1. Introduction
Wind energy is a mainstream source of electricity and continues to grow in usage to meet the future energy demands while reducing carbon emissions. At the end of 2019, the U.S. reached 105 GW wind energy capacity and continues to grow with 9143 MW capacity added in 2019 [
1]. The U.S. now aims to generate 20% of the nation’s electricity by 2030 and 35% by 2050 with wind energy [
2]. More scientific research is needed to optimize wind farm and operation to meet the global demand for clean energy [
3]. Wind turbine wakes, which reduce wind speed, are an important and ongoing challenge of wind energy research [
4]. Turbine wakes are responsible for the change of flow field within a wind farm that affects the local climate and reduces power generation [
3]. The effect of wind turbine wakes has been studied extensively using numerical, experimental, and analytical methods [
5,
6]. Although numerical and experimental studies can provide an accurate result in predicting the velocity and power deficit of the turbine arrays, they are costly and time-consuming. Analytical methods provide a simple, fast, and low-cost approach to model wind turbine wakes [
5,
7]. For these reasons, industry favors the analytical method for wind farm optimization [
8].
Until now, there are many types of analytical wake models, such as the Jensen model [
9,
10], the Larsen model [
11], and the Frandsen model [
12]. The most commonly used wake model in the industry is still the Jensen wake model, which was developed in the early 1980s and assumed a top-hat distribution in the wake region [
9,
10]. This model has been extensively used in commercial software for wind farm planning or design, such as WAsP, Wind Farmer, and OpenWind [
8]. The Jensen model is simplified by only considering the conservation of mass and does not consider the conservation of momentum [
13]. While known for its simplicity and its ability to represent a realistic wake, the Jensen model has the disadvantage of overestimating the wake deficit in the far-wake region [
14,
15].
Recent studies, using both experimental [
16] and numerical [
17] experiments, revealed the velocity distribution is better represented by a self-similar Gaussian wake profile. Bastankhah and Porté-Agel proposed an analytical wake model that assumes a Gaussian shape in the far-wake region that applies both mass and momentum conservation [
13]. To calculate the velocity deficit, the model requires an accurate determination of the wake growth rate and an estimation of the thrust coefficient. This model has been validated based on field measurements at a utility-scale wind turbine [
18].
A recent study by Niayifar and Porté-Agel demonstrated that the new model can represent turbine wakes and power losses in an uniform layout offshore wind farm, Horns Rev [
15]. The Horns Rev offshore wind farm has a regularly spaced wind turbine and exhibits a significant power deficit on actual measurements [
19]. Unlike regularly spaced offshore wind farms, many onshore wind farms, especially in the US, have irregularly spaced layouts. The power deficit at these wind farms may not be as significant as observed at the Horns Rev wind farm. Niayifar and Porté-Agel’s study calibrated the analytical model’s parameters with the numerical data generated from the Large Eddy Simulation (LES). The results were also compared against the LES results. In Fuertes et al.’s study [
18], the model was tested on a full-scale wind turbine and the parameter was calibrated based on LiDAR measurements made from the turbine’s nacelle. In both studies, the parameters were different and may suggest those parameters vary depending on local conditions.
For wind farm design or operational simulation, it is expensive and time-consuming to calibrate the model with experimental and LES numerical data. Furthermore, using existing parameters from previous numerical and experimental studies may cause inaccurate predictions due to the change of wind turbine specification and local variation. Previous studies have compared several analytical models’ prediction with wind turbine operation data [
20,
21,
22]. In Göçmen et al.’s study [
21], several wake models were compared without including the most recent Gaussian-based analytical wake model. In Archer et al.’s study [
22], a number of analytical model including two Gaussian-based analytical models, the Bastankhah and Porté-Agel model [
13] and the Xie and Archer model [
23], were evaluated for the first time against operation data from large wind farms. They concluded six models performed well in three studied wind farms, although the Xie and Archer model and Jensen model had higher prediction accuracy. However, the model’s key parameters, wake growth rate, in both Bastankhah and Porté-Agel model [
13] and Xie and Archer model [
23], were set constant values based on previous LES studies. The field experiment on a full-scale turbine [
18] and the LES study at the Horns Rev wind farm [
15] suggested that the wake growth rate should not assume to be constant; instead, it is a function of turbulence intensity. Furthermore, in the LES study at the Horns Rev wind farm [
15], the assumption of a constant wake growth rate in the Gaussian-based model produced an obvious underestimation of the power output. Allowing for variation in the wake growth rate makes the Bastankhah and Porté-Agel model superior to the Jensen model, which assumes a constant wake growth rate [
15].
