Forecasting of Energy Production for Photovoltaic Systems Based on ARIMA and ANN Advanced Models

Accurate forecasting of solar energy is essential for photovoltaic (PV) plants, to facilitate their participation in the energy market and for e ﬃ cient resource planning. This article is dedicated to two forecasting models: (1) ARIMA (Autoregressive Integrated Moving Average) statistical approach to time series forecasting, using measured historical data, and (2) ANN (Arti ﬁ cial Neural Network) using machine learning techniques. The main contributions of the authors could be synthetized as follows: (1) analysis and discussion of the experimental and simulated results regarding solar radiation forecast, as well as energy production prediction and forecasting based on ARIMA and ANN models for two case studies: (a) laboratory BIPV system developed at the Polytechnic University of Bucharest and (b) large PV park placed in a speci ﬁ c site of the south of Romania. A variability index of solar radiation was introduced for the model improvement; (2) comparison between the ARIMA and ANN results to highlight the ARIMA model which is more e ﬃ cient than the ANN one; (3) optimized method de ﬁ ned by the GMDH model (Group Method of Data Handling) proposed to provide a software program for calculation of the PV energy production.


Introduction
Solar energy is one of the most promising sources of power generation for residential, commercial, and industrial applications [1][2][3][4][5]. Photovoltaic solar energy, based on PV systems, has increased progressively in recent years due to its advantages of being abundant, inexhaustible, clean, and environmentally friendly [6][7][8].
Reliable and accurate forecasts play a key role in improving PV solar power plants [1][2][3][9][10][11][12]. The main challenging problem in the production of solar energy is the intermittent electricity generation using PV systems, due to weather conditions. A variation in temperature and solar irradiance can have a profound impact on the quality of electricity production, leading to a decrease of more than 20% in the PV energy production provided by real PV installations. This limits the integration of PV systems into the grid. Therefore, an accurate short-term forecast of photovoltaic energy is very useful for the efficient daily/hourly management of electricity production and storage in the grid [13].
Accurate forecasting of solar energy is essential for PV plants, in order to facilitate their participation in the energy market and for efficient resource planning [1][2][3]. Various methods have been reported in the literature for the forecast of PV energy [2,12]. These methods can be divided into four classes: (i) statistical approaches to time series forecasting, using measured historical data (ARIMA) [14]; (ii) machine learning techniques, in particular Artificial Neural Networks (ANN), based on machine learning approaches [15]; (iii) physical models based on numerical weather prediction and satellite imagery [16]; and (iv) hybrid approaches that represent a combination of the first three methods [17].
The beauty of the ARIMA model lies in its simplicity and can be applied only to stationary time series [14,18,19]. Therefore, our data from the time series, which are seasonal and nonstationary, are transformed into a stationary series for the application of the ARIMA model. The model can be developed using sophisticated statistical techniques [20]. The optimal approach is selected and validated using the Akaike Information Criterion (AIC) and the residual sum of squares (SSE).
Another statistical model is based on the seasonal analysis of the ARIMA time series (SARIMA) and is further improved by incorporating short-term solar radiation predictions derived from the NWP (numerical weather prediction) model [19]. Such a model can be used in energy forecast simulation software for buildings, such as Energy Plus, TRNSYS, or eQuest, both for existing buildings and in the design of new buildings [20,21].
On the other hand, data-based models are determined by mathematical models and measurements. They do not require detailed knowledge of buildings or equipment. Their forecasts are mainly based on historical data, which are available using the control systems implemented in buildings for (1) building automation systems (BAS) and (2) building energy management systems (BEMS). The accuracy of these models depends on the quality of the selected forecast model, as well as the quality and quantity of available data. Such models are easy to adapt to changing conditions and are relatively easy to be used. In most cases, the relationship between the predicted variable and its physical function is not clearly established [17,22].
