Wind power correction model designed by the quantitative assessment for the impacts of forecasted wind speed error

Abstract

The errors in wind power forecast will incur additional cost. It is critical to quantify the relationship between forecasting error in wind speed and power output. Unlike previous works that have rarely considered the speed error, we propose a comprehensive and repeatable wind power forecast correction model that quantitatively assess the impacts of speed error on power error, based on the power curves, speed predictions and distribution of speed forecast error. In this correction model, the power forecast error is obtained by calculating the mathematical expectation. The mathematical expectation of the wind power error is equal to the integral of the wind power error multiplied by its associated probability. Additionally, power forecast error and its probability are constructed as a function of speed forecast error and speed forecast error probability, respectively. To evaluate the model performance, numerical simulations are carried out in Guilin, Xiangyang and Xihai. The results suggest that the model can reduce the biases between observed and forecasted power, with the correlation coefficients increasing by over 15% in Guilin and Xihai. Furthermore, the root mean square error exhibits notable decline, with a reduction of over 35%, from 0.34 to 0.21 MW, from 0.42 to 0.27 MW and from 0.39 to 0.24 MW in the three aforementioned locations, respectively. This study contributes to enhancing the efficiency of wind power generation.

Headnote

Abstract

The errors in wind power forecast will incur additional cost. It is critical to quantify the relationship between forecasting error in wind speed and power output. Unlike previous works that have rarely considered the speed error, we propose a comprehensive and repeatable wind power forecast correction model that quantitatively assess the impacts of speed error on power error, based on the power curves, speed predictions and distribution of speed forecast error. In this correction model, the power forecast error is obtained by calculating the mathematical expectation. The mathematical expectation of the wind power error is equal to the integral of the wind power error multiplied by its associated probability. Additionally, power forecast error and its probability are constructed as a function of speed forecast error and speed forecast error probability, respectively. To evaluate the model performance, numerical simulations are carried out in Guilin, Xiangyang and Xihai. The results suggest that the model can reduce the biases between observed and forecasted power, with the correlation coefficients increasing by over 15% in Guilin and Xihai. Furthermore, the root mean square error exhibits notable decline, with a reduction of over 35%, from 0.34 to 0.21 MW, from 0.42 to 0.27 MW and from 0.39 to 0.24 MW in the three aforementioned locations, respectively. This study contributes to enhancing the efficiency of wind power generation.

Keywords: Correction model; Wind speed error; Wind power error; Quantitative computation

(ProQuest: ... denotes formulae omitted.)

1. Introduction

With huge potential for greenhouse gas (GHG) reduction, wind power serves as an important alternative to coal-fired power and has achieved a high penetration level in recent years (McElroy et al., 2009; Chen et al., 2021). However, in contrast to the conventional thermal power generation, largescale wind generation integration poses challenges to the safe and stable operation of grids. This is primarily due to the randomness, fluctuation, and intermittence of its output power (Ahmed et al., 2020; Albadi and El-Saadany, 2010). Wind speeds exhibit a high degree of variability, resulting in fluctuations in wind power output. This makes it challenging to align the supply of wind energy with the demand of the grid. Therefore, power dispatching authorities need to establish detailed scheduling plans and set spinning reserve capacity for power generation (Lei et al., 2009). Accurate power forecasts for future days are crucial (Lange, 2005), which mainly depend on the wind speed from numerical weather prediction models (Watson etal., 1994). Numerical prediction methods have been modified in previous studies with the objective of reducing wind forecast error. The near-surface wind speed and energy over Central Asia are projected using dynamical downscaling with bias-corrected global climate models (Zha et al., 2024). In addition, the application of reliability ensemble averaging helps to reduce the inherent uncertainties associated with surface wind speed projections over China in the 21st century (Zhang and Xu, 2024). Nevertheless, the forecasted wind speed is often inaccurate due to the limitations of the current weather forecasting skills (Alexiadis et al., 1998). Especially in regions with complex terrains and under extreme weather and climate events, the error in speed forecast becomes considerable owing to a series of unavoidable factors, including uncertainties in the weather systems, rough surface resolution, and the applicability of parameterization schemes in numerical modeling (Palmer, 2000; Liu et al., 2012, 2022; Guo et al., 2021). Notably, the forecasted error is magnified when transforming speed to power, due to the effective nonlinearity property in the power curve (Lange, 2005). Given the difficulties in substantially upgrading the accuracy of speed forecasts in a short period, it is critical to estimate the predicted power error caused by speed forecast error as accurately as possible for rational power generation planning and forecasted power correction.

