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Neural network approach to estimate 10-min solar global irradiation values on tilted planes
Renewable Energy 50 (2013) 576e584
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Neural network approach to estimate 10-min solar global irradiation values on tilted planes Gilles Notton a, *, Christophe Paoli b, Liliana Ivanova c, Siyana Vasileva c, Marie Laure Nivet b a
University of Corsica, UMR CNRS 6134, Research Centre of Vignola, Route des Sanguinaires, F-20000 Ajaccio, France University of Corsica, UMR CNRS 6134, Sciences Faculty, BP 52, F-20250 Corte, France c Faculty of Electrical Engineering, Technical University of Sofia, 8, Kl. Ohridski Blvd., 1000 Sofia, Bulgaria b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 20 February 2012 Accepted 26 July 2012 Available online 30 August 2012
Calculation of solar global irradiation on tilted planes from only horizontal global one is particularly difficult when the time step is small. We used an Artificial Neural Network (ANN) to realize this conversion at a 10-min time step. The ANN is developed and optimized using five years of solar data and the accuracy of the optimal configuration is around 9% for the RMSE and around 5.5% for the RMAE i.e. similar or slightly lower than the errors obtained with empirical correlations available in the literature and used for the estimation of hourly data. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Solar irradiation Artificial Neural Network Estimation
1. Introduction Vijayakumar et al. [1] showed that simulation studies of solar energy systems generally use hourly values to estimate long-term performance; considering that solar radiation fluctuates during an hour, using only hourly data results in inaccurate performance estimation. Indeed, some types of solar systems, such as photovoltaic systems, respond quickly and non-linearly to solar radiation. Thus, the distribution of short-term radiation within an hour results in greater utilizability, if the short-term data varies significantly. Generally, hourly solar data are used in quasi-state models for simulating the behaviour of renewable power systems. Nevertheless, the high variability of the load, of the renewable sources and of the fluctuating storage level (when it exists) requires that these data are known with a fine time step to take account more accurately the lag between the various renewable productions and the load. Thus, the sizing of a renewable system will be more correct and its reliability and cost-effectiveness will be higher. Notton et al. [2] studied the influence of the simulation time step in view to size a stand-alone photovoltaic system. They showed that daily data may lead to a significant under-sizing but there is a good agreement between the results obtained from minute and hourly data; the load profile used in this study varied throughout the day but not during the hour and we can reasonably * Corresponding author. Tel.: þ33 495524152; fax: þ33 495524142. E-mail address: [email protected] (G. Notton). 0960-1481/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2012.07.035
expect that the results are different with a more variable load. Hawkes and Leach [3] showed that it is important to use a fine temporal precision in energy system simulation when the effect of averaging will significantly impact how the model behaves. The knowledge of the variability of the solar source is important for improving the design of a system (by adding properly sized storage capabilities, for instance) and understanding the performance of a solar conversion system (e.g., understanding how the extremes can enhance or degrade system performance or during which season they occur most frequently) [4]. Halamay et al. [5] wrote that, as the penetration level of renewable systems increases, the added variability of the energy resource can cause greater ramp rates, greater inter-hour variability, and greater scheduling error. It is consequently necessary to take into account these short-term variations and consequently 10min data are used in their study for load, wind, solar, and wave power. Manz et al. [6] showed the necessity to develop a sub-hourly tool using 10-min time step for long-term dynamic simulation in view to quantify reserve violations, fast starts and load shed events caused by sub-hourly wind/solar/load changes. Corbus [7] applied this sub-hourly tool for Hawaii solar integration studies and future grid studies. Frank et al. [8] used the same time step of 10-min for grid integration study in Hawaii. Thus, as seen in the literature, 10 min solar data are particularly useful for studies of solar system integration. Solar radiation on tilted planes is very important for engineers and scientists in view to design flat plate collectors, photovoltaic
G. Notton et al. / Renewable Energy 50 (2013) 576e584
systems and other solar energy collecting devices. Although meteorological stations provide sometimes reliable solar radiation records, they usually measure radiation only for horizontal surfaces and generally the data on tilted surfaces are not available. Thus, they should be estimated with different models from those corresponding to a horizontal surface. In commercial software intended for the sizing of solar systems, photovoltaic or thermal systems, the inclination of solar collectors (and the optimization of this inclination) is an input parameter. From a meteorological data basis containing global horizontal solar radiations, the software “inclines” these radiations; but this conversion is realized with an average and often low accuracy. It is consequently useless to develop fine and precise models to describe the solar system behaviour if the input data are approximate. The relationships between the horizontal solar radiation and various other components of solar radiation are not linear and depend on many factors, both related to the geometric position of the sun and meteorological characteristics of the sky state (cloud density and position, water vapour concentration, etc). As underlined by Behr [9], three main reasons make impossible to develop a simple model with a view to converting horizontal global solar irradiation into inclined global solar irradiation: - the radiation incident on a tilted plane includes the radiation reflected by the environment; - when the plane is inclined, only a portion of the sky is “seen” and then the diffuse component of the sky depends not only on the inclination angle or the orientation of the collector, on the elevation of the sun as well as its azimuth but also on the sky conditions which are rarely uniform. This therefore induces some anisotropic effects complex and difficult to quantify [10]. There are two main problems relating to this sky anisotropy: firstly, the circum-solar brightness due to the solar radiation diffusion by some aerosols and concentrated in the sky area around the sun and secondly the brightness of the horizon concentrated near the horizon which is even more important in clear skies [11]; - at last, these inclined solar data are relatively rare. The anisotropy phenomenon especially affects the calculation of the tilted irradiation when the measure time step chosen is small, in fact, larger the time step is, more this anisotropy decreases (timeaveraging and compensating effect) and tends to move towards an isotropic distribution. Consequently, shorter will be the time step, more difficult will be to realize this conversion with a good accuracy. The conversion of horizontal solar irradiation in tilted solar irradiation is a complex issue often dealt with in the literature. Mostly, only monthly mean values of solar irradiation (daily or hourly) are converted using linear or polynomial interpolations; unfortunately, if these data allow to have a general vision of the solar potential, they are insufficient to model or to size a solar system. Numerous models have been developed at daily or hourly scales; though some empirical models allowing to determine directly the inclined solar irradiation from the horizontal one do exist [12], generally we must combine two types of models to reach our goal. The “conventional” method consists in considering separately the two components: the beam and the diffuse radiations: we combine a first model estimating the horizontal diffuse radiation from the horizontal global radiation and a second model calculating the inclined global solar radiation from the two previously mentioned ones i.e. the measured horizontal global radiation and the estimated horizontal diffuse radiation. The precision of such a method is relatively low at a hourly scale and depends on the meteorological conditions on site [11,13].
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The diffuse radiation is the solar component the more complicated to calculate. The state of the sky (clear, partially cloudy or cloudy) is characterized by the clearness index (ratio of the global radiation on the ground and the extraterrestrial one) and the anisotropy of the sky diffuse radiation is quantified by different complex index [14e20]. The diffuse fraction or the diffuse clearness index is calculated from linear, polynomial or more complicated relations. Behr [9] and Elminir [21] show that no simple and reliable relation exists between diffuse and global irradiation. The estimation of each solar component depends on the day number (or the declination), the solar hour, the solar elevation, the sky clearness, the clouds positions in the sky dome and their size, etc. For a same global irradiation, the distribution in the three components (beam, sky diffuse and ground diffuse) varies considerably depending on the parameters mentioned above. The reliability of empirical correlations is not very high [14,16] and Elminir et al. [21] suggested and proved that ANN models are particularly suitable for such applications. As said by Jiang [22], one of the limitations imposed by linear regression models is that they will underperform when used to model nonlinear systems which are the case for the problem we are interested in. All researches reported that the neural approaches out-performed the traditional linear methods. Moreover, these conclusions have been given for an estimation at a hourly scale and this fact is particularly true that the time step used in this study is smaller. The development of artificial intelligence processes, for several years, and more precisely of Artificial Neural Network (ANN) allows to estimate researched data from available ones when the relation between output and input data are complex and not linear. In this work, we present rapidly the ANN concepts and explain the choice of the various parameters influencing the ANN performances. We will realize a sensitivity analysis in view to optimize the ANN structure and the number of input parameters. 2. Data collection and processing The laboratory “Sciences for Environment” of the University of Corsica Pascal Paoli has a complete meteorological station in the gulf of Ajaccio (latitude: 41550 N; longitude: 8 550 E) situated at about 200 m from the sea and at an altitude of 70 m. The site is characterized by an insular Mediterranean climate. The surrounding topography is not very high but a small mask may appear in the winter when the sun rises. The data used in this work were measured and stored with a minute time step with a pyrheliometer Kipp & Zonen at normal incidence. This device is mounted on an automatic Solar Tracker Model Solys 2 (2 axis, azimuth/elevation device programmed to align direct beam instruments with the normal incidence of the sun) and measures the normal beam irradiance. Every two days, the pyrheliometer is cleaned and its alignment is verified. The global irradiance on horizontal, 45 and 60 tilted surfaces is measured by three Kipp & Zonen (CM11) pyranometers. The experimental data have been collected since January 2006 in a quasi-continuous manner. Thus, we have 5 years of minute data _ normal beam irradiance I_ , and of horizontal global irradiance I, b;n inclined global irradiance at 45 and 60 I_45 and I_60 . 45 and 60 are respectively the optimal inclinations for a maximum global irradiation on an annual period (inclination equal to latitude, here 41550 ) and on a winter period (latitude þ15 ). For each minute, we add three calculated parameters: - the horizontal extraterrestrial irradiance I_0 ; - the solar declination d; - the zenith angle qz .