In recent decades, researchers have been dedicated to improving efficiency and building energy use through various techniques and strategies. Energy forecasting in an existing building is essential for a variety of applications, such as demand response, fault detection and diagnosis, predictive model control, energy optimization, and management. Energy estimation models are a promising field of research, and this is true given the new advances in artificial intelligence and machine learning. Such models have been widely applied to both building energy systems and HVAC (heating, ventilation, and air conditioning) systems, as they can help with a variety of tasks. Most conventional approaches for solar energy forecasting are limited to discover the data correlation but are not able to analyze them in depth and discover relevant information. With many data in the modern power system, the use of conventional approaches is not appropriate to guarantee an accurate forecast [15,20,23].
Recently, Deep Learning (DL) approaches have emerged as powerful machine learning tools that allow shape recognition and regression analysis, as well as prediction applications [24][25][26]. DL approaches are becoming increasingly popular due to their real ability to describe time series data dependencies. Recently, many Deep Learning models have been proposed, including Boltzmann machines, Deep Belief Networks (DBNs), and Recurrent Neural Networks (RNNs). RNN is a type of neural network that exploits the sequential nature of input data. RNNs are used to model timedependent data and give good results, which have proven to be successful in several application fields [15,23,27]. Short-Term Memory Networks (STMN) are a type of RNNs, which are able to cope with the storage of information for much longer periods of time [16,[28][29][30]. They are considered to define one of the most used RNN models for predicting data from time series, which perfectly correspond to the problems of PV energy production.
Zhao and Magoulés published a review study in 2012, focusing on the main approaches to energy prediction and forecasting in buildings [20]. Specifically, the authors compared physical models with those based on machine learning, as well as with statistical ones. The authors noted that machine learning-based models had the highest accuracy and flexibility, especially compared to statistical models. Vector support machines have been developed to outperform ANN models; one of the areas recommended for future investigations is that of optimized applications.
In 2017, Wang and Srinivasan [23] explored the use of models based on artificial intelligence (AI) and overall models for predicting and forecasting the energy use of buildings. The authors provided a breakdown of how artificial intelligence as a whole was applied to predict the energy of the buildings. Most AI-based works were applied to an entire building load with hourly data. The authors also explored how overall methods were applied in predicting the energy of buildings. It was noted that such assemblies have been widely applied to fields outside the energy of buildings, and the results showed improved performance compared to unique forecast models.
Akhter et al. [31], Van Deventer et al. [32], Das et al. [33], and Seyedmahmoudian et al. [34] analyzed and compared different forecasting models for PV energy production, including the hybrid ones; their strengths and weaknesses were highlighted.
At the same time, the benefits of model optimization were discussed.
The main objective of this study was to apply the advanced techniques of ARIMA and ANN models for two cases: (1) a lab BIPV system and (2) a large PV park placed in the south of Romania. The obtained simulated and experimental results were compared and allowed to have interesting information regarding the energy production forecast on short term, as well as on long term; it could be very useful for improved performance of energy prediction and forecasting for buildings based on statistical approaches and artificial intelligence tools.

Conceptualization and Methodology
2.1. The Advanced Statistical ARIMA Model. The ARIMA (Autoregressive Integrated Moving Average) model establishes a remarkable method used in short-term solar radiation forecasting [14]. The functionality of the ARIMA model is represented in Figure 1 [35,36]. The software used to make the forecast, respectively, IBM SPSS, is a predictive analysis program, which offers a wide range of statistical procedures: linear regressions, Monte Carlo simulations, geospatial analysis, etc.
The model chosen in this article for the solar radiation forecast is of the ARIMA (1,0,0) or (1,1,0) type. This choice is due to the Akaike Informational Criterion (AIC) [19]. The AIC is used to evaluate the model, depending on the entered data; the aim is to determine the quality of the statistical model on a dataset, establishing whether this model is more efficient than other models. The ARIMA model provides a relative estimate of the information used to determine the processes that could generate the data.

2
International Journal of Photoenergy The Akaike statistical criterion for the ARIMA model can be calculated based on the following relation [35,36]: where L represents the maximum value given by an estimation function (MLE (Maximum Likelihood Estimation)) and k represents the number of estimated parameters. The minimum value of the AIC result defines that the corresponding model is the most efficient. The SPSS program [35,36] automatically calculates the AIC, which is a parameter for model fitting.
