How do you evaluate the accuracy of wind data measured by remote sensing equipment?

Introduction
Wind direction data correction
Regression coefficients and coefficients of determination
Warnings
Example :
Conclusion

Introduction

In wind observation for the purpose of wind power development, the use of remote sensing equipment such as Doppler LiDAR is rapidly expanding in order to obtain wind conditions at sites where observation masts are difficult to use.^1)2)3)4)shows statistics (regression coefficient and coefficient of determination) that can be obtained by referring to wind direction and speed measured by an observation mast as an index to quantitatively evaluate the accuracy of wind direction and speed measured by remote sensing equipment. It is known that there are various definitions of regression coefficient and coefficient of determination in statistics, but in this paper, we will use the guidelines, etc.^1)2)3)4)We will refer to this article to explain the statistics used in wind condition surveys, providing formulas and concrete examples.

Wind direction data correction

Wind direction, the direction from which the wind blows, takes on values ranging from 0° to 360°, but if each piece of data is used directly for accuracy verification, the two-dimensional positional relationship between the reference data 𝑥 (e.g., observation mast data) and the target data 𝑦 (e.g., Doppler lidar data) may result in the inclusion of data (raw data) that deviates significantly from the line 𝑦 = 𝑥 as shown in Figure 1, and therefore statistics will not be calculated correctly. Therefore, before evaluating accuracy, the following corrections are made to ensure that the two pieces of data align with the line 𝑦 = 𝑥.

here, 𝑥and the subscripts of 𝑦, 𝑖(=1,2,3,…,𝑁)As shown in Figure 1, the modified data can be obtained instead of the raw data, and the wind direction statistics can be calculated correctly.

Figure 1. Wind direction data manipulation

Regression coefficients and coefficients of determination

The wind direction and speed of the remote sensing equipment are explained using the data 𝑥𝑖 from the observation mast as explanatory variables and the data y_iIt is evaluated by linear regression with the objective variable.

The regression line of wind direction is expressed as y=ax+b with an intercept. The intercept of the regression line isbRoadmap for the commercial acceptance of floating LIDAR technology¹⁾According to the definition of x_iand y_iIt is expressed as the average difference between

here, 𝑥 ̅ The𝑥 _iThe average value ofy ̅ They_iThe slope of the regression line isais the interceptbBased on the least squares method, the following equation is obtained so that

The regression line for wind speed is expressed as 𝑦 = 𝑎′𝑥 with no intercept. The slope of the regression line 𝑎 can be calculated using the least squares method as follows:

As shown in equations (4) and (5), the slope of the regression coefficient with and without an intercept has different definitions.

Coefficient of determinationR²is how well the regression line fits, that is, the explanatory variablesxIt is an index that shows how well the objective variable y is explained. Tarald O. Kvalseth⁵⁾It is,R²Eight definitions forR²₁~R²₈), but this article focuses on the most common textbooks.⁶⁾It is published inR²₁is shown in the following equation.

here, f_iis the estimated value based on the regression line, and if the wind direction isf_i= 𝑎𝑥_i+b, if the wind speedf_i= 𝑎′𝑥_i.R²takes values between 0 and 1,R²When =1,𝑥_i y_iIt fully explains𝑥_iとy_iThis means that if you plot this on a graph, all the points will fall on a straight line.

Warnings

When calculating statistics in practice, you probably have many opportunities to use spreadsheet software (such as Microsoft Excel) or programming languages (such as Python and R). The regression analysis functions implemented in these programs are very useful for calculating statistics, but you need to fully understand how the statistics derived from the functions are defined before using them.

For example, a regression line obtained using a spreadsheet or programming languagey = 𝑎𝑥 + bThe intercept of is defined by the following equation based on the least squares method, which does not match equation (3) defined by the difference in means.

またR²It has been pointed out that the definition of (6) differs depending on the spreadsheet software and programming language, and on the presence or absence of an intercept.R²When calculatingR²₇is calculated,R²₁It has been confirmed that the value is larger than that⁷⁾⁸⁾.

Therefore, when calculating statistics based on equations (4) to (6) using a spreadsheet or programming language, we recommend that you build a program that faithfully follows the definitions by using your own functions instead of regression analysis functions.

Example :

As a concrete example of accuracy verification, Wind Farm Certification: Onshore Wind Power Plant Edition²⁾(hereinafter referred to as the NK Guidelines) and the Offshore Wind Observation Guidebook⁴⁾This section introduces the requirements shown in the .

The NK Guidelines require that correlations between observation data from observation masts and remote sensing instruments be verified according to the following conditions (1) and (2).³⁾.

(1) Combining observation data from an observation mast and remote sensing equipment to treat it as observation data at hub height and use it as input data for airflow analysis

(2) When observation data from an observation mast is used as input data for airflow analysis, and observation data from a remote sensing device is used only to verify the validity of the airflow analysis.

Table 1. Correlation between observation data from observation masts and remote sensing instruments³⁾

Offshore Wind Observation Guidebook⁴⁾defines KPIs (Key Performance Indicators) and acceptance criteria for the accuracy of wind direction and speed (Table 2). If the error of wind direction and speed measured by remote sensing equipment is below the minimum in the table, the measurement value is deemed to be of the same accuracy as that of a cup-type anemometer or arrow-type anemometer on an observation mast and can be used. The wind speed range is set to 4m/s or more and less than 16m/s because it is considered to have a large contribution to power generation and load for the purpose of calculating power generation forecasts and wind conditions.

Table 2 Accuracy KPIs and acceptance criteria

Conclusion

This article explained regression coefficients and coefficients of determination, which are used to evaluate wind condition data measured by remote sensing equipment. If you have any questions about the technical content, please feel free to contact RTI.

(Written by Takeyuki Misaki)

References

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Tarald O. Kvalseth, Cautionary Note about R2, The American Statistician Vol.39, No. 4, Part 1 (Nov., 1985), pp.279-285.
Department of Statistics, Faculty of Liberal Arts, University of Tokyo, Introduction to Statistics (Basic Statistics I), University of Tokyo Press.
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Iguchi Laboratory, Institute of Biological Sciences, Coefficient of Determination R²Differences: Using Excel, OpenOffice, LibreOffice and the statistical analysis software R, URL:
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