As the capacity of renewable energy resources increases, accurate forecasts of power production are becoming increasingly instrumental for efficient and effective management of the energy grid. In 2017, the worldwide wind power capacity grew by 10.8 % to a total capacity of 539 GW (World Wind Energy Association, 2018). This capacity covers only about 5 % of the total global energy demand, so continued growth of wind power generation capacity is expected. Large wind power plants that have tens to hundreds of turbines pose many challenges for forecasting, as the meteorological conditions, the topography, the array orientation, and the resulting wake effects may affect wind and power variability across the turbines at the farm. Ultimately, the variability in the wind power needs to be accounted for in farm-level, day-ahead wind power forecasts that are used in unit commitment and electricity market bidding strategies, as well as in intra-day wind power forecasts that are used for reliability, regulation, or sales on the spot market (Ahlstrom et al., 2013; Orwig et al., 2014).

There are two main sources of error in wind power forecasting: the error in the underlying weather forecast of wind speed and to a lesser degree air density (Pandit et al., 2018; Bulaevskaya et al., 2015), and the error in converting the wind speed to power. Past research has indicated an advantage in using machine learning methods for wind power conversion (Parks et al., 2011), and we further investigate this in the context of the super-turbine approach. In the super-turbine conversion methodology, the wind speed is forecast as a farm-average value, and that wind speed is converted to farm-level power. Bartlett (2018) pointed out that the super-turbine approach can result in substantial errors in power conversion, especially when variability exists across a wind farm. He further analyzed wind farm data to explore alternate methods to convert wind speed to power. The underlying assumption of the super-turbine approach ignores the variability in wind speed across the turbines and the nonlinearity of the power curve. Some methods, however, have added wind variability as an explanatory variable in wind power conversion (Pieralli et al., 2015). The issue of wind farm variability is illustrated in Fig. 1, where the blue line indicates the power conversion for the super turbine (i.e., farm-level mean wind speed), whereas the red distribution illustrates that each turbine may have different wind speeds centered around the mean value, which results in a distribution of power with a lower mean value than the super turbine for a 10 m s-1 wind speed. For this notional example, the super-turbine approach would predict an average power of approximately 1600 kW whereas the turbine-level approach would predict approximately 1500 kW.

The explanation for this phenomenon is in Jensen's inequality, which states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation, and vice versa for a concave transformation (Jensen, 1906). More generally, Jensen's inequality states that if you have a nonlinear function, the average of the function is not equivalent to the function of the average, and the magnitude of this inequality depends on the nonlinearity of the function and the variability (Pickett et al., 2015). Jensen published this mathematical proof over 100 years ago, and while there are some fields such as ecological physiology and evolutionary biology that have studied the impact of Jensen's inequality (Denny, 2017), the impact on wind power forecasting and methods that overcome the impact have not been studied. Denny (2017) provides details regarding the basic concepts of Jensen's inequality with specific examples in biology.

The impact of Jensen's inequality in wind power forecasting is best illustrated in the steep portion of the curve for converting wind speed to wind power, which is generally taken as a cubic function following the power density function. Thus, at low wind speed values the transformation is convex and at high wind speed values the transformation is concave, which is illustrated by the orange line in Fig. 1. Therefore, at low wind speeds we expect the super-turbine power conversion (i.e., the mean applied before) to be less than the turbine-level power conversion, but at high wind speeds we expect the super-turbine power conversion to be greater than the turbine level power conversion. The application of Jensen's inequality to wind power conversion is herein described as the super-turbine wind power conversion paradox.

Our goal is to quantify the error due to Jensen's inequality for a super-turbine power conversion using both simulated hypothetical wind farm data and empirical data. We show the expected difference using simulations for a hypothetical wind farm that has 100 turbines and empirical data from the Shagaya Renewable Energy Park in western Kuwait. In Sect. 2, we describe the methodology for the simulation of the hypothetical wind farm and present results. In Sect. 3, we present and discuss the impact of Jensen's inequality in an empirical analysis for the 10 MW Shagaya wind farm. In Sect. 4, we discuss the results of the hypothetical and empirical analysis. In Sect. 5, we propose a machine learning technique for predicting the total wind farm power and discuss the impact of those results in overcoming Jensen's inequality. Section 6 presents conclusions and suggestions for next steps.