If the forecast process contains seasonal fluctuations, as in this case, the process becomes SARIMA ðp, d, qÞ ðP, D, QÞ s, where p is the order of the AR process, d represents the differentiation term, q is the order of the moving average, P represents the order of AR seasonal processes, Q represents the MA order, D is the order of seasonal differentiation, and s represents the length of the seasonal period [19].
The peculiarities of the model used in this forecast are highlighted by defining a backshift operator B of the time series [9]: B Zt = z t−1 , where z t and z t−1 are two consecutive observations of the time series. Then, B j can be defined as B j Zt = z t−j . Using standard notation, ф ðBÞ is the autoregression operator, represented as a polynomial in the backshift operator: ф ðBÞ = 1 − ф 1 B − ф 2 B 2 − ⋯−ф p B p , and θ ðBÞ the mobile operator is represented by a polynomial in the operator backshift:   International Journal of Photoenergy degree of nonseasonal differentiation and at the white noise, from a mathematical point of view, the ARIMA model could be written as follows: where W is the differentiation operator which is the equivalent of result 1-B. We call this model ARIMA ðp, d, qÞ, where p, d, and q are process orders. In our model, the self-regression delay is determined separately for each month represented by the number of lighting hours, and the hours with zero irradiation (night hours) are neglected. As a consequence, we have used the ARIMA model ðx, 0, 0Þ, where x represents the number of hours in a day with solar radiation of those months (for example, for January it is 10, and for July it is 16).
ARIMA ðx, 0:0Þ was used to avoid the SARIMA model (seasonal model) [19,35,36]. The seasonality component is taken into account by the autoregression component, where the autoregression is equal to the period. 10 in January represents the fact that the morning value from 8 o'clock correlates with the value of the next day from 8 o'clock, which is in the 10th place from the first value, since we have used only 10 values each day. In other words, in the used experimental data string, there is an autocorrelation of 10 orders.

The Advanced ANN Model Based on Artificial Intelligence
(AI). Neural networks are a set of processing elements that have been developed separately from standard regression techniques. The ability to "mimic" natural intelligence by learning from experience makes this technique very attractive for solar radiation predictions. Functioning like a biological brain, a neural network is made up of a large number of interconnected neurons. Two main classes of neural network architecture can be identified, namely, (1) the architecture with the propagation of information from the input data to the output data, also called feed-forward architecture, and (2) the architecture of recurrent networks. The functionality of the neural network model is represented in Figure 2 (see [15,25]).
The ANN model is also used to estimate global solar radiation at an hourly interval, based on the following meteorological parameters: air temperature ðTÞ, relative humidity ðRHÞ, atmospheric pressure ðPÞ, wind speed ðWÞ, cloudiness ðCÞ, and hours of the day ðHÞ: The main parameters of the ANN model used in this study are as follows [35,36]: the number of neurons in the hidden layer is 1, the initial weight is 0.3, the learning rate is 0.3, and it used the Sigmund logarithmic function. The 10 days of meteorological data are included in the learning process. (a) The BIPV system from the Polytechnic University of Bucharest is a demo system connected to the grid; it was put into operation in July 2008 (see Figure 3) [11]. It consists of six PV panels, an inverter, equipment for monitoring and storing data, and a laboratory station for monitoring meteorological parameters. The power delivered by the PV panels is 615 Wp; they are semitransparent and have a power of 85 Wp each (type 1), respectively, and 120 Wp each (type 2). The system consists of three modules of type 1 and three modules of type 2 Also, to complete the BIPV system and make it functional, there is an inverter that makes the conversion between dc and ac, as well as monitoring and data storage equipment. The inverter is of the Sunny Boy SB700 type and achieves a high efficiency of conversion of direct current into alternating one, which it then delivers to the electricity distribution grid.
