Quantitative method for evaluating detailed volatility of wind power at multiple temporal-spatial scales

1 Introduction

High proportion of renewable energy is one of the key features of the future power system, and wind power will become the main source of the future power supply [1].Considering China as an example, by the end of 2018, the installed capacity of wind power was 184 million kW, accounted for 9.7% of the total installed capacity.The windpower generation was 366 billion kWh in 2018, accounted for 5.2% of the total power generation[2].However, the volatility of wind power presents great challenges to the electric power system, which aims to maintain secure [3], reliable [4], and economical [5] operation while supporting a large proportion of renewable energy.Therefore, it is significant to describe the volatility of wind power accurately.

Some scholars have conducted a lot of researches on the volatility of wind power.Novoa et al.[6] used statistical moments, such as the mean and variance, to characterize wind power variables.Widen et al.[7] surveyed the most common methods and models used to assess the volatility of wind power.Fogelberg et al.[8] used the weekly standard deviation to measure the fluctuation characteristics of wind energy.Zhang et al.[9] separated the wind power time series into higher and lower frequency sequences, and studied the volatility in different frequency domains.Han et al.[10] presented two indices to evaluate the volatility of wind power, including the random fluctuation between adjacent time slots and the ramp within continuous time windows.

The ramp event of wind power is also one of the important ways to reflect its fluctuation characteristics.Gallego-Castillo et al.[11] summarized a relatively complete definition of wind power ramp events that included the amplitude, rate, direction, duration, start time, end time, etc.Ren et al.[12] defined the wind power intermittency for the first time on the basis of the ramp event.Wang et al.[13] analyzed the wind power output characteristics in two aspects, one is the digital feature, including the maximum output that closed to the rated value and the range of ramp rate, another is the distribution, including the probability distribution function and the box plot.

The indexes that commonly used to evaluate the volatility of wind power include the average fluctuation amplitude, standard deviation, fluctuation frequency, ramp event, daily fluctuation amplitude, etc.However, as previously mentioned, the detailed volatility of wind power is ignored at present.To address this, a quantitative method for evaluating the detailed volatility of wind power at multiple temporal-spatial scales based on the frequency distribution of power variables is proposed in this paper.

The main contributions of this paper are as follows:

(1) The volatility indexes which can evaluate the detailed fluctuation characteristics of wind power are proposed based on the frequency distribution of power variables, and the presented indexes are compared with the traditional volatility indexes of the average fluctuation amplitudes.

(2) The detailed volatility of wind power under multiple temporal (year- season - month - day) and spatial scales (wind turbine - wind turbines - wind farm - wind farms) are studied via the presented indexes in four calculation time windows (10 min, 30 min, 1 h and 4 h).

(3) Finally, the relationships between the wind power forecasting error and the detailed volatility of wind power are analyzed, where the forecasting time horizons are consistent with the calculation time windows of the power variables.

The remainder of this paper is organized as follows.The evaluation indexes describing the detailed volatility of wind power are presented in Section 2.Then, the proposed indexes are used to evaluate the detailed fluctuation characteristics of wind power at multiple temporalspatial scales in Section 3.In Section 4, the relationships between the forecasting accuracy and the corresponding detailed volatility of wind power are analyzed.Finally, the conclusions drawn from this study are presented in Section 5.

2 Evaluation indexes of wind power detailed volatility

The definitions and calculation methods of the proposed volatility evaluation indexes are introduced in this section, which can describe the detailed fluctuation features of wind power in different temporal and spatial scales.The normalized power data obtained from a wind turbine in northern China is used as an example to illustrate the application of the proposed indexes.The time resolution of the data is 10 min.

The wind power variable is defined at first.

where, P(t) is the wind power at time t and Δt is the calculation time window.

Then, the wind power variables are calculated over the analysis time scale of month and the calculation time window of 10 min.The frequency distribution histogram of wind power variables is plotted, and fitted by the normal distribution (ND) model and the kernel density estimation (KDE) model, respectively, as shown in Fig.1.It can be seen that, the ND model cannot fit the distribution well cause the power variables are mostly concentrated around 0, but the KDE model which using the non-parametric method shows obvious advantages during the fitting.Therefore, the KDE model is used to fit the power variables distribution when calculating the proposed detailed volatility indexes in the following contents.

