Estimating Solar Irradiance on Tilted Surface with Arbitrary Orientations and Tilt Angles

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Article Estimating Solar Irradiance on Tilted Surface with Arbitrary Orientations and Tilt Angles

Hsu-Yung Cheng 1,*, Chih-Chang Yu 2, Kuo-Chang Hsu 3, Chi-Chang Chan 3, Mei-Hui Tseng 3 and Chih-Lung Lin 4 1 Department of Computer Science and Information Engineering, National Central University, 300 Zhongda Rd., Zhongli District, Taoyuan 32001, Taiwan 2 Department of Information and Computer Engineering, Chung Yuan Christian University, 200 Chung-Pei Rd., Zhongli District, Taoyuan 32023, Taiwan; [email protected] 3 Green Energy and Environment Research Laboratories, Industrial Technology Research Institute, 195 Sec.4, Chung-Hsing Rd., Chutung, Hsinchu 31057, Taiwan; [email protected] (K.-C.H.); [email protected] (C.-C.C.); [email protected] (M.-H.T.) 4 Department of Electronic Engineering, Hwa Hsia University of Technology, 111 Gon Jhuan Rd., Chung Ho dist., New Taipei City 23568, Taiwan; [email protected] * Correspondence: [email protected]

Received: 28 February 2019; Accepted: 10 April 2019; Published: 13 April 2019

Abstract: Photovoltaics modules are usually installed with a tilt angle to improve performance and to avoid water or dust accumulation. However, measured irradiance data on inclined surfaces are rarely available, since installing pyranometers with various tilt angles induces high costs. Estimating inclined irradiance of arbitrary orientations and tilt angles is important because the installation orientations and tilt angles might be different at different sites. The goal of this work is to propose a unified transfer model to obtain inclined solar irradiance of arbitrary tilt angles and orientations. Artificial neural networks (ANN) were utilized to construct the transfer model to estimate the differences between the horizontal irradiance and the inclined irradiance. Compared to ANNs that estimate the inclined irradiance directly, the experimental results have shown that the proposed ANNs with differential outputs can substantially improve the estimation accuracy. Moreover, the trained model can successfully estimate inclined irradiance with tilt angles and orientations not included in the training data.

Keywords: photovoltaics; inclined solar irradiance; artiﬁcial neural networks

1. Introduction To reduce the growing emission of carbon dioxide produced by fossil fuels, governments and international organizations have vigorously promoted sustainable energy [1]. The photovoltaic system (PV), which converts light to electrical energy, is one of the renewable energy technologies that have received growing attention in the last 32 years. PV installation has exhibited noticeable growth on a worldwide scale to provide clean and renewable electricity [2,3]. One of the most important factors that affect the amount of power generated by PV modules is solar irradiance [4]. Therefore, analyzing and predicting solar irradiance is an important issue for PV operators to perform effective energy management. In general, solar irradiance is measured on horizontal surfaces because installing pyranometers with various tilt angles results in high costs. However, PV modules are usually installed on inclined surfaces to improve performance and to avoid water or dust accumulation. Further, when PV modules are installed on top of buildings, they are commonly installed with an orientation angle and tilt angle so that they can fit the structure of the buildings or surrounding environments. Consequently, analyzing horizontal irradiance only is not enough. Since measured irradiance data

Energies 2019, 12, 1427; doi:10.3390/en12081427 www.mdpi.com/journal/energies Energies 2019, 12, 1427 2 of 14 on tilted surfaces are rarely available, obtaining the information of solar irradiance on tilted surfaces is necessary. With the importance of this research topic, the relationship between the inclined solar irradiance and horizontal solar irradiance has been investigated for many years in regions across the world, including Saudi Arabia [5], Spain [6,7], Singapore [8], Reunion Island [9], Pakistan [10], and Belgium [11]. Various empirical models have been developed to estimate inclined solar irradiance [12–15]. Among these methods, the correlation for estimating daily beam radiation on a tilted surface proposed by Liu and Jordan [12] has been used widely. However, Yang et al. [13] suggests that the Liu and Jordan method is only suitable to estimate solar irradiance on a tilted surface with an azimuth angle less than 15 degrees. The estimation error increases as the tilt angles and azimuth angles increase. Gueymard [14] explored the prediction process for the irradiance on the tilted surface in solar engineering applications. The relationship between the inclined solar irradiance and horizontal solar irradiance depends on many factors. For a tilted surface, the diffuse component of the sky depends on the tilt angle, the orientation angle, the solar azimuth angle, the solar zenith angle, and the sky conditions [16]. Since the sky conditions are rarely uniform, the resulting anisotropic effects would be complex and difficult to quantify [17,18]. Therefore, the transfer model from horizontal solar irradiance to inclined solar irradiance is not linear. The influential factors include the geometric position of the sun, meteorological characteristics of the current sky, and the reflectance of the surrounding surfaces. Therefore, these empirical models can only be considered as approximate methods. Since it is difficult to construct a simple and accurate model to calculate the inclined solar irradiance from horizontal irradiance using closed-form equations directly, researchers have worked on developing machine learning based methods to estimate solar irradiance on tilted surface from the measured horizontal irradiance [19–21]. Artificial neural networks (ANN) were utilized to construct the transfer model to estimate solar irradiance on tilted surfaces in References [19] and [20]. Support vector machines were also applied for the same purpose, and the results were compared with ANN in Reference [21]. Comparative analysis was performed to evaluate different transfer models on a benchmark dataset [22–24]. Moreover, some researchers have focused on how to set up optimum tilt angles for an inclined PV panel in a certain area [10,25,26]. However, these existing methods only train the transfer model with a particular tilt and orientation angle. They do not consider the problem of estimating inclined irradiance of arbitrary tilt and orientation angles. For example, if one trains a model to estimate inclined irradiance of a 30-degree tilt angle, they cannot obtain the inclined irradiance of a 15-degree tilt angle using the same model. They will need to train the model again using the data collected from a 15-degree tilted surface. In practice, it is not feasible to get training data of arbitrary angles and train numerous corresponding models. Nevertheless, estimating inclined irradiance of arbitrary orientations and tilt angles is also important because the desired installation orientations and tilt angles might be different at different sites. In this work, the goal of the proposed transfer model is to construct a unified transfer model that can obtain inclined solar irradiance of arbitrary orientations and tilt angles.

