Accounting for the speed shear in wind turbine power performance measurement

3.1 Evaluation of the wind speed profile

In most studies about the effect of wind shear on power performance, the wind speed shear is described by the shear exponent α obtained from the assumption of a power law profile:

(1)

where z₀ is a reference height, u₀ the wind speed at this height and u is the wind speed at height z. We therefore chose to use this model as a tool to investigate the various profiles measured in this campaign. This assumption is indeed very convenient as it enables us to represent a whole wind speed profile using only one parameter. A common practice is to derive the shear exponent from the wind speed measurements at hub height and below and to assume that the profile is continuous (i.e. it has the same shear exponent) above hub height. However, this assumption is not always wise since profiles with a local extremum or high shear above hub height cannot be adequately represented by a power law profile. Therefore, it is important to actually measure the wind speed profile covering the whole rotor swept area.

In our experiment, the lidar measured the wind speed at nine heights equally distributed between z_hub −0.4D and z_hub + 0.4D, where z_hub is the hub height and D is the rotor diameter. Here, a shear exponent was obtained by fitting the nine measurement points of each profile to a power law. The fit was forced through the point of coordinate (u_hub, z_hub) where z_hub is the hub height and u_hub the wind speed at that height:

(2)

This shear exponent (α_fit) is considered to be more representative of the profile than a shear exponent retrieved from only two measurement points.

3.2 Profile classification

We have chosen to split the available profiles into two groups, for which, using the conventional power curve method, two significantly different power curves are demonstrated. Later we investigate whether a modified wind speed definition, including the shear in some way, can unify the power curves for the two profile groups. The classification of the profiles into two groups could have been performed in many ways. We have chosen to classify according to the conformity of the profiles to a classic power law. This classification is not part of the equivalent wind speed method, merely a tool for demonstrating the need and assessing the effectiveness of the new method. To separate the profiles for which the power law assumption was acceptable from the profiles for which it was not, we evaluated the goodness of fit for each profile with the residual sum of squares (RSS) defined as follows:

(3)

where u_fit is the fit function defined in 2 and u_i the wind speed measured by the lidar at height z_i (i  =  1 to 9). Figure 3 shows two examples of measured profile with its shear exponent and RSS.

According to this classification, profile (a) in Figure 3 would be in group 1, and profile (b) in group 2. The value of 0.1 was chosen here as threshold for the RSS because it gave two groups of data showing two trends while being statistically comparable (as they count similar numbers of data: 511 in group 1 and 396 in group 2). It is recognized that this threshold is somewhat arbitrary; it depends on the absolute value of the mean wind speed, and it should subsequently be ‘fine–tuned’ using a large number of data sets.

Figure 4(a) presents the distribution of the two groups of profiles and shows that there is no significant dependance of the classification on the wind direction. According to Figure 4(b), the distributions of turbulence intensity for each group of data are pretty similar. They differ by 1% on average, which is a rather small difference when looking at influence on the power curve.

3.3 Conventional power curves for the two groups of profiles

Figure 5 shows the scatter plot of the power (a) and the C_p (b) as function of the wind speed at hub height. In this figure, C_p is defined as in the current IEC standard: 1

(4)

where P is the electrical power output of the turbine, ρ the air density (ρ  =  1.225 ×10⁻³ kg m⁻³), u_hub is the wind speed measured at hub height and corrected for air density and A is the rotor swept area. The two colours represent the two groups of profiles: points obtained with the group 1 profiles ( RSS  ≤ 0.1) are displayed in gray and those obtained with group 2 profiles ( RSS  >  0.1) are displayed in black. Thus, in Figure 5(a), (b), we can see two trends (one for each group) leading to two mean power curves and C_p curves (obtained after binning the data into 0.5 m s ⁻¹ wind speed bins and averaging as required by the IEC 614000-12-1 standard); see Figure 5(c), (d). The power output of the turbine for a given wind speed (at hub height) is smaller for data from group 2 (non–power law profiles) than for data from group 1, and the data from group 2 generally give a lower C_p.