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The declination angle d is the angle between the line joining the centres of the sun and the earth to the equatorial plane and it changes at every moment, but we can consider that it depends on the day number [23].
dðradÞ ¼ 0:006918 0:399912cos G þ 0:070257sin G 0:006758cos 2G þ 0:000907sin 2G 0:002697cos 3G þ 0:00148sin 3G
(1)
(2)
and dn is the day number. The zenith angle qz is the angle between the sun and a horizontal surface. It is given by [23]:
cos qz ¼ sin dsin 4 þ cos dcos 4cos u
(3)
with 4 the latitude and u the hour angle given by:
u ¼ 15ð12 tÞ
(4)
where t is the time expressed in true solar time. The extraterrestrial irradiance on a horizontal plane is calculated by [23]:
I_0 ¼ I_sc E0 cos qz
(5)
where Isc is the solar constant taken equal at 1367 W m2 according to the World Meteorological Organization. E0 is the eccentricity factor taking into account the variation of the earth-sun distance and calculated by [23]:
E0 ¼ ðr0 =rÞ2 ¼ 1:000110 þ 0:034221cos G þ 0:001280sin G þ 0:000719cos 2G þ 0:000077sin 2G
(6)
Some quality control tests are imposed on these minute data in order to extract outlier or missing data. The rejected data are those for which [24,25]: - the horizontal global irradiance is superior to the horizontal extraterrestrial irradiance; - the normal beam irradiance is higher to the normal extraterrestrial irradiance. Then the 10 min values are calculated: they are computed by integration on the 10 min period for energy data and for the zenith angle, the value is calculated at mid-period. We allowed 10% of missing data over the 10 min period i.e. one missing data for 10 min. Very often, the sunset and the sunrise induce some problems in the correlations, firstly because of the mask effect of the environment and secondly due to the bad response of pyranometers when the zenith angle is high (cosine effect) [18]. Thus, we extracted the 10 min period during which the sun rises or sets. Over the 5 years, we have 101,140 validated 10 min data i.e. about 6.46% of missing data. Consequently, we have several monthly files containing: -
the the the the
day; hour; 10 min period (from 1 to 6); declination d;
the the the the the the
zenith angle qz ; hourly horizontal global irradiation I; hourly normal beam irradiation Ib;n ; hourly horizontal extraterrestrial irradiation I0 ; hourly global irradiation on a 45 tilted plane I45 ; hourly global irradiation on a 60 tilted plane I60 .
3. Artificial Neural Network (ANN) 3.1. General presentation
where G is the day angle calculated by:
G ¼ 2pðdn 1Þ=365
-
The objective of this paragraph is to give some general information on the Artificial Neural Network (ANN). An ANN is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain process information. It is composed of a large number of highly interconnected processing elements (neurons) working in unison to solve specific problems. The goal of this network is to create a model that correctly maps the input to the output using historical data so that the model can then be used to produce the output when the desired output is unknown. An ANN has a parallel-distributed structure and consists in a set of processing elements called neurons. The ANN structure consists in: - an input layer which receives data; - an output layer to send computed information; - one or several hidden layers lying the input and output layers. According to the chosen architecture, all or a part of the neurons in a layer are connected with all or a part of the neurons of the previous and next layer. The number of hidden layers and the total number of neurons of each layer depends on the specific model, convergence speed, generalization capability, the physical process and the training data that the network will simulate [26]. A typical artificial neuron and the modelling of a multilayered neural network are illustrated in Fig. 1. The signal flow from inputs xk,1, xk,2,., xk,p of a layer k is considered here as unidirectional (feed forward configuration). The output signal O to the jth neuron of the following layer (kþ1) is given by [27]:
1 0 p X T O ¼ f ðnetÞ ¼ f w xk ¼ f @ wk;j xk;j A
(7)
j¼1
where wk;j is the synaptic weights (k is the layer index, j is the neuron index) and f ðnetÞ is referred to as an activation or transfer function. The variable net is the scalar product of the weight and input vectors. T is the transpose of a matrix. The result of this sum, net, is then transformed by a transfer function f which produces the output O of a neuron if this sum exceeds a certain threshold qk . The output is then distributed to other neurons as inputs. There are two main problems concerning the ANN implementation: specifying the network size (number of layers in the network and number of nodes in each layer) and finding the optimal values for the connection weights. An insufficient number of hidden nodes may be the cause of difficulties in learning data whereas an excessive number of hidden nodes might lead to unnecessary training time with marginal improvement as well as make the estimation for a suitable set of interconnection weights more difficult [28]. To determine the optimal number of hidden nodes; the common method used is trial and error based on a total error criterion. This method starts with a small number of nodes, gradually increasing the network size until the desired accuracy is achieved.
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Fig. 1. Architecture of an artificial neuron and a multilayered neural network.
One of the properties of ANNs is their ability to learn from their environment and to improve their performance through a learning process called also training process. Learning results in a change in the weights value wk;j connecting the neurons from one layer to another. The goal is to achieve equality between the actual output and simulated output. It is therefore necessary first to choose the learning algorithm and to define the part of the data used for learning in relation to the total amount of data available. The various steps in the implementation of an optimized ANN consist in selecting: -
ANN structure; transfer function type; ANN size (number of layers and number of neurons per layer); learning algorithm; training/test set; input data
3.2. Application of ANN to solar radiation ANN is a method already used to estimate or to predict the solar resource. Two types of works or studies have been realized: - forecast consisting in predicting what will happen at time t þ dt i.e. estimating Xðt þ dtÞ from the data at time t, XðtÞ and sometimes other information; - estimation of the value of a data YðtÞ from the knowledge of i others data Xi ðtÞ. This type of model allows for example to determine YðtÞ in locations where there is no way of measuring YðtÞ but where all the Xi ðtÞ are available. The approaches and methodologies developed concerning the structure for forecast or estimation are almost identical. The objective of this paper concerns the second type of study i.e. “estimation”. As we said in the introduction, ANN models have better performances than “conventional” empirical models for the solar irradiation estimation. The difficulty in the estimation of solar radiation comes mainly of the diffuse part reaching the tilted surface due to the non-uniformity of the sky state (anisotropy) and consequently there is not a one-to-one correspondence between the diffuse component and the global radiation. Jiang [22] stated that regression models underperform when used to model nonlinear systems, which is the case in this study. Mellit [29] presented a very complete state of art on the solar data forecast by utilization of artificial intelligence methods: ANN, fuzzy logic, genetic algorithm, expert system, hybrid method. More
than 80 papers have been presented. Notton et al. [30], after a bibliographical study on the utilisation of ANN to solar prediction and estimation, came to the same following conclusions: - more often only monthly average values are estimated from geographical and meteorological data such as temperature, humidity, solar fraction, wind speed, etc; - the estimated solar radiation component is often the horizontal global one, rarely the diffuse one and never the inclined global irradiation; - the time step is often daily (and often averaged on the month). Consequently, the estimation by ANN of tilted global solar irradiation is rarely studied and the time step of the estimated data is often average values and sometimes hourly data. 4. ANN implementation As previously specified in chapter 3.1, the ANN implementation consists in several stages: 1) structure: the chosen structure is the MultiLayer Perceptron (MLP) using feed-forward back-propagation; it is the most used in the literature for the solar radiation estimation [22,28,31e 40]; 2) transfer function: the combination of a sigmoid and a linear transfer functions is able to give a good approximation of several types of functions [37,40,41]. Several authors used this combination for the estimation of solar data [26,36,42]; 3) learning algorithm: the LevenbergeMarquardt algorithm (LM) is an approximation to the Newton method used during the training as suggested by literature [31,32,36,41,43]; 4) training/test set: among all the available data, we must use a percentage for the training and the rest will serve for the testing. As written by Dreyfus [41], we must use more than 50% of data for the training phase. Thus, we decide to take 3 whole years of data for training (60%) (2006e2008) and 2 whole years (40%) for testing (2009e2010); 5) ANN size: whatever the number of inputs, the ANN have to be as simple as possible (parsimony principle). Those configurations give the best results with a small error. The compromise between the number of hidden layers and number of neurons in each layer allows us to obtain a fast network, robust and that will give us the best results. Moreover, a large size architecture needs more data to be trained and it easily becomes overtrained, i.e. unable to generalize efficiently. Unfortunately, it does not exist any mathematical method for sizing optimally
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the ANN, the choice of the optimal structure can only be done only after testing different configurations and estimating their performance; 6) input data choice: the declination represents the Earth’s position with respect to the Sun and depends on the day of the year. The sun position influences on the quantity and the quality of the sun radiation. This position is characterized by the zenith angle (Eq. (3)). When the sun is high in the sky (low zenith angle), the solar radiation is maximal (in clear sky condition) and as the optical path is minimal therefore the incident radiation is less absorbed. For a same horizontal solar irradiation, correspond several values
of inclined solar irradiation according to the season, to the sky conditions, etc. In the diffuse radiation model, the clearness index or the diffuse index is used to take into account the sky condition; this sky conditions is an important parameter in the repartition of diffuse and beam radiations. Thus, more the clearness index is high, more the sky is clear and more the global radiation is constituted by beam radiation. The extraterrestrial irradiation is used as reference and should be useful as an input parameter (Eq. (5)). We chose also to introduce as input data the hour. However, it is possible that it is not useful to consider all the three parameters: time, declination and zenith angle. Indeed
Fig. 2. The three ANN models.