2. Proposed Method In this research, an artiﬁcial neural network (ANN) [4,27] was applied to construct the transfer model to estimate solar irradiance on tilted surfaces. A multilayer perceptron using feed-forward backpropagation was used in this work. The architecture of the ANN used in this work is illustrated in Figure1. There are M inputs x1, x2, ... xM. The neurons are fully connected. The output layer has only one neuron, which corresponds to the desired output value. Energies 2019, 12, 1427 3 of 14 Energies 2019, 12, x FOR PEER REVIEW 3 of 14

Figure 1. Architecture of the artiﬁcialartificial neuralneural networknetwork (ANN).(ANN).

ToTo applyapply thethe ANN,ANN, thethe ﬁrstfirst stepstep isis toto determine the network architecture, including the number ofof hiddenhidden layerslayers andand thethe numbernumber ofof neuronsneurons in in each each layer. layer. In In Figure Figure1, 1,x i푥denotes푖 denotes the theith ith component component of theof the input input feature feature vector vector and y andj(n )푦denotes푗(푛) denotes the output the output of the j ofth neuron the jth inneuron the n th in learning the nth iteration.learning Theiteration. second The step second of training step of ANNtraining is toANN initialize is to initialize the network, the network, including including setting setting the learning the learning rate η andrate the휂 and stop the criteria. stop criteria. The third The step third is step the feedforward is the feedforward stage. Instage. the nInth the learning nth learning iteration, iteration, the input the vectorinput vectorxi(n) is 푥 sent푖(푛) intois se thent into network. the network. The output The outputyj(n) of 푦 the푗(푛)j thof neuronthe jth inneuron the ﬁrst in the hidden first layerhidden is computedlayer is computed using Equation using Equation (1). In Equation (1). In Equation (1), the (1), term thevj (termn) denotes 푣푗(푛) denotes the net internal the net internal activity levelactivity of theleveljth of neuron the jth inneuron the nth in learningthe nth learning iteration. iteration. The function The functionϕ( ) denotes 휑(∙) thedenotes activation the activation function offunction the jth · neuron.of the jth The neuron. internal The activity internal level activityvj(n) levelis computed 푣푗(푛) is using computed Equation using (2). Equation In Equation (2). In (2), EquationP is the input(2), P dimensionis the input of dimension the jth neuron, of the whichjth neuron, does which not include does thenot thresholdinclude the term. threshold The term term.w jiThe(n) termdenotes 푤푗푖 the(푛) synapticdenotes th weightse synaptic from weights the ith neuronfrom the to i theth neuronjth neuron to the in j theth neuronnth learning in the iteration. nth learning Then, iteration. the output Then, of eachthe output layer isof calculated each layer through is calculated the hidden through layers the hidden with the layers feedforward with the network. feedforward network.

