Short-Term solarphotovoltaic power forecasting error distribution estimation

133 Tập 12, Số 5, 2018Tạp chí Khoa học - Trường ĐH Quy Nhơn, ISSN: 1859-0357, Tập 12, Số 5, 2018, Tr. 133-141 *Email: tuanhole@qnu.edu.vn Ngày nhận bài: 18/5/2018; Ngày nhận đăng: 30/6/2018 SHORT-TERM SOLARPHOTOVOLTAIC POWER FORECASTING ERROR DISTRIBUTION ESTIMATION TUAN-HO LE Department of electrical engineering, Faculty of Engineering and Technology, Quy Nhon University ABSTRACT Among the potential renewable energies, solar photovoltaic (PV) power has recently experienced enormous utilizationin electrical power generation due to the advances of solar PV technologies. Various studies have been conducted in order to forecast solar PV power generation in different perspectives. Unfortunately, there is always a mismatch between the forecasted solar PV power data from the proposed methodologies and the measured generation data. Therefore, the main purpose of this paper is to statistically investigate the short-term solar PV power forecasting errors of practical generation measurements. First of all, the Belgium short-term solar PV power generation data was analyzed. Secondly, the solar PV power forecasting errors were considered by using different statistical methods. Then, the analyzed error distribution of solar PV power was compared with the commonly assumed normal distribution. Finally, a correcting method was utilized to improve the assumption of solar PV power forecasting error distribution. Keywords: Error distribution, forecasting, short-term, solar photovoltaic power. TÓM TẮT Ước lượng thống kê hàm phân phối của sai số trong dự báo ngắn hạn về năng lượng mặt trời Các dạng năng lượng tái tạo bao gồm năng lượng gió và năng lượng mặt trời được xem như là nguồn năng lượng sạch và nguồn năng lượng vô tận trên thế giới. Do đó, gần đây, năng lượng mặt trời được sử dụng rộng rãi trong các nguồn điện của các hệ thống điện bởi các ưu điểm và sự tiến bộ về công nghệ của dạng năng lượng này. Việc dự báo về lượng điện năng được tạo ra bởi các tấm pin năng lượng mặt trời là một công việc khá quan trọng và đã có nhiều nghiên cứu đề xuất các phương pháp dự báo về lượng điện năng tạo ra. Tuy nhiên, các phương pháp dự báo này hầu như không thể hiện được sự vượt trội so với các phương pháp còn lại do có sai số lượng điện năng dự báo và đo đạc được. Do đó, các nhân viên vận hành hệ thống điện cần hiểu rõ về các sai số dự báo này để có thể vận hành hệ thống điện tốt hơn. Chính vì vậy, mục tiêu chính của bài báo này là nghiên cứu theo thống kê các sai số dự báo ngắn hạn của năng lượng mặt trời sử dụng các dữ liệu thực tế. Trước tiên, bài báo phân tích dữ liệu điện năng được tạo ra bởi năng lượng mặt trời trong hệ thống điện của Bỉ. Sau đó, các sai số dự báo năng lượng điện mặt trời được xem xét theo nhiều tiêu chuẩn thống kê khác nhau. Tiếp theo đó, hàm phân phối sai số này được phân tích và so sánh với hàm phân phối chuẩn. Cuối cùng, một phương pháp hiệu chỉnh được áp dụng trong bài báo này với mục đích là để giảm nhẹ giả thiết về hàm phân phối sai số trong dự báo ngắn hạn của năng lượng mặt trời. Từ khóa: Dự báo, hàm phân phối sai số, năng lượng mặt trời, ngắn hạn. 134 Tuan-Ho Le 1. Introduction The conventional energy resources from fossil fuels have been gradually reduced in electrical power generation systems because of several environmental issues about greenhouse, global warming, and climate change. In order to achieve this purpose, a transition from these traditional energies towards renewable energy resources such as wind power and solar power, is imperative. Particularly solar power generated through photovoltaics (PV) has seen tremendous growth over the last decade, with a total of 227.1 GW installed at the end of 2015 [1]. According to IEA’s highest projection, the installation of global solar PV power capacity could exceed 1700 GW by 2030 [2]. Unfortunately, high penetration of renewable energy into interconnected power systems is challenging to the balance of electricity due to the variability of these weather- dependent generation resources. The unpredicted solar PV power output adversely affects the stability, reliability, and scheduling of the power system operation, aside from the economic benefit [3,4]. An accurate forecasting of solar PV power generation may diminish the uncertainty of solar PV power on several electrical system issues, such as system reliability, power quality, and penetration level of PV systems. Therefore, a variety of useful forecasting methodologies and techniques were proposed to predict the solar PV power generation. According to time framework, solar PV power forecasting methodologies can be divided into four different types, namely very short-term, short-term, medium-term, and long-term predictions [5]. According to [5], very short- term solar PV power forecasting can refer to the generation information from a few seconds to minutes, and the application of this classification is for solar PV and storage control and electricity market clearing. Short-term prediction is used for the data from 1 hour up to 48-72 hours, and such forecasts arecrucial for different decision-making problems involved in the electricity market and