WindCurves: A Tool to Fit Wind Turbine Power Curves

Introduction:

This is a Vignettes of R package, WindCurves. The package WindCurves is a tool used to fit the wind turbine power curves. It can be useful for researchers, data analysts/scientist, practitioners, statistians and students working on wind turbine power curves. The salient features of WindCurves package are:

Fit the power curve with Weibull CDF, Logistic [1] and user defined techniques.
Comparison and visualization of the performance of curve fitting techniques.
Utilise as a Testbench to compare performace of user defined curve fitting techniques with user defined error measures.
Availability of Dataset [2] on the power curves of wind turbine from four major manufacturers: Siemens, Vestas, REpower and Nordex.
Feature to extract/capture Speed Vs Power discrete points from power curve image

Instructions to Use:

A power curve can be fitted with the WindCurves package from a discrete samples of wind turbine power curves provided by turbine manufacurers as:

library(WindCurves)
data(pcurves)
s <- pcurves$Speed
p <- pcurves$`Nordex N90`
da <- data.frame(s,p)
x <- fitcurve(da)
#>    Weibull CDF model
#>    -----------------
#>    P = 1 - exp[-(S/C)^k]
#>    where P -> Power and S -> Speed 
#> 
#>     Shape (k) = 4.242446 
#>     Scale (C) = 9.564993 
#>    ===================================
#> 
#>    Logistic Function model
#>    -----------------------
#>    P = phi1/(1+exp((phi2-S)/phi3))
#>    where P -> Power and S -> Speed 
#> 
#>     phi 1 = 2318.242 
#>     phi 2 = 8.658611 
#>     phi 3 = 1.366054 
#>    ===================================
x
#> $Speed
#>  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
#> 
#> $Power
#>  [1]    0    0    0   35  175  352  580  870 1237 1623 2012 2230 2300 2300 2300
#> [16] 2300 2300 2300 2300 2300 2300 2300 2300 2300 2300
#> 
#> $`Weibull CDF`
#>  [1]    0.0000    0.0000    0.0000   90.3871  175.0000  327.5161  563.9085
#>  [8]  882.3965 1253.7489 1623.0000 1929.1254 2134.6685 2242.6251 2285.2438
#> [15] 2297.3355 2299.6816 2299.9764 2299.9990 2300.0000 2300.0000 2300.0000
#> [22] 2300.0000 2300.0000 2300.0000 2300.0000
#> 
#> $`Logistic Function`
#>  [1]    0.00000    0.00000    0.00000   74.12834  148.99362  289.70750
#>  [7]  530.79747  884.98926 1303.20935 1686.50694 1964.36660 2133.40768
#> [13] 2225.51217 2272.70001 2296.11395 2307.54706 2313.08621 2315.75964
#> [19] 2317.04756 2317.66748 2317.96573 2318.10919 2318.17820 2318.21138
#> [25] 2318.22734
#> 
#> attr(,"class")
#> [1] "fitcurve"
#> attr(,"row.names")
#>  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
validate.curve(x)
#>   Metrics Weibull CDF Logistic Function
#> 1    RMSE  30.8761687        38.8753476
#> 2     MAE  15.1381094        29.3213877
#> 3    MAPE   3.9292946         5.9183675
#> 4      R2   0.9989322         0.9983073
#> 5     COR   0.9995413         0.9991591
plot(x)

User can utilize WindCurves package as a testbench so that a new curve fitting technique can be compared with above mentioned existing environment.
- The user defined function (similar to random()) should be saved in .R format and it should return the fitted values in terms of “Power values”. The random() function used in this example is stored as .R file with dump() function and made available in the WindCurves package and written as:

random <- function(x)
{
  data_y <- sort(sample(1:1500, size = 25, replace = TRUE))
  d <- data.frame(data_y)
  return(d)
}
dump("random")
rm(random)

A random() function is attached in the comparison the parameters MethodPath and MethodName as shown below, where MethodPath is a location of the function proposed by the user and MethodName is a name given to the function.

library(WindCurves)
data(pcurves)
s <- pcurves$Speed
p <- pcurves$`Nordex N90`
da <- data.frame(s,p)
x <- fitcurve(data = da, MethodPath = "source('dumpdata.R')", MethodName = "Random values")
#>    Weibull CDF model
#>    -----------------
#>    P = 1 - exp[-(S/C)^k]
#>    where P -> Power and S -> Speed 
#> 
#>     Shape (k) = 4.242446 
#>     Scale (C) = 9.564993 
#>    ===================================
#> 
#>    Logistic Function model
#>    -----------------------
#>    P = phi1/(1+exp((phi2-S)/phi3))
#>    where P -> Power and S -> Speed 
#> 
#>     phi 1 = 2318.242 
#>     phi 2 = 8.658611 
#>     phi 3 = 1.366054 
#>    ===================================