In order to use the Gaussian-based analytical model operationally, methods avoiding the use of time-consuming and costly experiments and computer simulations for model calibration are needed. Using previous numerical or experimental studies’ results may produce inaccurate predictions in different wind farm settings due to the changes in wind turbine specifications. Wind farms are equipped with Supervisory Control And Data Acquisition (SCADA) systems, which record basic meteorological and wind turbine operating conditions, such as wind speed, wind direction, and power generation. The data provides useful information, which has been used extensively in power prediction, wind turbine health assessment, and operation management [
24,
25]. Studies have established an understanding of the wake effect in existing wind farms using SCADA data to perform power predictions [
20,
21,
22,
26,
27]. The studies that are purely based on SCADA data and data-driven approaches generally can provide an accurate power deficit prediction and inform on wind turbine operation. However, machine learning methods cannot provide a description of the flow field and only gives predictions at locations where turbines are installed. Due to the inability to describe the flow field, machine learning methods cannot be used in wind farm expansion design, wind farm environmental risk assessment, and weather model improvements. A flow physic-based model is still needed. Thoroughly understanding the flow field within the wind farm aids in future engineering decisions and applications.
To improve analytical wake model accuracy for wind power plant simulation, the model must be calibrated based on the specific wind farm setting. Computer simulation and experiments are complex, expensive, and time-consuming, which is not feasible in industrial wind farm applications. In this study, our goal is to develop a simple procedure for calibrating the Gaussian-based analytical wake model to a specific wind farm using SCADA data assimilation. The proposed procedure utilizes a Gaussian-based analytical wake model with SCADA data to describe the flow field within the wind farm and power losses. A comparison between standard wake modeling and proposed wake modeling will be provided. This is the first demonstration of using SCADA data to calibrate the Gaussian-based analytical wake model without using extensive simulation and experiment.
The paper is as follows. In
Section 2 the analytical wake models used in this study are described. The wind farm wake modeling procedure with the Gaussian-based analytical model using SCADA data assimilation is proposed in
Section 3 and a case study at a large wind farm in Iowa, U.S. is shown. The results and discussion about the case study are included in
Section 4, and conclusion and recommendations are provided in
Section 5.
4. Case Study Results and Discussions
This section presented a case study demonstrating the proposed procedure for Gaussian-based analytical wake modeling with wind turbine SCADA data. The proposed procedure introduced in
Section 3 and its results were compared with the industry standard model, the Jensen model [
9,
10]. Since the study wind farm is an onshore wind farm and the surrounding land is mostly agricultural, a value of 0.075 was selected for the wake effect decay constant in the Jensen model. The time-window for this case study was 10 minutes.
To validate the model’s ability to analyze individual turbine wakes, an array of three turbines were selected from the study wind farm. Since the wind farm is irregularly spaced and prevailing wind directions are southeast and northwest, only an array of three turbines were available for the array of the wakes interaction demonstration. These three turbines are in the northeast corner of the wind farm with 630 m (6 rotor diameters) between the first and second turbines, and 690 m (6 rotor diameters) between the second and the third turbine, shown in
Figure 10.
Figure 11 shows the normalized power output generated from the proposed procedure and the Jensen model for the three-turbine-array under perfect alignment at the center of wake (a) and partial wake condition with 3° degree wind direction offset (b). The prediction of the power was calculated under a carefully selected 10 minutes average case. The wind direction and wind speed were constant for a period of time preceding the analyzed 10 minute time window to avoid the turbine’s response time affecting the results. The turbulence intensity was selected based on the suggestion from Fuertes et al.’s study [
18] to avoid the near-wake region, where the velocity deficit is not well represented by the Gaussian-based analytical model.
