Analysis of wind speed forecast errors

than 30% higher than measured. In the case of down-scaling, the variability is largely underestimated. The average positive 1-hour gradient for example is by 35% too low. This over- and underestimation of variability by scaling reflects the general analysis of the smoothing effect in Section 4.2.1. The presented simulation approach considers the changes of variability by taking into account the region size. Realistic generation profiles of single regions can thus be simulated.

0 10 20 30 40 0

0.5 1 1.5 2 2.5 3 3.5

Standard deviation (m/s)

Forecast horizon (h) (a) Standard deviation

0−4 4−8 8−12 12−16 0

0.5 1 1.5 2 2.5 3

Measured wind speeds (m/s)

Standard deviation (m/s)

(b) Conditional standard deviation

Figure 4.14.: Standard deviation of wind speed forecast errors

are similar at the different locations. The dash-dotted line indicates the assumed curve for the simulation in Section 4.3.2. The mean errors (not shown here) are either positive or negative over all forecast hours depending on the location, but no general over- or underestimation by the forecast can be observed. The mean errors will therefore balance each other out if forecasts at several locations are aggregated.

Figure 4.14-b shows to which extent the standard deviations depend on the actual wind speed. A hypothesis could be that the forecast errors increase in weather situations with high wind speeds. This hypothesis is analyzed by split-ting the measured wind speeds into different partitions (bins). For each bin, the standard deviations of the related forecast errors are calculated.¹¹ No systematic trend can be observed. Two locations show a clear increase of standard deviation with higher wind speeds but for the other four the standard deviations are simi-lar for all bins. In one case, the standard deviation related to the last bin is even slightly lower than the one related to the first bin. As the available data is limited, the results can only be indicative. The same observation is however also made in the literature. A similar analysis at two locations for example shows that there is no systematic trend between standard deviation and wind speed level [22]. The relative independence between the forecast error and the wind speed level does not apply to power forecasts. Due to the transformation to power by the nonlin-ear power curve, the power forecast errors systematically depend on the measured power level, as shown by the author, [154] and others, [17].

11Figure 4.14-b relates to the forecast error looking at all forecast horizons. Similar results can be seen looking at each forecast hour for itself.

0 10 20 30 40 0

2 4 6 8 10

Kurtosis

Skewness

Skewness and kurtosis

Forecast horizon (h) (b) Skewness and kurtosis

−200 −10 0 10 20

50 100 150 200

Speed forecast error (m/s)

Frequency

(b) Distribution example

Figure 4.15.: Distribution of speed forecast errors Distribution of forecast errors

Skewness and kurtosis are important parameters to characterize a distribution.

There are given in Figure 4.15-a. In all cases the kurtosis is higher than three so higher than in the case of a normal distribution. The kurtosis stays relatively constant at four locations, whereas it decreases with increasing forecast horizon at two locations. In some cases very high kurtosis values can be seen. Kurtosis values greater than three define leptokurtic distributions. Leptokurtic distributions have sharper peaks and longer, fatter tails. This means that the forecast errors are in general relatively low (high frequency around the average value), but extreme events occur more often than in the case of a normal distribution (fat tails). In other words, large errors happen less often but they are more extreme. Figure 4.15-b gives the distri4.15-bution of the forecast errors at one location in form of a histogram (ten-hour forecast). In addition, a normal distribution with the same mean value and standard deviation is given by the thin solid line. The example shows that the sharp peak around the mean of the error distribution is balanced by higher frequencies at the tails.

Figure 4.15-a shows that the skewness of the forecast errors is greater than zero and stays relatively constant for different forecast hours (the skewness of a normal distribution is zero). The forecast errors are therefore right-skewed indi-cating a higher concentration of values below the average. This means that the forecast in general underestimates the future wind speed but in some rare events a high overestimation occurs.¹²The example distribution in Figure 4.15-b illustrates the right-skewness of the forecast errors. The dash-dotted lines in Figure 4.15-a indicate the parameter assumptions for the simulation in Section 4.3.2.