With the mentioned technical resources of the studied BIPV system, data regarding the brightness and performance of the PV system were collected, in terms of power (Pac) for a period of five days. Three days were considered for the forecasting process, and the other two days were used to validate the results. Figure 4 illustrates the

Activation function
Output data Figure 2: Functionality of the ANN model for the energy production forecast of a photovoltaic system. 4 International Journal of Photoenergy evolution of the power and brightness parameters, over a period of one day of the chosen interval. The detailed comparison, which provides data every 15 minutes, allows us to obtain important information about system performance. An unexpected decrease in power can be seen in the second half of the day. The explanations can be the following: (i) The photovoltaic window is oriented to the east, so that the incident solar radiation is maximum in the first part of the day (ii) The system is shaded in the second part of the day, due to the architecture of the building The solar radiation data, obtained with the help of the lab weather station, were used for the short-term solar radiation forecast, using various methods, in the location where the BIPV system is placed [11].
The results of the ARIMA model are represented by the input dataset for 13.06-15.06 2012, and the model is checked by the data for 16.06-18.06 2012 [11,35,36] (see Table 1).    (b) Taking into account five meteorological variables, namely, air temperature, relative humidity, atmospheric pressure, wind speed, and sunshine duration, the forecast was made for 16.06.2012, using the method of neural networks (ANN) (see Figure 6) [11,35,36] A statistical relationship is defined between global solar radiation and energy production, taking into account the technical parameters of the system. The losses of the system due to its location were also taken into account.
The values of the predicted data for P ac were compared with the measured ones; this situation is represented in Figure 7. The error calculated between the measured and the predicted values of P ac using the ANN model is 8.89%.
3.2. Improving the Forecast Quality by the Variability Index of Solar Radiation. The meteorological data used in the study of the solar radiation forecast for a lab BIPV system come from the Archive of the National Meteorological Authority [9]. The parameters were measured in the Bucharest-Afumati weather station (44°30′N 26°13′E, h = 91 m, 8 km away from Bucharest, WMO code 15421) and represent data obtained every hour during 2008-2009. The meteorological tools and observations correspond to the WMO instructions (Global Observation System Manual, 2010). The meteorological variables included in this study are global solar irradiance in kJ/m 2 , average air temperature in°C at a height of two meters, atmospheric pressure in mbar, relative humidity, wind speed in m/s, and cloudiness in oktas.
At the Bucharest-Afumatzi meteorological station, the global solar irradiance is automatically measured on a horizontal surface, with the pyranometer CM11 Kipp&Zonen-  Delft. The pyranometer sensor is a thermopile protected from external effects by a double glass dome. It has the following features: a response time (95%) of 15 seconds, spectral range 305-2800 nm, sensitivity 4-6 μV/W/m 2 , nonlinearity < ±0:6% (<1000 W/m 2 ), this being installed inside the meteorological platform, facing south, at a height of 2 meters. In the solar radiation forecast of the BIPV lab system from the Polytechnic University of Bucharest, both studied models were used, namely, ARIMA and ANN [9,11]. The error of the solar radiation forecast depends on its daily variation. At the same time, the daily variation of solar radiation showed an important correlation with clouding; thus, cloud information will be needed to increase the accuracy of the forecast.
