The analysis of adjusted WRGs has allowed identifying that there is a limitation in the standard method for adjusting WRGs in very complex areas with significant wind rose variations. Due to the fact that the current method is run on “per directional sector” basis, it has been identified that problematic adjustments may be related with sectors having a small probability of occurrence, leading as a consequence to very large and meaningless “directional ratios”. This limitation can show, for example, severe discrepancies when doing cross-predictions between the different masts in a site.
This document presents a new proposed WRG adjustment method. This method is based on the following main hypotheses:
•The ratios hold true at the mast locations as they reflect the reality at these locations.
•When you move away from the mast, however, those ratios may lose part of their applicability in points in which the wind rose differs from the one at mast location. Following this, the method proposes relaxing these ratios at each modeled location depending on the following parameters:
1.the Direction Deviation (DD) index, described in Brower et al. (2015)2, that evaluates the similarity between the raw wind rose at the mast and the one at any other location.
1.A Replicability Factor (RF) that is associated to each sector aiming to evaluate how much the speed and frequency ratios of that sector are reliable to be applied in other points of the map.
•In the worst-case scenario in terms of WRG adjustment, i.e. at locations where the raw wind roses significantly differ with respect to that at the mast (DD close to 1) and for sectors with low replicability factor (RF close to 0), it is proposed to do the following adjustment:
oThe target wind speed is scaled by the mean wind speeds ratio at mast location (two different methods are proposed for wind speed ratios calculation)
oThe target wind frequency is the same of the raw WRG wind frequency.
•Example charts of the variation of the wind speed and frequency ratios as a function of DD and RF are shown in Figures 2, 3 and 4.
Location |
Acronym |
Name |
Values |
Interpretation |
---|---|---|---|---|
Mast location i |
Ri_d |
Wind speed ratio at mast location |
(0; >1] |
It is the ratio between mast and raw WRG wind speeds in sector d. If Ri_d=1 than the wind speed of the mast is the same of the raw WRG |
Fi_d |
Frequency ratio at mast location |
(0; >1] |
It is the ratio between mast and raw WRG frequencies in sector d. If Fi_d=1 than the frequency of the mast in a certain sector is the same of the raw WRG |
|
FFi_d |
Frequency Factor |
(0; 1] |
Indicates the minimum value of frequency in sector d between mast and raw WRG |
|
FRFi_d |
Frequency Ratios Factor |
(0; 1] |
Indicates how much the Fi_d deviates from 1 |
|
RFi_d |
Replicability Factor |
(0; 1] |
Calculated as the product of FFi_d and FRFi_d. Indicates how much we trust on the ratios Ri_d and Fi_d for a certain sector. Higher the RF value, higher is the reliability of those ratios |
|
Other WRG location y |
DDy_i |
Direction DeviationError! Bookmark not defined. |
[0; 1] |
Indicates how much the frequency wind rose in point y differs from the frequency wind rose in point i |
Nsect |
Number of sectors |
12, 16, etc.. |
Number of direction sectors of the raw WRG |
|
Sectorial distance |
[0; Nsect/2] |
Indicates the distance (in terms of number of direction sectors) between sectors s and d. = 0 if s = d |
||
[0; 1] |
Weight given to the wind speed ratio in a certain sector s in the calculation of the wind speed ratio in sector d. It depends on DDy and the sectorial distance between sectors s and d. For DDy = 0 no importance is given to sectors others than d, as higher the DDy higher will be their importance. See Figure 1. |
|||
[0; 1) |
Weight given to the ratio of mean wind speeds at mast location i (or to the frequency of the raw WRG at point y) in the calculation of the wind speed ratio Ry_d (and the frequency ratio Fy_d) at point y. Higher DDy_i (and lower the RFi_d) higher will be this weight |
Standard method
Currently, the wind speed ratio (Ri) and frequency ratio (Fi) for a direction sector d at mast location i are calculated as
Those factors are then applied to another point y of the map in order to scale the wind speed and frequency of the raw WRG:
However, the direct application of those ratios can create problems in the adjustment: for example if the frequency in a certain sector is really low (e.g. <1%) then the Ri_d and Fi_d obtained in that sector are not really representative and can hardly be applied in another location in which the frequency in that sector is much higher (e.g. > 10%).
In those cases, unrealistic wind speeds or frequencies can be obtained in some areas far from met mast.