12The error is here equal to the predicted value minus the measured one.

10 20 30 0

0.2 0.4 0.6 0.8 1

Lag = 1

Lag = 3

Lag = 8

Forecast horizon (h)

Autocorrelations

(a) Over forecast horizon

0 10 20 30 40

0 0.2 0.4 0.6 0.8 1

Lag (h)

Autocorrelations

(b) Over lag

Figure 4.16.: Autocorrelations of wind speed forecast errors

The results show that the speed forecast errors are generally not normal dis-tributed and extreme events occur more often. This observation is important as extreme forecast errors will cause more difficulties for the operation of the power systems. The distribution of the errors was therefore additionally tested by a chi-square test and a Jarque-Bera test. The hypothesis that the errors follow a normal distribution is thereby rejected for all forecast hours for a significance level of 0.1% (the probability that the errors follow a normal distribution is then lower than 0.1%).

Autocorrelations

To simulate forecast errors, it is also important whether the forecast error stays positive (or negative) for the total forecast horizon or whether it rather oscillates.

Autocorrelations indicate to which extent the forecast error depends on errors that are related to other forecast hours. Figure 4.16-a shows how the autocorre-lations develop with increasing forecast horizon. For example, the autocorrelation at forecast hour three looking at a lag of one hour gives the correlation between the errors of a three-hour forecast and the errors of a two-hour forecast.

The autocorrelations increase only slightly with increasing forecast horizon but they strongly depend on the lag. For a lag of one, the autocorrelations are mainly between 0.7 and 0.8, so relatively high, whereas they are significantly lower for greater lags. This can also be seen in Figure 4.16-b giving the correlation between the error of a one-hour forecast and the errors related to longer forecast horizons.

The autocorrelations decrease quickly with increasing lags and the autocorrela-tions are below 0.2 if the difference between the forecast horizons exceeds twelve hours. The errors of a forecast for one hour are therefore only similar to the

er-Figure 4.17.: Spatial correlations of speed forecasts and fitted mesh

rors of a forecast for another hour if the lag between the hours is small. The dash-dotted lines in Figure 4.16-a indicate the assumptions for the simulation in Section 4.3.2.

Geographical correlations

The forecasts of regional wind power generation are typically based on the forecasts at selected wind farm locations. Forecast errors at different locations balance each other to a certain extent (only perfectly correlated forecast errors would not level each other out). The correlations between errors at different locations are therefore decisive for the later simulation. The correlation between speed forecast errors and their dependence on the geographical distance is assessed in the following. For each possible pair of wind farms and each forecast hour the correlation between the forecast errors is calculated. The resulting correlation is related to the geographical distance between the locations. The grey mesh (solid lines) in Figure 4.17 shows a graphical representation of the correlations (the z-axis indicates the correlation level).

The results were fitted in order to obtain a general relation. The correlation value is given by a function with three exponential terms, see Equation 4.13, where ρ_i,jstands for the correlation between two locations,d_i,jfor their distance,hfor the forecast hour and a₁ to a₄ as fitting parameters. The combination of exponential functions is chosen for the following three reasons. First, the correlations should be zero when the distance is infinite, which is assured by the first exponential term. Secondly, the correlation should increase with the forecast hour, as, with

increasing forecast horizon, the weather situation at a specific location becomes less important and the general weather pattern of the region prevails. This is assured by the second exponential term. Thirdly, the correlation should be one if the distance is zero. The calculated correlation is equal to one if the distance is close to zero as the third exponential term (in the exponent) is then zero.

ρ_i,j = exp

−a₁ d_i,j

·(a₂−a₃·exp (−a₄·h))^1−exp(−d_i,j∗10⁵) (4.13)

The fitting parameters, found by a least-square optimization, are: a₁ =−0.0888, a₂ = 0.7998, a₃ = 0.5034 and a₄ = −0.1976. The blue mesh (dashed lines) in Figure 4.17 shows the correlations calculated by the fitting function.