power system operation, including economic dispatch, unit commitment. The medium-term forecasting indicates the forecasting up to one week ahead, and it is classified for maintenance scheduling of PV plants, conventional power plants, transformers, and transmission lines. The long-term prediction/estimation can be applied for long-term solar energy assessment and PV plant planning up to months to years. In this paper, the short-term forecasting is used to analyze the solar PV power. Similar to wind power forecasting domain, solar PV power forecasting methods can be generally classified in terms of physical models, traditional statistical methods, artificial intelligent (AI) approaches, and hybrid models [5, 6, 7]. Statistical approaches are based on data- driven formulation using historical measured data to forecast solar time series [8]. Several AI- based techniques were proposed to construct solar forecasters [9]. Physical models are based directly or indirectly on numerical weather prediction (NWP) or satellite images that predict solar irradiance and solar PV generation [10, 11]. The combination of two or three mentioned models can provide different hybrid models [12, 13]. All of the forecasting models and tools were proposed to estimatethe most accurate solar PV power generation values compared to the measured ones. Unfortunately, there are always errors between the predicted and practical data due to several factors, such as solar prediction, the path of sun, the atmosphere’s condition, the scattering process, and the characteristicsof a solar power plant. In most existing methods and 135 Tập 12, Số 5, 2018 statistical viewpoint, the forecast error distribution is often assumed to be normally distributed. This assumption may not always be guaranteed when the solar PV power forecast error distribution includes skewness and excess kurtosis. In this situation, the assumption of error distribution will have an impact on the final study results. Moreover, the error distribution is often assumed to be normal distribution with zero mean and constant variance. While assumed error distribution in solar power forecasting is normally distributed, the mean of this distribution does not always equal zero. Therefore, the primary motivation of this paper is to analyze the statistical properties of short-term solar PV power forecasting error distribution from Belgian solar power generation measurements. The comparisons were made between the investigated results and the commonly assumed normal distribution in most of electrical power system simulation studies. Finally, a correcting method was utilized to improve the normal distribution assumption of solar PV power forecasting error distribution. 2. Data analysis 2.1. Belgium solar PV power generation In this paper, solar PV power generation forecasting data from Belgian transmission system operator on their homepage was analyzed [14]. Belgium’s installed solar power capacity is currently 416.27 MW. The significant level of such solar PV capacity increases steadily, and this is also expected to progress in the following years. At every quarter of an hour, the unique quarter-hourly power generation value for upscaled measurements can be updated. This is also the amount of solar power generation equivalent to the running average evaluated for that particular quarterhour. In this paper, the available solar power data was monitored from April 26th, 2018 to May 03rd, 2018. The Belgian solar PV power generation and forecasting information was demonstrated in Figure 1. The solar power forecasting data was demonstrated by the solid line. The solar power forecasting data was illustrated by the dashed line. In the analyzed period, the mean value of generated solar power was 476.40 MW, the maximum generated solar power value was 2833.45 MW, and the minimum generated solar power generation was 0.00 MW. Also, the mean value of predicted solar power was 563.28 MW, the maximum predicted power generation was 2419.39 MW, and the minimum predicted solar power value was 0.00MW. 136 Tuan-Ho Le Figure 1. Belgian solar PV power generation and forecasting 2.2. Statistical background The Belgian solar power measurements were analyzed by several statistical techniquesin this paper. From statistical viewpoint, the range of values of a continuous random variable is normally illustrated by a probability density function (i.e., pdf) or density. The normal distribution is usually assumed to describe solar power forecasting error distribution. The first two standardized moments consisting of mean (i.e., µ ) and standard deviation (i.e.,σ ) or variance (i.e., 2σ ) are the primary information of solar PV power forecasting error distribution. In addition, the third moment of distribution (i.e., skewness) is the measurement of a probability distribution’s asymmetry, while kurtosis provides a measurement of the peakedness of a distribution as well as the weight of the tails of the distribution. If an observed data set is assumed to be normally distributed, the values of both skewness and kurtosis equal zero. In statistics, a histogram represented by a bar chart is normally used to demonstrate both the skewness and kurtosis