solar energy will have large effect on power system operation including in India, and incorporating its effects are important for future studies.

Data
The analysis was performed on hourly horizontal wind reanalysis product ERA5 from the European Centre for Medium-Range Weather Forecasts [22], for the summer monsoon (high-wind) months (June, July, August and September-JJAS) for the year 2010-2019 for the Indian region. Summer monsoon (JJAS) is selected for analysis as it is the high wind generation season in India [7]. We examined wind-speeds at 100 m above ground level, since the hub height of various wind turbines are around that height. The spatial resolution of the dataset is 0.25 × 0.25 degrees and the temporal resolution is hourly. In-situ, hourly 100 m wind measurement data from 39 weather stations from National Institute of Wind Energy (NIWE) is used to validate ERA5 wind speed pattern. IMDAA regional reanalysis dataset (high resolution (12 km, 1 hourly) over India) for JJAS 2014 [23] is also compared with ERA5 reanalysis.

Diurnal cycle
Step 1 We estimate the amplitude and phase angle of the diurnal cycle of monsoon wind speed over India based on the seasonal mean (JJAS) of hourwise wind speed pattern (24 h for years 2010-2019). The estimation is done for ERA5 reanalysis data. ERA5 based amplitude and phase angles are validated against direct observations from NIWE for 2014.
Given the presence of a substantial diurnal cycle in the monsoon wind speed over India [5], we fit equation (1) to the seasonal (JJAS) average wind pattern estimated based on 10 yr of ERA5 data (2010-2019): where ws(t) is the wind speed, A is the amplitude of diurnal cycle, ϕ (in radians) is related to timing of the diurnal cycle, c 1 is monsoon average wind speed, t denotes time in hours (Indian Standard Time (IST)) and e(t) is the residual error. The amplitude (A) describes the strength of the diurnal cycle at the location. For some locations, the estimated amplitudes of the diurnal cycles have negative values. Using a trigonometric identity, we convert these amplitudes to positive numbers by adding π to angle ϕ. The resulting ϕ, in radians, is related to the time of day when the maximum in the windspeed diurnal cycle occurs. We represent this in units of time (IST) of wind diurnal cycle peak which is henceforth called 'phase angle' .
Step 2 Aggregation of wind generation from plants with uncorrelated or weakly correlated wind speeds would result in the smoothing of aggregate power generation. Wind speed correlation for a pair of sites affects the variance of the combined generation output from wind plants at these sites. Therefore, the pairwise correlation coefficient is important to understanding smoothing achieved by aggregating generation from any two sites.
We choose 2500 (approximately) random pair of sites from the Indian region and estimate the correlation coefficient of hourly wind speed, r, between these pairs during the monsoon months (averaged for the years 2010-2019). These individual sites (latitudes, longitudes) are selected by independently sampling from uniformly distributed random numbers and then paired randomly. Correlation r is used as an indicator of smoothing that can be achieved by aggregating generation from that location pair, because it influences variance reduction in the aggregate generation.
Step 3 We estimate, using least-squares, three non-linear regression models for r given any location pair: using models involving separation distance, phase angle difference, and both distance and phase angle difference. The distance between the location pairs is calculated based on the Haversine equation [24] describing the great circle distance between two points on the Earth's surface.
(a and b are constants to be estimated in the regression). Model 2 -Model 2 considers that cosine of the difference in the time of diurnal cycle peak of wind (∆ϕ) (in radians) of a location pair can explain variations in r (theoretical calculation of correlation coefficient giving rise to this model is presented in the supplementary material). The phase angle difference ∆ϕ can vary from 0 to 2 π with phase angle difference of 2 π corresponding to time difference of 24 h. Hence, a phase angle difference of ± π means ±12 h difference in the time of the diurnal cycle maximum. As cos π = (−1), we expect the minimum value of r as per model 2 to occur when ∆ϕ = ± π or ∆ϕ = ± 12 h.
(c and d are constants to be estimated). Model 3 -For large separation distances (more than 200 km), distance and phase angle difference between location pairs are uncorrelated (supplementary material). Model 3 assumes that both the explanatory variables D and ∆ϕ are important in explaining the variations in r.