Power forecast error results in load loss and wind energy loss (Wu et al., 2014), and is essential for the optimal operation of wind farms (Ogimi et al., 2013). Previous studies attempt to calculate the power error by defining an appropriate probability density function (PDF) (Bludszuweit et al., 2008; Wu et al., 2014). The statistical distribution of power error has been proved to depend on the wind speed forecast (Menemenlis et al., 2012). For low wind speed, the forecast tends to underestimate the observed wind power, whereas for high speed, the forecast generally overestimates the observed wind power (Mauch et al., 2013; Ko et al., 2015). Moreover, several statistical distributions, such as the Gaussian distribution (Lange and Waldl, 2001) and the @-distribution (Fabbri et al., 2005), have been posed to describe the PDF.

Studies have attempted to correct power forecast (Wen et al, 2014), and machine learning also contributes to improving power forecast (Liang et al., 2016). For instance, the wavelet neural network utilizing multidimensional Morlet wavelet as the hidden layer neuron function and employing maximum correlation entropy criterion as the training criteria has been proposed (Chitsaz et al., 2015). Additionally, the weighting-based approaches are applied to combine various methods for correcting power forecasts (Xiao et al., 2015). Data preprocessing techniques are also used to decompose nonlinear wind power time series into regular elements (Liang et al., 2015). Most of the aforementioned works rely sorely on historical power forecast data and tend to ignore the impact of speed error, which directly determines power error (Watson et al, 1994). Moreover, although machine learning can enhance the accuracy of power predictions, the mechanisms by which meteorological factors affect power forecast error remain unclear, and the quantitative relationships between error in forecasted power and speed continue to warrant further investigation.

In elucidating the power forecast error, the uncertainties associated with speed error plays an important role (Menemenlis et al., 2012; Lange and Waldl, 2001; Rodríguez et al., 2015). Moreover, the relationship between power and speed is represented by the power curve. Consequently, power error is correlated with both power curve and wind speed error (Mauch et al., 2013). It is also noteworthy that, in contrast to the Gaussian distribution of speed error, power error presents strongly unsymmetric distribution, primarily due to the effective nonlinearity property that arises from transforming speed to power (Lange, 2005; Lei et al., 2009). Based on the power curve and normal distribution of speed forecast error, the 95% confidence interval for speed error is calculated to correct the forecasted power (Ko et al., 2015). Furthermore, by determining the speed error along with the elements involved in power estimation, and the corresponding uncertainties, the evaluation of wind resources can also be conducted (Rodríguez et al., 2015).

Although many studies have attempted to describe the power error in conjunction with speed error (Watson et al., 1994; Alexiadis et al., 1998; Parsons et al., 2004), most of them primarily focus on the PDF of power error. Investigations are notably lacking a comprehensive and repeatable model that elucidates the relationship between error in speed and power, which limits the understanding of the underlying physical mechanism, and complicates the correction of forecasted power for other wind farms based on their findings. Ko et al. (2015) attempt to correct power forecast by considering speed error; however, their studies constrained by specific circumstance, e.g., assuming a 95% confidence level for speed error and neglecting the shutdown of a turbine due to speed error, Which is a critical concern for the power system (Kay and MacGill, 2014). If the observed wind speed is slightly below cut-out wind speed and the speed forecast is slightly above it, speed error would be nearly zero, yet the power forecast could still exhibit large error. Therefore, a model for quantitatively computing power error based on the speed error Will present intuitive universal results, which will not only drive advancements in numerical weather prediction but also enhance the correction of wind power by subtracting the power error.

In consideration of the limitations of previous power forecast correction models, which are confined to particular scenarios and lack an understanding of the physical mechanisms, as well as an appreciation for the impacts of meteorological factors, we present the theoretical framework to obtain the power error as a function of speed error. On this basis, we put forth a comprehensive and repeatable wind power forecast correction model through a quantitative assessment of the impacts of speed forecast error on power forecast error. This work breaks through the limitations of traditional models by proposing a new theoretical framework that captures the intrinsic relationship between wind speed and power prediction error. By establishing a quantitative assessment method, our model achieves the correction of power forecast, enhancing the accuracy of wind power predictions, thereby contributing to the advancement of renewable energy generation, and strengthening the stability and reliability of power systems.

