Positive Correlations between Short-Term and Average Long-Term Fluctuations in Wind Power Output

: Wind power has been increasingly deployed in the last decade to decarbonize the electricity sector. Wind power output changes intermittently depending on weather conditions. In electrical power systems with high shares of variable renewable energy sources, such as wind power, system operators aim to respond ﬂexibly to ﬂuctuations in output. Here, we investigated very short-term ﬂuctuations, short-term ﬂuctuations (STFs), and long-term ﬂuctuations (LTFs) in wind power output by analyzing historical output data for two northern and one southern balancing areas in Japan. We found a relationship between STFs and the average LTFs. The percentiles of the STFs in each month are approximated by linear functions of the monthly average LTFs. Furthermore, the absolute value of the slope of this function decreases with wind power capacity in the balancing area. The LTFs reﬂect the trend in wind power output. The results indicate that the ﬂexibility required for power systems can be estimated based on wind power predictions. This ﬁnding could facilitate the design of the balancing market in Japan.


Introduction
For improved environmental sustainability, renewable energy generators have been widely deployed all over the world. At the 21st session of the Conference of the Parties, participating countries agreed to find a balance between sources and sinks of greenhouse gases in the second half of this century. These countries are required to meet aggressive targets to reduce greenhouse gas emissions [1].
Renewable energy generation is effective for reducing greenhouse gas emissions in the electricity sector. The International Energy Agency and the International Renewable Energy Agency reported that the share of renewable energy needs to increase from about 15% of the primary energy supply in 2015 to 65% in 2050 to accomplish the stated goals [2]. Wind and solar power provide a large proportion of variable renewable energy (VRE) [3]. In Japan, the installed capacity of wind power was 2.3 GW in 2010 and 3.9 GW in 2019 [4]. According to the Long-term Energy Supply and Demand Outlook report published by the Ministry of Economy, Trade and Industry, Japan, the target for wind power installation by 2030 is 10 GW [5]; according to a report by the Japan Wind Power Association, the target for 2050 is 75 GW [6].
VRE power output changes intermittently, depending on the weather conditions. Fluctuations in VRE power outputs can negatively impact the operation of a power system [7][8][9][10][11][12]. A power system operator requires flexibility to compensate for these fluctuations. Traditionally, the system operator controls power outputs from conventional power plants to maintain the balance between supply and demand, which is a source of flexibility.
Growth of the share of VRE in the power supply increases the impact of fluctuations in VRE output on system operation. Mitigation of these fluctuations and provision of flexibility are urgent issues that must be addressed to accelerate the use of VRE resources. The issue of power system flexibility has been discussed by committees within the Organization for Cross-regional Coordination of Transmission Operators, Japan (OCCTO) [13]. Recently, OCCTO has been studying the design of a new balancing market to ensure flexibility [14].
Understanding fluctuations in VRE outputs helps system operators to efficiently respond to them. From the perspective of system operation, fluctuations in demand and VRE outputs can be considered according to a few timescales. According to the European Network of Transmission System Operators and OCCTO, fluctuations are compensated by primary, secondary, and tertiary reserves [14,15]. In this paper, we focus on fluctuations in wind power output and distinguish among historical data on very short-term fluctuations (VSTFs), short-term fluctuations (STFs), and long-term fluctuations (LTFs). VSTFs and STFs have timescales of less than 1 min and several tens of minutes, respectively. LTFs indicate the trend of wind power output.
In Japan, there are 10 BAs. The northern BAs have a different pattern of fluctuations in wind power output than the southern ones. We analyzed fluctuations in three Japanese BAs, i.e., those of Hokkaido, Tohoku, and Kyushu, as shown in Figure 1. While the Hokkaido and Tohoku areas have many windy days in winter, the seasonal pattern is different in Kyushu. The STFs in the Hokkaido area have larger amplitudes than those in the other two BAs. The Tohoku area has smaller-amplitude STFs than the other two BAs. Although the seasonality and fluctuations differ among these three BAs, we found that they were similar with respect to the relationship between STFs and LTFs. The relationship is approximately a linear function, in which the STFs increase with the monthly average of the LTFs. In addition, the absolute value of the slope of each linear function decreases with the wind power capacity of the BA.
The relationship between the STFs and LTSs was revealed through this study and could influence the smoothing effect [8,16,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Thus, the large LTFs, which increase the STFs, could reduce the smoothing effect in the STFs. However, if the monthly wind power output is estimated using historical output and weather data, a system operator could more accurately estimate the reserves that are necessary to procure against the STFs in the wind power output of each BA. The secondary reserves respond to the STFs in wind power output and consist mostly of rapidly starting gas turbine power plants, hydro storage plants, and load shedding [37], while the hydro storage plants and power system planning cater to the LTFs. Information about STFs in wind power output could facilitate the design of a new balancing market that ensures the required reserves [14,38].  This paper is organized as follows. Sections 2 and 3 describe the historical data on wind power output used for our analysis of fluctuations and the methodology used to distinguish among the different sorts of fluctuations, respectively. Section 4 presents the results, and Section 5 is devoted to discussion and conclusions.