However, this type of calculation requires a large amount of input data and a huge computational ability. In most cases, acceptable results cannot be obtained. In order to take into account the influence of the clouding factor but at the same time reduce the number of input data, it is found that the affected situations are determined by an empirical index, as an input parameter, namely [11], where V DR is the variability index of the solar irradiance, χ MR is the average of the monthly global solar irradiance, and σ DR is the standard deviation of the daily global solar irradiance. In fact, this index quantifies the variation of solar irradiance for a given day, compared to the general situation in a given month; the variation of solar irradiance in the previous days is taken into account indirectly by the monthly average. This daily index provides more detailed information, as well as the classification based on synoptic situations, the latter variant including a longer period with various clouding factors. By default, the index provides an important correlation with the clouding factor, r = 0:66 in (1) Cloudy/foggy day on the forecast day and partly cloudy/clear sky on the previous day V DR < 0:5 (previous day V DR > 0:5) (2) Cloudy/foggy situation on the forecast day and cloudy/foggy sky situation on the previous day V DR < 0:5 (previous day V DR < 0:5)  We have analyzed the accuracy of the solar irradiance forecast developed in the four situations using a time series for the ARIMA and ANN models in order to quantify the significance of the forecast error in each case [22,27]. Considering the frequency of situations during the investigated period, the most common situation is the fourth, which represents 64% of cases. The least common situation is the second, with a frequency of occurrence of only 8-11% of cases. The results of the analysis are represented in Figure 9. By this method, the errors of daily solar radiation forecasts can be quantified even during the forecasting process, separating only three sit-uations, namely, the cloudy situation, the fog situation, and the one with partly overcast sky/clear sky conditions.   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 where F t represents the predicted value, A t represents the current value (measured), n is the number of predicted points, and G n represents the daily average of the current (measured) values. Tables 2 and 3 contain the results of the forecasts corresponding to the four situations. It is noted that the best results are obtained in the case of partly cloudy/clear sky on the forecast day and partly cloudy/clear sky on the previous day, where the monthly rRMSE varies between 6.2 and 53%, and the annual one is less than 26.7%. In the case of cloudy/foggy sky on the forecast day and cloudy/foggy sky the day before, as well as partly cloudy/clear sky on the forecast day and partly cloudy/foggy sky on the previous day, the accuracy of the forecasts is relatively similar, ranging between 52.6% and 96.8%. In the case of cloudy/foggy sky on the forecast day with partly cloudy/clear sky on the previous day, the forecasts show a huge error of over 250%.
Comparing the two models, ARIMA and ANN, it was observed that the overall amplitude of errors is relatively similar in each case, but at the same time, the ARIMA model offers better results by 2.9% in the fourth case. We can  9 International Journal of Photoenergy conclude that the accuracy of the forecasts depends closely on the daily variation of the solar radiation controlled by the clouding factor, and the weather series forecast provides acceptable results in the case of partly cloudy/clear sky on the forecast day with partly cloudy/clear sky the previous day (see [9,15]).
Using the models of solar radiation forecast, and establishing the most frequent situation of the days from the meteorological point of view, the solar radiation forecast for January 20, 2009, was made and is represented in Figure 10.
With the help of this forecast, another forecast was elaborated, that of the power delivered by the lab BIPV system, represented in Figure 11.
The error calculated in this case was 26.9% and is consistent with the errors calculated for the solar radiation forecast, namely, 26.7%. This result is due to the improvement of the solar radiation forecast by integrating the data related to different meteorological parameters (temperature, cloud cover, etc.), as well as by separating the synoptic situations according to the variability index.

Long-Term Solar Radiation
Forecast with Application for a BIPV Lab System. We have also analyzed the decade variation of solar radiation as long-term changes to be taken into account in solar energy applications. For the analysis of the change trend, we have applied the linear regression model. The annual data come from the World Radiation Data Center database (http://wrdc.mgo.rssi.ru) and represent data corresponding to the period 1975-2006 for the Bucharest weather station. The linear trend is significant at a probability level of 95%, thus identifying an increase in solar radiation. The magnitude of the absolute changes is 36.5 (±2.43) J/cm 2 day -1 /decade ( Figure 12).
Based on the analysis of multiannual changes in solar radiation (J/cm 2 day -1 ), the power forecast of the BIPV system for 2013 was made (see Figure 13).