2. Data and methods

2.1. Data

2.1.1. Observed and forecasted wind speeds

We employ observed and day-ahead predicted wind speeds from three wind turbines in China, utilizing three one-year datasets at 15-min intervals, amounting to approximately 3 x 2 x 35,040 (the number of moments for predicted and observed wind speed from three wind turbines). The aforementioned data serve to validate our proposed model, as shown in Table 1.

In contrast to the speed forecast error, the PDFs of wind speeds exhibit a Weibull distribution, as illustrated in Fig. 1. The discrepancy between the forecasted and observed wind speed distributions in Guilin is minimal, with the observations exceeding 10 m/s slightly more frequently than the forecasts. Notable discrepancies are evident in the forecasts for Xiangyang and Xihai, with the former showing an overestimation and the latter an underestimation of the observed values. Moreover, Guilin, Xiangyang, and Xihai are situated in the respectively. The discrepancies between the observed and predicted wind speeds, along with the locations of the three stations, provide an effective means of verifying the model.

2.1.2. Parameters of wind turbine

The wind turbine produces reactive power until the speed is larger than the cut-in speed (Veut-in). After reaching the rated speed (Vrated), the wind turbine maintains the rated power state until the speed reaches the cut-out speed (Veut-out). Accordingly, the wind speeds are divided into four regimes, according to the power curve, including low speed, variable speed, rated speed and high speed (Lange, 2005; Ko et al., 2015). We select the wind turbine, with a rated capacity of 1.6 MW, Veut-in of 3 m/s, Vratea Of 10 M/S, Veut-out Of 25 m/s, and the relatively fixed parameters (C) of 1600 to verify the model.

2.2. Methods

2.2.1. Data preprocessing

The collected data contain instances of missing values for certain observed wind speeds. To address the issue, we employ a linear regression approach to fill in these gaps. The wind speeds at the selected time point and five subsequent time points serve as learning target and input array, respectively, following Tawn et al. (2020) and Muhammad (2020). It is important to note that the purpose of this study is to correct the power forecast. The inputs to the model are historical observations and forecasted speeds, which are subsequently used to obtain the distribution of speed forecast error. It is of paramount importance to ascertain specific values for the average and standard deviation of speed forecast error. Therefore, data standardization may improve the correlation between modelderived and observed power error, but the model-derived wind power forecast correction. Accordingly, we do not standardize the data.

2.2.2. Correcting wind power forecast

The forecasted power error in the proposed correction model is obtained by calculating the mathematical expectation. The mathematical expectation of the wind power error is equal to the integral of the wind power error multiplied by the probability, as shown in Eq. (1).

...

where E, Perror, and f present the mathematical expectation, power forecast error and PDF of power forecast error, respectively.

The power forecast error and its PDF are constructed as a function of the speed forecast error and the PDF of the speed forecast error, respectively.

Forecasted and observed power can be described as piecewise functions of the corresponding forecasted and observed speed, respectively. Observed power and speed can be considered as forecasted power and speed minus the associated error, respectively. Notably, although day-ahead speed forecast is variable on from moment to moment, it can be obtained from numerical weather prediction a day in advance. When correcting wind power for a latter day, forecasted speed corresponds to a known value. Moverover, observed speed can be expressed as the forecasted speed minus its error. Consequently, when computing power error, the only unknown variable is the wind forecast error, and power error can be regarded as a function of speed forecast error, defined as function (F).

Due to the differing power curves associated with various speed regimes, power error can be regarded as four piecewise functions, based on the magnitude of wind speed forecast in relation to cut-in, cut-out, and rated wind speeds, as illustrated in Fig. 2.

It is important to note that the forecasted speed is variable at each moment, which implies that the function (F) is different for each moment. A detailed description of the determination of the expression for the function (F) can be found in the Appendix (Text Al).

The mathematical expectation of power forecast error can be regarded as the integral of the function (F) multiplied by the probability, by introducing function (7) into Eq. (1). The PDF of speed forecast error from NWP can be modeled using Gaussian distribution with mean (u) and standard deviation (o) of speed error (Lange, 2005; Ko et al., 2015). To increase the accuracy of the forecasted power, the forecasted power error is subtracted from the forecasted power in this correcting mode (Lange, 2005; Ko et al., 2015).