What might appear here as an underproduction of the wind turbine is actually due to an overestimation of the kinetic energy flux of the wind (KE_hub). Indeed, the kinetic energy flux approximation used in the C_p definition given by 4 is equivalent to assume that the wind speed is constant over the entire rotor swept area:

(5)

Or in other words, the wind speed shear is ignored.

4.1 For theoretical power law profiles

A better approximation of the kinetic energy flux than that given by 5 can be obtained by considering the wind speed profile between lower and higher rotor tip. For example, if we assume a power law profile and a horizontally homogeneous surface layer (which is fair over flat terrain), then the wind speed only depends on the height z and the profile can be described by 1. For such a theoretical profile, the kinetic energy flux is given by the following:

(6)

where z_LT is the lowest tip height, z_HT the highest tip height and R the turbine rotor radius.

For a given turbine (fixed hub height and rotor diameter), the ratio is then independent of the wind speed but varies with the shear exponent α. The variation of this ratio as a function of the shear exponent is shown by a line with diamonds in Figure 7. According to this analytical analysis, the ratio between the two kinetic energy approximations is below 1 for a shear exponent between 0 and 0.3 with a minimum around 0.2. However, the ratio remains quite high as it is 0.988 for α  =  0.2. For negative shear exponents and shear exponents higher than 0.3, the ratio is larger than 1. This shows that an error of up to 1% is made in the evaluation of the energy available in the wind for a power law profile with a usual shear exponent (between 0 and 0.4).

4.2 For measured profiles

The kinetic energy flux for each profile measured by the lidar can be approximated by the following:

(7)

where u_i is the wind speed measured at the i–th height in the profile and corrected for the air density and A_i is the area of the corresponding segment of the rotor swept area (see Figure 6).

The ratio is displayed in Figure 7. The profiles from group 1 (gray points) follow rather well the analytical results showing a moderate error due to the constant profile assumption. The non–power law profiles (group 2), on the other hand, do not follow the analytical curve at all and demonstrate much larger difference between the two ways of evaluating the kinetic energy flux. The approximation of a constant wind speed over the whole rotor swept area, made in 5, overestimates the kinetic energy flux for most of the data of group 2 and underestimates it for a few of them.

Two wind speed profiles can have the same wind speed at hub height but different kinetic energy. In a standard power curve, such profiles would have the same abscissa (hub height wind speed), whereas they would almost certainly result in different power outputs. This is partially why the two groups of profiles give two different power curves. The kinetic energy flux overestimation has even more impact on C_p, which explains why C_p for the group 2 profiles is generally lower than for group 1.

Another contribution to the differences between the power curves can be the real influence of the wind speed shear on the power output due to the aerodynamics effect. Indeed, two profiles resulting in the same kinetic energy may give different turbine power output because for some wind speed shear conditions (e.g. a power law profile with a large shear exponent), the turbine is not able to extract as much energy as in other shear conditions (e.g. a constant profile).

5.1 Definition

Intuitively, the power output of the turbine should be more closely related to the kinetic energy flux derived from the whole profile than that derived from the wind speed at hub height. We should then consider the power output of the turbine as a function of the kinetic energy flux. To do so, we suggest an equivalent wind speed based on the measured kinetic energy flux as defined in 7:

(8)

Then C_p becomes

(9)

which is a better representation of the efficiency of the turbine to extract the energy from the wind.

5.2 Application to the classified profiles

Figure 8 shows plots comparable with the plots in Figure 5. In Figure 8, the power and C_p are plotted as a function of the equivalent wind speed 8 (instead of the wind speed at hub height in Figure 5), and C_p is calculated according to 9 (instead of 4 in Figure 5). In Figure 8, profiles from both groups follow the same trend. The mean power and C_p curves obtained with each group of points overlap each other. This shows that the difference in power curves between the two groups shown in Figure 5 was mainly due to the error in kinetic energy flux.

5.3 Application to the unified data set

In a conventional power performance measurement, the data would not be grouped according to the profile shapes, but all the data would be considered together indifferently. The results obtained for the unified data set are shown in Figure 9.