In addition to the simulated hypothetical data, we examined data from a 10 MW wind farm located at the Shagaya Renewable Energy Park in Kuwait. The location of the wind farm is labeled “Shagaya” in Fig. 5 and the turbines are located at an elevation of 240 m, a latitude of approximately 29.22∘ N, and a longitude of approximately 47.05∘ E. The local topography is flat and the climate is characterized by persistent arid conditions with large temperature differences between summer and winter. There are five 2 MW turbines currently located at Shagaya with data available at a 10 min frequency from 1 September 2017 until 31 May 2018, which comprised 34 393 instances in the initial dataset. The SCADA dataset includes the mean power produced by each turbine over a 10 min period as well as the standard deviation, minimum, and maximum of the 1 min raw data over this 10 min period.

The data were preprocessed for quality control beginning with removing instances with missing data that occurred in approximately 22.5 % of the original dataset. Negative power observations in the dataset were set equal to 0 MW power as there were small negative values recorded when the wind turbine was not generating power, likely a result of the turbine consuming a small amount of power. Finally, if the wind speed at the turbine was measured at greater than 3 m s-1 but no power was reported, those instances were removed from the dataset as they reflected times of possible maintenance or other forced shutdown of the turbine, which occurred in 2.09 % of the original dataset. We converted the measured wind speed to a converted wind power using the 10th-order polynomial and all wind power values converted from wind speeds that were above 2020 kW were replaced with 2020 kW, as that was approximately the maximum observed. The total dataset size after quality control comprised 23 679 instances that included measured power at all five turbines. Over this period of time, the average power at each turbine ranged between 787 and 813 kW and the average wind speed ranged between 7.00 and 7.13 m s-1 with a standard deviation across the turbines of 0.26 m s-1. To quantify the correlation between the wind speed and the power measured at each turbine, we computed the R2 between wind speed and power for each turbine independently. Using all data, the wind speed to power R2 was in the range of 0.76–0.86 for each of the turbines; when limiting the data to the range of 3–12 m s-1, we found that the R2 was in the range of 0.79–0.90 for each of the turbines.

Next, we quantified the difference in nacelle wind speed, measured wind power, and converted wind power among the turbines at the Shagaya wind farm. Table 3 shows the mean wind speed (second column) and measured mean power (third column) differences between turbine 1 and all of the other turbines. The mean difference in wind speed varied from 0.03 to 0.13 m s-1 and the mean difference in power ranged between 19.27 and 27.19 kW. We then computed the farm-level power with the super-turbine approach using the mean wind speed across the turbines and the polynomial fit to convert to power. We also computed the turbine-level total wind farm power by converting the wind speed at each turbine to power and taking the sum across all turbines. Comparing both power conversion techniques to the actual power produced we found a mean absolute difference of 2.63 kW per 2 MW turbine, or a total wind farm power difference of 13.15 kW. We then computed a mean absolute error of 68.83 kW per 2 MW turbine between the super-turbine power conversion and the measured power and a mean absolute error of 68.52 kW per 2 MW turbine between the turbine-level power conversion and the measured power.

The differences between the power conversion using a polynomial fit to the wind speed data and the measured power are not only due to the effect of Jensen's inequality, but are also due to the Shagaya wind speeds measurements by nacelle anemometers. These measurements occur behind the blades of a turbine and the wind speeds are consequently impacted by wake effects. St. Martin et al. (2017) showed that there is a substantial difference at wind speeds of greater than 9 m s-1 and accounted for the wake effects of using nacelle wind speeds for power conversion by applying a fifth-order polynomial fit between an upwind met tower and the nacelle wind speed data. We avoid the use of a transfer function to map between a met tower and the nacelle wind speeds because we want to isolate the impact from Jensen's inequality; however, an operational power conversion methodology should attempt to take the impact of using nacelle wind speeds to convert to power into account and should therefore either include the met tower observations as a predictor in the power conversion machine learning or should apply a transfer function to the nacelle wind speed data.