(a, b and d are constants to be estimated).
Step 4 We compare these predicted values of r (from each of the models fitted in step 3) with the calculated values of r based on ERA5 (from step 2). This helps us understand the relative importance of separation distance and difference in time of wind speed diurnal peak during monsoon months over India. We compare the fit between the predicted r, from models in step 3, and the computed values in step 2, by evaluating the correlation coefficient between these two quantities across the random pairs.
Step 5 Finally, we estimate the threshold diurnal cycle amplitude and separation distance beyond which the relative performance of the models changes.

Case study
Thus far in our approach, the evaluation of the importance of the diurnal cycle for smoothing wind generation is based on wind speed data. Next, we investigate if these findings are valid for simulated wind generation from randomly chosen location pairs over India. The power curve of Suzlon S.88-2100 [25] model turbine with a rated power output of 2.1 MW is used to convert ERA5 wind speed to wind power generation. We have assumed copper plate connectivity between these simulated wind plants, without any transmission bottlenecks.
We compare the normalised variance (σ norm ) 2 (equation (3)) of aggregate generation for the pairs with similar timing of wind diurnal cycle peak (phase angle) with that of the location pairs having widely separated phase angles (12 h apart) to estimate the improvement that can be achieved from diurnal smoothing. supplementary material illustrates benefits of diurnal smoothing in further detail: where, σ norm -Normalised aggregate standard deviation σ -Aggregate standard deviation σ max -Maximum possible standard deviation of aggregate generation, if wind-speed were perfectly correlated.

Diurnal cycle
Equation (1) could capture the observed monsoonal wind speed diurnal cycles for two wind monitoring stations, Akkanayakanpatti and Lingampalli with considerable accuracy (figure 1). The R square values for the curve fits are 0.9 and 0.84 for Lingampalli and Akkanayakanpatti respectively. If the diurnal pattern were due to effects of maximum solar forcing alone, a lag in the diurnal pattern would only be expected when the locations are separated in longitude. Akkanayakanpatti and Lingampalli are located at similar longitude, but still the wind diurnal cycle peak occur at very different times of the day, about 12 h apart. The diurnal phase angles are different, because as we noted earlier the phenomena giving rise to diurnal cycle differences are complex. We validate the diurnal cycle of ERA5 100 m wind speed that is used for analysis against in-situ wind speed measurements (NIWE) at 100 m above ground level (supplementary material). The average diurnal cycle of wind speed estimated from the in-situ observations for different weather monitoring stations (39 stations) and the average diurnal cycle of wind speed estimated from ERA5 analysis for the corresponding, i.e. nearest, grid point for JJAS, 2014 are in good agreement (supplementary material). ERA5 gridded reanalysis data underestimates the amplitude of diurnal cycles for point observations at a few locations. It is likely that this underestimation in ERA5 is due to partial smoothing of the diurnal cycle in the gridded ERA5 analysis, from spatial variation in phase, as amplitude is estimated after fitting the sinusoidal model to the data. Since much of the analysis of this paper is based on this gridded data, the diurnal smoothing effect will be underestimated as a result. We have also compared the IMDAA regional reanalysis dataset (high resolution (12 km, 1 hourly)) over India for JJAS 2014 with ERA5 reanalysis, finding that the wind diurnal cycle peak is similar to ERA5 (supplementary material).