G. Notton et al. / Renewable Energy 50 (2013) 576e584
they are linked together according to Eq. (3). To remove this ambiguity, we decided to take a sensitivity analysis in order to find which of the potential inputs could have significant influence on the results. Then, before the sensitivity analysis, the input parameters are i) the time, ii) the declination, iii) the zenith angle, iv) the horizontal global solar irradiation, v) the extraterrestrial horizontal solar irradiation. To estimate the accuracy of the models developed, we use different indicators as suggested by Iqbal [23]. yi and xi are the estimated and measured values and the respective average values PN PN are given by x ¼ i ¼ 1 xi and y ¼ i ¼ 1 yi where N is the data number. The Mean Absolute Value (MAE) and its Relative value RMAE are calculated by:
MAE ¼
N X
jyi xi j=N
RMAE ¼
i¼1
N X
jyi xi j=Nx
(8)
i¼1
The Mean Bias Error MBE is expressed by:
MBE ¼
N X
ðyi xi Þ=N
(9)
i¼1
The Root Means Square Error (RMSE) and its Relative value RRMSE are given by:
RMSE ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðyi xi Þ2 =N
RRMSE ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ðyi xi Þ2 =N
(10)
The correlation coefficient R2 is:
"
N X R ¼ ðyi yÞðxi xÞ
#,("
2
i¼1
N X
#" ðyi yÞ
i¼1
2
N X ðxi xÞ2
#)1=2
i¼1
(11)
5. Results 5.1. Choice of architecture We developed three ANN (Fig. 2) 1. the first one for the estimation of 10-min global solar irradiation on a 45 tilted plane; 2. the second one for the estimation of 10-min global solar irradiation on a 60 tilted plane; 3. the third one for the estimation of 10-min global solar irradiation on a 45 or 60 tilted planes. Table 1 Results for Model 1: 45 . Hidden neurons 1st layer
2nd layer
6 6 6 6 12 12 12 12 24 24 24
0 6 12 24 0 6 12 24 0 6 12
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Table 2 Results for Model 2: 60 . Hidden neurons 1st layer
2nd layer
6 6 6 6 12 12 12 12 24 24 24
0 6 12 24 0 6 12 24 0 6 12
MAE Wh/m2
MBE Wh/m2
RMSE Wh/m2
RRMSE %
RMAE %
R2
4.731 4.826 4.832 4.798 4.041 3.998 3.912 3.789 3.279 3.456 3.801
0.255 0.125 0.156 0.112 0.165 0.154 0.152 0.148 0.253 0.275 0.278
7.715 8.231 8.322 7.912 6.415 6.321 6.295 6.242 5.690 5.941 6.123
13.85 14.58 14.65 14.02 11.52 11.48 11.42 11.38 10.22 10.32 11.12
8.49 8.56 8.56 8.53 7.26 7.18 7.12 6.99 5.89 6.23 6.98
0.989 0.992 0.993 0.993 0.992 0.994 0.994 0.995 0.994 0.994 0.994
In the case 3, the number of data used to train and validate the model are doubled because each value of global irradiation is introduced two times, the first one with b ¼ 45 and the second one with b ¼ 60 . Moreover, the inclination b (equal to 45 or 60 ) is added as an input data. Our objective is to find the simplest and accurate MLP model and to estimate its accuracy. First we have to choose the number of hidden layers. In practice one hidden layer can be sufficient, but it depends on the application and sometime two hidden layers are necessary to improve the reliability. We limit the number of hidden layers to two because as said previously the ANN must be as simple as possible (parsimony principle). We tested various configurations of monolayer and double layer ANN for the three models presented above and some results are shown in Tables 1e3. The best ANN structure for each model is highlighted in grey. The decision criteria are applied to the whole data (training set and test set). The models being nonlinear, the training results depend on the initial values of the model inputs. Thus, each configuration has been tested 6 times and only the best results are given in Tables 1e3. We varied the number of neurons in the first hidden and second hidden layers from 6 to 24 and we noted that by increasing the number of neurons in each hidden layer, the error stabilizes more quickly during the ANN training and requires a smaller number of samples. The two most significant errors are RRMSE and RMAE and our comments will be made mainly on the basis of these two indicators. Several remarks can be made: - in the three models, the best results are obtained for a monolayer ANN composed of 12 neurons in its hidden layer for model 1 and 3 and with 24 neurons for model 2 with an RRMSE between 8.18% and 10.22% and an RMAE between 3.65% and 5.89%; Table 3 Results for Model 3: 45 þ60 .
MAE Wh/m2
MBE Wh/m2
RMSE Wh/m2
RRMSE %
RMAE %
R2
3.240 4.510 3.110 3.321 3.113 3.230 3.232 3.356 3.641 3.724 3.952
0.121 0.065 1.002 1.235 0.156 0.078 0.098 0.118 0.140 0.152 0.172
5.290 6.481 5.290 5.462 5.152 5.370 5.302 5.458 5.612 5.782 6.211
8.85 10.12 8.45 8.68 8.18 8.24 8.21 8.45 8.99 9.23 9.45
3.21 5.89 2.21 2.45 4.94 3.00 2.89 3.26 4.21 4.48 4.54
0.996 0.994 0.996 0.995 0.996 0.996 0.996 0.996 0.997 0.997 0.997
Hidden neurons 1st layer
2nd layer
6 6 6 6 12 12 12 12 24 24 24
0 6 12 24 0 6 12 24 0 6 12
MAE Wh/m2
MBE Wh/m2
RMSE Wh/m2
RRMSE %
RMAE %
R2
2.621 4.231 2.523 2.679 2.129 2.456 2.623 2.665 2.232 2.323 2.456
0.031 1.001 0.046 0.010 0.002 0.004 0.012 0.054 0.012 0.102 0.105
5.449 5.477 5.246 5.326 4.270 4.652 4.723 4.956 4.508 4.612 4.942
12.30 12.35 11.85 12.03 9.63 10.49 10.65 11.19 10.18 10.40 11.15
4.49 7.25 4.32 4.59 3.65 4.22 4.49 4.56 3.82 3.98 4.22
0.996 0.994 0.994 0.994 0.998 0.998 0.997 0.997 0.997 0.997 0.996
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Table 4 Results for optimized multilayer perceptron ANNs.
Table 5 Comparison of various model performances for hourly data.
MBE RMSE RRMSE % RMAE % R2 First Second MAE layer layer Wh/m2 Wh/m2 Wh/m2 Model 1 qz and h Only h Only qz Model 2 qz and h Only h Only qz Model 3 qz and h Only h Only qz
12 12 12
0 0 0
3.113 3.052 3.162
0.156 5.152 0.142 5.139 0.174 5.095
8.18 8.16 8.10
4.94 4.84 5.02
0.996 0.996 0.996
24 24 24
0 0 0
3.279 3.822 3.205
0.253 5.690 0.233 6.224 0.201 5.629
10.22 11.17 10.11
5.89 6.86 5.75
0.994 0.993 0.994
12 12 12
0 0 0
2.129 3.381 3.565
0.002 4.270 0.066 5.686 0.099 5.613
9.63 9.58 9.46
3.65 5.70 6.01
0.998 0.994 0.995
References
Inclination
Site
RMAE
RRMSE
Notton et al. [30] Notton et al. [30] Notton et al. [30] Olmo et al. [12] Notton et al. [13] Notton et al. [13] Padovan & Delcol [11]
45 60 45 e60
Ajaccio Ajaccio Ajaccio Granada Ajaccio Ajaccio Padova
2.79% 3.42% 3.93% 10% [12]
5.28% 6.24% 6.62% 27% [44] 8.11% 10.71% 6.4%e8.7%
45 60 20 e30
6%e7%
- the errors are higher for an inclination of 60 than 45 and generally in the literature, this observation is also realized for empirical model and is probably due to the greater influence of the anisotropy of the sky diffuse radiation when the inclination angle is high; - model 3, which proposes to generalize the first two models, gives the best results. 5.2. Optimization of input data number
Fig. 3. Modelled 10-min values versus experimental data for Model 1 (45 ).