2.2.3. Software implementation and computational efficiency

The model runs in python v3.10.7 on a local machine, with the CPU of 13th Gen Intel(R) Core(TM) 17-13700К. The code comprises multiple modules, including variable definitions for reading inputs and saving outputs, as well as custom functions for obtaining the distribution of speed forecast error, calculating the mathematical expectation of the power error and correcting power forecast. The inputs to the code include the speed forecast at a given time point, along with the observed and forecasted speed over the entire historical time period. The output of the code is the corrected power forecast at the given time point. The time required to correct wind power for a wind turbine at a given moment is less than 0.001 s on average, demonstrating the high computational efficiency of this model.

3. Results

3.1. Comparisons between model-derived and observed wind power forecast error

In order to derive the power forecast error using the proposed model, an analysis is conducted on the observed and forecasted speeds of wind turbines in the three aforementioned locations (Fig. 3a-c). The observed wind speeds at the these locations exhibit distinct seasonal patterns. The speeds at Guilin demonstrate higher values during winter and lower values during summer. In constrast, the speeds at Xihai show an inverse pattern. Additionally, the speed of the Xiangyang turbine does not exhibit discernible seasonal patterns.

The forecasted speeds at the three locations exhibit a similar seasonal pattern to that observed in the actual speeds, with the correlation coefficient between observed and forecasted wind speed of 0.78, 0.78 and 0.81 in Guilin, Xiangyang and Xihai, respectively. However, a discrepancy is evident in the magnitude of forecast error. The average of error at Guilin is -0.45 m/s', which is relatively low. Conversely, the speeds at Xiangyang and Xihai are underestimated and overestimated, respectively, on average by up to 2.12 and 1.47 m/s. It is also notable that all the three PDFs of speed forecast error can be modeled using Gaussian distributions (Fig. 3d-f).

The model-derived and observed power error are presented in Fig. 4. Despite the varying distributions of wind speed prediction error observed at the three sites, a notable degree of similarity exists between the modeled and observed power error at each location, with correlation coefficients of 0.83, 0.91, and 0.77 in Guilin, Xiangyang, and Xihai, respectively. This evidence substantiates the efficacy of the model in replicating the characteristics of power forecast error.

3.2. Comparisons between wind power forecast before and after correction

The comparisons between the observed, forecasted, and corrected wind power at three locations are presented in Fig. 5. The power demonstrates a similar seasonal pattern to wind speeds in Guilin, Xiangyang and Xihai. However, whereas the correlation coefficient between forecasted and observed speeds is high (r = 0.78), that power correlation in Xiangyang is only 0.41. This indicates the fundamental impact of the power curve in amplifying error in speed prediction (Lange, 2005).

After the subtraction of the model-derived power error, the correction model also serves to reduce the bias between observed and forecasted power. The correlation coefficients demonstrate an increase from 0.79 to 0.91 in Guilin, from 0.41 to 0.43 in Xiangyang and from 0.79 to 0.91 in Xihai. The root mean sguare error (RMSE) exhibits a notable decline, from 0.34 to 0.21 MW, from 0.42 to 0.27 MW, and from 0.39 to 0.24 MW in Guilin, Xiangyang, and Xihai, respectively, as illustrated in Table 2. It is noteworthy that although the increase in the correlation coefficient in Xiangyang is relatively lower than that observed in other locations, the reduction in RMSE of over 35% implies the efficacy of the model in correcting power forecasts, even when the correlation between predicted and observed power is low.

To further verify the model performance, the observed and forecasted wind speeds of a wind turbine over half a year (from 1 January to 30 June 2021), with a temporal resolution of 15 min and a total of 2 x 17,520 data points, are employed (Fig. A1). The correlation coefficient between model-derived and observed power forecast error is 0.87. After subtracting the model-derived power error, the correlation coefficient increases from 0.75 to 0.94, and the RMSE is reduced from 0.38 to 0.17 MW. These findings demonstrate that the proposed model is effective in correcting wind power forecasts, provided that the wind speed forecast error follows a normal distribution over the designated time period.