The proposed indexes, including lower confidence limit (LCL), upper confidence limit (UCL) and confidence interval (CI) of power variables under the certain confidencelevel (CL) can evaluate the detailed characteristics of positive, negative and overall wind power variables, respectively, as shown in Fig.1.The proposed volatility indexes can be applied to different temporal and spatial scales.In addition, the CL can be selected according to the specific application.

Fig.1 Confidence bounds to characterize the detailed vola tility of wind power

Two cases are employed to illustrate the effectiveness and advantages of the proposed indexes in comparison to the traditional indexes, including the mean backward fluctuation amplitude (MBFA), mean forward fluctuation amplitude (MFFA), and mean absolute fluctuation amplitude (MAFA).The normalized power data used in these two cases are obtained from the same wind turbine in northern China, as mentioned before.The analysis time scale is month and the calculation time window is 10 min in two cases.

The traditional volatility indexes for Case 1 are MBFA = -0.056, MFFA = 0.058, and MAFA = 0.055, while for Case 2 they are MBFA = -0.056, MFFA = 0.057, and MAFA = 0.055.The actual frequency distributions of wind power variables in Case 1 and Case 2 are shown in Fig.2.It can be seen that, even though the fluctuation characteristics of wind power are different in two cases, the values of the traditional volatility indexes for the two cases are very similar.In contrast, the values of the proposed indexes which listed in Table 1 reflect the differences well.

Table 1 Detailed volatility of wind power in Case 1 & Case 2

CL = 90% CL = 95% CL = 99%LCL UCL LCL UCL LCL UCL Case 1 -0.140 0.135 -0.174 0.195 -0.297 0.317 Case 2 -0.123 0.121 -0.161 0.165 -0.355 0.273

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CL = 90% CL = 95% CL = 99%CI CI CI Case 1 0.276 0.368 0.613 Case 2 0.243 0.327 0.628

Fig.2 Frequency distributions of power variables in Case 1 & Case 2

3 Case study

The actual wind power data obtained from a location in northern China is taken as an example to illustrate the application method of analyzing the detailed volatility of wind power at multiple temporal (year-season-monthyear) and spatial scales (wind turbine-wind turbines-wind farm-wind farms) based on the proposed volatility indexes defined in Section 2.The calculation time windows are 10 min, 30 min, 1 h, and 4 h.

3.1 Data

The actual data of a certain place in northern China is used in this study.This region consists of three wind farms, which referred to as wind farm 1 (WF1), wind farm 2 (WF2), and wind farm 3 (WF3), respectively, in this paper, as shown in Fig.3.Where, the WF1 contains 13 wind turbines (WTs).The existing dataset includes the output data of each wind turbine in WF1, the respective output data of WF2 and WF3.The length of data is two years and the temporal resolution of data is 10 min.

To study the detailed volatility of wind power under different spatial scales, the measured output data should be normalized in this study, the calculation formula is shown in Eq.(2).

where, P ( t) is the normalized wind power at time t, P(t) is the measured wind power at time t, and Pmax, Pmin is the maximum and minimum of the measured wind power series, respectively.

Fig.3 The location of 3 wind farms used in this paper

3.2 Detailed volatility of wind power at multiple temporal scales

Wind Turbine 1 in WF1, which is referred to as WT1 in this paper, is taken as an example to study the detailed volatility of wind power at multiple temporal scales (year-season- month-day) by using the proposed volatility indexes.

(1) Temporal scale of year

First, the UCL, LCL and CI of WT1 output variables at the CLs of 90%, 95%, and 99% are calculated, respectively, under four calculation time windows at the temporal scale of year, as shown in Table 2.It can be seen that the frequency distribution of WT1 output variables is basically symmetrical at the temporal scale of year.When the CL is constant, the coefficient bound of WT1 output variables is increased with the calculation time window.

