Interval evaluation of wind power prediction based on skew-slash t distribution

Wind resources have characteristics of stochastic volatility, resulting in wind power forecast errors greatly. The short and medium-term wind power forecast error distribution of the actual wind farm operating system is analyzed, and a sharp peak and thick tail, with heavy tail and skewness characteristics, are presented. A skew-slash t distribution (SST distribution) is proposed to describe the error distribution of short-term wind power prediction, and the parameter estimation is carried out by probability density curve fitting. The fitting results of the wind power forecast error historical data of the Wind power plant based on SST distribution, t distribution, and t Location-slash distribution (TLS distribution) are obtained, and the fitting goodness and prediction accuracy are compared. The results show that the SST distribution can more effectively estimate the wind power prediction interval.


Introduction
Wind power forecasting technology can effectually enhance the output stability of wind farms, raise the power generation capacity of wind farms, and facilitate the arrangement of maintenance and fault treatment of wind turbines [1].
At present, the accuracy of domestic wind power point prediction is about 10%~30%, which makes it difficult to meet the application demand [2].Meanwhile, from the perspective of wind power application, a single-point prediction value is not enough.A more accurate estimation of the fluctuation range of wind power forecast is required.
It is found that the prediction error of short-term wind power does not completely follow the normal distribution [3], and due to the heavy tail and bias characteristics of the prediction error distribution, if the forecast error is assumed to follow a normal distribution, there will be an unreasonable differential term after differentiating the objective function of optimization problem [4].
Liu et al. [5] proposed a beta-based wind power interval estimation, and an optimized beta function was used to describe the probability distribution.Based on this, the wind power fluctuation interval was obtained through probabilistic prediction, but the solution process was very complicated.Tewari [6] and Hodge [7] et al. used the Cauchy and Laplace symmetrical distribution model.Due to fewer adjustment parameters, the probability density curve was not flexible enough.Liu et al. [8] added two adjustment parameters to the t-distribution model, which can flexibly describe the symmetrical distribution with heavy-tailed characteristics, but did not consider the skewness in the distribution  [9] used a piecewise exponential distribution model to describe the heavy-tailedness and skewness of wind power prediction errors, but there were disadvantages such as complicated calculation and poor peak fitting.
To predict wind power more accurately, the distribution characteristics of short and medium-term wind power forecast errors in actual wind power systems are studied, and based on this, an SST distribution is used to describe the prediction error distributions.Compared to the goodness of the prediction errors of the three distributions, the prediction accuracy is evaluated.

Prediction error and distribution characteristics
At present, the widely used short-term wind power forecasting system is mainly based on numerical weather forecast (NWP) prediction method, which is affected by factors such as prediction model error, NWP data accuracy, wind power unit output power dispersion, and wind turbine failure and shutdown uncertainty influences [10].As shown in Figure 1, it is the comparison between the measured power and predicted power of a wind farm on January 1-2, with a time interval of 15 minutes.Among them, the prediction data was obtained by the wind power prediction system, and the NWP prediction method was used.Figure 1 shows that the actual power of the wind turbine does not completely match the predicted power.Prediction error e ୧ is: where p ୧ is the measured power when i wind farms, and p ୧ ᇱ is i wind power prediction power.Moreover, p ୧ >0, because when the actual power is sharply reduced or shut down due to natural or man-made reasons, negative power is formed.At this time, p ୧ 0. It is meaningless to perform error analysis under this condition.
Further, the relative error of ɂ ୧ describes the statistical characteristics of the error.P n is the wind farm capacity rating, and the wind power relative prediction error ⍊ i is: Figure 2 shows a wind farm power prediction error frequency density histogram.Figure 2 has the following characteristics: IOP Publishing doi:10.1088/1742-6596/2771/1/0120413 1) Heavy tail.In Figure 2, the right distribution is concentrated in the region of 0 ~ 0.2, but the attenuation is slow in the region of 0.2 ~ 0.8, and the proportion is relatively high, showing a heavy tail.
2) Skewness.The distribution of the sample data is steep, with sharp peaks, asymmetrical distribution on both sides, shifted to the right and tilted, the skewness coefficient is 0.6442, and the kurtosis coefficient is 3.5333.
Visibly, the forecast error does not obey the normal distribution.From Figure 2 it can be seen that the main characteristic of the forecast error distribution is that the peaks have thick tails, that is, heavy tails, followed by slight skewness.

SST distribution and probability prediction
According to M C Jones, Saralees Nadarajsh, and Samuel Kotz [11], if the density function of the x is: then x obeys the SST distribution of the parameter Ȝ (SST distribution), and is calculated as x ‫‬ xt(ɋ, ɉ), where parameter Ȝ is the shape parameter of the skew distribution.f(x) and F(x) are: SST distribution is where ቁ is the hypergeometric function.
The SST distribution model has the thick tail characteristic of the t distribution model.At the same time, after considering the skewness of the t distribution, the distribution kurtosis skewness adjustment parameter Ȝ is added to achieve the fitting of asymmetric distribution data.So the SST distribution model has the characteristics of spikes, thick tails, and skewness, which is very suitable for describing the heavy-tailedness and skewness of short-term forecast error distributions.Figure 3 shows that the fitting curves of the three distribution models are given, including the t distribution, TLS distribution, IOP Publishing doi:10.1088/1742-6596/2771/1/0120414 and SST distribution model.Table 1 presents the parameters of the fitted curves for the three distribution models.
Figure 3 shows that when ȝ is the same, the peak shape of the SST distribution is more prominent.