It was observed that the maximum power production of the BIPV system is achieved in May. The total production forecast for 2013 is 163 kW, a result that is in line with the power achieved in the previous year, of 157 kW. We can conclude that the best forecast result can be obtained in situation (4) of partly cloudy/clear sky on the forecasted day and partly cloudy/clear sky on the previous day V DR > 0:5, rRMSE-the annual error being 22.3 in the case of ARIMA and 25.3 in the case of ANN. In this case, ARIMA gives the best result in line with Reikard [9,23] who showed that at 60-minute time resolution, the ARIMA model forecast dominates all time series forecasting methods in four from the six test stations used in the study. In the other synoptic situations, the errors are large. Given the frequency of occurrence of the four synoptic    International Journal of Photoenergy situations presented above, the situation in which we obtain the best results appears in 65.56% of the days of a year, so acceptable forecasts can be made in 65% of that year. In the case of the other situations, namely, (1), (2), and (3), physical and/or satellite models are required [24]. The results of the study contributed to the improvement of the forecasts of the photovoltaic systems, which will be used for the analysis of an experimental PV park [31,32].  11 International Journal of Photoenergy radiation data are acquired by 2 pyranometers and 10 calibration solar cells. The pyranometers are located in the meteorological station within the PV park, and the solar calibration cells are placed at each transformation center, thus being arranged on the entire surface of 33 ha of the park. The pyranometers and calibration cells are placed in a plane that has the same inclination to the horizontal as PV modules. There are a total of 20 Green Power PV500 inverters. The energy produced is measured at the connection point, and the furthest inverter is located at 400 m from this connection point. Each transformation center contains two inverters. The PV module is made of 72 polycrystalline silicon solar cells.
A view of the PV park from Grojdibodu is presented in Figure 14, and its electrical diagram is shown in Figure 15 [35,36].
The main meteorological parameters of the Grojdibodu location defining the PV park are indicated in Table 4 [35,36].
The technical parameters of the studied PV system/PV module, used in the forecast, are presented in Tables 5(a) and 5(b). Figure 16 shows the I-V (current-voltage) characteristics for the Suntech Power PV module used in the componence of the studied PV park.
The technical performance indicators of the Grojdibodu PV system for one year are designated in Table 6.

Results and Discussion
: Energy Production Forecast for the Experimental PV Park. The purpose of the forecast was to size an experimental PV park located in the south of Romania, at Grojdibodu. Thus, a comparison was made for the results obtained by numerical modeling, with those measured in the PV park, as well as a comparative analysis of the programs used in terms of their efficiency. For simulation, we  Figure 15: Electrical diagram of the studied PV park.

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International Journal of Photoenergy have used the data measured by the meteorological station within the PV park, considering the optimization of the PV system parameters, the increase of its energy efficiency, and the improvement of its global efficiency [26,35,36].
With the help of the ARIMA and ANN models, the forecast of surface solar radiation (SSR) was made, using data obtained from the PV park. Forecasts for 10 days have been developed based on the previous time series. The periods were chosen randomly, and the forecasts were checked for clear or partially clear days, which have the variability index of direct solar radiation V DR > 0:5. The variability index refers to the classification of synoptic situations (clear or cloudy). These situations are not included in the forecasts made. The index helps to validate forecasts; thus, statistical forecasts are more effective in the case of partly cloudy/clear sky on the forecast day and partly cloudy/clear sky on the previous day V DR > 0:5. In the case of "cloudy" situations (1), (2), and (3), the rRMSE has large errors; for this reason, the statistical/empirical method for forecasting is not useful, and physical models are needed to improve it.
The correlation between the variability index and nebulosity was calculated in order to validate the index. The index shows the cloudiness on that day, without using cloudy data.
The aim was to obtain information about nebulosity from solar radiation data. The validation of the index was performed for a shorter period; for example, in cases where the index had a value > 0:5, we checked whether the nebulosity was low or not, and vice versa. Following the verifications, the correlation was significant; as a result, we have used V DR instead of nebulous data.
Short-term solar radiation forecasts were developed using the ARIMA and ANN models. The efficiency of the forecasted time series is quantified by rRMSE. The model was improved by separating the days with clear sky from the days with a high degree of cloudiness and using the variability index of solar radiation, V DR . The separation of synoptic situations was performed on time series, not on predicted values. It has been established that the most common cases for the use of the V DR index are days with clear sky on the forecast day and the day before. These days represent more than 75.4% of the total number of days of the year.
For the rest of the days when there are situations of cloudy sky or high fog, the forecast provides very large errors. The forecasted results conclude that the ARIMA model is more efficient than the ANN model. The statistics are significant in the case of ARIMA (1,0,14), this being the reason why this variant was chosen for analysis. Also, comparing the results of the forecast with the measured values, we noticed that the ARIMA model (1,0,14) is more efficient than the ARIMA (1,0,7). The ARIMA (1,0,7) and ARIMA (1,0,14) models were selected after performing several iterations. The statistical test used to identify the most significant model is Box Ljung Statistics. This test is applied in the case of ARIMA on the residues of a fitted model, not on the original series, and verifies that these residues do not show autocorrelation.
