Solar Irradiance Forecasting UsingTriple Exponential Smoothing

Soumyabrata Dev†1, Tarek AlSkaif†2, Murhaf Hossari1, Radu Godina3, Atse Louwen2, and Wilfried van Sark21The ADAPT SFI Research Centre, Trinity College Dublin, Dublin, Ireland

2 Copernicus Institute of Sustainable Development, Utrecht University, Utrecht, The Netherlands3 Department of Electromechanical Engineering (C-MAST), University of Beira Interior, Covilha, Portugal

Abstract—Owing to the growing concern of global warmingand over-dependence on fossil fuels, there has been a huge interestin last years in the deployment of Photovoltaic (PV) systems forgenerating electricity. The output power of a PV array greatlydepends, among other parameters, on solar irradiation. However,solar irradiation has an intermittent nature and suffers fromrapid fluctuations. This creates challenges when integrating PVsystems in the electricity grid and calls for accurate forecastingmethods of solar irradiance. In this paper, we propose a tripleexponential-smoothing based forecasting methodology for intra-hour forecasting of the solar irradiance at future lead times. Weuse time-series data of measured solar irradiance, together withclear-sky solar irradiance, to forecast solar irradiance up-to aperiod of 20 minutes. The numerical evaluation is performedusing 1 year of weather data, collected by a PV outdoor testfacility on the roof of an office building in Utrecht University,Utrecht, The Netherlands. We benchmark our proposed approachwith two other common forecasting approaches: persistenceforecasting and average forecasting. Results show that the TESmethod has a better forecasting performance as it can capturethe rapid fluctuations of solar irradiance with a fair degree ofaccuracy.

Index Terms—Photovoltaic, Solar energy analytics, Clear skymodel, Intra-hour forecasting, Exponential smoothing.

I. INTRODUCTION

Due to the increasing concerns on greenhouse gases, thetransition towards a more renewable energy system is becom-ing more visible. This trend has manifested itself by a sharpincrease in the deployment of Photovoltaic (PV) systems forgenerating electricity in both standalone and grid-connectedresidential and large-scale systems due to their economicaland environmental benefits [1].

Although PV systems generally operate reliably and overlifetimes of 20+ years, the integration of PV in electricitygrids is hampered by their intermittent power output. Thepower output of PV systems can be highly variable due to thevariability of solar global irradiation and the effect of differentmeteorological variables, such as temperature, humidity level,air pressure and cloud cover. This variability can create seriousissues for the electricity grid needs, such as power systemcontrol, grid integration and power planning, especially at highpenetration levels of PV [2], [3].

† Authors contributed equally.This work is supported by the the Joint Programming Initiative (JPI) Urban

Europe project: PARticipatory platform for sustainable ENergy managemenT(PARENT) and the Netherlands Science Foundation (NWO).

Send correspondence to T. AlSkaif, E-mail: t.a.alskaif@uu.nl

In order to ensure reliable and stable operation of theelectricity grid, it is important to accurately forecast the solarirradiance and predict its variability over time. As a result, thefield solar irradiance forecasting has received an increasingattention among researchers over the past decade [2], [4], [5].The literature is rich with various methods for forecasting solarirradiance. A comprehensive review on the recent advances inPV power forecasting are presented in [2], [4]. These papersprovide overviews on: i) the factors that affect the PV powerforecasting, ii) the different methods of forecasting; includingpersistence methods, physical models and machine learningmodels, and iii) forecasting performance evaluation metrics.

Solar irradiance forecasting can be classified into differentcategories, depending on the forecasting horizon, rangingfrom intra-hour or intra-day forecast to days or weeks aheadforecast. Very short term solar irradiance forecasting (i.e.,intra-hour forecast) is very important to assure grid stability,power quality, to reformulate bids in sub-hourly markets andto correctly plan reserve capacity and demand response, espe-cially in locations or applications where high solar penetrationis present [4]. In intra-hour PV solar irradiance forecasting,the forecasting horizon ranges from a few seconds to 1 hourand it has been addressed using different forecasting methodsin [6]–[12]. The work in [6] uses a forecasting model based onsatellite images and a support vector machines (SVM) learningscheme. A PV power forecasting in a network of neighboringPV systems is proposed in [7]. The work is based on thecross-correlation time lag between clear-sky index of those PVsystems that are influenced by the same cloud. An estimationof solar irradiance using ground-based whole sky imagersis proposed in [8], [9]. An analysis of different clear skymodels for the evaluation of PV output forecast is proposedin [10], [11]. A Markov chain-based forecasting approach isused in [12] for calculating the expected value of future PVoutput power.