SST distribution and probability prediction
The D-test is used to verify the coincidence between the actual and theoretical distribution of the samples.
The K-S function in the MATLAB program can be used for verification, and the calling format is [݄, p, ksstats, cv] = kstest(x, CDF, alp݄a, type) where the parameter h is the test decision, p is the minimum significance probability of rejecting the null hypothesis, ksstats is the value of the test statistic, and cv is the threshold of the reject domain; the input parameter x is the sample data, and CDF is specified by the null hypothesis of the test distribution form, alpha is to test the significant level, and type is the labeled value of the selected hypothesis type.When the sample data can pass the K-S test of multiple hypothetical distribution functions, the best C 2 fit curve of the sample data cannot be determined, so C 2 parameters are introduced to further test the good and bad of the fit.The range C 2 is [0,1], the smaller the value is, the better the fit is.C 2 is as follows: where ‫ݕ‬ is the actual value, ‫ݕ‬ ො is the curve fitting value, and ‫ݕ‬ ത is the actual average value.

Confidence interval estimation of power prediction error
If F(x) is: where m i n; is the probability corresponding to the prediction error.
The prediction error interval is expressed as: where ‫ܨ‬ ‫)ݔ(‬ is the inverse function of ‫,)ݔ(ܨ‬ p is wind power prediction, and alpha is the given confidence level, is power prediction.Type (10) is Used to calculate the confidence interval of the predicted value.

Power forecast error probability distribution fitting and fluctuation interval estimation
Using 1 -June as wind field data, the probability distribution of wind power prediction error and fluctuation interval estimation are analyzed.The sampling period is 15 minutes, and the rated capacity of a single fan is 2000 kW.
From the practical point of view of engineering application, the sample error probability interval is counted, the histogram curve is obtained, and then the nonlinear fitting of the SST distribution function is used to obtain the estimated value of the SST distribution model parameters quickly.Related parameters for the optimized SSLT distribution function argument are Ȟ = 1.2086 and Ȝ = 0.7053.
To verify the validity of SST distribution, the t distribution model and TLS distribution model are compared.Figure 5 shows the prediction error fitting curves for the three distribution models.The output results and values of the goodness-of-fit test using the K-S function are shown in Table 2.The longitudinal and transverse errors of the probability density distribution of forecast error in SST distribution are significantly smaller than those in the t distribution and TLS distribution model, and the fitting effect is better.At the same time, the value of the SST distribution model is the smallest, and the goodness of fit is higher.The SST distribution can be adjusted to make the shape skew, which is more flexible and applicable, so the SST distribution model has higher accuracy than other models.

Evaluating indicator
Two indicators PICP and PIAW are used to evaluate the accuracy of the forecasting model.PICP value, the greater the actual wind that falls into the prediction range, the better the prediction effect is.
PICP is expressed as: where n represents the actual number of wind power; n out represents the actual amount of wind power falling outside the predicted interval.The smaller the PIAW value is, the closer the predicted interval is to the actual interval.PIAW is expressed as: where ⌨p i represents the difference between the upper and lower prediction intervals at time i.
By calculation, Table 3 lists the three kinds of distribution model prediction accuracy evaluation indexes.The SST distribution model has the largest coverage of the prediction interval, and its average bandwidth is less than the t distribution and TLS distribution models.The obtained prediction interval is better than the probability prediction based on the t distribution and TLS distribution.

Conclusions
This paper studies the distribution characteristics of short and medium-term wind power forecasting errors in the actual wind power system and proposes an SST distribution model to describe the wind power forecasting errors.At the same time, three distribution models, SST distribution, t distribution, and TLS distribution, are compared and analyzed for the goodness of fit and prediction accuracy.The conclusions are as follows: 1) The forecast error of wind power does not obey the normal distribution, showing the shape characteristics of sharp peaks and thick tails, and has obvious heavy tails and skewness.
2) A prediction model is put forward based on SST distribution, and SST distribution under the fitted curve has more obvious peak features.
3) Using wind farm operation data, the short-term power prediction error fitting results of the SST distribution model, t distribution model, and TLS distribution model are obtained.
4) By comparing the goodness of fit and prediction accuracy of the three distribution models, the SST distribution model can better describe the probability distribution of short-term wind power prediction error than the t distribution and TLS distribution.

Figure 1 .
Figure 1.Comparison of actual and predicted wind power.Prediction error e ୧ is:

Figure 2 .
Figure 2. Histogram of frequency density of wind power short-term forecast error.

Figure 3 .
Figure 3. Curves and effect of different parameters on three different distribution models.

Figures 6 -
Figures 6-8 show the estimation results of the wind power fluctuation interval of the SST distribution, t distribution, and TLS distribution at the confidence level of 95%.

Figure 6 .
Figure 6.Power interval estimated with SST distribution at 95% confidence level.

Figure 7 .
Figure 7. Power interval estimated with t distribution at 95% confidence level.

Figure 8 .
Figure 8. Power interval estimated with TLS distribution at 95% confidence level.

Table 1 .
Parameters of the distribution curves.

Table 2 .
results of goodness-of-fit test.

Table 3 .
Comparison of indexes among different models.