Both models tended to produce errors in the power prediction of the second turbine, but the Gaussian-based wake model outperformed the Jensen model in overall accuracy. As shown in
Figure 11, the Jensen model tended to underestimate the power output of the third turbine, which was also observed in the Horns Rev wind farm study [
15]. Under partial wake conditions, the Gaussian-based analytical wake model had better accuracy in predicting the power output because the Gaussian-based wake model can realistically predict the maximum velocity deficit, shown in previous studies [
13,
15,
18]. As shown in
Figure 12, the Jensen model produces an unrealistic velocity deficit in the wake, which has a uniform velocity distribution in the spanwise direction. This means the Jensen model underestimates the velocity deficit in the far wake region. The Gaussian-like velocity deficit assumption enables the model to accurately predict a more realistic maximum velocity deficit under partial wake condition. The wake growth rate parameter and turbulence intensity are adjusted according to the local streamwise turbulence intensity, so the wake recovery prediction is also more accurate than the Jensen model.
To demonstrate the performance of the Gaussian-based analytical wake model for a wind farm, we conducted a sitewide average wind speed and sitewide total power prediction assessment by using three different methods: a SCADA calibrated Gaussian-based model with a variable wake growth rate , the standard Jensen model, and a Gaussian-based model with constant . In this case study, ten-minutes averaged SCADA was used, and all turbines within the wind farm were considered. The sitewide average wind speed prediction assessment is done by comparing three models’ prediction of the average wind speed across all the operating turbines. The sitewide total power prediction assessment is done by comparing three models’ prediction of the total power of the wind farm generation. To assess the three models’ performance, Mean Absolute Percentage Error (MAPE), Root Mean Square Error (RMSE), and Mean Squared Error (MSE) were reported.
The cases were selected from a period of 4 months. A consistent data selection process was implemented. The chosen cases had stationary wind speed and wind direction for over a half-hour period. In this period, the wind speed for each operating turbine had to remain in the wind speed range of 5 – 10 m/s with at least 49 of the 51 turbines in the study wind farm operating. Situations where nearby patches of trees made determining the incoming wind speed too hard to determine were discarded. Overall, 100 cases were selected from a period of 4 months.
Figure 13 shows the percent relative error of the sitewide total power prediction for three models.
Table 1 shows models’ prediction errors of the selected 100 cases. In the comparison, a constant wake growth rate,
were used, which is the same as the previous study [
22]. The Gaussian-based analytical wake model with constant
has similar performance with the Jensen model, but the Jensen model performed slightly better. After calibrating the Gaussian-based analytical model with SCADA data using the proposed procedure, the Gaussian-based analytical model is found to outperform the Jensen model and produced less error for predicted wind farm total power. A variable wake growth rate for each turbine significantly improves the prediction. The variable wake growth rate can simulate a more realistic wind farm condition, where the wake recovers faster further away from the front row of the wind farm due to a higher turbulence intensity compared to ambient conditions. Increasing the number of cases did not significantly impact the results.
Furthermore, in sitewide average wind speed prediction, the Gaussian-based model with variable k* outperformed the Jensen model and Gaussian-based model with constant
.
Figure 14 shows the percentage relative error of the sitewide averaged wind speed prediction for 100 selected cases, and
Table 2 shows the models’ prediction errors of the 100 selected cases. The sitewide averaged wind speed was obtained by averaging all the wind speed at each turbine. In most cases, the sitewide averaged wind speed was less than uniform incoming wind speed. In terms of sitewide average wind speed prediction, the Gaussian-based model with variable
had less error, according to
Table 2. The Jensen model tended to overestimate the velocity deficit because of the assumption of a top-hat shape velocity deficit profile, which overestimated in the far-wake region.