푦푗(푛) = 휑푗(푣푗(푛)) (1) yj(n) = ϕj(vj(n)) (1) 푃 P 푣푗(푛) =X∑ 푤푗푖(푛)푦푖(푛) (2) ( ) ( ) ( ) vj n = 푖=0wji n yi n (2) i=0 The forth step is the back-propagation stage. The local gradient function 훿푗(푛) in the output The forth step is the back-propagation stage. The local gradient function δ (n) in the output layer layer in the nth learning iteration is computed using Equation (3): j in the nth learning iteration is computed using Equation (3): 훿푗(푛) = 푒푗(푛)휑′(푣푗(푛)). (3) δj(n) = ej(n)ϕ0(vj(n)). (3) In Equation (3), the error term 푒푗(푛) is defined as 푑푗(푛) − 푂푗(푛), where 푑푗(푛) is the desired output of the jth neuron in the nth learning iteration. If the activation function is a sigmoid In Equation (3), the error term ej(n) is deﬁned as dj(n) Oj(n), where dj(n) is the desired output function, 푦푖(푛) can be expressed as Equation (4): − of the jth neuron in the nth learning iteration. If the activation function is a sigmoid function, yi(n) can be expressed as Equation (4): 1 푦푖(푛) = , −∞ < 푣푗(푛) < ∞. (4) 1 + exp (−푣푗(푛)) 1 yi(n) = , < vj(n) < . (4) The term 휑′(푣푗(푛)) can be expressed1 + exp asv 푦푗((n푛)) (1−∞− 푦푗(푛)). Then,∞ the local gradient function − j 훿 (푛) of each layer is calculated through back-propagating from the last hidden layer to the first 푗 hiddenThe layer term inϕ accordance(v (n)) can with be expressed Equation as (5)y: (n) 1 y (n) . Then, the local gradient function δ (n) 0 j j − j j of each layer is calculated through back-propagating from the last hidden layer to the ﬁrst hidden layer 훿 (푛) = 휑′(푣 (푛)) ∑ 훿 (푛) 푤 (푛) = 푦 (푛) (1 − 푦 (푛)) ∑ 훿 (푛) 푤 (푛). in accordance with푗 Equation푗 (5): 푘 푘푗 푗 푗 푘 푘푗 (5) 푘 푘 X X The fifth stepδ (n is) =to ϕadjust0(v (n the)) synapticδ (n) w weight(n) = ofy (eachn) 1 neurony (n) in accordanceδ (n) w ( nwith). Equation (6). (5) j j k kj j − j k kj The adjustment value to the synaptick weight 푤푗푖(푛) in the training phasek is denoted as ∆푤푗푖(푛).

푤푗푖(푛 + 1) = 푤푗푖(푛) + ∆푤푗푖(푛) = 푤푗푖(푛) + η훿푗(푛)푦푗(푛) (6)

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The ﬁfth step is to adjust the synaptic weight of each neuron in accordance with Equation (6). The adjustment value to the synaptic weight wji(n) in the training phase is denoted as ∆wji(n).

Energies 2019, 12, x FOR PEER REVIEW 4 of 14 Energies 2019, 12, x FOR PEERw REVIEWji(n + 1 ) = wji(n) + ∆wji(n) = wji(n) + ηδj(n)yj(n) 4 of(6) 14

TheThe sixthsixth stepstep isis toto continuecontinue iteratingiterating thethe learninglearning steps from stepstep 33 untiluntil thethe stopstop criteriacriteria areare satisfiedsatisfied oror thethe numbernumber ofof iterationsiterations reachesreaches thethe predefinedpredefined maximummaximum numbernumber ofof iterations.iterations.. TheThe inputinput featuresfeatures ofof thethe ANNANN inin thisthis workwork includeinclude thethe horizontalhorizontal irradiance,irradiance, solarsolar elevationelevation angle,angle, CosineCosine valuevalue ofof thethe solarsolar azimuthazimuth angle,angle, SineSine valuevalue ofof thethe solarsolar azimuthazimuth angle,angle, CosineCosine valuevalue ofof thethe azimuthazimuth angleangle ofof thethe pyranometer,pyranometer, SineSine valuevalue ofof thethe azimuthaziazimuth angleangle ofof thethe pyranometer,pyranometer, thethe tilttilt angleangle ofof thethe pyranometer, pyranometer, and and the the incidence incidence angle angle of solarof solar ray. ray. The The definitions definitions of the of solar the zenithsolar zenith angle andangle the and solar the azimuthsolar azimuth angle angle are illustrated are illustrated in Figure in Figure2. To be2. ableTo be to able deal to with deal tilted with surfacestilted surfaces with arbitrarywith arbitrar orientationsy orientations and tiltand angles, tilt angles, the azimuth the azimuth angle angle and tiltand angle tilt angle of the of pyranometer the pyranometer are also are includedalso included in the in input the input feature feature vector. vector. Instead Instead of using of using the angle the angle values values directly, directly, the Cosine the Cosine and Sineand valuesSine values are used are used for azimuth for azimuth angles. angles. The The reason reason is that is that the azimuththe azimuthimuth angles anglesangles can rangecancan rangerange from fromfrom 0 to 00 360 toto degrees.360 degrees. Although Although 0 and 0 and 360 degrees360 degrees are farare from far from each each other other in terms in terms of angle of angle values, values, they arethey both are describingboth describing the north the north direction. direction. Therefore, Therefore, expressing expressing the azimuth the azimuth angles angles with Cosine with Cosine and Sine and values Sine canvalues solve can this solve problem. this pro Thereblem. is There no need is no to need express to express the tilt angle the tilt using angle Cosine using and Cosine Sine and values Sine because values thebecause range the of the range tilt angles of the is tilt practically angles is limited practically to smaller limited values. to smaller The tilt values. angle of The the tilt pyranometer angle of the or thepyranometer PV panel andor the the PV incidence panel and angle the of incidence solar ray angle are illustrated of solar ray in Figure are illustra3. Alltedted the inin input FigureFigure features 3.3. All arethe normalizedinputinput featuresfeatures to theareare rangenormalizednormalized of 0 to toto 1. thethe rangerange ofof 00 toto 1.1.