In this paper, we address the problem of intra-hour solarirradiance forecasting by asking the following question: canwe predict the very short term future solar irradiance values,using historical measured values of solar irradiance? To answerthis question, we propose a new method of solar irradianceforecasting, using Triple Exponential Smoothing (TES), thatexploits the seasonality of the measured solar irradiance topredict future values up-to a period of 20 minutes. We usetime-series data of measured- and theoretical clear-sky- solar

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irradiance values for the prediction of future solar irradiancevalues. To the best of our knowledge, this is the first time theseasonality of the measured solar irradiance values using TESis exploited for very short term solar irradiance forecasting. Webenchmark our proposed approach with other methods and weobtain competitive forecasting results.

The structure of the paper is organized as follows. Section IIdescribes the clear sky model that is used in this paper.The triple exponential smoothing method is described inSection II. Section IV presents and discusses the numericalresults. Finally, we conclude the paper and give pointers forfuture directions in Section V.

II. CLEAR SKY MODEL

The total solar irradiation falling on the earth’s surface inthe event of a clear sky day is popularly referred as the clearsky solar irradiance. There are several theoretical models thatestimates the amount of clear sky irradiance for a particular lo-cation on the earth’s surface. This is dependent on a number offactors, including the time of the day, azimuth- and elevation-angles of the sun, and other atmospheric conditions. In ourprevious work [13], we have provided a comparative analysisof various clear-sky models for a particular experimentallocation in the tropical region of Singapore.

In this paper, we use the popular Bird’s clear sky model,proposed by Bird and Hulstrom [14]. It estimates the totalclear sky irradiance by using the results from radiative transfercodes. This model is widely used by solar analytics expertsbecause it considers the various atmospheric conditions, in-cluding the amount of aerosols, ozone and water vapor. Theclear sky radiation [15], φs (W/m2) is given by:

φs =(φd + φl)

(1−Rgrs), (1)

where φd denotes the direct solar irradiance, φl is the scatteredsolar irradiance, Rg is the water surface reflectivity, and rs isthe atmospheric albedo. The direct solar irradiance φd, alsomeasured in W/m2, is computed via:

φd = 0.9662φextTATWTUMT0TR, (2)

where the parameter φext is the extraterrestrial solar irradiance,TA is the transmittance of aerosol absorptance and scattering,TW is the transmittance of water vapor, TUM is the trans-mittance of uniformly mixed gases, T0 is the transmittanceof ozone content, and TR is the transmittance of Rayleighscattering.

The scattered solar irradiance φl, is also measured in W/m2,and is given by:

φl = 0.79φextTAATWTUMT00.5(1− TR) +Ba(1− TA

TAA)

1−mp +m1.02p

,

(3)where the TAA is the transmittance of aerosol absorptance,Ba is a dimensionless constant, and mp is the relative opticalair mass. We use the python implementation of Bird’s clear

sky model for estimating the total received solar irradiance,on a clear sky day [16]. This assists us in estimating the solarirradiance on a particular location of the earth, for a clear skyday.

However, in most of the days, the actual solar irradiancevaries from this theoretical clear sky irradiance value. This ismainly because of the clouds and other atmospheric eventsin the atmosphere – the actual solar irradiance differs fromthe clear sky irradiance value. We define the clearness index(k) as the ratio of measured solar irradiance (φact) and thetheoretical clear sky irradiance (φs) data.

k = φact/φs. (4)

We model this clearness index (k) based on the actual- andclear-sky- irradiance data. This provides us an estimate on thecondition of the sky (clear sky or overcast) at a particular timeinstant.

III. TRIPLE EXPONENTIAL SMOOTHING

In order to estimate the future values of solar irradiance, weuse the Triple Exponential Smoothing (TES) method. In gen-eral, the exponential smoothing function is a straightforwardmethod for smoothing time series data. It assigns exponentiallydecreasing weights over time [17]. When dealing with highfrequency signals, it is common to to use a TES function. Itcan be used three times to smooth a three-signals time seriesand remove high frequencies encountered [18].

Suppose we have a sequence of clear sky indices (i.e., timeseries data) kt, for a cycle of seasonal change that has thelength L and in case we are starting at t = 0. The TES functionprovides a best estimate of the future clearness index value attime t+1 (kt+1) (i.e., the smoothed value) using the clearnessindex value at time t as an input (kt). The TES computes aline that follows the trend of the data (i.e., clear sky indices).Similarly, it calculates the seasonal indices which are used toweigh the corresponding values in the trend line by fitting thetime point to the cycle of length L.

For a given time t, we can represent the smoothed valueof the clearness index by st. To represent for a given lineartrend the sequence of estimates we use bt. Those estimateswould be the best estimates that are placed on the seasonalchanges. Seasonal correction factors can be represented by ct.For a give time t mod L in the cycle where we took theobservations on, ct stands for the expected proportion of thetrend that was predicted.