Table 2 Detailed volatility of WT1 output at the temporal scale of year

Δt 10 min 30 min CL 90% 95% 99% 90% 95% 99%LCL -0.11 -0.16 -0.28 -0.20 -0.27 -0.44 UCL 0.11 0.16 0.28 0.19 0.27 0.45 CI 0.22 0.32 0.56 0.39 0.54 0.89 Δt 1 h 4 h CL 90% 95% 99% 90% 95% 99%LCL -0.27 -0.36 -0.57 -0.50 -0.57 -0.84 UCL 0.26 0.37 0.56 0.45 0.65 0.83 CI 0.53 0.73 1.13 0.95 1.22 1.67

(2) Temporal scales of season and month

Second, the season is taken as the analysis temporal scale, and the seasonal detailed fluctuation characteristics of WT1 output are studied under the four calculation time windows by applying the proposed volatility indexes, as depicted in Fig.4.The confidence intervals of wind power variables at the confidence level of 95% (referred to as CI95% in this paper) in four seasons are shown in Table 3.As can be seen that, 1) the detailed volatility of WT1 output follow the same trend over the four seasons in all calculation time windows.The CI is the smallest in autumn and the largest in spring at the same calculation time window, whereas the CL, which means the detailed volatility of WT1 output is the most severe in spring, followed by summer and winter, and the least in autumn.For the calculation time window of 1 h, the CI95% of WT1 seasonal output variables is 0.77, 0.71, 0.63, and 0.75, respectively.2) The UCL and LCL of WT1 seasonal output variables are similar at the same CL for different calculation time windows, i.e., the frequency distribution of the WT1 seasonal output variables is essentially symmetrical.

Then, the CI95% of WT1 output variables is used as an evaluation index to study the detailed volatility of WT1 output in different months, as depicted in Fig.5.It can be seen that, the monthly fluctuation characteristics of WT1 output are essentially the same under different calculation time windows, and the detailed volatility of WT1 output is more severe in September and less severe in December.

Fig.4 Detailed volatility of WT1 output at the temporal scale of season

Table 3 CI95% of the WT1 seasonal output variables

CI95% 10 min 30 min 1 h 4 h Spring 0.346 0.559 0.766 1.309 Summer 0.317 0.540 0.706 1.181 Autumn 0.287 0.492 0.629 1.062 Winter 0.314 0.554 0.747 1.246

Fig.5 Detailed volatility of WT1 output at the temporal scale of month

(3) Temporal scale of day

The detailed volatility of WT1 output at the daily temporal scale is studied in this part by analyzing the fluctuation characteristics of wind power in different typical days.The improved K-means clustering algorithm (X-means) [14] is adopted to extract the typical days in this study.Compared with the K-means algorithm, the X-means algorithm only need to give the range of clustering number (k) in advance, not the specific value of k.The clustering results will be more scientific and effective with the X-means algorithm.

The WT1 daily output are divided into 32 categories through the X-means algorithm, and 9 typical-day sets are selected randomly to show the clustering results, as depicted in Fig.6.In each sub-figure, the dotted lines and the solid line, respectively, represent the actual daily wind power series and their clustering center per typical-day set, respectively.To further analyze the detailed volatility of WT1 output at the daily temporal scale, the CI95% of WT1 output variables in each typical-day set are computed under the four calculation time windows and the results are listed in Table 4.

It can be seen that the volatility of WT1 daily outputis much more affected by the calculation time window, compared with the temporal scales of year, season and month.Taken the typical-day 2 and typical-day 13 sets as an example, when the calculation time windows are no more than 1 h, the CI95% of WT1 output variables in typicalday 13 set are larger, but it is smaller at the calculation window of 4 h.These results are consistent with the variation trends of WT1 output series in the two typicalday sets shown in Fig.6, where it can be seen that the amplitudes of WT1 output variables are larger in typicalday 13 set when the calculation time window is small, but the main variation trend in typical-day 13 set weakened the fluctuation amplitude at the calculation time window of 4 h, which leads to smaller CI95%.The detailed fluctuation characteristics of WT1 output under four calculation time windows are essentially consistent with the variation trends of WT1 output series in 9 typical-day sets, which illustrates the effectiveness of the proposed volatility indexes.