ERA5 reanalysis does broadly capture the phase angle and amplitudes during the JJAS 2014. Therefore, ERA5 wind speed analysis at 100 m above ground level based on 10 yr average wind speed is used to determine the spatial patterns of amplitudes and phase angles of seasonal mean wind diurnal cycles during JJAS. Figure 2(A) shows the spatial pattern of the average amplitude of the seasonal mean diurnal cycle (m/s −1 ) over Indian region. Much of the Arabian sea and Bay of Bengal that are far from any land region show very low diurnal amplitude. Similar findings are shown in previous literature [8]. The areas near orography and near land-sea boundaries such as the Western Ghats, Himalayas and the Tibetan plateau, and around the Gulf of Mannar between India and Sri Lanka as well as Khambat Bay in the western coast, show the largest diurnal amplitudes. Figure 2(B) shows the spatial variation of the average timing of wind diurnal cycle peak or phase angle over the region during JJAS. India experiences a large variation of phase angle of the wind-speed diurnal cycles. The high-terrain regions such as Western Ghats, the Himalayan region and Tibetan Plateau show similar phase angle. The bay regions that are surrounded by land (Dhanuskodi and Khambat Bay) also show a similar diurnal pattern. The patterns in figure 2 remain largely similar across years (supplementary material). Thus, the patterns of figure 2 reflect systematic differences in the diurnal cycle and influence the persistent correlation of hourly wind-speed between sites. The simulated wind energy over India also shows similar patterns (supplementary material). Worldwide, strong-amplitude diurnal cycles appear in tropical latitudes, especially where the elevation is high or above deserts and many coastal zones. The diurnal cycle in wind-speed, as measured by the amplitude, is high, not only over land but also over the coastal ocean. The phenomenon of diurnal smoothing is therefore relevant beyond our present context, to grid-integrate windpower in such areas.

Effect of distance and diurnal cycle on the correlation coefficient
We use approximately 2500 random pairs of sites (shown in supplementary material) in the Indian region for our analysis, to estimate and compare effects of geographical smoothing versus time of wind speed diurnal cycle peak based smoothing or diurnal smoothing. Out of 10,881 grid locations we have  chosen random location pairs using uniformly distributed random numbers. We found that increasing the number of random location pairs (10,000) does not affect the findings (supplementary material). Diurnal cycle has been estimated from the 10 year average of ERA5. The hourly wind speed correlation coefficients have been calculated for each of these pairs, for JJAS, 2010-2019. The separation distance and phase angle difference between location pairs are generally uncorrelated (supplementary material), and thus the effects of diurnal smoothing described below are independent. We use three regression models (section 3) to estimate the dependence of wind speed correlation coefficients on the distance and the phase angle difference between the pairs. We compare the models' performance based on their ability to predict the wind speed correlation coefficient values. Figure 3(A) shows the mean decay of the hourly wind speed correlation coefficient between the pairs with increasing separation distance. A large part of variation in correlation coefficients as observed from ERA5 remains unexplained by model 1. Model 1 (with distance as the only explanatory variable) can capture the general decay of correlation with increasing separation distance. However, the median of predictions made by model 1 underpredicts the correlation coefficients, especially at smaller distances where the exponential model does not grow as rapidly as the observations with decreasing distance. The model predicted variation (yellow band) of model 1 decreases with the separation distance between the pairs. Figures 3(B) and (C) shows predictions from model 2 and model 3. The variance in correlation coefficients is better explained by models 2 and 3 respectively compared to model 1. Phase angle plays a larger role for larger distance. As distance increases, the variations (models 2 and 3) grow. Model 2 (based on the phase angle difference between the pair) and model 3 (based on separation distance and phase angle difference) capture a larger share of the variation in wind speed correlation coefficients than model 1. Further, model 3 captures a wider share of the distribution of correlation coefficients compared to model 2, especially at smaller separation distances. However a significant fraction of variation in correlation coefficients at given separation distance, which could result from weather variability on various timescales, remains unexplained by any of these models. Using 10 yr averaged diurnal cycles gives similar findings for correlation coefficients in other years (supplementary material). Figure 4 explores large variability of the hourly wind speed correlations of location pairs. Although there is clear decay in the correlation coefficients with increasing separation distance (explained by model 1), there is additional variation along with this decaying trend for the location pairs separated by similar distance. Although shorter separation distances can predict a large portion of the variability in the hourly wind speed correlation coefficients of location pairs, the variation around the decaying trendline remains unexplained without considering the effects of the diurnal cycle. Further analysis (supplementary material) reveals that the variation in hourly wind speed correlation appears as the cosine of the phase angle difference (basis for model 2). For pairs within a given distance bin, hourly wind speeds are more likely to be uncorrelated for pairs that have difference of ±12 h in local time of wind maximum. This is in line with our expectation as per the theoretical model, based on which model 2 was introduced.