We wondered if there was no redundancy between some input data, particularly between temporal and geometrical data. The extraterrestrial irradiation cannot be removed because it is used as a reference of maximum available irradiation at a given time and it is used in all empirical models for the solar estimation. The declination is a characteristic of the day number. However, the zenith angle depends on the declination and on the hour angle and it is likely the same information that is contained in the hour and the zenith angle. To verify this hypothesis, we decided to repeat the previous study, by removing respectively the hour and zenith angle and in comparing the results with those of the previous study. The results of the sensitivity analysis are presented in Table 4. We found that reducing the number of input data does not change
Fig. 4. Validation of the Model 1 on a 5 days period.
G. Notton et al. / Renewable Energy 50 (2013) 576e584
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Fig. 5. RRMSE and RMAE of various models at hourly scale and comparison with our models at 10-min scale.
the optimal structure of the network i.e. the number of neurons in hidden layers. Whatever the model is, we note that removing the hour as input data improves the network performances. This confirms that there was a redundancy between the input data. The sensitivity study is simple in its application, and not timeconsuming regarding the benefits, it shows that all entries have not the same influence on the outputs of the neural network. Some entries have little or no influence and reduce the performance of the neural network. To illustrate the good reliability of the developed models, we show, as an example, in Fig. 3 the estimated data versus the experimental one for the first model without the hour h as an input data. We randomly chose a periods of 5 days for which we plotted the experimental and calculated data by Model 1(Fig. 4). 5.3. Comparison with other models Some studies have been performed to determine the global solar irradiation on tilted plane from horizontal ones, but all these methods have been applied to hourly data. There are two possibilities to achieve this goal: directly or by combination of two successive models. The Olmo model estimates directly the global irradiation on inclined surfaces [12]. Another method to calculate tilted global irradiation consists in: - first, calculating the horizontal diffuse solar radiation from the measured horizontal global solar radiation; - then, determining the global solar radiation on tilted planes from the measured global and calculated diffuse solar radiations on a horizontal plane. Following this method, 94 combinations have been tested (for inclinations equal to 45 and 60 ) in Ajaccio by Notton et al. [13]. The RRMSE obtained were around 10% and the best coupling allows to obtain the hourly inclined data with an RRMSE equal to 8.11% for an inclination of 45 and 10.71% for an inclination of 60 .
Padovan and Delcol [11] tested 12 combinations from meteorological data of Padova (Italy) for 20 and 30 tilted surfaces. The RMAE varied from 6% to 7.2% and the RRMSE between 6.4% and 8.7%. At last, Notton et al. [30] applied ANN methodologies for estimating hourly 45 and 60 tilted global solar irradiations from horizontal one and obtained an RRMSE between 5.28% and 6.62% for an RMAE between 2.79% and 3.93%. A synthesis of the various studies is presented in Table 5 for hourly data and a comparison of the accuracy of the various models is illustrated in Fig. 5. We note that the accuracy of the developed models is in the same order of magnitude or just a few percents higher than empirical models for hourly data available in the literature. Smaller is the time step, more it is difficult to estimate the data with a good accuracy. Compared with same models applied for hourly data collected in the same meteorological site, the two errors increase from 2 to 3% for a 6 times lower time step. We must keep in mind that these optimized developed models must be tested for other meteorological sites in view to confirm or not their applicability to other sites all around the World and particularly in the Mediterranean area. 6. Conclusions We developed three models of solar global irradiation on tilted plane from horizontal ones using Artificial Neural Networks. This study was performed by using five years of 10-min solar radiation data (101,140 validated data) collected in the Mediterranean site of Ajaccio, France. The choice of the ANN type and transfer functions were realized from a bibliographical study. Successively, using a sensitivity analysis, we optimized the ANN architecture i.e. the number of layers and the number of neurons per layer. Five input parameters were used in these models (six for the third model) and we showed that the deletion of one of them improved the accuracy of the model.
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The adequacy of the models developed in this work with experimental data is very satisfying and in the same order than that obtained with empirical models of the literature for a higher time step (hourly time step). This ANN approach out-performs the traditional methods for this type of estimation and allows to obtain more reliable input data for modelling and optimizing solar energy systems for both thermal or photovoltaic applications. In a future work, we will develop such ANN models for estimating hourly or 10-min normal beam irradiations values from global horizontal ones for solar applications using concentrating energy system. References [1] Vijayakumar G, Klein SA, Beckman WA. Analysis of short-term solar radiation data. Solar Energy 2005;79(5):495e504. [2] Notton G, Muselli M, Poggi P, Louche A. Autonomous photovoltaic systems: influence of some parameters on the sizing: simulation timestep, input and output power profile. Renewable Energy 1996;7(4):353e69. [3] Hawles A, Leach M. Impacts of temporal precision in optimisation modelling of micro-combined heat and power. Energy 2005;30:1759e79. [4] Wilcox S, Gueymard CA. Spatial and temporal variability of the solar resource in the United State. Phoenix, AZ: ASES e Proc. Solar; 2010. [5] Halamay DA, Brekken TKA, Simmons A, McArthur S. Reserve requirement impacts of large-scale integration of wind, solar and ocean wave power generation. IEEE Transactions on Sustainable Energy 2011;2(1):321e8. [6] Manz D, Miller N. Data requirements for wind & solar integration studies. Portland, Oregon: UWIG Solar Integration User Group Meeting; April 14, 2010. [7] Corbus D. Hawaii Solar Integration Study Technical Review Committee. RSWG, October 26, 2011, Available at: http://www1.eere.energy.gov. [8] Frank JD, Brower M, Barton M, Beaucage P. Solar Photovoltaic power output data sets for grid-integration studies in Hawaii. Denver, CO: ASES; 13e17 May 2012. [9] Behr HD. Solar radiation on tilted south-oriented surfaces: validation of transfer-models. Solar Energy 1997;61(6):399e413. [10] Robledo L, Soler A. Modelling irradiance on inclined planes with an anisotropic model. Energy 1998;23(3):193e201. [11] Padovan A, Del Col D. Measurement and modeling of solar irradiance components on horizontal and tilted planes. Solar Energy 2010;84:2068e84. [12] Olmo FJ, Vida J, Foyo I, Castro-Diez Y, Alabos-Arboledas L. Prediction of global Irradiance on inclined surfaces from horizontal global irradiance. Energy 1999;24:689e704. [13] Notton G, Poggi P, Cristofari C. Predicting hourly solar irradiations on inclined surfaces based on the horizontal measurements: performances of the association of well-known mathematical models. Energy Conversion and Management 2006;47:1816e29. [14] Orgill JE, Hollands KGT. Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy 1977;19(4):357e9. [15] Erbs DG, Klein SA, Duffie JA. Estimation of the diffuse radiation fraction for hourly daily and monthly-average global radiation. Solar Energy 1982;28(4): 293e304. [16] Hollands KGT. A derivation of the diffuse fraction_s dependence on the clearness index. Solar Energy 1985;35(2):131e6. [17] Hollands KGT, Crha SJ. An improved model for diffuse radiation: correction for atmospheric back-scattering. Solar Energy 1987;38(4):233e6. [18] De Miguel A, Bilbao J, Aguiar R, Kambezidis H, Negro E. Diffuse solar irradiation model evaluation in the North Mediterranean belt area. Solar Energy 2001;70(2):143e53. [19] Skartveit A, Olseth JA. A model for the diffuse fraction of hourly global radiation. Solar Energy 1987;38(4):271e4. [20] Maxwell EL. A quasi-physical model for converting hourly global horizontal to direct normal insolation. Report SERI/TR-215e3087. Golden. CO: Solar Energy Research Institute; 1987.