Historical Data of Wind Power Outputs
Historical data on the total wind power output in three BAs were used: Hokkaido, Tohoku, and Kyushu. These three BAs are considered to have high potential for wind power resources [39]. The wind power capacities of these BAs are listed in Table 1. The capacity of the Hokkaido area was smallest, being less than half that of Tohoku, which has the largest capacity and highest number of WPPs. Kyushu is intermediate in terms of capacity and number of WPPs. The time periods of the data are 1 fiscal year (FY), FY2012, in the Hokkaido and Kyushu areas, and 3 years, FY2010-FY2012, in the Tohoku area. A FY in Japan is from 1 April to 31 March. The data have small time resolutions, less than 10 s, allowing analysis of the VSTFs and STFs, whose periods are less than 1 min and several tens of minutes, respectively.
Some of the raw data for the wind power outputs were missing. When the period of missing data was less than 3 h, the missing data were linearly interpolated. Missing data with longer periods were removed from the time series, as listed in Appendix A.
The data were analyzed in terms of per unit (p.u.) values, which indicate the ratio of wind power output to capacity. While the ideal maximum output corresponds to 1 p.u., the actual total output never reaches this level because of the smoothing effect. Figure 2 shows the duration curves of the wind power output in the Hokkaido, Tohoku, and Kyushu areas in FY2012. The maximum output was approximately 0.9 p.u. The Kyushu area had smaller outputs than Hokkaido and Tohoku.

Methods
Traditionally, in Japanese electrical power systems, STFs in electricity demand have been defined within 20-min windows. STFs in wind power output have been analyzed in terms of standard deviation (SD), differences among several timeframes, power spectra, and wavelets; their static features have also been discussed [16,19,26,32,[40][41][42][43]. While analysis methods based on SDs and differences in timeframes are computationally cheap, they cannot separate time series of wind power outputs into time series of different sorts of fluctuations. Methods based on power spectra and wavelets can provide detailed information about fluctuations; however, the computational costs of these methods are too large for applications in real-time system operation.
In this paper, we focus on STFs in wind power output to promote the use of wind power in power systems, while avoiding any negative impact of this power source on system operation. The methods used to distinguish STFs in wind power outputs must be computationally cheap and provide detailed information. A method with centered moving averages (CMAs) is adopted, which is described later [26].
Regardless of the differences in the methods mentioned above, some common characteristics of STFs have been reported: smoothing effects mitigate STFs [16,32] and STFs have non-Gaussian distributions with heavy tails [26,43].

Separation Method for LTFs, STFs, and VSTFs in Wind Power Outputs
The method using CMAs is summarized in Figure 3. The time series of wind power output is separated into VSTFs, STFs, and LTFs by this method. The timescale of VSTFs is determined by a parameter, X; the time resolution of the wind power output must be shorter than this timescale. Likewise, the timescales of STFs and LTFs are determined by X and Y. X has only positive values and must be sufficiently smaller than Y. The X and Y CMAs,p X (t) andp Y (t), respectively, are defined as follows: where p(t) represents the wind power output at time t and N X and N Y are the numbers of data points in time periods X and Y (in minutes), respectively. VSTFs, STFs, and LTFs are represented byp X (t) andp Y (t), respectively, as follows: VSTF(t), STF(t), and LTF(t) can reproduce the original time series as follows: . Therefore, no information is lost during the separation process. Figure 4 shows the time series of VSTFs, STFs, and LTFs. X and Y in Figure 3 and Equations (1)-(3) are fixed to 1 and 30 min, respectively. The validity of this parameter setting is discussed in the next subsection. Figure 4a includes the time series of p(t), LTFs, STFs, and VSTFs. LTFs represent the trend of p(t) and change smoothly; they have larger amplitudes than those of VSTFs and STFs. In Figure 4b, note that the STFs and VSTFs change around a value of zero. STFs tend to have larger amplitudes and longer cycles than VSTFs.  Figure 5 shows the power spectra of the time series of p(t), LTFs, STFs, and VSTFs in the Tohoku area in FY2012. Comparing Figure 5a,b indicates that the power spectrum of the LTFs has a similar trend to that of p(t). However, the power spectrum of the LTFs has a steeper slope than that of p(t) because the separated VSTFs and STFs are removed. The power spectra of the VSTFs and STFs have peaks after around several tens of seconds and minutes, as can be seen in Figure 5c,d, respectively. Microscopically, the time series of STFs consist of multiple half-cycles. Each half-cycle has an amplitude, ∆p, and a period, τ, as shown in Figure 6, which contains more than three half-cycles. The timescale of STFs is double the length of τ, thus, the period of a whole cycle.  In this paper, STFs are defined as fluctuations that have a timescale of several tens of minutes. The parameters were set to X = 1 min and Y = 30 min in Figure 3 and Equations (1)-(3), and analyzed STFs based on the power spectra, ∆p, and τ. In our first analysis, there were a number of STFs with τ values of several tens of minutes. Other analyses showed that STFs with the highest ∆p have periods of approximately 30 min. We consider that appropriate parameter settings can provide STFs consistent with this definition.