In order to intialize a set of seasonal factors, we need aminimum of 2L periods, which translates to two full seasonsof historical data.

The TES method’s output (Ft+m) is a predicted estimationfor the value of k at a time t+m,m > 0. This predication isbased on that data seen before and up to the time t. The TESwith additive seasonality is given by the following formulas[19]:

s0 = k0, (5)st = α(kt − ct−L) + (1− α)(st−1 + bt−1), (6)bt = β(st − st−1) + (1− β)bt−1, (7)ct = γ(kt − st−1 − bt−1) + (1− γ)ct−L, (8)

Ft+m = st +mbt + ct−L+1+(m−1) mod L, (9)

where the data smoothing factor is α, 0 < α < 1, the trendsmoothing factor is β, 0 < β < 1, and the seasonal changesmoothing factor is γ, 0 < γ < 1.

b0 =1

L

(kL+1 − k1

L+kL+2 − k2

L+ . . .+

kL+L − kLL

).

(10)

Setting the initial estimates for the seasonal indices ci fori = 1, 2, . . . , L is a bit more involved. If N is the number ofcomplete cycles present in our data, then:

ci =1

N

N∑j=1

kL(j−1)+i

Aj∀i = 1, 2, . . . , L, (11)

where in jth cycle of our data, Aj stands for the average valueof x and it can be calculated as follows:

Aj =

∑Li=1 kL(j−1)+i

L∀j = 1, 2, . . . , N. (12)

In summary, our proposed forecasting algorithm can besummarized in Algorithm 1.

Algorithm 1 TES based solar irradiance forecasting

Require: Time series data of measured solar irradiance, lati-tude and longitude of the region of interest.

1: Ensure that the geographical co-ordinates of the place isaccurate;

2: Set the period of historical data (at least 2L time periods)for training the TES model;

3: Compute the Bird’s clear sky model for the consideredhistorical time period;

4: Generate the values of the clearness index k, for the sametime period, as described in Section II;

5: Train the TES model using the computed k values of thehistorical time period;

6: Estimate the future values of k values for the consideredlead time;

7: return Estimated k values, required in computing fore-casted solar irradiance.

IV. RESULTS AND DISCUSSION

A. Data Collection

The data used here is collected from a rooftop PV testfacility at Utrecht University: the Utrecht Photovoltaic OutdoorTest facility (UPOT), shown in Figure 1. The facility is located

Fig. 1: Photograph of the Utrecht Photovoltaic Outdoor Testfacility (UPOT). Photo courtesy of Arjen de Waal. (Bestviewed in color)

at Utrecht University’s campus in the center of the Netherlandson top of an eight story building. It is equipped with 24PV modules of different brands and technologies, of whichIV curves and back-of-module temperature are measured ata time-resolution of 2 minutes. The test facility has an arrayof solar irradiance sensors measuring direct normal irradiance(DNI) and diffuse horizontal irradiance (DHI), global horizon-tal irradiance (GHI), global in-plane irradiance, and spectralirradiance. Furthermore, air temperature, wind speed and othermeteorological data are measured. These measurements aretaken at a time-resolution of 30 seconds. More details aboutUPOT test facility can be found in [20].

(a) Clear-sky day

(b) Overcast day

Fig. 2: Illustration showing the measured solar irradiance andtheoretical clear-sky irradiance, on the event of (a) clear-skyday and (b) overcast day. (Best viewed in color)

In this paper, we use the measured solar irradiance data for1 year, collected at the UPOT site. We use the Bird’s clearsky model to compute the theoretical clear sky irradiance fora particular date. Figure 2(a) illustrates the measured solarirradiance and the Bird’s clear sky irradiance for a sample clearsky data. We observe that the theoretical model closely followsthe actual solar irradiance measured at the site. Interestingly,on a non-clear sky day as shown in Fig. 2(b), there arerapid fluctuations in the measured solar irradiance. Thesefluctuations occur because of the clouds and other atmosphericevents that impact the received solar irradiance.

Fig. 3: Scatter plot between measured solar irradiance andcomputed clear sky solar irradiance, for all the clear sky daysin 2015. The Pearson degree of correlation is 0.99.

In order to evaluate the performance of Bird’s clear skymodel at our Utrecht solar irradiance data, we analyze therelationship between measured- and clear sky- solar irradiancedata. We manually list all the clear sky days in the year of2015 from the solar irradiance measurements. We compute thecorresponding clear-sky irradiance values for all the availablesolar irradiance recordings of clear sky days. We observe thatthere are a total of 22 clear sky days in the year 2015 atUtrecht. Figure 3 shows the scatter plot between the measured-and Bird’s clear sky- irradiance for all the 22 clear sky days.We compute the Pearson degree of correlation between thetwo values and obtain a high value of 0.99. This high degreeof correlation indicate that the Bird’s clear sky can model thesolar irradiance value (on a clear day) with a high degree ofaccuracy.

