OUTPUTS AND ERROR INDICATORS FOR SOLAR FORECASTING MODELS

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1 OUTPUTS AND ERROR INDICATORS FOR SOLAR FORECASTING MODELS Mathieu DAVID Hadja Maïmouna DIAGNE Philippe LAURET PIMENT University of La Reunion Saint Denis Cedex 9 Reunion Island mathieu.david@univ-reunion.fr hdiagne@univ-reunion.fr philippe.lauret@univ-reunion.fr ABSTRACT The power output of photovoltaic systems is strongly dependent from the solar availability. Unfortunately, the solar energy fluctuates without correlation with the electricity demand. Thus, solar irradiance forecasting is necessary in order to achieve the large-scale integration of solar renewables into electricity grids. The solar irradiance is a particular physical quantity and it cannot be treat like one of the other weather parameters. The development and the assessment of forecasting models must take into account the differences between the long-term behavior and the short-term behavior of the solar energy. On the other hand, the energy production forecasting of solar renewables must be done in accordance with the electricity grid management. The day ahead forecasting of the solar energy permits a better scheduling of the other means of production of electricity. It must focus on the daily amount of solar energy and the daily profile. A shorter forecast time horizon is helpful for the unit commitment. The shortterm fluctuation range and occurrence must also be forecasted. A large number of error indicators have been used in the literature to assess the accuracy of the solar forecasting models. But only some of them could be useful to evaluate the quality of a solar irradiance forecasting. If the mean relative error (MrE) and the mean relative absolute error (MrAE) must be banished to deal with the solar energy or power, the mean bias error (MBE) seems to be more pertinent. 1. INTRODUCTION There is a growing concern about the potential of photovoltaic (PV) power output variability having a negative effect on utility grid stability. High levels of high frequency variability during partly cloudy conditions have been reported at some central PV generating stations and have contributed to create an awareness of this issue to the point where some in the utility industry believe it could constrain the penetration of grid connected PV (1). The solar irradiance forecasting is recently investigated in order to improve the management of electricity grids where the rate of PV power increases (2)(3). The solar forecasts will permit to better schedule the means of productions and to warn the grids operators from high amplitude ramping events. This paper presents a transversal reading of recent works that deals about the solar forecasting. The aim is to focus on the relevant parameters to forecast and the associated measures of error. These two aspects are analyzed inside eighteen articles published during the last ten years. In order to better present the studied concepts, an example of forecasting is implemented all along this paper. It concerns the first ten days of 2012, between the 1 st and the 10 th of January. The global horizontal irradiance was recorded at the weather station of Saint-Pierre, Reunion Island, with a sampling rate of 1 minute. Two forecasts of the hourly profile of irradiance of the following day are provided. The first one is derived from the GFS total cloud cover using the Perez and al. formula (4) between the sky cover and the global horizontal irradiance. The second one is the reference persistence model. The hourly profile of the clear sky index (2) corresponds to measured profile of the previous day. This example is not given to assess the performance of these two models (Fig. 1). We can notice that the GFS forecast is not able to catch the local weather of the site of measure. The Reunion Island experiences a lot of specific microclimates due to its relief. The altitude of its culminant point, Piton des Neiges, is 3070m. 1

2 Fig. 1: Measured and forecasted hourly profile of the global horizontal irradiance 2. FORECAST HORIZONS The variations of the solar irradiance that reaches the ground on a specific location are the result of two phenomena: The daily and annual rotations of the earth generate seasonal changes. The local weather, specially the clouds, causes intra-day fluctuations. The variations due to the sun path in the sky can be considered as the deterministic part of the solar radiation. Clear sky models are widely used in order to deseasonalize the time series of solar measurements (2) (4)(5)(6). The local weather conditions can be considered as the random part of solar radiation. They influence the instantaneous solar power but also the amount of energy. The dynamic of clouds produces a high frequency variability of the irradiance. Previous study (7)(8) found 1-min data to have different statistics from longer frequency (hourly, daily, monthly, etc.). Different works quantified the level of variability. They introduced new indices in the field of solar radiation analysis: Fluctuation power index (9), Ramp rates or first derivative (Fig. 2)(10), Standard deviation and relative standard deviation (1). Time scales required by the electricity value chain participants define the most critical requirements to solar power forecasting. Kostylev and Pavlovski (11) define most common industry-requested operational forecasts and their corresponding granularity: Intra Hour: 15 minutes to 2 hours ahead with 30 seconds to 5 minute granularity (relates to ramping events, variability related to operations) Hour Ahead: One to 6 hours ahead with hourly granularity (related to load following forecasting) Day Ahead: One to 3 days ahead with hourly granularity (relates to unit commitment, transmission scheduling, and day ahead markets) The table 1 gives a synthetic view of the forecast horizons used in works dealing with the solar radiation forecasting. Only one publication about the state of the art of solar forecasting includes all the needed time horizons of forecast (19). Actually, no model forecasts the solar radiation from 30 seconds to several days. 3. RELEVANT PARAMETERS TO FORECAST Actually, the developed methods predict the global horizontal irradiance (W.m -2 ) or its associated energy (Wh.m -2 ). It is clearly the most relevant parameter because it is proportional to the power or the energy produced by a PV system. To take into account the uncertainty associated to the forecast, a confidence interval around the mean value must also be provided. The standard deviation of the error would be interesting, but it is not proven that the statistic distribution of errors follows a normal law (Fig. 3). 2