3.3. Sensitivity analysis of the model to input parameters

To ascertain the sensitivity of the model to the input parameters, six scenarios of random disturbances are introduced to the speed forecast at each moment for three wind farms. The scenarios entail a random increase or decrease of 0-10%, 0-20%, and 0-30% in the speed. The mean speed forecast error for the six scenarios in Guilin is found to be -0.42, -0.24, -0.06, -0.78, -0.96, and - 1.14 m/s", with a standard deviation of 2.08, 2.09, 2.11, 2.10, 2.13, and 2.17 m/s, respectively, as shown in Table 3. The correction model demonstrates efficacy in reducing the bias between observed and forecasted power for all six scenarios, as evidenced by the increased correlation coefficients and reduced RMSEs, as illustrated in Table 3 and Fig. 6. Furthermore, sensitivity analyses in Xiangyang and Xihai yield analogous results (Tables Al and A2), thereby substantiating the robustness of the model.

4. Discussion and conclusions

In contrast to previous power forecast correction models, which are limited in scope and did not account for meteorological factors, this study presents a comprehensive and repeatable wind power forecast correction model that provides an understanding of the underlying physical mechanisms. Based on the Gaussian distribution of wind forecast error from NWP and the power curve, the model is proposed through a quantitative assessment of the impact of speed forecast error on power forecast error.

The power forecast error in this correction model is obtained by calculating the mathematical expectation. The mathematical expectation equals the integral of the wind power error multiplied by the probability, and is constructed as the integral of function of the speed forecast error multiplied by the probability of the speed forecast error. The critical process is providing the power error as a function of speed error. Furthermore, the forecasted power is corrected by subtracting the power error to improve power prediction accuracy. The results show that our model can not only accurately capture the relationships between error in forecasted power and speed, but also effectively reduce the bias between observed and forecasted power. The correlation coefficients increase by over 15% in Xiangyang and Xihai. The RMSE exhibits notable declines, with reductions of over 35% in all the locations.

Although an explicit quantitative assessment and correction model is carried out here, the noises caused by the small sample size lead to the asymmetric distributions of speed forecast error, which then influence the accuracy of the modelderived error. In addition, the distribution of speed error depends on the location of the wind farm, local meteorological conditions, and different seasons. Due to the data limitations, only three one-year datasets are obtained from three wind farms. The interannual and seasonal variations of wind speed may result in disparate distributions of speed forecast error, which warrant further investigations in practical applications when the quantity of data is sufficiently robust. Furthermore, alterations to numerical weather prediction methods, e.g., the transition from WRF v4.1 to WRF v4.3, may impact the distribution of speed forecast error, thereby introducing uncertainty into the results of the correction model. Overall, our model not only quantifies the effects of speed forecast error but also has the potential to reduce the forecasted power error and facilitate the prediction of wind power as accurately as possible, which will be beneficial for the development and utilization of wind energy. Notably, uncertainties may remain for a prolonged period at the forecasted wind speed (Palmer, 2000). Accordingly, in addition to the observed wind resources, the potential power loss due to wind forecast uncertainties can be evaluated by our model, which will assist in the selection of optimal locations for future wind farm construction.

Declaration of competing interest

The authors declare no conflict of interest.

CRediT authorship contribution statement

Zhi-Qi Xu: Writing - review & editing, Writing - original draft, Methodology, Investigation, Funding acquisition, Data curation, Conceptualization. Tong Xue: Writing - original draft, Visualization, Resources, Methodology, Data curation, Conceptualization. Xin-Yu Chen: Resources, Methodology, Investigation, Funding acquisition, Data curation, Conceptualization. Jin Feng: Methodology, Investigation, Funding acquisition, Data curation, Conceptualization. Gu-Wei Zhang: Writing - review & editing, Writing - original draft, Visualization. Cheng Wang: Writing - review & editing. Chun-Hui Lu: Project administration, Formal analysis, Data curation, Conceptualization. Hai-Shan Chen: Project administration, Formal analysis, Data curation, Conceptualization. Yi-Hui Ding: Project administration, Formal analysis, Data curation, Conceptualization.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (42205040, 42275009 and 42205170), the Beijing Meteorological Bureau (202201007), Key Laboratory of Meteorological Disaster, Ministry of Education & Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters open research project (202303 and 202306), and China Meteorological Administration Training Centre Youth Research Program (2023 CMATCQNOS).

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.accre.2024. 12.006.

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