Finally, we compared the super-turbine power conversion to the turbine-level power conversion from the mean wind speed and nacelle wind speeds at the individual turbines. The data are plotted in Fig. 6 with the super-turbine mean power per 2 MW turbine on the x axis and the turbine-level mean power per 2 MW turbine on the y axis. The scatter along the 1 : 1 line aligns with the hypothetical data analysis: for wind speeds less than 8 m s-1 the super-turbine mean underestimated the power, and for wind speeds greater than 8 m s-1 the super-turbine mean overestimated the power.

The greater difference for the power conversion in the 2 m s-1 standard deviation hypothetical wind speed variability scenario of the simulated data is a result of a higher frequency of turbine-level power conversions further from the mean of the wind speed. This is illustrated in Fig. 7 where the blue distribution on the x axis indicates the mean wind speeds used for the super-turbine power conversion, which is shown as the blue distribution on the y axis. The red distribution on the x axis indicates the turbine-level wind speeds used in the conversion to power, which is shown as the red distribution on the y axis. This analysis is for the 1000 simulated instances in the 2 m s-1 variability scenario. The turbine-level power conversion draws from a wider distribution whereas the mean value in the super-turbine power conversion draws from a narrower distribution. This result occurs because taking the mean of the individual simulations narrows the distribution via the law of large numbers where the mean of a large number of simulations or observations should approach the expected value as the number of simulations or observations increases (Wilks, 2011). The wind speed and wind power are asymmetric around the mean of the distribution of the wind speed at 10 m s-1, as illustrated in Fig. 1 where the orange line is the polynomial fit to the data for the cubic power transformation. At wind speeds of 12 m s-1 and greater, the power stays approximately constant at the maximum value of 2000 kW. However, below wind speeds of 10 m s-1 on the left-hand side of the wind speed distribution, the power decreases according to the orange line and does not hit a minimum value in the same way the power achieves a maximum value on the right-hand side of the wind speed distribution. This simulated dataset illustrates the impact of Jensen's inequality on the wind speed to power conversion and how larger variability will introduce greater differences due to more samples drawn from the distribution further from the mean.

The empirical data from the Shagaya wind farm in Kuwait highlights the same structural differences between the super-turbine wind power conversion and the turbine-level wind power conversion. Although the magnitude of the differences is less than the magnitude of the simulated hypothetical data, the wind farm in Kuwait is characterized by less variability among the turbines than would be expected from a wind farm that covers a larger spatial area, that is located in more diverse geography, or that experiences more variable weather which could produce greater wind speed variability among turbines. Wind farms may not measure wind speed and wind power at each turbine individually; therefore, a technique to predict the total wind power at the connection node given information about the mean wind speed and the variability across the wind farm would be valuable, and machine learning may provide an alternative solution.

Machine learning has been used to convert wind speed to power for wind farms where data are available (Mahoney et al., 2012; Parks et al., 2011). Machine learning is best utilized when there is a nonlinear relationship among the predictors and the predictand and the true relationships can be found in the dataset, which is a characteristic of this wind power conversion problem. The machine learning model used here is the random forest supervised learning method (Breiman, 2001). The random forest represents an ensemble of regression trees where the final prediction is an average of the prediction from each of the trees. Figure 8 illustrates the structure of the random forest: the final prediction is an average of the predictions from each tree in the forest where each tree is given a subset of the available predictors and training data. Regression trees utilize the predictive power of dividing a dataset into smaller subsets based on the predictive relationships between the predictor and the predictand until the subsets minimize the cost function (Witten and Frank, 2005). Regression trees do not search for the most important predictor in order to split a node, but rather exclusively search for the best predictor among a random subset of the predictors. This technique results in a final model that reduces overfitting the training data and ultimately generalizes better (Witten and Frank, 2005). The random forest used here is the python package “scikit-learn” random forest regressor (Pedregosa et al., 2011). Note that we opted to use the random forest method because it is a machine learning method that captures nonlinear relationships between predictors and the predictand, and has the added benefit of avoiding overfitting as it is an ensemble approach. Other machine learning methods such as the artificial neural network or gradient boosted regression trees may work similarly well, however, our goal is not to find the most optimal machine learning approach but rather to highlight that machine learning can be used to learn the impact of Jensen's inequality in this application.