Figure 2. Solar zenith angle and azimuth angle. Figure 2. Solar zenith angle and azimuth angle.

Figure 3. Tilt angle of pyranometer and incidence angle of solar ray.ray. Figure 3. Tilt angle of pyranometer and incidence angle of solar ray.

The existing works using ANN,, such as thatthat inin Reference [19][19],, estimate the inclined irradiance 퐼̂푟푟푖푛푐푙푖푛푒푑 as the desired output in their work. However, the output of the ANN in the proposed 퐼̂푟푟푖푛푐푙푖푛푒푑 as the desired output in their work. However, the output of the ANN in the proposed method is the difference between the horizontal irradiance 퐼푟푟ℎ표푟푖푧표푛푡푎푙 and the inclined irradiance method is the difference between the horizontal irradiance 퐼푟푟ℎ표푟푖푧표푛푡푎푙 and the inclined irradiance 퐼푟푟푖푛푐푙푖푛푒푑 , which is denoted as ∆퐼푟푟 = 퐼푟푟ℎ표푟푖푧표푛푡푎푙 − 퐼푟푟푖푛푐푙푖푛푒푑 . Instead of predicting inclined 퐼푟푟푖푛푐푙푖푛푒푑 , which is denoted as ∆퐼푟푟 = 퐼푟푟ℎ표푟푖푧표푛푡푎푙 − 퐼푟푟푖푛푐푙푖푛푒푑 . Instead of predicting inclined irradiance 퐼̂푟푟푖푛푐푙푖푛푒푑 directly, one of the main characteristics of the proposed ANN model is that the irradiance 퐼̂푟푟푖푛푐푙푖푛푒푑 directly, one of the main characteristics of the proposed ANN model is that the

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The existing works using ANN, such as that in Reference [19], estimate the inclined irradiance Irrˆ inclined as the desired output in their work. However, the output of the ANN in the proposed method is the difference between the horizontal irradiance Irrhorizontal and the inclined irradiance Irrinclined, which is denoted as ∆Irr = Irr Irr . Instead of predicting inclined irradiance Irrˆ horizontal − inclined inclined directly, one of the main characteristics of the proposed ANN model is that the differential value ∆Irr is used as the output. The cost function of network training is based on the mean square error between ∆Irr and ∆Irrˆ . The estimated inclined irradiance would be the horizontal irradiance plus the output estimated differential value ∆Irrˆ , i.e., Irrˆ inclined = Irrhorizontal + ∆Irrˆ . We demonstrate in the experimental results that the design of differential output could reduce the estimation error.

3. Data Acquisition Two datasets were used in the experiments. The pyranometers used to collect the datasets in the experiments are LP PYRA 03 AC. We recorded one irradiance reading per minute in the dataset. Technical characteristics of the pyranometers are listed in Table1. Dataset A contains solar irradiance data of a fixed orientation and tilt angle. It includes the horizontal and tilted solar irradiance at three different locations in Taiwan for a one-year duration. The pyranometers that measured the tilted solar irradiance were with a tilt angle of 20 degrees facing south. The second dataset, dataset B, contains data of various orientations and tilt angles. To collect the second dataset, we set up five pyranometers with adjustable azimuth angles and tilt angles, as shown in Figure4. The center pyranometer is horizontal. Each of the other four pyranometers has a tilt angle facing a certain orientation. The tilt angles and azimuth angles of these four pyranometers are adjusted periodically to specific values so that we can collect the desired experimental data. In total, there were inclined solar irradiances with four different tilt angles (10◦, 15◦, 20◦, 30◦) and sixteen different azimuth angles collected in dataset B. The sixteen azimuth angles included in dataset B are illustrated in Figure5. For the 0-, 90-, 180-, and 270-degree orientations, the inclined pyranometers were adjusted to a 30-degree tilt angle for 20 days. The inclined pyranometers were adjusted to the other 3 tilt angles every 5 days. For all other orientation angles (other than 0◦, 90◦, 180◦, and 270◦), the inclined pyranometers were adjusted to each of the four tilt angles every 5 days. After collecting data for all the tilt angles of an orientation, the pyranometers were adjusted to a new orientation angle. With four inclined pyranometers, we can collect four orientations at the same time. It takes 95 days for one rotation. The process of rotation took place 3 times, taking 285 days. The data were collected from 08:00 to 17:00 for 9 h and recorded each minute. Therefore, there were 540 data points collected in a day. For a 30-degree tilt angle of 0, 90, 180, 270-degree orientations, we have 60-day data with 32,400 data points. For all other angle combinations, we have 15-day data with 8100 data points for each angle combination. Note that the horizontal pyranometer has all the data for a 0-degree tilt angle throughout the entire data collection period.