In the following, the models’ predictions of individual turbines will be discussed. The study wind farm is located in central Iowa, USA. As shown in
Figure 15, the main direction of the wind was northwest and southeast. The average wind speed near the ground was approximately 10.4 mph (4.6 m/s).
Two cases that represent the main wind directions, 186° and 307° were selected. The flow field prediction of the calibrated Gaussian-based analytical wake model and mean average percentage error (MAPE) of wind speed prediction for all turbines is shown in
Figure 16.
Figure 16a shows the flow field of the 186° wind direction and
Figure 16b shows the flow field of the 307° direction. In both directions, by comparing predictions and true values, many errors can be observed in a certain region of the wind farm. Errors occurred at the front row turbine and the turbine in the far end region due to the uniform wind flow assumption. In addition, the Gaussian-based wake model is very sensitive to the wind direction due to the assumption of Gaussian shape, especially in the far-wake region. Assuming a uniform wind direction can result in inaccurate predictions of the velocity deficit because the model does not account for wind direction variation. This has observed for some areas of the wind farm. However, when the condition was ideal and ambient condition was stationary for some time, the prediction accuracy increased. The Gaussian-based wake model can accurately predict the power and velocity deficit of the downstream turbines.
In general, when predicting the individual turbine’s wind speed and power generation, the calibrated Gaussian-based model exhibits higher accuracy than the Jensen model, shown in
Table 3. The Gaussian-based model has a lower mean average percentage error (MAPE) and mean square error (MSE) for both wind speed and power prediction of individual turbines within the wind farm. The range of the percent error in power and wind speed prediction for individual turbines in each case study are shown in
Table 4. In comparison with Jensen model, the Gaussian-based model has a smaller range of percent error in both power and wind speed prediction.
Even though the accuracy of the individual turbine may vary according to the wind direction and other atmospheric conditions because of the complexity in nature, the model is still able to provide a reasonable prediction for sitewide average wind speed and total power generation.
Figure 17 shows a timeseries of sitewide average wind speed and sitewide average yaw direction from SCADA data as well as prediction errors from the Gaussian-based analytical wake model. The errors were the largest when the wind speed and yaw direction were not stationary. As the wind speed and yaw direction remained constant for a period of time, both wind speed and power prediction errors reduced.
Figure 17 provides evidence that the proposed procedure, and the Gaussian-based wake analytical model has the potential to accurately model the wind farm wake effects and power prediction for real-time operation of a wind farm.
Future improvement is needed to reduce the error in the period when the inflow conditions are evolving. Considering wind/wake transport time across the wind farm and variation in wind direction for different regions would improve the model’s accuracy in real-time modeling. Assuming a uniform incoming wind speed is a simple approach but not realistic in all wind farm conditions. Taking a wind speed gradient across the wind farm into account can provide more realistic modeling results. In addition, turbine yaw misalignment and turbine control should also be considered. Misalignment in the yaw direction is inevitable in the real wind farm operation. The yaw error can cause wake deflection, so a simple wake model may not accurately represent the behavior of wakes. Future analysis should consider wind farm yaw error and incorporate a wake deflection model.
The proposed procedure for calibration of the Gaussian-based analytical wake model for wind farm modeling provides a simple approach for wind farm operation analysis using only SCADA data. The proposed procedure enables the Gaussian-based wake model to simulate streamwise velocity on a horizontal plane at the hub height level using real-time data. The Gaussian-based wake model can model sitewide power generation and average wind speed, and it can provide insight into the power deficit at varied wind direction. With the proposed procedure for calibration of the model, the model has the potential to perform a real-time analysis of the wind turbine wake effects in a wind farm, identify waked regions, and determine turbines, which may be shut-off due to low wind speed. Incorporating turbulence intensity, the Gaussian-based model has the potential to model power based on specific operating conditions.
Understanding the wake behavior within the wind farm is very important for future applications. The wake model can help biologists to determine low wind speed regions within and near wind farms for wildlife impact mitigation. The model can also help to improve weather models, such as Weather Research and Forecasting Model (WRF), to account for wind farm wake effects in real-time.