Artigo ICREPQ ARIMA Method Applied to Solar Predictability

Prévia do material em texto

International Conference on Renewable Energies and Power Quality (ICREPQ’17) 
Malaga (Spain), 4th to 6th April, 2017 
Renewable Energy and Power Quality Journal (RE&PQJ) 
 ISSN 2172-038 X, No.15 April 2017 
 
 
 
 
ARIMA Method Applied to Solar Predictability 
 
Marcello Anderson F. B. Lima1, Paulo C. M. Carvalho1 Deivid Matias de Freitas1 
Josileudo R. Leite2, Luiz J. de Bessa Neto2, Gessica Kelly Laureano Rodrigues2 
 
1Department of Electrical Engineering 
 Federal University of Ceará, Fortaleza, Brazil 
e-mail: marcello@ifce.edu.br, carvalho@dee.ufc.br, deivid_ce@oi.com.br 
2Department of Oil and Gas 
Federal Institute of Education, Science and Technology, Tabuleiro do Norte, Brazil 
e-mail: josileudorodrigues@hotmail.com, luizbessa2015@hotmail.com, gessicakelly07@gmail.com 
 
 
 
 
Abstract. Studies in recent years show the great population 
growth and the need for energy demand, and the search for 
sources that cause less impact to the environment. In this 
scenario, emerged renewable energy sources, including solar, 
which has been increasing in several countries, especially in 
Brazil, but a disadvantage of this energy is present up 
intermittently. The characteristics of solar energy makes it 
unpredictable, so the balance between production and 
consumption of it is difficult for operators. Currently there are 
several mathematical models applied predictability of solar 
radiation on the earth's surface, among them stand out the 
methods of Artificial Neural Networks (ANN), and Regression 
models and Self-Regressive (SR). This work suggests the 
adaptation of ARIMA method, which are modern techniques for 
high precision using correlations between data collected in 
different periods, allowing the description systematically both 
stationary behavior in the non-stationary of the series in 
intermittent renewable electricity sector, aiming to make 
predictions of solar irradiance from data obtained through an 
anemometer station located in Maracanaú - CE. 
 
Key words. ARIMA method, solar irradiance, 
renewable electricity. 
 