Fig.6 WT1 daily output series and their clustering centers in 9 typical-day sets

Table 4 CI95% of WT1 output variables in 9 typical-day sets

Typical-day set 10 min 30 min 1 h 4 h 2 0.353 0.607 0.735 1.138 6 0.394 0.623 0.886 1.482 11 0.335 0.552 0.724 0.896 13 0.379 0.641 0.898 1.077 15 0.356 0.643 0.780 0.829 16 0.158 0.208 0.265 0.403 25 0.198 0.339 0.471 0.782 26 0.142 0.231 0.319 0.547 28 0.237 0.398 0.531 0.613

The CI95% of WT1 output variables in 32 typical-day sets are calculated under the four calculation time windows, and the results are shown in Fig.7.It can be seen that, 1) the detailed volatility of WT1 output varies greatly in different typical-day sets for the same calculation time window.2) The variation range of WT1 output variables in different typical-day sets is increased with the calculation time window, the changing amplitude of CI95% is 0.308, 0.546, 0.759, and 1.484 when the calculation time window is 10 min, 30 min, 1 h, and 4 h, respectively.These findings suggest it is important to develop different operational and maintenance strategies for the wind turbine, according to the detailed fluctuation characteristics of wind power output in each typical-day, so as to reduce the adverse impact of wind power integration.

Fig.7 Detailed volatility of WT1 output in each typical-day set

3.3 Detailed volatility of wind power at multiple spatial scales

In Section 3.2, the detailed volatility of wind power are systematically studied under multiple temporal scales (year- season - month - day), but the analysis objective is only for the single wind turbine.In this part, the spatial scales are extended to wind turbines, wind farm and wind farms to research on the detailed volatility of wind power in four calculation time windows, at the temporal scale of year.

(1) Spatial scales of wind turbine, wind turbines and wind farm

First, the analysis spatial scales are extended to wind turbines and wind farm, and the detailed volatility of different number of WTs output in WF1 are analyzed based on the proposed evaluation indexes.As shown in Fig.8, 1) the detailed volatility of WTs output is decreased with the number of WTs, although the calculation time windows are different.The smoothing effect of wind power trends to be stable in the wind farm when the number of WTs reaches about eleven.2) Thesmoothing effect of WTs output is basically independent with the calculation time window.When the analysis spatial scale increases from wind turbine to wind farm, the variation range of the CI95% of wind power variables is basically the same in four calculation time windows, about 0.14, 0.16, 0.17, and 0.16, respectively.

Fig.8 Detailed volatility of different number of WTs output in WF1

(2) Spatial scales of wind farm and wind farms

Then, the spatial scale is further extended to wind farms, the proposed volatility evaluation indexes are used to study the detailed fluctuation characteristics of wind farm and wind farms output in four calculation time windows, as shown in Fig.9.Compared with the detailed volatility of different number of WTs output in WF1 (as shown in Fig.8), the smoothing effect among wind farms output is weaker.Taken the calculation time window of 1 h as an example, the CI95% of WF1, WF1&2, WF1&2&3 output variables are 0.56, 0.54, and 0.53, respectively.The CI95% of wind power variables only changes 0.03 from one wind farm to three wind farms.There are two reasons for this weaker smoothing effect.One is the distance between wind farms is much larger than the distance between wind turbines in the same wind farm, which weakens the temporal-spatial coupling characteristics of the wind conditions.Another is the installed capacities of WF2 and WF3 are both smaller than that of WF1, leaded to the smaller output of WF2 and WF3, which further weakens the smoothing effect among wind farms.