of the probability distribution of a continuous variable. Another useful statistical criterion is quantile-quantile (Q-Q) plot which can be employed to compare the probability distribution of the Belgian solar power data set to the associated normal distribution by drawing their quantiles against each other. Furthermore, a probability plots is a graphical representation for comparing the similarities and differences between the distribution of observed data and the corresponding normal distribution. In this given plot, the non-normality of the observed data set with the associated 95% confidence interval (i.e., significance level α=0.05) is investigated by using Anderson-Darling (i.e., AD) test. Finally, a plot of empirical cumulative distribution function (i.e., cdf) is a statistical criterion used to evaluate the fitness of a distribution to solar power data, to estimate percentiles, and to compare different sample distributions. While the fitted distribution in a probability plot is a straight line, the fitting distributionin an empirical cdf plot is a highly nonlinear curve. In order to perform all statistical analysis in this paper, the MINITAB software and MATLAB software were utilized. 3. Results The impact of solar PV power generation forecasting error distribution on operations of power system will be considered in this section. The distribution of solar PV power generation 137 Tập 12, Số 5, 2018 forecasting errors observed in seven days from Belgian transmission power system was investigated. In this paper, error solar PV power value (i.e., forecastP ) is calculated by subtracting the forecast value (i.e., forecastP ) to the realized value (i.e., actualP ) as follows: (1) Figure 2 shows a histogram plot of the observed solar PV power forecast errors from Belgian transmission system in 2018. In this figure, the dotted line demonstrates a normal distribution with the mean value (86.76 MW) and standard deviation (351.5 MW) as the observed errors. The mean value (86.76 MW) shows the positive mean bias. In this figure, the number of observed data points is shown by N (i.e., 721). The observed error distribution is highly peaked, with the extremely right tail than the corresponding normal distribution. The error distribution of Belgian solar power generation forecasting is positive skew and leptokurtic. This histogram plot shows that the solar power generation error distribution is highly non-normal. power forecast actualP Pε = − Figure 2. Histogram of the observed solar PV power generation forecast errors in the Belgian transmission system in MW In statistics, a Q-Q plot is also used to examine the normality of the distribution. A normal Q-Q plot of Belgian solar power forecast errors generated by MATLAB software was shown in Figure 3. The solid line in the graph get sover the first and third quantiles of the planes of the observed data. This QQ plot also confirms non-normality of the errors distribution because all observations (i.e., the star symbols) do not fit the straight line. The probability plot of Belgian solar PV power generation forecasting errors with the corresponding 95% confidence intervals was drawn by using statistical MINITAB software as shown in Figure 4. Three solid lines in Figure 4 indicate a normal distribution of the observed data (i.e., the middle line) with the same mean (i.e., 86.76 MW) and standard deviation (i.e., 351.5 MW) with the associated 95% confidence intervals (i.e., two border solid lines). 138 Tuan-Ho Le Figure 4. Probability plot of Belgian solar PV power generation forecasting errors with the associated 95% confidence intervals The solar power errors shown as the star symbols in Figure 4 (i.e., outside the 95% confidence intervals) indicate the non-normality of the solar power error distribution. Both AD test value (132.808 greater than 0.641) and p-value (less than 0.005) describe the null hypothesis of the forecast error data coming from a normal distribution that was rejected at a statistical significance level of α = 0.05. A cdf plot of the observed errors and the corresponding normal distribution is shown in Figure 5. In this figure, the dashed line shows the fitted normal distribution with the same mean and standard deviation and the solid line show the empirical cdf of the observed errors. This figure also confirms the non-normality of the errors distribution because the dashed line is different with the solid line. Figure 3. Q-Q plot of Belgian solar PV power generation forecast errors 139 Tập 12, Số 5, 2018 Figure 5. A cdf plot of Belgian solar PV power generation forecasting errors Obviously, Figures (2) - (5) showed that the solar PV power forecasting errors are non- normal distribution which is usually assumed in the existing solar PV power generation forecasting studies. The estimated mean value of errors in the fitted normal distribution (i.e., 86.76 MW) as shown in Figures (2) - (5) are not zero. The accuracy of solar PV power forecasting models and methods can be increased by improving this mean value. The corrected solar PV power generation forecast can be transformed by subtracting the mean value of solar PV power forecast errors from solar PV power forecast as