[21] Eliminir HK. Experimental and theoretical investigation of diffuse solar radiation: data and models quality tested for Egyptian sites. Energy 2007;32(1): 73e82. [22] Jiang Y. Prediction of monthly mean daily diffuse solar radiation using artificial neural networks and comparison with other empirical models. Energy Policy 2008;36:3833e7. [23] Iqbal M. An introduction to solar radiation. Canada: Academic Press, ISBN 012-373752-4; 1983. [24] Li DHW, Lam JC. Evaluation of slope irradiance and illuminance models against measured Hong Kong data. Building Environment 2000;35:501e9. [25] Lam JC, Wan KKW, Yang L. Solar radiation modelling using ANNs for different climates in China. Energy Conversion and Management 2008;49:1080e90. [26] Krishnaiah T, Srinivasa Rao S, Madhumurthy K, Reddy RS. Neural network approach for modelling global solar radiation. Journal of Applied Sciences Research 2007;3(10):1105e11. [27] Abraham A. Artificial neural networks. In: Sydenham Peter, Thorn Richard, editors. Handbook for measurement systems design. London: John Wiley and Sons Ltd., ISBN 0-470-02143-8; 2005. p. 901e8. [28] Alam S, Kaushik SC, Garg SN. Assessment of diffuse solar energy under general sky condition using artificial neural network. Applied Energy 2009; 86:554e64. [29] Mellit A. Artificial Intelligence technique for modelling and forecasting of solar radiation data: a review. International Journal of Artificial Intelligence and Soft Computing 2008;1-1:52e76. [30] Notton G, Paoli C, Vasileva S, Nivet ML, Canaletti JL, Cristofari C. Estimation of hourly global solar irradiation on tilted planes from horizontal one using artificial neural network. Energy 2012;39:166e79. [31] Mellit A, Benghanem M, Hadj Arab A, Guessoum A. A simplified model for generating sequences of global solar radiation data for isolated sites: using artificial neural network and a library of Markov transition matrices approach. Solar Energy 2005;79:469e82. [32] Mellit A, Pavan AM. A 24-h forecast of solar irradiance using artificial neural network: application for performance prediction of a grid-connected PV plant at Trieste, Italy. Solar Energy 2010;84:807e21. [33] Voyant C, Muselli M, Paoli C, Nivet ML. Optimization of an artificial neural network dedicated to the multivariate forecasting of daily global radiation. Energy 2011;36(1):348e59. [34] Mohandes M, Rehman S, Halawani TO. Estimation of global solar radiation using artificial neural networks. Renewable Energy 1998;14:179e84. [35] Dorvlo ASS, Jervase JA, Al-Lawati A. Solar radiation estimation using artificial neural networks. Applied Energy 2002;74:307e19. [36] Sozen. A, Arcakly E, Ozalp M. Estimation of solar potential in Turkey by artificial neural networks using meteorological and geographical data. Energy Conversion and Management 2004;45:3033e52. [37] Tymvios FS, Jacovides CP, Michaelides SC, Scouteli C. Comparative study of Angstroms and artificial neural networks methodologies in estimating global solar radiation. Solar Energy 2005;78(6):752e62. [38] Behrang MA, Assareh E, Ghanbarzadeh A, Noghrehabadi AR. The potential of different artificial neural network (ANN) techniques in daily global solar radiation modeling based on meteorological data. Solar Energy 2010;84: 1468e80. [39] Elminir HK, Azzam YA, Younes FI. Prediction of hourly and daily diffuse fraction using neural network, as compared to linear regression models. Energy 2007;32:1513e23. [40] Parizeau M. Réseaux de neurones. Educational report, University of Laval GIF21140-GIF-64326, 2004. [41] Dreyfus. G, Martinez J, Samuelides M, Gordon M, Badran F, Thiria S. Apprentissage statistique: Réseaux de neurones e Cartes topologiques e Machines à vecteurs supports. 3rd ed. Eyrolles; 2008. [42] Reddy KS, Manish R. Solar resource estimation using artificial neural networks and comparison with other correlation models. Energy Conversion and Management 2003;44:2519e30. [43] Al-Alawi SM, Al-Hinai HA. An ANN-based approach for predicting global radiation in locations with no direct measurement instrumentation. Renewable Energy 1998;14(1e4):199e204. [44] Ruiz E, Soler A, Robledo L. Comparison of the Olmo model with global irradiance measurements on vertical surfaces at Madrid. Energy 2002;27: 975e86.
Contents lists available at SciVerse ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Neural network approach to estimate 10-min solar global irradiation values on tilted planes Gilles Notton a, *, Christophe Paoli b, Liliana Ivanova c, Siyana Vasileva c, Marie Laure Nivet b a
University of Corsica, UMR CNRS 6134, Research Centre of Vignola, Route des Sanguinaires, F-20000 Ajaccio, France University of Corsica, UMR CNRS 6134, Sciences Faculty, BP 52, F-20250 Corte, France c Faculty of Electrical Engineering, Technical University of Sofia, 8, Kl. Ohridski Blvd., 1000 Sofia, Bulgaria b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 20 February 2012 Accepted 26 July 2012 Available online 30 August 2012
Calculation of solar global irradiation on tilted planes from only horizontal global one is particularly difficult when the time step is small. We used an Artificial Neural Network (ANN) to realize this conversion at a 10-min time step. The ANN is developed and optimized using five years of solar data and the accuracy of the optimal configuration is around 9% for the RMSE and around 5.5% for the RMAE i.e. similar or slightly lower than the errors obtained with empirical correlations available in the literature and used for the estimation of hourly data. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Solar irradiation Artificial Neural Network Estimation
1. Introduction Vijayakumar et al. [1] showed that simulation studies of solar energy systems generally use hourly values to estimate long-term performance; considering that solar radiation fluctuates during an hour, using only hourly data results in inaccurate performance estimation. Indeed, some types of solar systems, such as photovoltaic systems, respond quickly and non-linearly to solar radiation. Thus, the distribution of short-term radiation within an hour results in greater utilizability, if the short-term data varies significantly. Generally, hourly solar data are used in quasi-state models for simulating the behaviour of renewable power systems. Nevertheless, the high variability of the load, of the renewable sources and of the fluctuating storage level (when it exists) requires that these data are known with a fine time step to take account more accurately the lag between the various renewable productions and the load. Thus, the sizing of a renewable system will be more correct and its reliability and cost-effectiveness will be higher. Notton et al. [2] studied the influence of the simulation time step in view to size a stand-alone photovoltaic system. They showed that daily data may lead to a significant under-sizing but there is a good agreement between the results obtained from minute and hourly data; the load profile used in this study varied throughout the day but not during the hour and we can reasonably * Corresponding author. Tel.: þ33 495524152; fax: þ33 495524142. E-mail address: [email protected] (G. Notton). 0960-1481/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2012.07.035
expect that the results are different with a more variable load. Hawkes and Leach [3] showed that it is important to use a fine temporal precision in energy system simulation when the effect of averaging will significantly impact how the model behaves. The knowledge of the variability of the solar source is important for improving the design of a system (by adding properly sized storage capabilities, for instance) and understanding the performance of a solar conversion system (e.g., understanding how the extremes can enhance or degrade system performance or during which season they occur most frequently) [4]. Halamay et al. [5] wrote that, as the penetration level of renewable systems increases, the added variability of the energy resource can cause greater ramp rates, greater inter-hour variability, and greater scheduling error. It is consequently necessary to take into account these short-term variations and consequently 10min data are used in their study for load, wind, solar, and wave power. Manz et al. [6] showed the necessity to develop a sub-hourly tool using 10-min time step for long-term dynamic simulation in view to quantify reserve violations, fast starts and load shed events caused by sub-hourly wind/solar/load changes. Corbus [7] applied this sub-hourly tool for Hawaii solar integration studies and future grid studies. Frank et al. [8] used the same time step of 10-min for grid integration study in Hawaii. Thus, as seen in the literature, 10 min solar data are particularly useful for studies of solar system integration. Solar radiation on tilted planes is very important for engineers and scientists in view to design flat plate collectors, photovoltaic
G. Notton et al. / Renewable Energy 50 (2013) 576e584
systems and other solar energy collecting devices. Although meteorological stations provide sometimes reliable solar radiation records, they usually measure radiation only for horizontal surfaces and generally the data on tilted surfaces are not available. Thus, they should be estimated with different models from those corresponding to a horizontal surface. In commercial software intended for the sizing of solar systems, photovoltaic or thermal systems, the inclination of solar collectors (and the optimization of this inclination) is an input parameter. From a meteorological data basis containing global horizontal solar radiations, the software “inclines” these radiations; but this conversion is realized with an average and often low accuracy. It is consequently useless to develop fine and precise models to describe the solar system behaviour if the input data are approximate. The relationships between the horizontal solar radiation and various other components of solar radiation are not linear and depend on many factors, both related to the geometric position of the sun and meteorological characteristics of the sky state (cloud density and position, water vapour concentration, etc). As underlined by Behr [9], three main reasons make impossible to develop a simple model with a view to converting horizontal global solar irradiation into inclined global solar irradiation: - the radiation incident on a tilted plane includes the radiation reflected by the environment; - when the plane is inclined, only a portion of the sky is “seen” and then the diffuse component of the sky depends not only on the inclination angle or the orientation of the collector, on the elevation of the sun as well as its azimuth but also on the sky conditions which are rarely uniform. This therefore induces some anisotropic effects complex and difficult to quantify [10]. There are two main problems relating to this sky anisotropy: firstly, the circum-solar brightness due to the solar radiation diffusion by some aerosols and concentrated in the sky area around the sun and secondly the brightness of the horizon concentrated near the horizon which is even more important in clear skies [11]; - at last, these inclined solar data are relatively rare. The anisotropy phenomenon especially affects the calculation of the tilted irradiation when the measure time step chosen is small, in fact, larger the time step is, more this anisotropy decreases (timeaveraging and compensating effect) and tends to move towards an isotropic distribution. Consequently, shorter will be the time step, more difficult will be to realize this conversion with a good accuracy. The conversion of horizontal solar irradiation in tilted solar irradiation is a complex issue often dealt with in the literature. Mostly, only monthly mean values of solar irradiation (daily or hourly) are converted using linear or polynomial interpolations; unfortunately, if these data allow to have a general vision of the solar potential, they are insufficient to model or to size a solar system. Numerous models have been developed at daily or hourly scales; though some empirical models allowing to determine directly the inclined solar irradiation from the horizontal one do exist [12], generally we must combine two types of models to reach our goal. The “conventional” method consists in considering separately the two components: the beam and the diffuse radiations: we combine a first model estimating the horizontal diffuse radiation from the horizontal global radiation and a second model calculating the inclined global solar radiation from the two previously mentioned ones i.e. the measured horizontal global radiation and the estimated horizontal diffuse radiation. The precision of such a method is relatively low at a hourly scale and depends on the meteorological conditions on site [11,13].