Effectiveness of diurnal smoothing
In this section we explore distance and amplitude thresholds beyond which models 2 and 3, accounting for phase angle difference, outperform model 1, which is based on distance alone. These models are created based on equations (2a)-(2c) (section 3.1). We compare these models' performance with respect to association between each of their predictions of the hourly wind-speed correlation coefficient, r (step 3 in methodology) and observed correlation directly estimated from the hourly ERA5 dataset  (step 2 in methodology). The three models' relative performance varies with separation distance. Model 1 performs better than model 2 for location pairs separated by distances less than 200 km. However, model 3 generally performs better than the other models, since this model considers both distance and phase angle difference as predictors. The impact of separation distance reduces exponentially with increasing distance. Thus, beyond separation distance of 450 km, model 2 and model 3 perform much better and their performance is comparable to each other ( figure 5(A)). The effectiveness of the models also depends on the amplitude of the diurnal cycle. Amplitude here denotes the mean of the amplitudes of a location pair. model 2 and model 3 perform better than model 1 for location pairs whose mean amplitude of seasonal mean diurnal cycle is larger than about 0.5 m s −1 . This threshold is met for most of the Indian region (figure 2(A)). Below this threshold value, model 1 performs better than the other models ( figure 5(B)).

Case study
To demonstrate diurnal smoothing for power generation, we compare the normalised variance (equation (3)) of aggregate generation from a large ensemble of randomly chosen pairs of sites having a strong diurnal cycle ( figure 6). While the normalised variance decreases with increasing separation distance, owing to geographic smoothing, there is a clear benefit to diurnal smoothing as well: the normalised variance is smaller for pairs of sites that have different phase angle (12 h apart) compared to sites similar phase angle. Thus, for a pair of sites, separated by similar distance, more smoothing can be expected if they have different times of wind diurnal cycle peak. This effect is evident across separation distance. The diurnal smoothing achieved at shorter distances is comparable to the geographical smoothing benefit achieved from sites with similar phase. As the effect of the diurnal cycle on correlation coefficients is prominent with separation distances larger than 200 km and average diurnal amplitude of more than 0.5 m s −1 (figure 5), we choose location pairs that have stronger diurnal cycles (⩾0.75 m s −1 ) for the figure.

Discussion and Conclusion
Smoothing of wind generation variability is vital for grid integration of large-scale wind power plants.
Prior studies of smoothing have focused on geographical smoothing, based on distance. Earlier studies have pointed out that geographical smoothing, based on a fixed relationship with distance, is not enough to explain the variable extent of smoothing achieved by aggregating generation from different wind power plants.
In this paper, we introduce a concept, 'diurnal smoothing' , that depends on spatial variations in the timing of the seasonal-mean diurnal cycle of wind speed ('phase angle'). Although meteorologists have examined the diurnal cycle in wind speed, a detailed study of the heterogeneity in the timing of the diurnal cycle peaks is new. This heterogeneity can be important for wind integration, but a fuller understanding depends on variability of other power sources as well as demand. We show how spatial heterogeneity in timing of the wind-speed diurnal peak over India can be exploited to smooth wind power variability over and above geographic smoothing. This study, for the first time, quantifies benefits arising from systematic differences in the time of peak of seasonal mean diurnal cycle of wind-speed. Diurnal cycle properties are estimated from multiyear averages, and show that differences in timing of wind diurnal cycle peak is important for accounting for variation in hourly wind speed correlation coefficients between randomly chosen pairs of sites that are separated by a similar distance. The impact is prominent for regions having stronger diurnal cycle and separated by distance more than 200 km. Therefore, aggregating generation from two nearby plants can yield additional smoothing benefits if the diurnal cycles are different. Conversely, the smoothing achieved by aggregating wind-plants separated by a large distance, but having similar times of wind-speed diurnal cycle peak, may be modest and comparable to the diurnal smoothing alone that is achieved at shorter distances (figure 6).