Fig.9 Detailed volatility of wind power at spatial scales of wind farm and wind farms

4 Relationships between wind power forecasting accuracy and its detailed volatility

To further validate the effectiveness of the proposed volatility evaluation indexes, the relationships between wind power forecasting accuracy and its detailed volatility are studied in this section.Back propagation neural network (BPNN) [15] is employed to predict the wind power at multiple spatial scales under the ultra-short-term time scale, based on the internal relationship among wind power series.The forecasting time horizons in this analysis are 10 min, 30 min, 1 h, and 4 h, which corresponded to the calculation time windows of wind power variables.

The operating environment of wind power forecasting in this section is MATLAB 2017a, and the main parameters of BPNN in several spatial scales are listed in Table 5, where lr refers to the learning rate, layer is the number of hidden layers in the neural network, n1 is the number of nodes in the first hidden layer, and n2 is the number of nodes in the second hidden layer.

Table 5 Main parameters of BPNN in several spatial scales

WT1 10 min 30 min 1 h 4 h lr 0.04 0.03 0.05 0.05 layer 2 2 2 2 n1 4 2 2 3 n2 2 3 5 2 7WTs 10 min 30 min 1 h 4 h lr 0.01 0.03 0.01 0.04 layer 2 2 2 2 n1 2 2 3 4 n2 4 4 5 4 WF1 10 min 30 min 1 h 4 h lr 0.03 0.02 0.05 0.01 layer 2 2 2 2 n1 2 2 2 2 n2 4 4 2 5 WF1&2&3 10 min 30 min 1 h 4 h lr 0.03 0.02 0.04 0.04 layer 2 2 2 2 n1 2 4 3 3 n2 5 5 4 3

The root mean square error (RMSE) [16] is used as the evaluation criterion to evaluate the wind power forecasting accuracy in this study, and the calculation formula is shown as follows.

where, Pf (t ) is the forecasted wind power at time t and n is the number of forecasted samples.

First, the RMSE of WT1 output in four forecastingtime horizons are calculated, the results are listed in Table 6.As shown, the forecasting error is increased with the time horizon in the four seasons due to the previous wind speed has less influence on the subsequent wind conditions.The forecasting error is the largest in spring, followed by summer and winter, and the smallest in autumn under different time horizons.The seasonal trend observed in the forecasting results is essentially identical to the trend observed in the CI95% of WT1 output variables in four seasons (as listed in Table 3).

Table 6 RMSE of WT1 output under four forecasting time horizons

RMSE 10 min 30 min 1 h 4 h Spring 7.6% 10.1% 12.4% 20.0%Summer 7.4% 10.0% 12.0% 18.6%Autumn 6.5% 8.6% 10.7% 17.4%Winter 6.8% 9.5% 12.1% 18.5%

Then, the RMSE of WT1 output in 32 typical-day sets under four forecasting time horizons are calculated, the scatter diagrams between RMSE of WT1 output and CI95% of WT1 output variables are shown in Fig.10.The linear regression method is used to fit these two variables in different time horizons.As shown, 1) the RMSE of WT1 output is positively correlated with the CI95% of WT1 output variables, when the calculation time window of wind power variables is the same as the forecasting time horizon, i.e., the power forecasting error is increased with the power detailed volatility under the ultra-short-term time scale.2) The slope of the linear regression equation is decreased with the forecasting time horizon.In other words, as the forecasting time horizon increased, the influence of the wind power detailed volatility on the forecasting accuracy becomes gradually weaker under the ultra-short-term time scale.The slope of the linear regression equation is 0.195, 0.142, 0.134, and 0.09 under the forecasting time horizons of 10 min, 30 min, 1 h, and 4 h, respectively.