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The diffuse radiation is the solar component the more complicated to calculate. The state of the sky (clear, partially cloudy or cloudy) is characterized by the clearness index (ratio of the global radiation on the ground and the extraterrestrial one) and the anisotropy of the sky diffuse radiation is quantified by different complex index [14e20]. The diffuse fraction or the diffuse clearness index is calculated from linear, polynomial or more complicated relations. Behr [9] and Elminir [21] show that no simple and reliable relation exists between diffuse and global irradiation. The estimation of each solar component depends on the day number (or the declination), the solar hour, the solar elevation, the sky clearness, the clouds positions in the sky dome and their size, etc. For a same global irradiation, the distribution in the three components (beam, sky diffuse and ground diffuse) varies considerably depending on the parameters mentioned above. The reliability of empirical correlations is not very high [14,16] and Elminir et al. [21] suggested and proved that ANN models are particularly suitable for such applications. As said by Jiang [22], one of the limitations imposed by linear regression models is that they will underperform when used to model nonlinear systems which are the case for the problem we are interested in. All researches reported that the neural approaches out-performed the traditional linear methods. Moreover, these conclusions have been given for an estimation at a hourly scale and this fact is particularly true that the time step used in this study is smaller. The development of artificial intelligence processes, for several years, and more precisely of Artificial Neural Network (ANN) allows to estimate researched data from available ones when the relation between output and input data are complex and not linear. In this work, we present rapidly the ANN concepts and explain the choice of the various parameters influencing the ANN performances. We will realize a sensitivity analysis in view to optimize the ANN structure and the number of input parameters. 2. Data collection and processing The laboratory “Sciences for Environment” of the University of Corsica Pascal Paoli has a complete meteorological station in the gulf of Ajaccio (latitude: 41550 N; longitude: 8 550 E) situated at about 200 m from the sea and at an altitude of 70 m. The site is characterized by an insular Mediterranean climate. The surrounding topography is not very high but a small mask may appear in the winter when the sun rises. The data used in this work were measured and stored with a minute time step with a pyrheliometer Kipp & Zonen at normal incidence. This device is mounted on an automatic Solar Tracker Model Solys 2 (2 axis, azimuth/elevation device programmed to align direct beam instruments with the normal incidence of the sun) and measures the normal beam irradiance. Every two days, the pyrheliometer is cleaned and its alignment is verified. The global irradiance on horizontal, 45 and 60 tilted surfaces is measured by three Kipp & Zonen (CM11) pyranometers. The experimental data have been collected since January 2006 in a quasi-continuous manner. Thus, we have 5 years of minute data _ normal beam irradiance I_ , and of horizontal global irradiance I, b;n inclined global irradiance at 45 and 60 I_45 and I_60 . 45 and 60 are respectively the optimal inclinations for a maximum global irradiation on an annual period (inclination equal to latitude, here 41550 ) and on a winter period (latitude þ15 ). For each minute, we add three calculated parameters: - the horizontal extraterrestrial irradiance I_0 ; - the solar declination d; - the zenith angle qz .
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The declination angle d is the angle between the line joining the centres of the sun and the earth to the equatorial plane and it changes at every moment, but we can consider that it depends on the day number [23].
dðradÞ ¼ 0:006918 0:399912cos G þ 0:070257sin G 0:006758cos 2G þ 0:000907sin 2G 0:002697cos 3G þ 0:00148sin 3G
(1)
(2)
and dn is the day number. The zenith angle qz is the angle between the sun and a horizontal surface. It is given by [23]:
cos qz ¼ sin dsin 4 þ cos dcos 4cos u
(3)
with 4 the latitude and u the hour angle given by:
u ¼ 15ð12 tÞ
(4)
where t is the time expressed in true solar time. The extraterrestrial irradiance on a horizontal plane is calculated by [23]:
I_0 ¼ I_sc E0 cos qz
(5)
where Isc is the solar constant taken equal at 1367 W m2 according to the World Meteorological Organization. E0 is the eccentricity factor taking into account the variation of the earth-sun distance and calculated by [23]:
E0 ¼ ðr0 =rÞ2 ¼ 1:000110 þ 0:034221cos G þ 0:001280sin G þ 0:000719cos 2G þ 0:000077sin 2G
(6)
Some quality control tests are imposed on these minute data in order to extract outlier or missing data. The rejected data are those for which [24,25]: - the horizontal global irradiance is superior to the horizontal extraterrestrial irradiance; - the normal beam irradiance is higher to the normal extraterrestrial irradiance. Then the 10 min values are calculated: they are computed by integration on the 10 min period for energy data and for the zenith angle, the value is calculated at mid-period. We allowed 10% of missing data over the 10 min period i.e. one missing data for 10 min. Very often, the sunset and the sunrise induce some problems in the correlations, firstly because of the mask effect of the environment and secondly due to the bad response of pyranometers when the zenith angle is high (cosine effect) [18]. Thus, we extracted the 10 min period during which the sun rises or sets. Over the 5 years, we have 101,140 validated 10 min data i.e. about 6.46% of missing data. Consequently, we have several monthly files containing: -
the the the the
day; hour; 10 min period (from 1 to 6); declination d;
the the the the the the
zenith angle qz ; hourly horizontal global irradiation I; hourly normal beam irradiation Ib;n ; hourly horizontal extraterrestrial irradiation I0 ; hourly global irradiation on a 45 tilted plane I45 ; hourly global irradiation on a 60 tilted plane I60 .
3. Artificial Neural Network (ANN) 3.1. General presentation
where G is the day angle calculated by:
G ¼ 2pðdn 1Þ=365
-
The objective of this paragraph is to give some general information on the Artificial Neural Network (ANN). An ANN is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain process information. It is composed of a large number of highly interconnected processing elements (neurons) working in unison to solve specific problems. The goal of this network is to create a model that correctly maps the input to the output using historical data so that the model can then be used to produce the output when the desired output is unknown. An ANN has a parallel-distributed structure and consists in a set of processing elements called neurons. The ANN structure consists in: - an input layer which receives data; - an output layer to send computed information; - one or several hidden layers lying the input and output layers. According to the chosen architecture, all or a part of the neurons in a layer are connected with all or a part of the neurons of the previous and next layer. The number of hidden layers and the total number of neurons of each layer depends on the specific model, convergence speed, generalization capability, the physical process and the training data that the network will simulate [26]. A typical artificial neuron and the modelling of a multilayered neural network are illustrated in Fig. 1. The signal flow from inputs xk,1, xk,2,., xk,p of a layer k is considered here as unidirectional (feed forward configuration). The output signal O to the jth neuron of the following layer (kþ1) is given by [27]:
1 0 p X T O ¼ f ðnetÞ ¼ f w xk ¼ f @ wk;j xk;j A
(7)
j¼1
where wk;j is the synaptic weights (k is the layer index, j is the neuron index) and f ðnetÞ is referred to as an activation or transfer function. The variable net is the scalar product of the weight and input vectors. T is the transpose of a matrix. The result of this sum, net, is then transformed by a transfer function f which produces the output O of a neuron if this sum exceeds a certain threshold qk . The output is then distributed to other neurons as inputs. There are two main problems concerning the ANN implementation: specifying the network size (number of layers in the network and number of nodes in each layer) and finding the optimal values for the connection weights. An insufficient number of hidden nodes may be the cause of difficulties in learning data whereas an excessive number of hidden nodes might lead to unnecessary training time with marginal improvement as well as make the estimation for a suitable set of interconnection weights more difficult [28]. To determine the optimal number of hidden nodes; the common method used is trial and error based on a total error criterion. This method starts with a small number of nodes, gradually increasing the network size until the desired accuracy is achieved.