## The user can specify .R files from other locations as:
# x <- fitcurve(data = da, MethodPath = "source('~/WindCurves/R/random.R')", MethodName = "Random values")
validate.curve(x)
#>   Metrics Weibull CDF Logistic Function Random values
#> 1    RMSE  30.8761687        38.8753476   992.9108117
#> 2     MAE  15.1381094        29.3213877   851.0000000
#> 3    MAPE   3.9292946         5.9183675   106.9830660
#> 4      R2   0.9989322         0.9983073    -0.1042121
#> 5     COR   0.9995413         0.9991591     0.8871836
plot(x)

Also, WindCurves allows user to add more error measures with validate.curve() function as explained below:

Consider error() is a function which uses two vectors as input and returns a error value with a specific error measure, such as RMSE or MAPE as shown below:

# PCV as an error metric
error <- function(a,b)
{
d <- (var(a) - var(b)) * 100/ var(a)
d <- as.numeric(d)
return(d)
}
dump("error")
rm(error)

The effect of this function can be seen in the results obtained with Validate.curve() function as:

library(WindCurves)
data(pcurves)
s <- pcurves$Speed
p <- pcurves$`Nordex N90`
da <- data.frame(s,p)
x <- fitcurve(da)
#>    Weibull CDF model
#>    -----------------
#>    P = 1 - exp[-(S/C)^k]
#>    where P -> Power and S -> Speed 
#> 
#>     Shape (k) = 4.242446 
#>     Scale (C) = 9.564993 
#>    ===================================
#> 
#>    Logistic Function model
#>    -----------------------
#>    P = phi1/(1+exp((phi2-S)/phi3))
#>    where P -> Power and S -> Speed 
#> 
#>     phi 1 = 2318.242 
#>     phi 2 = 8.658611 
#>     phi 3 = 1.366054 
#>    ===================================
validate.curve(x = x, MethodPath = "source('dumpdata.R')", MethodName = "New Error")
#>     Metrics Weibull CDF Logistic Function
#> 1      RMSE  30.8761687        38.8753476
#> 2       MAE  15.1381094        29.3213877
#> 3      MAPE   3.9292946         5.9183675
#> 4        R2   0.9989322         0.9983073
#> 5       COR   0.9995413         0.9991591
#> 6 New Error  -1.8141636         0.4127417
plot(x)

Similarly, user can compare various techniques used for wind turbine power curve fitting.

Dataset [2] on the power curves of wind turbine from four major manufacturers: Siemens, Vestas, REpower and Nordex. These datsset represent wind turbine power output in ‘kW’ against wind speed in ‘metres per second’ as shown below:

data(pcurves)
pcurves
#>    Speed Vestad V80 Vestad V164 Siemens 82 Siemens 107 Repower 82 Nordex N90
#> 1      1          0           0          0           0          0          0
#> 2      2          0           0          0           0          0          0
#> 3      3          0           0          0           0          0          0
#> 4      4          2         101         42          80         64         35
#> 5      5         97         461        136         238        159        175
#> 6      6        255         902        276         474        314        352
#> 7      7        459        1595        470         802        511        580
#> 8      8        726        2513        727        1234        767        870
#> 9      9       1004        3737       1043        1773       1096       1237
#> 10    10       1330        4988       1394        2379       1439       1623
#> 11    11       1627        5987       1738        2948       1700       2012
#> 12    12       1772        6698       2015        3334       1912       2230
#> 13    13       1797        6984       2183        3515       2000       2300
#> 14    14       1802        6985       2260        3577       2040       2300
#> 15    15       1802        6995       2288        3594       2050       2300
#> 16    16       1802        6995       2297        3599       2050       2300
#> 17    17       1802        6995       2299        3600       2050       2300
#> 18    18       1802        6995       2300        3600       2050       2300
#> 19    19       1802        6995       2300        3600       2050       2300
#> 20    20       1802        6995       2300        3600       2050       2300
#> 21    21       1802        6995       2300        3600       2050       2300
#> 22    22       1802        6995       2300        3600       2050       2300
#> 23    23       1802        6995       2300        3600       2050       2300
#> 24    24       1800        6995       2300        3600       2050       2300
#> 25    25       1800        6995       2300        3600       2050       2300

Further, WindCurves package provides feature to extract/capture Speed Vs Power discrete points from power curve image. This can be achieved with the following simple instruction:

#img2points("image.jpeg")

where, image.jpeg is the name of power curve image from which discrete points are to extracted. The procedure of extraction is as follows:

With the command img2points("image.jpeg"), an image will get appeared in the Viewer.
Click on leftmost point of the X-axis of the curve image and assign the respective appropriate value.
click on rightmost point on X-axix, downmost point of Y-axis and topmost point of Y-axis, and assign their values, respectively.
click of the points on the curve image. By default, 15 points can be captured from the curve image, but user can set the desired number of points with n parameter in img2points() function.
This function returns a data.frame with two columns, i.e., wind speed and wind power, which can be used as input data for fitcurve() function.

References

[1] D. Villanueva and A. E. Feij´oo, “Reformulation of parameters of the logistic function applied to power curves of wind turbines,” Electric Power Systems Research, vol. 137, pp. 51–58, 2016.(via)

[2] Iain Staffell, “Wind turbine power curves, 2012” (via)