The results of the measurements and forecasted values for August and September 2013 are presented in Figures 17  and 18. The accuracy of the predictions is quantified using relative absolute mean error (rMAE) and relative mean square error (rRMSE): where the following notations are introduced: F t is the predicted value, A t is the measured value, n is the number of predicted points, and G n is the daily average of the measured values. Table 7 shows the measured and forecasted values for 10 days in August 2013, while Table 8 shows the same data in September 2013 for 10 days.  The two models offer good efficiency in terms of relative errors, but forecasts with errors below 40% are present only in the summer periods, especially in July, August, and September characterized by high values of solar radiation. In the case of low SSR, this methodology produces very large errors.
Based on the predicted values of solar radiation, the electrical power that would be inserted in the national power distribution grid could be calculated taking into account the efficiency of the PV park. Most likely, a global relationship for estimating the energy generated was [35,36] where E represents the delivered energy (kWh), A represents the total surface area covered by the PV modules, H represents the average annual solar irradiance on the inclined   PV panels (Wh/m 2 ), and PR (performance ratio) represents the efficiency of the PV system. The comparison between the forecasted energy and the energy inserted in the grid is made between 01.08.2013 and 09.08.2013 for the analyzed experimental PV park (see Table 9).
Considering the data collected within the PV park, the monthly exported energy shows that in the period chosen for testing 01-09.08.2013, a quantity of 513.98 MWh was delivered in the distribution grid. The share of the measured solar irradiance, corresponding to this value of the delivered energy, was 63165.75 Wh/m 2 .   Figure 19: Difference between the measured values of solar irradiance and predicted ones based on the ARIMA model. 15 International Journal of Photoenergy Using the ARIMA model (1,0,14), the solar irradiance for this period was forecasted at 63091 Wh/m 2 , respectively, and at 63891 Wh/m 2 using the ARIMA model (1,0,7). Taking into account the technical parameters of the system and the forecast for solar irradiance, the energy forecast with the ARIMA type model will be 513.39 MWh, respectively, and 519.90 MWh (Figure 19).

Comparative Analysis of Different Forecasting Models
A comparison of different forecasting models (algorithms) developed in various countries is presented in Table 10.
The results obtained in the present article were compared based on ARIMA and ANN tools with new advanced forecasting models (algorithms), like PSO, GA, SVM, MLR, and RT [31][32][33][34]. The accuracy was much better in these cases. However, the authors have studied two interesting PV installations, namely, a BIPV lab system and an experimental PV park; both were placed in the south of Romania. Based on our results, it was established that the ARIMA (1,0.14) model is more efficient than the ANN one and is very close to the experimental case (see Figure 19). Although our approach is dedicated especially to short-term forecast, the long-term solar radiation forecast was discussed too with application for a BIPV system developed at the Polytechnic University of Bucharest. This feature is different from the other more advanced forecasting models presented in Table 10.

Optimized Forecasting Method for Energy
Production Based on GMDH Model: Case Study-PV Park in Romania The GMDH (Group Method of Data Handling) represents a family of inductive algorithms for the mathematical modeling of multiparametric datasets that are able to automatically optimize the used models [37]. In the GMDH, continuous or discrete input variables can be introduced, and depending on their type, optimal parametric or nonparametric algorithms can be identified. In order to optimize the energy production developed by a Romanian PV park, a specific approach was conceived to render efficient solar radiation prediction on the surface of PV panels for a PV park. As the prediction method involves a large amount of data and a great computing power, Romania's territory was divided into 28 regions, considering that each region possesses at least one actinometric station.
For the created regions, important meteorological data will be provided for solar radiation prediction and will be introduced into advanced models for prediction and evaluation (ARIMA, ANN). The goal is to improve prediction efficiency in both effective results and computing time.