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Fig. 1. Architecture of an artificial neuron and a multilayered neural network.
One of the properties of ANNs is their ability to learn from their environment and to improve their performance through a learning process called also training process. Learning results in a change in the weights value wk;j connecting the neurons from one layer to another. The goal is to achieve equality between the actual output and simulated output. It is therefore necessary first to choose the learning algorithm and to define the part of the data used for learning in relation to the total amount of data available. The various steps in the implementation of an optimized ANN consist in selecting: -
ANN structure; transfer function type; ANN size (number of layers and number of neurons per layer); learning algorithm; training/test set; input data
3.2. Application of ANN to solar radiation ANN is a method already used to estimate or to predict the solar resource. Two types of works or studies have been realized: - forecast consisting in predicting what will happen at time t þ dt i.e. estimating Xðt þ dtÞ from the data at time t, XðtÞ and sometimes other information; - estimation of the value of a data YðtÞ from the knowledge of i others data Xi ðtÞ. This type of model allows for example to determine YðtÞ in locations where there is no way of measuring YðtÞ but where all the Xi ðtÞ are available. The approaches and methodologies developed concerning the structure for forecast or estimation are almost identical. The objective of this paper concerns the second type of study i.e. “estimation”. As we said in the introduction, ANN models have better performances than “conventional” empirical models for the solar irradiation estimation. The difficulty in the estimation of solar radiation comes mainly of the diffuse part reaching the tilted surface due to the non-uniformity of the sky state (anisotropy) and consequently there is not a one-to-one correspondence between the diffuse component and the global radiation. Jiang [22] stated that regression models underperform when used to model nonlinear systems, which is the case in this study. Mellit [29] presented a very complete state of art on the solar data forecast by utilization of artificial intelligence methods: ANN, fuzzy logic, genetic algorithm, expert system, hybrid method. More
than 80 papers have been presented. Notton et al. [30], after a bibliographical study on the utilisation of ANN to solar prediction and estimation, came to the same following conclusions: - more often only monthly average values are estimated from geographical and meteorological data such as temperature, humidity, solar fraction, wind speed, etc; - the estimated solar radiation component is often the horizontal global one, rarely the diffuse one and never the inclined global irradiation; - the time step is often daily (and often averaged on the month). Consequently, the estimation by ANN of tilted global solar irradiation is rarely studied and the time step of the estimated data is often average values and sometimes hourly data. 4. ANN implementation As previously specified in chapter 3.1, the ANN implementation consists in several stages: 1) structure: the chosen structure is the MultiLayer Perceptron (MLP) using feed-forward back-propagation; it is the most used in the literature for the solar radiation estimation [22,28,31e 40]; 2) transfer function: the combination of a sigmoid and a linear transfer functions is able to give a good approximation of several types of functions [37,40,41]. Several authors used this combination for the estimation of solar data [26,36,42]; 3) learning algorithm: the LevenbergeMarquardt algorithm (LM) is an approximation to the Newton method used during the training as suggested by literature [31,32,36,41,43]; 4) training/test set: among all the available data, we must use a percentage for the training and the rest will serve for the testing. As written by Dreyfus [41], we must use more than 50% of data for the training phase. Thus, we decide to take 3 whole years of data for training (60%) (2006e2008) and 2 whole years (40%) for testing (2009e2010); 5) ANN size: whatever the number of inputs, the ANN have to be as simple as possible (parsimony principle). Those configurations give the best results with a small error. The compromise between the number of hidden layers and number of neurons in each layer allows us to obtain a fast network, robust and that will give us the best results. Moreover, a large size architecture needs more data to be trained and it easily becomes overtrained, i.e. unable to generalize efficiently. Unfortunately, it does not exist any mathematical method for sizing optimally
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the ANN, the choice of the optimal structure can only be done only after testing different configurations and estimating their performance; 6) input data choice: the declination represents the Earth’s position with respect to the Sun and depends on the day of the year. The sun position influences on the quantity and the quality of the sun radiation. This position is characterized by the zenith angle (Eq. (3)). When the sun is high in the sky (low zenith angle), the solar radiation is maximal (in clear sky condition) and as the optical path is minimal therefore the incident radiation is less absorbed. For a same horizontal solar irradiation, correspond several values
of inclined solar irradiation according to the season, to the sky conditions, etc. In the diffuse radiation model, the clearness index or the diffuse index is used to take into account the sky condition; this sky conditions is an important parameter in the repartition of diffuse and beam radiations. Thus, more the clearness index is high, more the sky is clear and more the global radiation is constituted by beam radiation. The extraterrestrial irradiation is used as reference and should be useful as an input parameter (Eq. (5)). We chose also to introduce as input data the hour. However, it is possible that it is not useful to consider all the three parameters: time, declination and zenith angle. Indeed
Fig. 2. The three ANN models.
G. Notton et al. / Renewable Energy 50 (2013) 576e584
they are linked together according to Eq. (3). To remove this ambiguity, we decided to take a sensitivity analysis in order to find which of the potential inputs could have significant influence on the results. Then, before the sensitivity analysis, the input parameters are i) the time, ii) the declination, iii) the zenith angle, iv) the horizontal global solar irradiation, v) the extraterrestrial horizontal solar irradiation. To estimate the accuracy of the models developed, we use different indicators as suggested by Iqbal [23]. yi and xi are the estimated and measured values and the respective average values PN PN are given by x ¼ i ¼ 1 xi and y ¼ i ¼ 1 yi where N is the data number. The Mean Absolute Value (MAE) and its Relative value RMAE are calculated by:
MAE ¼
N X
jyi xi j=N
RMAE ¼
i¼1
N X
jyi xi j=Nx
(8)
i¼1
The Mean Bias Error MBE is expressed by:
MBE ¼
N X
ðyi xi Þ=N
(9)
i¼1
The Root Means Square Error (RMSE) and its Relative value RRMSE are given by:
RMSE ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðyi xi Þ2 =N
RRMSE ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ðyi xi Þ2 =N
(10)
The correlation coefficient R2 is:
"
N X R ¼ ðyi yÞðxi xÞ
#,("
2
i¼1
N X
#" ðyi yÞ
i¼1
2
N X ðxi xÞ2
#)1=2
i¼1
(11)
5. Results 5.1. Choice of architecture We developed three ANN (Fig. 2) 1. the first one for the estimation of 10-min global solar irradiation on a 45 tilted plane; 2. the second one for the estimation of 10-min global solar irradiation on a 60 tilted plane; 3. the third one for the estimation of 10-min global solar irradiation on a 45 or 60 tilted planes. Table 1 Results for Model 1: 45 . Hidden neurons 1st layer
2nd layer
6 6 6 6 12 12 12 12 24 24 24
0 6 12 24 0 6 12 24 0 6 12
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Table 2 Results for Model 2: 60 . Hidden neurons 1st layer
2nd layer
6 6 6 6 12 12 12 12 24 24 24
0 6 12 24 0 6 12 24 0 6 12
MAE Wh/m2
MBE Wh/m2
RMSE Wh/m2
RRMSE %
RMAE %
R2
4.731 4.826 4.832 4.798 4.041 3.998 3.912 3.789 3.279 3.456 3.801
0.255 0.125 0.156 0.112 0.165 0.154 0.152 0.148 0.253 0.275 0.278
7.715 8.231 8.322 7.912 6.415 6.321 6.295 6.242 5.690 5.941 6.123
13.85 14.58 14.65 14.02 11.52 11.48 11.42 11.38 10.22 10.32 11.12
8.49 8.56 8.56 8.53 7.26 7.18 7.12 6.99 5.89 6.23 6.98
0.989 0.992 0.993 0.993 0.992 0.994 0.994 0.995 0.994 0.994 0.994
In the case 3, the number of data used to train and validate the model are doubled because each value of global irradiation is introduced two times, the first one with b ¼ 45 and the second one with b ¼ 60 . Moreover, the inclination b (equal to 45 or 60 ) is added as an input data. Our objective is to find the simplest and accurate MLP model and to estimate its accuracy. First we have to choose the number of hidden layers. In practice one hidden layer can be sufficient, but it depends on the application and sometime two hidden layers are necessary to improve the reliability. We limit the number of hidden layers to two because as said previously the ANN must be as simple as possible (parsimony principle). We tested various configurations of monolayer and double layer ANN for the three models presented above and some results are shown in Tables 1e3. The best ANN structure for each model is highlighted in grey. The decision criteria are applied to the whole data (training set and test set). The models being nonlinear, the training results depend on the initial values of the model inputs. Thus, each configuration has been tested 6 times and only the best results are given in Tables 1e3. We varied the number of neurons in the first hidden and second hidden layers from 6 to 24 and we noted that by increasing the number of neurons in each hidden layer, the error stabilizes more quickly during the ANN training and requires a smaller number of samples. The two most significant errors are RRMSE and RMAE and our comments will be made mainly on the basis of these two indicators. Several remarks can be made: - in the three models, the best results are obtained for a monolayer ANN composed of 12 neurons in its hidden layer for model 1 and 3 and with 24 neurons for model 2 with an RRMSE between 8.18% and 10.22% and an RMAE between 3.65% and 5.89%; Table 3 Results for Model 3: 45 þ60 .