Table 1. Speciﬁcations of the pyranometer.

Detailed Speciﬁcations of the Pyranometer LP PYRA 03 AC Typical Sensitivity 10 µV/(W/m2) Measuring Range 0~2000 W/m2 Output Signal 4~20 mA, 4 mA = 0 W/m2, 20 mA = 2000 W/m2 Impedance 33~45 Ω Operating Temperature 40 °C~80 °C − Response Time <30 s Energies 2019, 12, 1427 6 of 14 Energies 2019, 12, x FOR PEER REVIEW 6 of 14 Energies 2019, 12, x FOR PEER REVIEW 6 of 14

Figure 4. Pyranometers for Data Collection. Figure 4. Pyranometers for Data Collection.

Figure 5. Diﬀerent Azimuth Angles of Pyranometers. Figure 5. Different Azimuth Angles of Pyranometers. Figure 5. Different Azimuth Angles of Pyranometers. 4. Experimental Results 4. Experimental Results 4. ExperimentalWe used Dataset ResultsA and Dataset B described in the previous section to evaluate the proposed We used Dataset A and Dataset B described in the previous section to evaluate the proposed framework.We used The Dataset results A on and Dataset DatasetA and B described Dataset B inare the elaborated previous in section Sections to 4.1evaluate and 4.2 the, respectively. proposed framework. The results on Dataset A and Dataset B are elaborated in Subsections 4.1 and 4.2, framework.In the experiments, The results we used on the Dataset network A and with Data threeset hidden B are layers. elaborated Each hidden in Subs layerection hass 4 six.1 and neurons. 4.2, respectively. In the experiments, we used the network with three hidden layers. Each hidden layer Therespectively. evaluation In metricsthe experiments, of the estimation we use resultsd the network are mean with absolute three percentagehidden layers. error Each (MAPE) hidden and layer root has six neurons. The evaluation metrics of the estimation results are mean absolute percentage error hasmean six square neurons. percentage The evaluation error (RMSPE) metrics as of shown the estimation in Equations results (7) are and mean (8), respectively. absolute percentage Mean absolute error (MAPE) and root mean square percentage error (RMSPE) as shown in Equations (7) and (8), (MAPE)error measures and root the absolute mean square diﬀerence percentage between error the ground(RMSPE) truth as inclined shown irradiance in EquationIrrsinclined (7) andand (8),the respectively. Mean absolute errorˆ measures the absolute difference between the ground truth inclined respectively.estimated inclined Mean absolute irradiance errorIrrinclined measures. Root the mean absolute square difference error is between the square the rootground of the truth average inclined of irradiance 퐼푟푟푖푛푐푙푖푛푒푑 and the estimated inclined irradiance 퐼̂푟푟푖푛푐푙푖푛푒푑. Root mean square error is the irradiancesquared errors. 퐼푟푟푖푛푐푙푖푛푒푑 Both and of these the estimated error metrics inclined are presentedirradiance in퐼̂푟 terms푟푖푛푐푙푖푛푒푑 of. Root percentage mean square in contrast error tois the square root of the average of squared errors. Both of these error metrics are presented in terms of squareoriginal root ground of the truth average values. of In squared Equations errors. (7) and Both (8), ofNum thesedenotes error metrics the number are presented of data samples in terms being of percentage in contrast to the original ground truth values. In Equations (7) and (8), Num denotes the percentageevaluated. Thein contrast time scale to the of theoriginal metrics ground is one truth minute. values. In Equations (7) and (8), Num denotes the number of data samples being evaluated. The time scale of the metrics is one minute. number of data samples being evaluated. The time scale of the metrics is one minute. PNum ( ) ˆ ( ) 푁푢푚k=1 Irrinclined k Irr̂ inclined k MAPE =∑푁푢푚푘=1 |퐼푟푟푖푛푐푙푖푛푒푑(푘) − ̂퐼푟푟푖푛푐푙푖푛푒푑(푘)| 100% (7) MAPE =∑푘=1 |퐼푟푟푖푛푐푙푖푛푒푑PNum (푘) − 퐼푟푟푖푛푐푙푖푛푒푑(푘)| × 100% (7) 푁푢푚 Irrinclined(k) × MAPE = ∑푁푢푚푘k==11 퐼푟푟푖푛푐푙푖푛푒푑 (푘) × 100% (7) ∑푘=1 퐼푟푟푖푛푐푙푖푛푒푑 (푘) q 1 2 √ 1 1 ∑P푁푢푚Num(퐼푟푟 ((푘) − ˆ퐼̂푟푟 ( ()푘))2 √ 푁푢푚Num 푁푢푚푘=k1=1 Irr푖푛푐푙푖푛푒푑inclined k ̂Irrinclined푖푛푐푙푖푛푒푑k 2 ∑푘=1 (퐼푟푟푖푛푐푙푖푛푒푑(푘) −−퐼푟푟푖푛푐푙푖푛푒푑(푘)) (8) 푅푀푆푃퐸RMSPE푓푐f c== 푁푢푚 P ×100%100% (8) 푅푀푆푃퐸 = 11 푁푢푚Num ×× 100% (8) 푓푐 1 ∑ = 퐼푟푟Irrinclined(k()푘) 푁푢푚Num∑푁푢푚푘k=11퐼푟푟 푖푛푐푙푖푛푒푑(푘) 푁푢푚 푘=1 푖푛푐푙푖푛푒푑