1. Introduction 
 
With the population growth of recent years, where 
esteemed that the world population will exceed 11 billion 
in 2100 [1]. While the population should have an 
estimated increase of 121% by 2030, energy consumption 
has increased 219% [1]. 
Along with population growth and increasing energy 
demand, another factor that is gaining prominence is the 
search for ways to minimize impacts to the environment, 
where there are impacts caused by use of non-renewable 
fossil fuels such as oil, and high consumption renewable 
resources such as the use of forest timber and fishing 
activities. 
The sources that stand out are solar and wind power, 
which has remarkable growth in global energy matrix, 
where in 2015, 147 new renewable energy gigawatts are 
now generated in the world [2]. 
In the case of solar energy, it has been highlighted in 
several countries. In moderate scenario, it is expected that 
solar will be responsible for about 11% of the world 
supply of electricity in 2050, data from the International 
Energy Agency (IEA), which corresponds to values close 
to 5,000 TWh. Despite its high cost, solar energy is still 
prominent among renewable energy, an example is 
Germany, which in 2014 generated 18.8% of solar energy 
in the world, followed by China which generated 15.7% 
and Italy which yielded 12.7% [3]. 
The renewed energy has been growing in recent years 
in Brazil, thanks to natural resources, such as high solar 
irradiance, mainly in the Northeast, which has the highest 
overall index, so solar energy has great prominence [3]. 
In 2018, Brazil will is among the 20 countries with the 
highest solar power generation, considering the power 
already contracted and the scale of the expansion of other 
countries according to the Ten Year Plan for Energy 
Expansion (PDE 2024), expected still that by 2050, 18% 
of the country's homes will have solar power generation 
[3]. 
The substantial increase in the share of renewables in 
the global energy matrix has caused impacts on 
electricity generation that need to be studied, known and 
where necessary prevented. Current models used in 
Brazil for the expansion planning and operation need to 
be revisited to reflect this new reality [4]. The Brazilian 
electric system, hydroelectric base in the near future will 
have a large installed capacity of solar parks and wind 
farms, which bring different characteristics of the current 
generating capacity, as the short-term variability in 
generation and greater unpredictability. 
Photovoltaic solar energy although it is a source of 
abundant energy, it has the disadvantage of presenting up 
intermittently, that is, not concentrated and difficult to 
capture in comparison with hydroelectric power by being 
subject to climate variability, mainly due to performance 
of the clouds and the movement of the planet. So to be 
harnessed, are studies demanded that allow better 
understand their regional availability, temporal variability 
and predictability [5]. 
The characteristics of solar generation makes it 
unpredictable, that is, not fully controllable by the human 
being, transforming thus the balance between generation 
and consumption of electricity a challenging task for 
operators. Therefore, the higher the share of this source 
in the global energy matrix, the greater the stochasticity 
and short term intermittency in electricity generation and 
minimize these impacts has become an important subject 
of study [6]. 
Currently, there are several mathematical models that 
allow to simulate the solar radiation reaching the Earth's 
surface. However, to make such predictions, should take 
into account the processes that occur when solar radiation 
interacts with the atmosphere. Among the factors that 
attenuate solar radiation the presence of clouds is the 
most important or even the only, in the case of specific 
regions such as the Northeast. The attenuation caused by 
the same process is a stochastic nature, which makes its 
prediction a complex task [7]. 
In this sense, there have been employed various 
techniques in order to provide solar radiation at the 
earth's surface, among them include the use of Artificial 
Neural Networks (ANNs), which are systems in parallel 
distributed and constituted by processing units capable of 
modelling from simple to the most complex problems 
and, once trained, can make predictions and 
generalizations instantly [7]. 
Another commonly used technique for predicting 
solar irradiance in the medium and long term, are the 
regression models and self-regressive (ARs) with 
coefficients that vary over time. This method consists in 
constructing a matrix L(i,k) of 52 rows by 24 columns, 
where each row corresponds to one week of the year and 
each column at a time of day. Thus, it is concluded that in 
this model there is a self-regressivity first order in the 
determination of estimates, i.e., the amount of radiation 
in the preceding time has significance for the calculation 
of the radiation over the next hour. 
The images obtained from satellites have also been 
continuously used to make estimates and predictions of 
solar radiation in the medium term, because of its 
excellent spatial and temporal resolution. In this type of 
prediction, it is considered that there is a linear 
relationship between the cloudiness index, which refers 
to the fraction of the partly covered by the sky at a given 
moment with the transmissivity of the atmosphere, by 
which is the ratio of terrestrialsolar radiation and the 
alien [8]. The global irradiance on Earth, is then 
calculated by combining these weather features with 
solar radiation calculation model for the conditions of a 
day with clear skies. 
However, the satellites have some limitations such as 
the high cost of implementation and high duration of the 
acquisition of two images, which makes it impossible to 
make predictions very short time, or for time scales of 
less than 30 minutes [7]. Furthermore, the use of this 
technology is not the most appropriate when you want to 
run a forecast at the local level. 
In the mid of 1970, the statistical George Box and 
Gwilym Jenkins, developed a methodological body, 
called ARIMA (Self-Regressive Integrated and Moving 
Averages) to identify, assess and diagnose dynamic time 
series models, in which the time variable plays a role key. 
An important part of this methodology was designed in 
order to release the investor of economic models 
specification task, leaving the very time of the variable 
data in the study, indicating the characteristics of the 
underlying probabilistic structure [9]. 
The ARIMA's models are highly accurate modern 
techniques that use correlations between data collected in 
different times and could describe systematically both 
stationary behaviour as the non-stationary of the series 
[10]. Thus, it can be said that this is a flexible modelling 
methodology that forecasts based on these models are 
made from the current and past values of the time series. 
The ARIMA method has been regularly used in the 
administrative and food sector to manage, control and 
estimates of commodity prices in shops, malls and large 
restaurants, with a high margin of efficiency and 
reliability. Moreover, it has been used in stationary time 
series for forecasting and indication of prices of cattle 
prices in the state of Paraná, in Brazil, aiming assist 
sectors involved in beef cattle to achieve a future price 
reference that enables management efficient market risk 
as well as the decision-making process, cost management 
and definition of the best marketing strategies [11]. 
This work suggests the adaptation of ARIMA method 
in intermittent renewable electricity sector, aiming to 
make predictions of solar irradiance and wind speed from 
data obtained through an anemometer station located in 
Maracanaú - CE, located 18 km from Fortaleza, 
considered the largest industrial center of the state. 
 
2. Literature review 
 
In the last decades has found several lines of research 
on the application of ARIMA method in energy planning 
sector, estimates for electricity generation costs and 
electricity consumption. Priority is thus to minimize the 
alternative/intermittent sources of energy forecasting 
errors, diversify the global energy mix. 
Search by means of this method adaptation to the 
electrical system to the variability of solar photovoltaic 
generation, as well as facilitate the integration of 
renewable technologies variables efficiently in terms of 
electricity production costs. One of the first scientific 
papers to adapt the ARIMA method on temporary 
electricity series, aiming to provide financial figures for 
electricity consumption with a day in advance, was 
published in 2003 by the Spanish researcher Javier 
Contreras. 
The table I presents a summary of the works found in 
the world regarding the adaptation of the ARIMA 
method[7-12]-[20]. 
 