MAE Wh/m2
MBE Wh/m2
RMSE Wh/m2
RRMSE %
RMAE %
R2
3.240 4.510 3.110 3.321 3.113 3.230 3.232 3.356 3.641 3.724 3.952
0.121 0.065 1.002 1.235 0.156 0.078 0.098 0.118 0.140 0.152 0.172
5.290 6.481 5.290 5.462 5.152 5.370 5.302 5.458 5.612 5.782 6.211
8.85 10.12 8.45 8.68 8.18 8.24 8.21 8.45 8.99 9.23 9.45
3.21 5.89 2.21 2.45 4.94 3.00 2.89 3.26 4.21 4.48 4.54
0.996 0.994 0.996 0.995 0.996 0.996 0.996 0.996 0.997 0.997 0.997
Hidden neurons 1st layer
2nd layer
6 6 6 6 12 12 12 12 24 24 24
0 6 12 24 0 6 12 24 0 6 12
MAE Wh/m2
MBE Wh/m2
RMSE Wh/m2
RRMSE %
RMAE %
R2
2.621 4.231 2.523 2.679 2.129 2.456 2.623 2.665 2.232 2.323 2.456
0.031 1.001 0.046 0.010 0.002 0.004 0.012 0.054 0.012 0.102 0.105
5.449 5.477 5.246 5.326 4.270 4.652 4.723 4.956 4.508 4.612 4.942
12.30 12.35 11.85 12.03 9.63 10.49 10.65 11.19 10.18 10.40 11.15
4.49 7.25 4.32 4.59 3.65 4.22 4.49 4.56 3.82 3.98 4.22
0.996 0.994 0.994 0.994 0.998 0.998 0.997 0.997 0.997 0.997 0.996
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Table 4 Results for optimized multilayer perceptron ANNs.
Table 5 Comparison of various model performances for hourly data.
MBE RMSE RRMSE % RMAE % R2 First Second MAE layer layer Wh/m2 Wh/m2 Wh/m2 Model 1 qz and h Only h Only qz Model 2 qz and h Only h Only qz Model 3 qz and h Only h Only qz
12 12 12
0 0 0
3.113 3.052 3.162
0.156 5.152 0.142 5.139 0.174 5.095
8.18 8.16 8.10
4.94 4.84 5.02
0.996 0.996 0.996
24 24 24
0 0 0
3.279 3.822 3.205
0.253 5.690 0.233 6.224 0.201 5.629
10.22 11.17 10.11
5.89 6.86 5.75
0.994 0.993 0.994
12 12 12
0 0 0
2.129 3.381 3.565
0.002 4.270 0.066 5.686 0.099 5.613
9.63 9.58 9.46
3.65 5.70 6.01
0.998 0.994 0.995
References
Inclination
Site
RMAE
RRMSE
Notton et al. [30] Notton et al. [30] Notton et al. [30] Olmo et al. [12] Notton et al. [13] Notton et al. [13] Padovan & Delcol [11]
45 60 45 e60
Ajaccio Ajaccio Ajaccio Granada Ajaccio Ajaccio Padova
2.79% 3.42% 3.93% 10% [12]
5.28% 6.24% 6.62% 27% [44] 8.11% 10.71% 6.4%e8.7%
45 60 20 e30
6%e7%
- the errors are higher for an inclination of 60 than 45 and generally in the literature, this observation is also realized for empirical model and is probably due to the greater influence of the anisotropy of the sky diffuse radiation when the inclination angle is high; - model 3, which proposes to generalize the first two models, gives the best results. 5.2. Optimization of input data number
Fig. 3. Modelled 10-min values versus experimental data for Model 1 (45 ).
We wondered if there was no redundancy between some input data, particularly between temporal and geometrical data. The extraterrestrial irradiation cannot be removed because it is used as a reference of maximum available irradiation at a given time and it is used in all empirical models for the solar estimation. The declination is a characteristic of the day number. However, the zenith angle depends on the declination and on the hour angle and it is likely the same information that is contained in the hour and the zenith angle. To verify this hypothesis, we decided to repeat the previous study, by removing respectively the hour and zenith angle and in comparing the results with those of the previous study. The results of the sensitivity analysis are presented in Table 4. We found that reducing the number of input data does not change
Fig. 4. Validation of the Model 1 on a 5 days period.
G. Notton et al. / Renewable Energy 50 (2013) 576e584
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Fig. 5. RRMSE and RMAE of various models at hourly scale and comparison with our models at 10-min scale.
the optimal structure of the network i.e. the number of neurons in hidden layers. Whatever the model is, we note that removing the hour as input data improves the network performances. This confirms that there was a redundancy between the input data. The sensitivity study is simple in its application, and not timeconsuming regarding the benefits, it shows that all entries have not the same influence on the outputs of the neural network. Some entries have little or no influence and reduce the performance of the neural network. To illustrate the good reliability of the developed models, we show, as an example, in Fig. 3 the estimated data versus the experimental one for the first model without the hour h as an input data. We randomly chose a periods of 5 days for which we plotted the experimental and calculated data by Model 1(Fig. 4). 5.3. Comparison with other models Some studies have been performed to determine the global solar irradiation on tilted plane from horizontal ones, but all these methods have been applied to hourly data. There are two possibilities to achieve this goal: directly or by combination of two successive models. The Olmo model estimates directly the global irradiation on inclined surfaces [12]. Another method to calculate tilted global irradiation consists in: - first, calculating the horizontal diffuse solar radiation from the measured horizontal global solar radiation; - then, determining the global solar radiation on tilted planes from the measured global and calculated diffuse solar radiations on a horizontal plane. Following this method, 94 combinations have been tested (for inclinations equal to 45 and 60 ) in Ajaccio by Notton et al. [13]. The RRMSE obtained were around 10% and the best coupling allows to obtain the hourly inclined data with an RRMSE equal to 8.11% for an inclination of 45 and 10.71% for an inclination of 60 .
Padovan and Delcol [11] tested 12 combinations from meteorological data of Padova (Italy) for 20 and 30 tilted surfaces. The RMAE varied from 6% to 7.2% and the RRMSE between 6.4% and 8.7%. At last, Notton et al. [30] applied ANN methodologies for estimating hourly 45 and 60 tilted global solar irradiations from horizontal one and obtained an RRMSE between 5.28% and 6.62% for an RMAE between 2.79% and 3.93%. A synthesis of the various studies is presented in Table 5 for hourly data and a comparison of the accuracy of the various models is illustrated in Fig. 5. We note that the accuracy of the developed models is in the same order of magnitude or just a few percents higher than empirical models for hourly data available in the literature. Smaller is the time step, more it is difficult to estimate the data with a good accuracy. Compared with same models applied for hourly data collected in the same meteorological site, the two errors increase from 2 to 3% for a 6 times lower time step. We must keep in mind that these optimized developed models must be tested for other meteorological sites in view to confirm or not their applicability to other sites all around the World and particularly in the Mediterranean area. 6. Conclusions We developed three models of solar global irradiation on tilted plane from horizontal ones using Artificial Neural Networks. This study was performed by using five years of 10-min solar radiation data (101,140 validated data) collected in the Mediterranean site of Ajaccio, France. The choice of the ANN type and transfer functions were realized from a bibliographical study. Successively, using a sensitivity analysis, we optimized the ANN architecture i.e. the number of layers and the number of neurons per layer. Five input parameters were used in these models (six for the third model) and we showed that the deletion of one of them improved the accuracy of the model.
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