The power curve obtained with the equivalent wind speed is shifted to the left (towards lower wind speeds) compared with the power curve obtained with measurements at hub height (see Figure 9(a)). However, this does not mean that the turbine has produced more power since these two power curves were obtained with the same data set. The equivalent wind speed method modifies the wind speed used in the abscissa in the power curve, but the wind turbine power output remains unchanged. This shift in the mean power curve actually shows that for this data set, the equivalent wind speed was on average smaller than the wind speed at hub height. Consequently, the mean C_p curve was shifted upwards (see Figure 9(c)). Note that at another site with larger wind shear above hub height than below, such as in Antoniou et al. 4 and VanLuvanee et al., 5 the equivalent wind speed may be larger than the wind speed at hub height so that the power curve would be shifted to the right.

However, the main result of using an equivalent wind speed accounting for the shear concerns the scatter in the power curve and C_p curve. Indeed, ignoring the wind speed shear in the kinetic energy flux estimation appears as a large scatter in the power and C_p curves. The use of the equivalent wind speed results in the reduction of this scatter. This can be seen directly by comparing Figure 8(a) (respectively (b)) to Figure 5(a) (respectively (b)).

To make the difference clearer, we quantified the scatter as the mean residual error relative to the mean power curve. As the mean power curve is actually a succession of line–segments, we calculated the mean residual error for each line–segment. If the j–th line–segment is defined by the two points of coordinates (v_i−1,P_i−1) and (v_i,P_i), where v_i is the i–th wind speed in the mean power curve and P_i the corresponding average power, then the error for the j–th line–segment is defined as follows:

(10)

where N_j is the number of data in the bin corresponding to the j–th line–segment (i.e. v_i−1  ≤  u_j,k  ≤  v_i  ∀ k), u_j,k the measured wind speed of the k–th point in the j–th bin, y_j,k the corresponding power output and

(11)

is the equation of the j–th line–segment. However, this error depends on the slope of the mean curve that for the power curve gets higher as the wind speed increases up to the rated wind speed. So to observe the variation of the error independently of the mean curve trend, we normalized the error defined in 10 by the segment slope . Table 1 shows the mean value of this normalized error for the power curve obtained with the cup anemometer, the lidar measurements at hub height and the equivalent speed (derived from the lidar profile measurements).

Table 1. Average of the normalized scatter for various power curve definitions.

Wind speed	Cup anemometer	Lidar	Lidar with
definition	at hub height	at hub height	equivalent speed
Scatter
	76	72	61

Similarly, the residual error for the C_p is defined by 10, but with

(12)

As the C_p curve has a low slope for the medium load range of wind speeds, there is no need to normalize it. Figure 9(b), (d) shows these two residual errors as a function of wind speed bin. It appears clearly that the use of the equivalent wind speed results in a reduction of the scatter in both the power curve and the C_p curve.

The scatter in the power curve is directly related to the category A power uncertainty in the power curve measurement uncertainty according to the IEC 61400-12-1 standard, as it is defined by the standard uncertainty in each speed bin:

(13)

where is the power standard deviation and N_i the number of points in the i–th bin. Contrary to the standard uncertainty, the quantity defined by 10 is independent of the segment slope and therefore enables us to estimate the reduction of scatter in a more subtle way. Indeed, the reduction of the scatter around the mean curve means that the power curve is less sensitive to shear and therefore more repeatable.

6.1 The equivalent wind speed method

We showed that the assumption of a constant wind speed over the rotor swept area leads to a misinterpretation of the power curve and the C_p. A better approximation of the kinetic energy flux was obtained by accounting for the vertical wind shear with measurements of wind speed at nine heights covering the vertical range between lower and higher rotor tips. The comparison of these two kinetic energy flux approximations showed that for most of the measurements presented here, the constant profile assumption overestimated the kinetic energy flux, especially for the non–power law profiles. Indeed, for those profiles, the underestimation was on average up to 5%, whereas it was only of 1% for the power law profiles. In the standard performance measurement, this appeared as if less power was produced from non–power law profiles, and the C_p was globally lower for power law profiles.