Table I. - Literature of review 
 
AUTHORS APPLICATION METHOD 
 
GENERATION 
SOLAR 
Melo, 
2016 
Forecast of 
solar potential 
in the Brazilian 
Northeast 
ARIMA Yes 
Hussain; 
Alili, 2016 
Forecast solar 
daily in Abu 
Dhabi 
ARIMA Yes 
Colak et al, 
2015 
Forecast solar 
of multiple 
periods 
ARIMA Yes 
Passos, F., 
2015 
Forecast of 
consumption of 
electricity 
ARIMA Yes 
Christo, E., 
2013 
Forecast of 
electric charges 
ARIMA Yes 
Silva, T., 
2012 
Forecast of 
electric charges 
ARIMA Yes 
Mohamad,A. 
2012 
Forecast of 
demand of 
electricity 
ARIMA Yes 
Wu; Chan, 
2012 
Forecast solar 
monthly 
ARIMA Yes 
Jesus, T., 
2008 
Forecast of 
consumption of 
electricity 
ARIMA Yes 
Contreras, 
2003 
Forecast of 
renewable 
electric 
resource 
ARIMA Yes 
 
3. ARIMA method of adaptation to forecast 
solar and wind data 
 
To achieve greater flexibility in real time series 
assembly with a greatly reduced number of parameters, it 
is advantageous and include both Autoregressive Moving 
Average. Thus, there is a ARMA combination which is 
given by the estimation of the regression of dependent 
variable Y depending on the lags of the variable itself Y, 
indicated by “p” autoregressive terms, according to the 
random error, indicated by “q” terms moving average, 
with the model indicated as ARMA. Thus, the ARMA 
(1,1) model can be written as (1) [12]. 
 
1 1 1 1 0 Yt Yt pYt p Ut Ut qUt q                (1) 
 
wherein Δ is the difference operator. 
 
However, most of the time series is naturally non-
stationary, then the model application requires processing 
thereof by differences to make them through the 
stationary method ARIMA (p, d, q) in equation (2) [21]. 
 
     0 t t tB z B dz B a      
 
at where: 
 
 
1 2 2
1 2 2
 1 
 1 
p
p
q
q
B B B B
B B B B
   
   
    
    
 
 
ϕ(B) is the autoregressive operator; It is stationary, 
where ϕ(B) = 0 are outside the unit circle. 
φ(B) = φ(B)∇d is the generalized autoregressive 
operator; It is a non-stationary operator d of the φ(B) = 0 
equal to unity, that is, unit roots of d. 
θ(B) is the moving average operator; inverted, that is, 
where roots θ(B) = 0 they are outside the unit circle. 
When d = 0, this one model is a stationary process. The 
stationarity and invertibility requirements apply 
independently, and, in general, the operators φ(B) and 
θ(B) will not be of the same order. 
However, many series have a seasonal component 
and it may be necessary to model these in order to obtain 
a more reliable model. In this sense, SARIMA models 
were developed (Seasonal ARIMA), which are seasonal 
ARIMA models. Such a model is represented in equation 
(3) as SARIMA (p, d, q) (P, D, Q)s, where s is the period 
corresponding to seasonality, Φ(Bs) are seasonal 
coefficients autoregressive; ϕ(B) are the coefficients of 
the seasonal moving averages and (1-Bs)D is the order 
difference operator D of seasonal differentiation [12]. 
 
            1 1 Bs B Bs d Bs D Xt Bs B Zt     
 
 
The ARIMA methodology for modeling a process 
consisting of four steps according to [15] be seen in 
figure 1 below [21]-[22]. 
 I - Identification 
of the model
II - Parameter 
estimation
III - Diagnostic 
checking
IV - Forecasting
No (Return to Step 1)
Yes (Go to Step 4)
 
Fig. 1. Steps ARIMA methodology (p, d, q). 
 
a) Identification: the first step consists in 
obtaining the appropriate values for p, q and d. 
For this, one can employ the correlogram and 
partial correlogram, which are graphical 
representations of the autocorrelation function 
(CAF) and Partial Autocorrelation Function 
(FACP) against the size of the gap. The aim is to 
get the index (p, d and q) in which the series has 
stationarity; 
b) Estimation: It is estimated model parameters, 
which can be done by means of least squares or 
non-linear estimation method; 
c) Diagnostic check: it is checkedwhether the 
estimated model fits the data well. One way to 
do this check is to see if the model residuals are 
white noise. If so, whether to accept the model. 
Otherwise, the process must be restarted in order 
to identify another model; 
d) Forecast: the model is used to make forecast. 
The ARIMA processes have the general 
characteristic of their predictions revert to the 
mean when the forecast horizon increases. In 
this sense, the potential for prediction of these 
models is limited to short-term horizons. 
 