Statistical Properties of STFs
Some statistical properties of STFs in the Hokkaido, Tohoku, and Kyushu areas in FY2012 are listed in Table 2. Comparing the three BAs, the Hokkaido one has the highest SD and percentile values. The Kyushu area has a smaller SD and the 1st and 99th percentile values than Tohoku. These results are consistent with the trend shown in Figure 7. Table 2. Statistical properties of short-term fluctuations: standard deviation (SD) and correlation coefficients between short-and very short-term fluctuations (CC VSTF ) and long-term fluctuations (CC LTF ). The Pearson product-moment correlation coefficients between STFs with VSTFs and LTFs provide information about correlations at the same time and are defined as follows:

Hokkaido Area
VSTF, STF, and LTF are the averages of VSTF(t), STF(t), and LTF(t) during N∆t, where ∆t is the time step, 3 s in the Hokkaido area, 10 s in the Tohoku area, and 2.5 s in the Kyushu area; N is the number of data. Correlation coefficients of STFs with VSTFs and LTFs, CC VSTF and CC LTF , respectively, in Table 2, indicate that the correlations are small; thus, STFs are effectively not correlated with VSTFs or LTFs. On the other hand, we can deduce that there is some relationship between LTFs and STFs, as follows. When LTFs in p(t) are small, p(t) itself must also be small. When p(t) is small (large), the possible decrement (increment) of STFs at the next time step is limited by the wind power output at that time. Therefore, we infer that the LTFs affect STFs.

Distributions of STFs and LTFs
To determine if there was any relationship between STFs and LTFs, we investigated their distributions. Scatter diagrams of STFs versus LTFs in FY2012 are shown in Figure 8. In the three BAs, when LTFs are small, the ∆p of the STFs are also small. The high-density areas, which are colored red, are approximately elliptical in shape in all three BAs, with the height and width of the ellipse differing according to the BA. The ellipse is largest for the Hokkaido area, indicating that it had the largest ∆p for STFs. Although the Tohoku area has LTFs of approximately the same width of those of the Hokkaido area, the ∆p of its STFs tended to be smaller than those in Hokkaido, except for a few long lines around the ellipse. In Figure 8, there are several long lines around the ellipses. Each continuous line indicates a single event during which p(t) increased or decreased drastically. In the Hokkaido and Tohoku areas, these events tended to occur in winter. In the Tohoku area, two events in November induced the largest and smallest STFs; these events are depicted in Figure 9. In both cases, multiple low-pressure systems passed near the Tohoku area [44] and we infer that these caused the events. The event on 27 November, shown in Figure 9a, resulted from the wind turbines in multiple WPPs cutting out [26]. Wind turbines cut out to avoid damage when the wind speed exceeds 25 m/s.  Figure 10 shows the maximum, minimum, and percentiles of SFTs in each month in FY2012. While the maximum and minimum in each month are strongly affected by single significant events, the percentiles depend on multiple events. Therefore we focus on these in this paper, especially the 10th and 90th ones. In the Hokkaido and Tohoku areas, the percentiles have larger amplitudes in winter, from November to February. The percentiles for the Kyushu area are approximately the same across FY2012. The STFs in the Tohoku area in FY2010 and FY2011 also tended to be larger in winter, as shown in Figure 11. From FY2010 to FY2012, its STFs were smaller from June to August.