These observations were only possible because the wind speed profiles were measured at several heights between lower tip height and higher tip height and were not extrapolated. Indeed, if the profiles from group 2 had been extrapolated assuming a power law profile for example, the kinetic energy would probably have been overestimated, leading once more to an erroneous value for C_p.

To reduce the error due to an incorrect evaluation of the kinetic energy flux in the performance measurement, we defined an equivalent wind speed, on the basis of the new approximation of the kinetic energy flux. With the standard method, where the hub–height wind speed is used as the abscissa, the two groups of profiles gave significantly different power and C_p curves. By using the equivalent wind speed instead, very similar power and C_p curves were obtained. Therefore, a part of the differences in performance curves obtained with the wind speed at hub height was not due to a poor performance of the turbine but to an error in the evaluation of the kinetic energy flux.

Thus, the use of this equivalent wind speed reduced the scatter because of the fact that the effect of the wind speed profile on the kinetic energy flux was not taken into account in the standard power curve. Further analysis with the same data set (not shown in this paper but fully described in Wagner 13) showed that a considerable decrease of the scatter in the power curve could be obtained with measurements at fewer heights (but at least three) with the proviso that one measurement was taken above hub height. According to the current IEC standard, this scatter would have been interpreted as a power uncertainty of category A. In that sense, the use of such an equivalent wind speed reduces the uncertainty due to the wind shear. On the other hand, a complete uncertainty calculation would require the uncertainty in wind speed measurements achieved by the particular instrumentation used.

Moreover, the equivalent wind speed definition based on the kinetic energy flux gives a more meaningful C_p than the use of the wind speed at hub height and in that sense enables us to measure the true efficiency of the turbine. Indeed by using a better approximation of the kinetic energy flux, the C_p then shows how much energy the turbine actually extracts from the wind. Only by using the equivalent wind speed it is possible to see if the turbine efficiency actually depends on the shear situation, i.e. if the turbine extracts less power in some shear situations than in others whereas the energy available in the wind is the same.

It should be noted that this equivalent wind speed definition allows us to correct for the wind speed shear in the kinetic energy flux but does not correct for the actual effect of the wind speed shear on the aerodynamics of the turbine. Indeed, two different wind speed profiles can have the same kinetic energy flux but present differences in the aerodynamic performance of the turbine, resulting in different power outputs. Therefore, a power curve calculated with this equivalent wind speed is still shear dependent but much less than the power curve obtained with the wind speed at hub height. Differences in power curves obtained with the equivalent wind speed (for the same type of turbine operating at two different sites for example) reveal differences in the turbines performance and not differences in the wind energy resource related to shear. For really comparable power curves to be obtained, corrections for other wind characteristics, such as wind direction shear and turbulence, need to be accounted for too.

Finally, the use of an equivalent wind speed accounting for the shear in the power performance measurement can be a little confusing as the resulting power curve is not directly comparable with the standard power curve. Indeed, these two power curves have two different quantities in abscissa. For this reason, the most logical way of estimating the annual energy production with the equivalent wind speed is to combine it with the distribution of the equivalent wind speed. This means that the wind speed profile should be measured not only during the power curve measurement but also during the wind resource assessment.

6.2 Use of remote sensing

Another important conclusion we can draw from this experiment is that the pulsed lidar has successfully measured a power curve using the equivalent speed method. As it was noted in the second section, the scatter in the power curve obtained with the lidar measurements at hub height was approximately equal to the scatter obtained with the cup anemometer measurements. Consequently, the scatter obtained with the lidar with the equivalent wind speed was lower than for the conventional cup anemometer method (see Table 1).

Generally, as the measuring noise of a remote sensing instrument increases, the scatter in the power curve would also increase, as we suspect it to be the case for the sodar measurements in VanLuvanee. 5 If the increased uncertainty of the speed measurement is larger than the reduction in uncertainty because of the added wind shear information, we would get no net improvement. Therefore, the equivalent wind speed method can be applied with remote sensing measurements; however, this requires an instrument with both high absolute accuracy and low noise.