3 Fig. 2: Measured global horizontal irradiance and ramp rates for the TABLE 1: FORECAST HORIZONS PRESENTED IN THE SET OF REFERENCES Ref. 1 day and more 3 hours to 24 hours 1 hour to 3 hours 30 sec.to 1 hour (2) X (3) X X X (4) X X (5) X X (6) X X (12) X X (13) X (14) X (15) X (16) X (17) X X X (18) X X X (19) X X X X (20) X (21) X (22) X (23) X (24) X When the high frequency fluctuations of the solar radiation experience successive strong ramp rates, up to several hundreds of W.m -2 in a minute, the performance of the forecasting models decreases strongly (18)(21). These variations weakly influenced the total amount of the produced energy, but they lead to strong ramp rates of power. These events can create instabilities in small-scale grids or in local areas. The forecasting of the nature of the irradiance fluctuation is relevant in order to manage smoothing methods (e.g. energy storage) in order to decrease the risk of instabilities. In the financial domain, high frequency fluctuations are also called the volatility. The aim of the very short-term forecasting (i.e. forecast time horizons shorter than several hours) is to assess the volatility of the solar irradiance. In the table 2, we propose some parameters to predict according to common short-term industry-requested operational forecasts, from several minutes to several days ahead. TABLE 2: PARAMETERS TO FORECAST FOR THE DIFFERENT HORIZONS Horizons Objectives Relevant parameters Day ahead Intra-day Intra-hour Scheduling and unit commitment Monitoring of the production and adjustments of scheduling Volatility and ramping events 4. RELEVANT ERROR METRICS Daily amount of energy Hourly profile of the irradiance Corrected hourly profile of irradiance Hourly fluctuation power index Mean and maximum ramp rates Frequency of the ramping events Standard deviation The basic error indicators assess the bias between two values. For the assessment of model performances, these two values are commonly the measured and the modeled data. These error indicators are detailed in the following equations 1 and 2. (1) ""#" " "#$ () = "#$% "#$ (2) "#$%&'( ""#" (") = "#$% "#$ Fig. 3: Statistic distributions of the error of the two forecasting models of the example Theses indices of error are useful in order to show the correlation between the bias of a model and a parameter that can be an input or another variable. The Error (1) and the Absolute Error (2) are particular interesting because they are expressed with same units as modeled and measured data. A large number of works propose relative versions of these basic error indicators (equations 3 and 4). 3