4.1. Results on Dataset A 4.1. Results on Dataset A Dataset A was used to test the accuracy of the transfer model for the training and testing data Dataset A was used to test the accuracy of the transfer model for the training and testing data with a fixed orientation and tilt angle. The purpose of Dataset A is to show that the design of with a fixed orientation and tilt angle. The purpose of Dataset A is to show that the design of

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differentialDataset outputA was usedvalues to could test the reduce accuracy the of estimation the transfer error model compared for the training with existing and testing ANN data transfer with amodels. fixed orientation If horizontal and and tilt tilted angle. irradiance The purpose data of two Dataset or moreA is years to show are that available, the design it is suggested of differential that outputthe data values of one could year reduceare used the for estimation training and error the compared data of the with following existing years ANN are transfer used for models. testing. If horizontalSince the solar and tiltedelevation irradiance angle, datathe solar of two azimuth or more angle, years and are available,the incidence it is suggestedangle of solar that theray datawould of oneexhibit year a aredifferent used forrelationship training and in different the data ofmonths the following of a year, years it would are used be better for testing. to construct Since thea model solar elevationin January angle, and use the the solar model azimuth to test angle, the data and thein January incidence in the angle following of solar years. ray would However, exhibit in a this diff erentcase, relationshipit would require in di collectingfferent months measurement of a year, data it would for a bewhole better year to before construct the asystem model can in January be deployed. and use In thecase models where to the test training the data data in January are limited in the in following the initial years. deployment However, phase, in this a case,solution it would is to use require the collectingtraining data measurement from the previous data for month a whole to year construct before a the model system and can apply be deployed. the transfer In model cases where to obtain the trainingtilted irradiance data are limitedin the following in the initial month. deployment This solution phase, ahelps solution with is faster to use deployment the training datawith from limited the previousdata. To further month toanalyze construct the ainfluence model and of the apply time the period transfer of modelthe training to obtain data tilted used irradianceto construct in the followingtransfer model, month. we This divide solutiond the helps experiments with faster on deployment dataset A into with two limited parts. data. The To first further experiment analyze theon influencedataset A ofused the a time fixed period time ofdistance the training between data the used training to construct and testing the transfer data. T model,he second we dividedexperiment the experimentson dataset A onconsidered dataset A variousinto two time parts. distances The first between experiment the training on dataset andA testingused a fixeddata and time observe distanced betweentheir impact the training on the estimationand testing accuracy. data. The Notesecond that experiment the experiments on dataset onA Datasetconsidered A train various separate time distancesmodels for between separate the locations. training and testing data and observed their impact on the estimation accuracy. NoteFor that the the fir experimentsst experiment on Dataseton dataset Atrain A, the separate samples models in each for month separate were locations. tested using the ANN modelFor constructed the first experiment by the data on dataset from theA, theprevious samples month. in each For month example, were testedthe data using of theFebruary ANN model were constructedtested usingby the the transfer data from model the constructed previous month. by the For data example, of January. the dataThe data of February of March were were tested validated using theusing transfer the transfer model modelconstructed trained by by the the data data of January. of February. The dataThe of estimation March were MAPE validated and RMSPE using the on transferdataset modelA using trained data byfrom the the data previous of February. month The as estimation training MAPE data are and shown RMSPE in on Figure datasets 6A andusing 7, datarespectively. from the The previous MAPE month and RMSPE as training of the data proposed are shown method in Figures and 6the and method7, respectively. in Reference The MAPE[19] are andcompared. RMSPE The of the settings proposed of the method ANN and implementation the method in in Reference Reference [ 19[19]] are are compared. the same The as the settings best ofparameters the ANN recommended implementation in Reference in Reference [19]. [19 We] are can the observe same asthat the the best proposed parameters method recommended exhibits lower in Referenceerror in terms [19]. of We both can MAPE observe and that RMSPE the proposed compared method with exhibitsReference lower [19]. error The inmain terms reason of both is that MAPE the andrange RMSPE of the compareddifference withbetween Reference the horizontal [19]. The and main inclined reason irradiance is that the rangeis much of smaller the difference than the between range theof inclined horizontal irradiance and inclined. Therefore, irradiance the is design much smaller of the thanproposed the range method of inclined helps to irradiance. reduce estimation Therefore, theerrors. design of the proposed method helps to reduce estimation errors.