However, in most of the days, the measured solar irradiancefluctuates from the theoretical clear sky data (cf. Fig. 2b).These fluctuations are detrimental in the context of solarenergy analytics, as it is very difficult to track and predictthe fluctuations. In this paper, we attempt to forecast thesefluctuations in the future using the method proposed in Sec-tion III, based on historical data of measured solar irradianceand theoretical clear sky model.

B. Statistical Results

In this paper, we consider the meteorological measurementsfor the entire year of 2015, as recorded by the UPOT testfacility. All measurements are resampled in intervals of 5

minutes. Therefore, we set the seasonal period parameter inHolt-Winters as L = 288, as repeatability occurs after every288 observations (i.e., one day). In this section, we perform astatistical analysis of our proposed solar forecasting method-ology. We use a total of 1728 observations (or 6L periods)for training our model, and attempt to predict the clearnessindex values up to a period of 20 minutes. We compute theMean Absolute Error (MAE) between the predicted- and theactual- k values to evaluate the performance of the model.These samples of timestamps are sampled randomly over theperiod of all the observations in the year of 2015. We perform150 such experiments, sampled at random time stamps of theyear. Figure 4 shows the box-plot of the error values for thedifferent lead times. The distributions of the errors for differentlead times show a clear trend – the error increases graduallywith increasing lead times.

5 mins 10 mins 15 mins 20 mins

Lead times

0.0

0.5

1.0

1.5

2.0

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Fig. 4: Trend of average error (i.e., MAE) in estimating kfor various lead times. We perform 150 experiments for aparticular lead time.

C. Benchmarking

We compare the performance of our proposed approachusing TES with other two common forecasting approaches,viz. persistence forecasting and average forecasting. The per-sistence forecasting method assumes that the conditions forthe forecast do not change (especially for shorter lead times).This method works well in conditions when weather mapschange very slowly with time. We also compare our approachwith the average forecasting method. Such method assumesthat the average value remains the same while the responsevalue fluctuates randomly for the shorter time period. Suchtechnique assumes that the forecast value is the average of theprevious observed recordings.

TABLE I: Benchmarking of several forecasting methods.

Methods Average Error (100 experiments)Persistence 0.41Average 0.36Proposed TES 0.13

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Fig. 5: Prediction of future k values using our proposed approach. We observe that the trend of the predicted k values closelyfollows the trend of the actual k values. (Best viewed in color)

We compare the performance of these methods using ourUtrecht solar irradiance data. We compute the forecast valuesof clearness index k, upto lead times of 20 minutes, at intervalsof 5 minutes. We use historical data of 1152 observations (or4L periods) for the purpose of forecasting. The entire datasetis sampled at random time stamps to perform the forecastingaccuracy. We perform a total of 100 experiments to removesampling bias. Table I discusses the results. As expected, theperformance of the persistence method is poor. The averagemethod has a better performance, as it can capture the trendof the k values. However, our proposed approach using TEShas the best performance, as it can also capture the rapidfluctuations of k values and predict it with a good degree ofaccuracy.

D. Time Series AnalysisAs a final comparison, we visualize the performance of our

proposed approach in the form of clearness index time series.Figure 5 shows the time-series results for two distinct timeseries. We observe that for a shorter lead time, as shownin Fig. 5(a), the predicted clearness index values has the

similar trend as the actual index values. This is an interestingobservation, as our proposed approach can capture the rapidfluctuations of solar irradiance with a fair degree of accuracy.Similarly, for a longer lead time (cf. Fig. 5b), the trend of thek values is captured well even for the next day timestamps.However, the predicted k values missed the very short-termvariations. This is promising for us, as our proposed approachcan provide a reliable methodology for estimating the futuresolar irradiance.

V. CONCLUSIONS AND FUTURE WORK

In this paper, we have proposed a methodology for intra-hour forecasting of solar irradiance, based on the historicaldata of the clearness index. Our method is based on a tripleexponential smoothing function that captures the seasonality ofour dataset with high degree of accuracy. It takes into accountboth the trend and seasonality to forecast future intra-hoursolar irradiance values.

Our future work involves the validation of our proposed ap-proach using more than 1 year of meteorological data. We alsoplan to use images captured by ground-based sky cameras [21],

[22] to further enhance the prediction results. Our initial resultsin [9] indicate that image-based features obtained from skycameras can accurately estimate the solar irradiance fallingon earth’s surface. Unlike point-measurement devices, skycameras provide additional information about cloud movementand coverage. This additional information will equip us witha multi-modal approach in predicting future solar irradiancewith a high degree of accuracy.

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