Alongside corresponding solar radiation data and associated factors, for an efficient prediction, it will be introducing also the PV park's characteristics [38]. System loss categories are present in all PV systems but differ from one system to another. It is important to mention the fact that the PV park loss is not an aggregate of particular losses. These losses affect the system individually, their impact being calculated for each component. The losses are due to the dust deposits on the module's surfaces, shadows within the emplacement zone, snow deposits, electrical installation's asymmetry, losses between module's connections (wire harness loss), connection loss, etc. Also, the degrading agent must be considered; it represents the production loss during the maintenance period. Considering the PV park's parameters and also the system loss, the energy produced by the PV park was determined both during the days 04, 08, and 14 of July for 2013 and 2014 and during the days 29, 30, and 31 of July 2013 and 2014. The results are presented in Table 11.
Forecasted values show acceptable results correlated with the errors for solar radiation forecasting. Since the forecasts were made using data from previous days without taking into account future forecasts, this method could be optimized by introducing additional parameters that  16 International Journal of Photoenergy meteorologically describe the following day. Research continues to identify correlations between meteorological parameters that could accurately be forecasted for a given day and their impact on solar radiation.

Conclusions and Prospects
Short-term solar radiation forecasts were developed using the ARIMA and ANN models. The efficiency of the forecasted time series is quantified by rRMSE. The models were improved by separating the days with clear sky from the days with a high degree of cloudiness using the variability index of solar radiation, V DR . The separation of the synoptic situations was performed on time series, not on the predicted values. It has been established that the most common cases for the use of the V DR index are days with clear sky on the forecast day and the day before, where the day "forecast" is represented by the day "tomorrow" and the day "before" is represented by "today" or the day on which the forecast is made. These days represent more than 75.4% of the total number of days of the year, so only these cases were chosen. For the rest of the days when there are situations of cloudy sky or high cloud cover, these forecast solutions offer very large errors. The results of the forecast attest to the fact that the ARIMA model is more efficient than the ANN model. The statistics is significant in the case of ARIMA (1,0,14), this being the reason why this variant was chosen for analysis. Also, comparing the results of the forecast with the measured values, we notice that the ARIMA model (1,0,14) is more efficient than the ARIMA (1,0,7). The ARIMA (1,0,7) and ARIMA (1,0,14) models were selected after performing several iterations. The statistical test used to identify the most significant model is Box Ljung Statistics. This test is applied for ARIMA on the residues of a fitted model, not on the original series, and verifies if these residues do not show autocorrelation. The Box-Ljung model, also known as the statistically modified Box-Pierce model, provides guidance on whether the model is specified correctly. A value significantly lower than 0.05 implies that within the observed data there is a structure that is not relevant for the model. A value greater than 0.05, as determined in the case, indicates that the chosen model is correct (see [9,35,36]. The future work will be based on a new optimized forecasting method for PV energy production, presented in Section 6. Taking into account that the optimized prediction and forecasting method would involve a large amount of data and great computing power, Romania's territory could be divided into 28 regions; each region would possess one actinometric (solar radiation) station. A polynomial neural network based on the GMDH (Group Method of Data Handling) approach would be used. It would be represented by a family of inductive algorithms (parametric and nonparametric), which could optimize automatically the energy production of the PV park.

PV:
Photovoltaic ARIMA: Autoregressive Integrated Moving Average ANN: Artificial Neural Network BIPV: Building Integrated Photovoltaics GMDH: Group Method of Data Handling AIC: Akaike Information Criterion SARIMA: Seasonal ARIMA NWP: Numerical weather prediction TRNSYS: Transient system simulation tool eQuest: Quick energy simulation tool Energy Plus: Open source software BAS: Building automation systems BEMS: Building energy management systems HVAC: Heating, ventilation, and air conditioning DL: Deep Learning DBN: Deep Belief Networks RNN: Recurrent Neural Networks STMN: Short-Term Memory Networks AI: Artificial intelligence IBM SPSS: Predictive analysis program MLE: Maximum Likelihood Estimation