With the aim of obtaining the solar data were 
considered information obtained from the study 
"CHARACTERIZATION AND PREDICTION OF 
SOLAR POTENTIAL: CASE STUDY FOR 
PARNAÍBA (PI) Maracanaú (CE) and PETROLINA 
(PE)" Authorship [12]. Thus, we performed a study of 
information The study of solar data was done with 
information obtained in northeast Brazil, more precisely 
in Maracanaú - CE. The city was chosen due to this being 
one of the three sites with tower data collection 
belonging to CNPq project "wind potential forecast 
aiming optimal integrated operation of generating 
electricity units: a case study for the Northeast of Brazil", 
which finances the study presented. By way of 
(2) 
(3) 
information, the other two municipalities included in the 
project are Parnaíba (PI) and Petrolina (PE) [12]. 
The equipment installed in the station used to obtain 
the observed data have an acquisition system NRG 
Symphonie-plus, which enables data storage at 10 min 
intervals (obtained by arithmetic means from data 
processed every 2 s). The parameters recorded directly by 
the datalogger, are stored in MMC type memory cards 
(Multi Media Card), with capacity of 32 M, which 
ensures a range of more than one year of recorded data 
[12]. 
The present equipment at the weather station can be 
seen by Figure 2. 
 
Fig. 2. Datalogger NRG Symphonie-plus installed. Source: 
Adapted of [17]. 
 
This study made use of the XLSTAT software, as this 
is modulated to Microsoft Excel for data entry and 
presentation of results, but the calculations are performed 
with the components of own software independently. We 
used this tool for achieving solar forecast data, since it 
takes on characteristics of an automatic statistics 
software. 
The XLSTAT generates the confidence limits based 
prediction graph is also shown graphically validation 
media (sized) and the average prediction as a straight line 
in the graph. Figure 3 shows the main elements of the 
graph produced by the software [12]. 
 
Fig. 3. Explanation of the XLSTAT forecast charts. Source: 
Adapted of [12]. 
 
Thus, after analysis of data were carried 23970 4485 
hourly averages between 5:00 am to 17:00 and forecast 
solar irradiance data 10 in 10 minutes by selecting 42 
days and obtaining 3318 data. 
These surveys are formed as time series and therefore 
for their predictions, require specific statistical methods 
for their treatment. As seasonality is a characteristic 
present in data time series solar irradiance, 
characterization and prediction is made using the 
seasonally low solar irradiance component, the 
atmospheric transparency index, Kt. 
The use of this component is justified by the fact of 
providing results more accurate and reliable forecasts, 
low interference from components trends present in 
temporal data series, the forecasting process. The 
forecasts made in this study using the ARIMA method, 
exponential smoothing method in Simple mode (AES) 
and the method of Moving Averages [23]. 
Thus, after the selection of monthly time series for 
each time band period information collected it was 
carried out analysis of 23 series time corresponding to the 
5:00 hours of tracks to 17:00 hours every day of the 
period of may 2012 to April 2013. Analyzed by XLSTAT 
looked at the range of twelve months each series and 
obtained from the series generates a graph for each series. 
Thus, there were the forecast of 546 hours of solar 
irradiance. 
 
4. Results and discussions 
 
The behavior of prediction of solar resource, with a 
comparison between the predicted and observed values of 
irradiance solar, can be seen by Figure 4, which shows 
the values in W / m² obtained forecasts acquired by 
ARIMA method, and the irradiance solar measured by 
the anemometer station. Were realized forecasts of 546 
hours of solar radiation were performed. 
 
Fig. 4. Comparison between the solar irradiance measured 
and observed. 
 