Ratio Weighting and Ratio Relaxation Methods
The idea is that the wind speed and frequency ratios for each direction sector are weighted by their reliability (evaluated by a factor called Replicability Factor) and the Directional Deviation (DD) in a certain point (with respect to mast location). Thus new ratios are obtained for each sector and at each map location.
The proposed process for ratios calculations follows a 4-steps calculation:
At Mast location (i):
1. Calculate the Ri and Fi for each sector such as in current calculation
2. Evaluate the replicability of these ratios: an index called Replicability Factor (RF) is assigned to each sector. This factor depends on the probability of that sector (higher the frequencies of one sector more reliable are the ratios) and on the frequency ratio of that sector (more that ratio deviates from 1, lower will be the reliability of the ratios). Thus, RF can be calculated as the product between the probability of that sector (Frequency Factor – FF, calculated as the minimum value between Pi_d and Pi_d_RawWrg) and on how much Fi_d deviates from 1 (Frequency Ratio Factor - FRF).
Where:
and
The factor RF is also an indicator of how good the raw WRG catch the wind rose at mast location, in comparison with mast data.
At another WRG point (y):
In the proposed method, the ratios of wind speed and direction frequency vary across the map, thus the adjusted WRG wind speed and frequency in a certain point y are calculated as
The calculation of wind speed and frequency ratios in point y (Ry_d and Fy_d respectively) are reported in the following (steps 3 and 4).
3.Calculate the wind speed ratios to be applied
For the calculation of wind speed ratios Ry we propose the following two methods.
Ratios Weighting Method. Wind speed ratios Ry_d, depend on
1.Wind speed ratio in sector d (Ri_d)
2.Replicability factor (RFi_d)
3.Direction Deviation of the point y with respect to mast location i (DDy_i)
4.Wind speed ratios in closer sectors (Ri_d1) where d1≠d
Using this formula, for calculating the wind speed ratio in a certain sector d, a weight is given to the ratios of each sector as function of the Replicability Factor (RF) of that sector and a parameter that depends on the distance to the analyzed sector d and the Direction Deviation of point y:
Where:
is a normal distribution centered in d with standard deviation = . See Figure 184.
Nsect is the number of sectors of the WRG (12, 16, etc.)
Figure 185 shows the variation of Ry_d for a sector d with Ri_d = 0.935, while the mean value of the Ri_d weighted for their RFi_d is 0.845. It should be noted that a closer sector s as a high replicability Factor and an Ri_s = 0.77. Analyzing the curve for RFi_d = 0.01 we can see that:
•For null DDy_i, Ry_d is equal to Ri_d,
•For low DDy_i values (but higher than 0), the effect of the closer sector is prevailing
•When DDy_i tends to 1, Ry_d tends to equal the mean values of the ratios at mast location weighted by their replicability factors.
These trends are smoother when RFi_d is higher (greed and red curves).
Figure 185: Example of the variation of Ry_d (calculated using the first method) as a function of DDy_i and RFi_d |
Ratios Relaxation Method. Wind speed ratios Ry_d, depend on
1.Wind speed ratio in sector d (Ri_d)
2.Replicability factor (RFi_d)
3.Direction Deviation of the point y with respect to mast location i (DDy_i)
Where
is the ratio between mast and raw WRG mean wind speeds at point i
Figure 186 shows the variation of Ry_d for the same sector previously described (Figure 2). It should be noted that the ratio between mean wind speeds at mast location is 0.87.
Analyzing the curve for RFi_d = 0.01 we can see that:
•For null DDy_i, Ry_d is equal to Ri_d,
•As DDy_i, increases, Ry_d tends to equal the ratio between mean wind speeds at mast location
These trends are smoother when RFi_d is higher (greed and red curves).
Figure 186: Example of the variation of Ry_d (calculated using the ratio relaxation method) as a function of DDy_i and RFi_d |
4.Calculate the frequency ratios to be applied
Frequency ratio Fy_d, as a function of
a. Replicability factor (RFy_d)
b. Direction Deviation of the point y with respect to mast location i (DDy_i)
Figure 187 shows the variation of Fy_d for the same sector previously described, in which Fi_d = 0.78. Analyzing the curve for RFi_d = 0.01 we can see that:
•For null DDy_i, Fy_d is equal to Fi_d,
•As DDy_i increases, Fy_d tends to 1
These trends are smoother when RFi_d is higher (greed and red curves).