follows: (2) where correctforecastP and ( )powerE ε are the corrected solar PV power forecast and the average of solar PV power forecast errors, respectively. Based on Equation (2), the corrected solar power forecast errors (i.e., correctpowerε ) can be calculated as follows: . (3) From Equation (3), the fitted normal distribution of the corrected solar power forecast errors can be drawn in Figure 6. As shown in Figure 6, the fitted normal distribution of corrected errors has zero mean value and the same standard deviation value (i.e., 351.5 MW) with the previous error value as shown in Figure 2. ( )correctforecast forecast powerP P E ε= − correct correct power forecast actualP Pε = − 140 Tuan-Ho Le Figure 6. Histogram of the corrected solar PV power generation forecast errors in the Belgian transmission system in MW Obviously, the biased mean of error in the commonly assumed normal distribution showed that the existing difference between the forecasted and practical values in most actual forecasting methods. This normal assumption may cause the errors of forecasting results. Based on the proposed method, the error of solar power forecasting in common assumption is expected to be reduced when the mean of errors of the normal distribution equals to zero. Hence, the accuracy of the solar power forecasting can be increased. Therefore, the errors of solar power generation scheduling will be diminished. 4. Conclusions In this study, the solar PV power generation information including the forecast values and practical measurements from Belgian transmission system was investigated. In addition, the short-term solar PV power generation errors were analyzed by using various statistical methods including histogram plot, quantile-quantile plot, probability plot, Anderson-Darling test, and empirical cumulative distribution function plot. The errors distribution of solar PV power generation forecastingis not normally distributed as frequently assumed in several existing solar PV power forecasting studies. Moreover, the mean value of the corresponding fitted normal distribution does not equal zero. Therefore, a corrected method was applied in order to transform the mean value of the fitted normal distribution into zero. The applied method was presumed to reduce the solar PV power forecasting errors of existing estimation models and methods. For further studies, the Belgian short-term solar PV power generation forecasting errors will be statistically analyzed with other renewable resources in the same electrical power system. The actual distribution of renewable generation forecasting errors also can be investigated. 141 Tập 12, Số 5, 2018 REFERENCE 1. International Energy Agency, Snapshot of global photovoltaic markets, Technical Report, (2016). 2. International Energy Agency, Technology roadmap: solar photovoltaic energy, Technical Report, (2014). 3. Strzalka, A., Alam, N., Duminil, E., Coors, V., and Eicker, U., Large scale integration of photovoltaics in cities, Applied Energy, 93, 413-421, (2012). 4. Woyte, A., Van Thong, V., Belmans, R., and Nijs, J., Voltage fluctuations on distribution level introduced by photovoltaic systems. IEEE Trans Energy Convers, 21 (1),202–209, (2006). 5. Wan, C., Zhao, J., Song, Y., Xu, Z., Lin, J., and Hu, Z., Photovoltaic and solar power forecasting for smart grid energy management, CSEE Journal of Power and Energy Systems, 1 (4), 38-46, (2015). 6. Tung, D. D., and Le, T. H., A statistical analysis of short-term wind power forecasting error distribution. International Journal of Applied Engineering Research, 12 (10), 2306-2311, (2017). 7. Liu, Y., Li, Z., Bai, K., Zhang, Z., Lu, X., and Zhang, X., Short-term power-forecasting method of distributed PV power system for consideration of its effects on load forecasting, The Journal of Engineering, 13, 865-869, (2017). 8. Bacher, P., Madsen, H., and Nielsen, H. A., Online short-term solar power forecasting. Solar Energy, 83(10), 1772-1783, (2009). 9. Sfetsos, A., and Coonick, A. H., Univariate and multivariate forecasting of hourly solar radiation with artificial intelligence techniques. Solar Energy, 68 (2), 169-178, (2000). 10. Perez, R., Lorenz, E., Pelland, S., Beauharnois, M., Van Knowe, G., HemkerJr, K., ...and Steinmauer, G., Comparison of numerical weather prediction solar irradiance forecasts in the US, Canada and Europe. Solar Energy, 94, 305-326, (2013). 11. Geraldi, E., Romano, F., and Ricciardelli, E., An advanced model for the estimation of the surface solar irradiance under all atmospheric conditions using MSG/SEVIRI data. IEEE transactions on geoscience and remote sensing, 50(8), 2934-2953, (2012). 12. Nanou, S. I., Papakonstantinou, A. G., and Papathanassiou, S. A., A generic model of two-stage grid-connected PV systems with primary frequency response and inertia emulation. Electric Power Systems Research, 127, 186-196, (2015). 13. Bouzerdoum, M., Mellit, A., and Pavan, A. M., A hybrid model (SARIMA–SVM) for short-term power forecasting of a small-scale grid-connected photovoltaic plant. Solar Energy, 98, 226-235, (2013) 14. Belgian solar power generation and forecasts, Available online: power-generation/ Solar-power-generation-data/ Graph, (2018).