It appears that the forecast of solar irradiance are 
monitoring the characteristic behavior of solar 
availability throughout the day, but it is clear that 
situations of climate interference, such as rainfall and 
periods cloudy for example, were not understood and 
anticipated by the method ARIMA forecasting. Yet the 
method was able to present values close to forecasts and 
observations at different times of the analysis period. 
Recalling that the night period was disregarded for 
predictions because they are null values. 
For situations where the values obtained from the 
time series of anticipation method ARIMA did not follow 
satisfactory way the solar irradiance values observed, 
there is a highlight for the time 156 of the forecast, where 
the predicted value for the solar resource availability for 
this time was 24.54 W/m² and the observed value was 
0.76 W/m², presenting as the biggest difference between 
predicted and observed values for all 546 hours forecast, 
causing a large overestimation of the amount available 
solar resource for time analysis. 
The smallest difference in predicted and observed 
values was found in time 318 with the predictive value of 
683.79 W/m² and observed value of 683.23 W/m². 
The greatest value of solar irradiance observed was 
996.54 W/m² while the highest value of planned solar 
irradiance was 769.26 W/m². The lowest solar irradiance 
observed was 0.57 W/m² while lower value provided to 
solar irradiance was 3.15 W/m². 
The forecast did not get results faithful regarding 
climate variations over the hours, but managed to make a 
good accompaniment of natural variations of amplitudes 
in the course of hours. 
The Figure 5 shows the comparison between the data 
obtained from the forecasts obtained by ARIMA method 
for solar irradiance measurements and for the first 24 
hours of forecast. The separation of this period facilitates 
viewing and comparing the predicted and measured 
values. 
 
Fig. 5. Comparison between measured solar irradiance and 
observed for the first 24 hours of forecast. 
 
The behavior of the errors can be viewed by Figure 6. 
In the image prediction with ARIMA method presents a 
clear tendency to overestimate the availability of wind 
resource. 
 
Fig. 6. Error solar of forecast. 
 
The forecast with the highest proportional difference 
between irradiance values measured solar and collected 
was carried out at the time 156, where it obtained the 
mistake of 3.115, 2%. In other six situations the errors 
were also higher than 1.000%, specifically in 442 hours 
with 1.918,18%, 456 with 1.354,7%, 520 with 
1.815,93%, 539 with 1.384,82%, 540 with 2.968,12% 
and 546 with 1.310,41%. The Figure 7 shows the 
combination of the collected solar irradiance and the 
forecast errors. 
 
Fig. 7. Comparison between the collected solar irradiance and 
the forecast errors. 
 
In hour 144 there was greater underestimation solar 
resource availability, where obtained a forecast of error 
of -86.04% forecast. The smallest difference in predicted 
and observed values was found in time 318,with the of 
forecast value showing 0,08% of error. Finally, the 
average error of the solar forecast was of 82,14% and the 
mean of the errors module is of 91.86%. 
For a better understanding of the behavior of forecast 
errors, a separation of hours the prediction was made, 
related to Increasingly, demonstrating their respective 
scope of departure from the ideal, wind and solar 
prediction. Thus, it is possible to separate negative and 
positive errors and understand the amount of hours that 
was among the tracks of -5% a 5% e -10% a 10%. The 
figure show the separation hours predictability of errors 
for this one resource. 
 
Fig. 8. Hours of sun forecast in relation to forecasting errors 
 
The Predictability of solar resource obtained a total of 
underestimation of 179 hours, 367 hours of positive error 
and no null error. Related to the departure from the ideal 
forecast for the error range of -5% a 5% of error, a total 
of 61 hours remained within the range and 127 hours for 
the range of -10% to 10% error. The forecast ranged of -
86,04% to 3,115,2%. 
 
5. Conclusions 
The Brazilian energy matrix, composed mostly of 
electricity from hydraulic resource, must be diversified 
due to the non implementation of new large hydroelectric 
plants. From this fact, management methodologies and 
potential analysis of implementation of complementary 
sources for the electrical grid should be developed. 
Due to the high availability of solar resource, 
northeast Brazil presents itself as a contributor to 
expanding the diversification of the Brazilian energy 
matrix. This potential contribution can best be exploited 
provided that adequate investment and technical policy 
be developed for understanding the impacts caused by the 
implementation of alternative energy sources in the 
energy matrix of the country. The generation 
intermittency characteristic solar source must be worked 
towards developing techniques that can reduce the 
additional costs to the production of electricity; of these 
costs is related to the expected availability of solar and 
wind resources due to climatic factors. 
The forecast solar irradiance could follow the 
irradiance behavior throughout the day, but could not 
foresee climate variations such as cloud cover and rain. 
The value found in solar forecast of errors module 
was of 91,86%, however, the large previously presented 
forecast errors occurred in situations with low solar 
irradiance, more precisely at sunrise or sunset. This 
characteristic minimizes the forecast errors, because 
precisely the moment where the solar forecast errors were 
higher, a possible electricity production system from the 
sun resource would be at a time of low energy 
production, resulting in less impact the energy matrix. 
 
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