The wind power forecasted at the farm level is of the utmost importance for a utility or system operator; however, the variability at the farm level is a function of the variability across the turbines at the farm. In this study we use both hypothetical simulated data and empirical data to analyze the effect of Jensen's inequality on the application of a super-turbine power conversion where a mean wind speed is used to convert to power. We showed that there are systematic nonlinear differences between a turbine-level power conversion and a super-turbine power conversion at a range of wind speeds from 5 to 14 m s-1. The effect of Jensen's inequality was found to be most pronounced at approximately 7 and 11 m s-1 in the simulated hypothetical wind farm, where the curvature of the power curve is the greatest. Understanding the impact of Jensen's inequality on the total power at a wind farm, or the super-turbine wind conversion paradox, allows a utility to choose a power conversion methodology that incorporates this effect for a more accurate power conversion estimate.

In the empirical data analysis, we were similarly able to show differences between the turbine-level power conversion and a super-turbine power conversion even for a relatively small 10 MW wind farm consisting of five individual turbines in flat desert terrain. One would expect that a wind farm with more turbines in a larger area would exhibit more variability, especially if there were local terrain or wake effects. In the hypothetical data we showed there is a larger effect of Jensen's inequality as the wind variability increases, as would be expected for a larger wind farm.

Finally, we showed that the random forest machine learning method is able to reduce the error in the wind speed to power conversion when provided with predictors that quantify the differences due to Jensen's inequality. This was first done using the hypothetical simulated data where the error was reduced to under 2.5 kW per 2 MW turbine for all wind speeds from 5 to 14 m s-1. In the empirical data analysis, we were able to reduce the error from an average difference of over 68 kW per 2 MW turbine to a little over 44 kW per 2 MW turbine, which represents an error reduction of greater than 35 %.

In this study, we focused on utilizing machine learning to isolate and remediate the differences caused by Jensen's inequality on wind speed to power conversion. We did not try to find the lowest error methodology for converting wind speed to power, as a machine learning model would likely have lower error when including other meteorological variables such as wind direction, temperature, and humidity. The error for the machine learning method would also be impacted by measurement error including the error caused by using nacelle wind speeds without a transfer function, so we would not expect any method to produce an error of 0 kW per 2 MW turbine in this study. The super turbine approach will typically use power measured at a meter for the entire farm whereas the turbine approach will use the power measured at each turbine; however, there can be discrepancy between the sum of the turbine power values and the power measured at the farm's meter due to losses in transmission. Ultimately, utilities are interested in the power measured at the farm's meter or at a meter on the transmission line away from the farm, and the machine learning method should produce accurate predictions of power at that meter considering the effect of Jensen's inequality.

Jensen's inequality can produce significant differences in wind power conversion between a super-turbine approach and a turbine-level approach to power conversion, which we have named the super-turbine wind power conversion paradox. This analysis suggests that forecasters responsible for predicting power for a utility should perform power conversion at the turbine level or use machine learning to reduce the effects of Jensen's inequality in power conversion. Additionally, if the temporal standard deviation of wind speed is known, machine learning can incorporate both the effects from Jensen's inequality on the spatial and temporal variability of wind speeds across wind turbines at a wind farm.