4 This normalization permits to have relative information that are independent of the values of the data. (3) "#$%&'" ""#" (") = "#$% "#$ "#$ (4) "#$%&'" "#$%&'( ""#" ("#) = "#$% "#$ "#$ When the data represents a large set, mean error indicators offer the possibility to assess the performance of a model with only one value. These mean indices are derived from the basic error indices (equations 5 to 7) (5) "#$ "#$ ""#" ("#) = "#$, (6) "#$ "#$%&'( ""#" ("#) = "#$ (7) ""# "#$ "#$%& ""#" ("#$) = "#$%, "#$, "#$%, "#$% As for the basic indices, relative versions of the mean error indicators are commonly used. Equations 8 to 11 give the most popular mean relative indices used by the scientific community. (8) "#$%&'" "# "#$ = "# "#$ (9) "#$ "#$%&'( "#$"%&'(" ""#" ("#$) = "# "#$ (10) "#$%&'" "#$ ("#$%) = "#$ "#$ (11) "#$ "#$"#$ ""#" ("#) = "#$%, "#$, "#$ (12) "#$ "#$%&'" "#$%&'( ""#" ("#$) = "#$%, "#$, "#$ The calculated values of the mean measures of error of the example are given in table 3. The table 4 presents the different indices of errors used in works dealing with the solar radiation forecasting. Some of them give the formula of the measures of errors. Even if the definitions of the main part of these indices of error are well known, it is important to define them in order to clearly inform the reader. For example, in the abstract of the article of Mathiesen (5), the MBE is called the bias. Reikard (18) uses the MrAE but calls it the MAPE. In their work, Mellit and Pavan (20) give a MAE in percent smaller, in absolute value, than the rmbe. It is normally not possible. So we can wonder us if they provided the MrAE instead of the MAPE. Faced with the large number of indicators proposed in the different works it would be interesting to select some of the most pertinent ones to avoid these misunderstandings. TABLE 3: MEASURES OF ERROR OF THE TWO FORECASTING MODELS OF THE EXAMPLE GFS Persistence MBE (W.m -2 ) MAE (W.m -2 ) RMSE (W.m -2 ) rmbe -38.5% 6.6% MAPE 46.5% 26.8% rrmse 65.3% 40.7% MrE -33.0% 24.5% MrAE 45.2% 45.7% MBE, RMSE and their relative versions (i.e. nmbe and nrmse) are the most used measures of error. In order to assess the efficiency of models to fit to the measured data, the RMSE is indicated. Glassley and al. (19) claim that that the RMSE metric is problematic as it is dominated by large errors. Thus if a forecast model is usually correct but occasionally off by a large amount it may score worse than a model that is always slightly off but never way off. The value of the RMSE has no physical meaning and cannot be used by the industrial operators to quantify the bias of the models. The MBE gives the mean error of the models with the same dimension as the data. It is particularly relevant to assess the hourly or daily amounts of energy. The value of the MBE can be used by the operators to evaluate the average risk associated to the scheduling of the means of production. For shorter time-scales, the MBE is not able to quantify the accuracy of models to forecast the fluctuations of the solar irradiance. A succession of variations upper and lower than the forecasted values generates a small MBE whereas the real efficiency of the model is bad. For forecast horizons lower than a day, the MBE must be complemented with the RMSE or the MAE. Glassley and al. (19) recommend adding the mean absolute error (MAE) or mean absolute percentage error (MAPE) as a standard evaluation metric since it is less sensitive to large errors than the RMSE. The MrE and the MrAE are sometimes used to quantify the mean error of the models. In the example, although the GFS forecast is less accurate than the persistence, its MrAE is smaller (table 2). These metrics are problematic. The value of these relative measures of error not depends on the value of the solar radiation. A bias of 5 W.m -2 presents a relative error of 100% if the irradiance is 5 W.m -2 and only 0.5% if irradiance is 1000 W.m -2. The 4

5 level of the solar radiation is highly correlated to the solar angle and so the error of the forecasting models (Fig. 4 and 5). These two indices are clearly not relevant to assess the efficiency of the forecasting methods. TABLE 4: MEASURES OF ERROR USED IN THE SET OF REFERENCES Basic error Mean error Mean relative error Ref. MrE Definition E AE RE MBE MAE RMSE rmbe MAPE rrmse MrAE (2) X X X X X (3) X X (4) X X (5) X X X X X X (6) X X (12) X X X X X (13) X X X (14) X X X (15) X (16) X X (17) X X X X (18) X X (19) X X X (20) X? X? (21) X X X (22) (23) X X X X X X (24) X X X X X a b Fig. 4: Influence of the solar zenith angle on the error (a) and relative error (b) of the two forecasting models of the example 5. CONCLUSION The outputs parameters of solar forecasting model and the associated measures of error mainly depend on the forecast horizon. For the short-term predictions, from several hours to more than one day, an accurate assessment of the hourly and daily energy is relevant. For the very short-term, from several minutes to several hours, the forecast of the solar irradiance and its volatility are the most important parameters to forecast. The relative mean error (MrE) and the mean relative absolute error (MrAE) are clearly not relevant to deals with solar energy. The mean bias error (MBE) or the relative mean bias error (rmbe) are interesting in order to assess the ability of the models to predict the amount of solar energy for time-scales greater than an hour. For the comparison of the accuracy of the models, the root mean square error (RMSE) or the mean absolute error (MAE) remain the best indicators. 6. REFERENCES (1) T. E. Hoff, R. Perez, Quantifying PV power Output Variability, Solar Energy, Vol. 84(10), pp ,

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