Relationship between STFs and Average LTFs
Although the correlation coefficient between STFs and LTFs is small, as shown in Table 2, in the Hokkaido and Tohoku areas, the ∆p of STFs tended to increase in winter when p(t) increased [45], as shown in Figures 10 and 11. Hence, we inferred that there is some relationship between STFs and p(t).
To reveal the relationship between STFs and LTFs, which indicate the trend in p(t), in Table 3, we list the correlation coefficients between the monthly 10th and 90th percentiles of STFs and average LTFs. These coefficients in each BA are sufficiently large to indicate that there is a correlation. In Figure 12, we show the relationship between the 10th and 90th percentiles of STFs and the average LTFs in each month, for the three BAs in FY2012. Each data point represents one monthly percentile for one BA, thus, there are 72 dots in the figure, with those above (below) the horizontal axis representing the 90th (10th) percentile. For all BAs, the STF percentiles increased with the monthly average LTFs, with the dashed lines indicating functions obtained by least-squares approximation of the percentiles in each BA. The slopes of the lines for the Hokkaido area have the largest absolute values, with those of Tohoku being the smallest. The absolute values of the slopes tend to decrease with wind power capacity, as shown in Figure 13. While the capacity in the Hokkaido area was less than half that in the Tohoku area, the absolute values of the slopes for Tohoku were larger than half of those for Hokkaido. We expect that the absolute values of these slopes would not become zero even if the capacity were exceptionally large. Therefore, the absolute values of the slopes would decrease asymptotically to nonzero values.

Discussion and Conclusions
A high share of wind power generation in a power system can amplify the negative impact of fluctuations in p(t) on system operation. To respond to these fluctuations, a system operator prepares reserves with different timescales. While the fluctuations in p(t) are mitigated though the smoothing effect [8,16,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], large fluctuations can remain when a BA is restricted to a small area and has insufficient interconnections with other BAs. Japan has 10 BAs and an inhomogeneous distribution of potential wind resources. Therefore, to promote wind power generation in Japan, mitigation of the fluctuations in p(t) is an urgent issue.
The fluctuations are separated into VSTFs, STFs, and LTFs, depending on their timescale. VSTFs and STFs have timescales of less than 1 min and several tens of minutes, respectively. LTFs have the longest timescale, of more than 1 h. In this paper, time series of p(t) are separated into VSTFs, STFs, and LTFs via a method based on CMAs [26].
We analyzed historical data for p(t) in three BAs, i.e., Hokkaido, Tohoku, and Kyushu, in Japan in terms of their fluctuations. The Hokkaido and Tohoku areas are in the northern region on the main island of Japan, while the Kyushu area is mainly on Kyushu Island, which is south of the main island. The northern BAs, Hokkaido and Tohoku, have a different pattern of fluctuations from Kyushu, where the p(t) and ∆p of their STFs tend to be large in winter and decrease in summer.
The LTFs of p(t) indicate the trend in p(t). We investigated the relationship between STFs and LTFs and found that the 10th and 90th percentiles of the STFs in each month increased with the monthly average LTFs in all three BAs. The relationship between the STF percentiles and the average LTFs in each BA can be approximated by a linear function. Additionally, the absolute value of the slope of this function decreases with the wind power capacity in a BA. This result provides an indication of the reserves required to respond to STFs in p(t) in each BA.
Recently, OCCTO has been working on a new balancing market for Japan [14]. Information about fluctuations in p(t) could contribute to efficient procurement of reserves for the planned market, to promote the penetration of wind power while avoiding the negative impact of fluctuations in p(t) on the power system. We consider that information about fluctuations is essential for designing the new balancing market. According to the relationship between the STFs and LTFs, the STFs can amplify and require more reserves when wind power output increases, thus, when the LTFs are expected to increase. The new balancing market should be designed to ensure required reserves, and system operators use the reserves to respond to the STFs. In future work, we plan to develop a wind power control method to mitigate STFs therein. Funding: This paper is based on results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO).

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available because they include information about the non-public corporate profits of WPP owners.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:

Appendix A. Missing Data List
Sequences of missing data longer than 3 h were removed from the original time series of p(t). The missing data are listed in Table A1. We defined the rate of loss as the proportion of 1 year that the missing data covered. In the Hokkaido area, the missing data were caused by signal transfer errors from WPPs to the data aggregation system. In the Tohoku area, there were two sequences of missing data: from 11 to 12 March 2011 and from 7 to 8 April 2011. Earthquakes caused these losses; the former began just after the Great East Japan Earthquake on 11 March 2011, and the latter began just after an aftershock of the earthquake.