Figure 6. 6. EstimationEstimation mean mean absolute absolute percentage percentage error error (MAPE) (MAPE on dataset) on datasetA using Adata using from data the from previous the monthprevious as month training as data. training data.

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Figure 7.7. Estimation root mean square percent percentageage error ( (RMSPE)(RMSPE)) on dataset A using data from the previous monthmonth asas trainingtraining data.data.

For the second experiment on dataset A,, wewe useduseused trainingtraining andand testingtesting datadata fromfrom variousvarious timetime periods. The estimation MAPE and RMSPE on dataset A usingusing training training and and testing testing data data with with different diﬀerent distances are shown in FiguresFigures8 8 and and9 ,9, respectively. respectively. In In Figures Figure8s and8 and9, the9, the x axis represents the time distancedistance betweenbetween thethe trainingtraining and andtesting testing data. data. The The unit unit of of the the time time distance distance is is month. month. For For example, example, if theifif thethe transfer transfertransfer model modelmodel constructed constructedconstructed by byby the thethe data datadata of ofof February FebruaryFebruary is usedisis usedused to to estimateto estimateestimate the thethe inclined inclinedinclined irradiance irradianceirradiance of March,of March, the the time time distance distance between between the the training training and and testing testing data data is is one one month. month. IfIf thethe transfertransfer modelmodel constructed by the data of JanuaryJanuary is usedused toto estimateestimate thethe inclinedinclined irradianceirradiance of March, the time distance between the training and testing datadata isis twotwo months.months. Notice that the time distance between JanuaryJanuary andand DecemberDecember is is one one instead instead of of eleven. eleven. The The maximum maximum time time distance distance between between the trainingthe training and testingand testing data isdata six. is We six. can We observe can observe that the that error the of error estimation of estimation increases increases as the time as the distance time becomesdistance larger.becomes Therefore, larger. Therefore, it is recommended it is recommended that the that time the distance time distance between between the training the training and testing and testing data shoulddata should not be not too be large.too large. In practice, In practice, a transfer a transfer model model can can be be constructed constructed each each month. month. TheThe transfertransfer model can be used toto obtainobtain thethe estimatedestimated inclinedinclined irradianceirradiance ofof thethe succeedingsucceeding month.month.

Figure 8. EstimationEstimation MAPE MAPE on on dataset A using training and testing data with different diﬀerent distances.distances.

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Figure 9. EstimationEstimation RMSPE RMSPE on on dataset dataset AA using training and testing data with different different distances. 4.2. Results on Dataset B 4.2. Results on Dataset B B Dataset B was used to validate the proposed transfer model when the tilt and azimuth angles were not in thethe trainingtraining set.set. As described in Section 33,, DatasetDataset BB containscontains irradianceirradiance datadata ofof fourfour differentdifferent tilt angles (10(10◦°, 15◦°, 20 °◦,, 30°◦)) and and sixteen sixteen different different azimuth angles. To To establish the transfer model, the training data are listed in Table2 2.. TheThe horizontalhorizontal irradianceirradiance isis regardedregarded asas trainingtraining datadata with a zero-degreezero-degree tilt angle. In the experiments, we use usedd inclined irradiance of 0 and 30 degrees as the training samples in order to infer inclined irr irradianceadiance of all the other tilt angles between 0 and 30 degrees. We We use usedd training data of four orthogonal azimuth angles to infer the inclined irradiance of all the other azimuth angles in the testing data. The combinations of the four tilt angles and sixteensixteen azimuth angles in the testing data are shown in Table3 3.. Table 2. Training data. Table 2. Training data. Azimuth Angle Tilt Angle Azimuth Angle Tilt Angle - 0 - 0° ◦ 0◦ 30◦ 90◦ 0° 30°30 ◦ 180◦ 90° 30°30 ◦ 270 30 ◦ 180° 30° ◦ 270° 30° Table 3. Testing data.