We find that ERA5 analysis based on average summer monsoon wind speed captures the spatial pattern of time of wind diurnal cycle peak reasonably well with reference to in-situ observation. For the seasonally-averaged diurnal cycle during the summer monsoon (June-September) of 2014, the correlation between the in-situ observation and ERA5 average is reasonably high for the phase angle, while the amplitudes have a lower correlation. This may suggest the likely limits of predictability that current global numerical weather prediction models can achieve. In most of the observation sites, ERA5 underestimates the diurnal cycle amplitudes, likely due to the effects of spatial smoothing in the gridded dataset as amplitude is estimated after fitting the sinusoidal model to the data. Previous literature has also found underestimation of wind speed in ERA5 due to spatial resolution in US and Iran mountain sites [26]. Jourdier [27] shows that apart from a few mountainous areas, ERA5 accurately estimates the diurnal cycle in France, even in complex environments. Stronger diurnal cycles would increase the benefits of 'diurnal smoothing' . Overall, our paper implies that diurnal cycles could be slightly larger than we estimate from ERA5, with concomitantly larger benefits of diurnal smoothing. Notwithstanding errors in ERA5, it is clear that the diurnal wind signal we find is large and spatially extensive enough to make diurnal smoothing relevant.
In ERA5, we found that areas near Gulf of Mannar between India and Sri Lanka, Khambat Bay on the western coast, Rajasthan, Himalayas, and Tibetan Plateau show the largest diurnal amplitude of 100 m wind speed. Generally over this region, high diurnal cycle amplitudes are associated with elevation and land-sea contrast. Some of these findings concur with the earlier findings by Dai and Deser [8], who report that diurnal wind variations are larger over land than ocean and strongest over high terrains such as the Rocky Mountains, the Andes, and the Tibetan Plateau. The diurnal cycle amplitude over the ocean is generally much attenuated as compared to overland except in a few oceanic regions such as those adjacent to Sri Lanka and Madagascar, which occur near coastlines.
The diurnal smoothing effect can be understood in terms of its impact on wind-speed correlation between sites. Our analysis shows that, although for shorter distances (lower than around 200 km), the separation distance between the two sites can predict a small but significant part of the variability in the wind speed correlation coefficients, the variations around the decay (with increasing distance) remains unexplained by the distance-based model. Diurnal phase angle differences between the two sites account for the systematic part of this variation, and beyond around 200 km separation distance, the phase angle difference can predict more of the variation in hourly wind speed correlation coefficients.
For location pairs whose average diurnal cycle amplitude is larger than about 0.5 m s −1 , which is found in western and southern parts of India, the combined model of separation distance and cosine of phase angle difference as explanatory variables explain more of the hourly wind-speed correlation coefficient than distance alone. Moreover, distance has a negligible explanatory effect for location pairs with large separation distances and strong diurnal cycle, making the model based on the phase angle difference adequate. The benefits of diurnal smoothing extend beyond wind speed to wind power generation (case study). Depending on the regional load demand pattern, heterogeneous wind-speed diurnal cycles can provide benefits for load following, but this analysis is beyond the scope of the present study. As grid integration depends on the combined variation of all sources, future studies must consider diurnal variation of combined wind and solar generation in relation to load patterns.
As India plans to double wind capacity, many new plants are expected and strategically locating new wind plants considering diurnal phase and amplitude along with various factors including geographical constraints would help manage the impacts of regular wind speed variations on the grid. However, according to the present tariff policies in India, considering benefits of diurnal smoothing might not directly favour wind developers. Grid operations would benefit from diurnal smoothing, and, policy measures to incentivize wind developers to consider the diurnal smoothing aspect while siting new plants would have clear benefits for the grid, especially in those regions with a large diurnal cycle. The general benefits of diurnal smoothing are likely to be relevant in other regions of the world having strong windspeed diurnal cycles.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https://cds.climate.copernicus.eu/cdsapp#!/home.