Finally, the forecasting spatial scale is extended to further study the relationships between the RMSE and the detailed volatility of wind power.The wind power forecasting error in the different spatial scales (wind turbine-wind turbines-wind farm-wind farms) at the temporal scale of year are shown in Fig.11, where it can be seen that the RMSE of wind power is decreased as the number of WTs is increased for a given forecasting time horizon, although the rate of reduction is decreased as the number of WTs is increased.The forecasting accuracy of WTs output in WF1 approached stability when the number of WTs reaches about eleven.The variation rule describing the wind power forecasting accuracy versus the number of WTs in this figure is essentially the same as that describing the variation in the detailed volatility of wind power with the number of WTs in Fig.8.The RMSE of wind power is improved by 2.92%, 2.82%, 2.63%, and 2.46%, respectively, under four forecasting time horizons, from one wind turbine (WT1) to one wind farm (WF1).As the forecasting spatial scale is further expanded, the improvement of the forecasting accuracy among wind farms is smaller than the improvement among wind turbines.The corresponding law describing these changes is essentially the same as that describing the trend of the output detailed volatility with the number of wind farms in Fig.9.The RMSE of wind power is improved by 0.44%, 0.48%, 0.47%, and 0.25%, respectively, under four forecasting time horizons, from one wind farm (WF1) to three wind farms (WF1&2&3).

Fig.10 Relationships between RMSE and detailed volatility of WT1 output

Fig.11 RMSE of the wind power in different spatial scales

These results indicate that the wind power forecasting accuracy is closely related to the detailed fluctuation characteristics of wind power under the ultra-short-term time scale, and further verify the effectiveness of the proposed volatility indexes.

5 Conclusions

This study proposes the volatility indexes which can evaluate the detailed fluctuation characteristics of wind power based on the frequency distribution of power variables.Data obtained from a particular location in northern China is used to illustrate the method of applying the presented indexes at multiple temporal (year-seasonmonth-day) and spatial scales (wind turbine-wind turbines -wind farm-wind farms).In addition, the relationships between wind power forecasting accuracy and its detailed volatility are analyzed to confirm the effectiveness of the proposed indexes.The following conclusions can be drawn from the results of this study.

(1) The proposed indexes are shown to effectively describe the detailed fluctuation characteristics of wind power at multiple temporal-spatial scales.

(2) The detailed volatility of WT output is the most severe in spring, followed by summer and winter, and the least in autumn in four calculation time windows at the temporal scale of season.When the temporal scale descends to day, the X-means algorithm can provide an effective classification of typical days.The detailed volatility of WT output varies greatly in different typical-day sets.It is important to develop different operational and maintenance strategies for the wind turbine, according to the detailed fluctuation characteristics of wind power output in each typical-day.

(3) As the number of WTs is increased, the detailed volatility of WTs output becomes weaker, and the rate of weakening is decreased as the number of WTs is increased.When the number of WTs reaches about 11, the decreasing rate of WTs output detailed volatility is essentially zero, and the smoothing effect tends to be stable.The variation of the CI95% of power variables under the four calculation time windows is 0.14, 0.16, 0.17, and 0.16, respectively, when the analysis spatial scale changes from one wind turbine (WT1) to one wind farm (WF1).Affected by the temporalspatial coupling characteristics of the wind conditions and different installed capacities of the wind farms, the smoothing effect among wind farms is weaker than that within the single wind farm.For example, the variation of the CI95% of power variables under the calculation time window of 1 h is only 0.03 when the analysis spatial scale changes from one wind farm to three wind farms.

(4) When the time series method is employed to wind power forecasting under the ultra-short-term time scale, the forecasting error is essentially positively correlated with the detailed volatility.The results further demonstrate the effectiveness of the proposed indexes.Furthermore, as the forecasting time horizon is increased, the influence of the power detailed volatility on the forecasting accuracy is gradually weakened.The slope of the linear regression equation between the RMSE of WTs output and the CI95% of WTs output variables is 0.195, 0.142, 0.134, and 0.09 under the forecasting time horizons of 10 min, 30 min, 1 h, and 4 h, respectively.

In terms of limitations, as weather conditions, power grid conditions, and wind turbine operating conditions, etc., vary from location to location, the understanding developed in this study of the wind power detailed volatility at multiple temporal-spatial scales in one location is somewhat limited.Even so, the proposed indexes and the analysis methods are valid and can be applied to investigate the wind power detailed volatility at multiple temporal-spatial scales in other locations globally.

Acknowledgements

This work was supported in part by the National Key R&D Program of China (No.2017YFE0109000), and the project of China Datang Corporation Ltd.