Azimuth Angles of PyranometersTable 3. Testing data. Tilt Angle 0 10 , 15 , 20 Azimuth Angles◦ of Pyranometers Tilt◦ Angle◦ ◦ 90◦ 10◦, 15◦, 20◦ 180◦ 0° 10°,10◦, 1515°,◦, 20 20°◦ 270◦ 90° 10°,10◦, 1515°,◦, 20 20°◦ 22.5◦ 10◦, 15◦, 20◦, 30◦ 45◦ 180° 1010°,◦, 15 ◦15°,, 20◦ ,20° 30◦ 67.5270° 1010°,◦, 15 ◦15°,, 20◦ ,20° 30◦ 112.5◦22.5° 10°,10◦, 15°, 15◦, 2020°◦, 30, 30°◦ 135◦ 10◦, 15◦, 20◦, 30◦ 157.545° 10°, 10◦, 15°, 15◦, 2020°,◦, 30 30°◦ 202◦ 67.5 10°,10◦, 15°, 15◦, 2020°,◦, 30 30°◦ 225◦112.5° 10°,10◦, 15°, 15◦, 2020°,◦, 30 30°◦ 247.5◦ 10◦, 15◦, 20◦, 30◦ 292.5◦135° 10°,10◦, 15°, 15◦, 2020°,◦, 30 30°◦ 315◦ 157.5 10°,10◦, 15°, 15◦, 2020°,◦, 30 30°◦ 337.5◦202° 10°,10◦, 15°, 15◦, 2020°,◦, 30 30°◦ 225° 10°, 15°, 20°, 30° 247.5° 10°, 15°, 20°, 30°

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292.5° 10°, 15°, 20°, 30° Energies 2019, 12, 1427 315° 10°, 15°, 20°, 30° 10 of 14 337.5° 10°, 15°, 20°, 30°

The purpose of the experiments on dataset B is to show show that that although although the tilt angles angles and and azimuth azimuth angles areare not not included included in in the the training training data, data, the trained the trained model model can still can successfully still successfully estimate estimate the inclined the inclinedirradiance irradiance of all the of tilt all angles the tilt and angles azimuth and angles azimuth in theangles testing in the data testing with adata low with error a rate. low Theerror MAPE rate. Theand MAPE RMSPE and of estimated RMSPE of inclined estimated irradiance inclined on irradiance dataset B onare dataset shown B in are Figures shown 10 in and Figure 11, respectively.s 10 and 11, respectively.Note that the Notesamples that of the 30-degree samples tilt of angles 30-degree on the tilt four angles orthogonal on the orientations four orthogonal 0◦, 90 ◦ orientations, 180◦, 270◦ 0°are, 90° not, 180° evaluated, 270° are because not evaluated these angle because combinations these angle are includedcombinations in the are training included data. in the The training overall data.averaged The MAPEoverall andaveraged RMSPE MAPE on dataset and RMSPEB are 6.79% on dataset and 8.02%, B are 6.79 respectively.% and 8.02%, From respectively. the experimental From theresults, experimental we can observe results, that we using can horizontal observe that irradiance using horizontal and 30-degree irradiance inclined and irradiance 30-degree in inclined the four irradianceorthogonal in directions the four asorthogonal training data, directions the established as training model data, can the estimate established inclined model irradiance can estimate with inclinedarbitrary irradiance azimuth angles with arbitrary and tilt angles azimuth between angles 0 and to 30 tilt degrees. angles between 0 to 30 degrees.

Figure 10. Estimation MAPE on dataset B.

Figure 11. Estimation RMSPE on dataset B.

To further further evaluate evaluate the the eﬀ ectiveness effectiveness of the of proposed the proposed method, method, we compared we compare the proposedd the proposed method withmethod a baseline with a methodbaseline using method the using skill score the skill (SS) score [28] metrics(SS) [28] deﬁned metrics in defined Equation in (9).Equation A method (9). Ais

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Energies 2019, 12, x FOR PEER REVIEW 11 of 14 better than the baseline method if the skill score is positive. The degree of improvement is better when themethod skill is score better is closerthan the to one.baseline method if the skill score is positive. The degree of improvement is better when the skill score is closer to one. MAPEproposed RMSPEproposed SSMAPE = 1 푀퐴푃퐸푝푟표푝표푠푒푑, SSRMSPE = 1 푅푀푆푃퐸푝푟표푝표푠푒푑 (9) 푆푆푀퐴푃퐸 = 1 − MAPEbaseline , 푆푆푅푀푆푃퐸 = 1 −− RMSPEbaseline (9) 푀퐴푃퐸푏푎푠푒푙푖푛푒 푅푀푆푃퐸푏푎푠푒푙푖푛푒 Existing methods that construct transfer models via training such as those in References [[1818–2121]] cannot estimate inclined irradiance for tilt and orientation angles that are not included in the training dataset using a unifiedunified transfer model. Therefore, for dataset B, we comparedcompared the proposed method with a baseline method that uses horizontal irradiance as the estimation result. The SSMAPE and with a baseline method that uses horizontal irradiance as the estimation result. The 푆푆푀퐴푃퐸 and SSRMSPE are illustrated in Figure 12. We can observe that the skill scores of the proposed method are 푆푆푅푀푆푃퐸 are illustrated in Figure 12. We can observe that the skill scores of the proposed method are positive under all conditions. As the tilt angle becomes larger, the proposed method exhibits better skill scores in contrast to the baseline. This means that the proposed proposed transfer model has more obvious improvement when the didifferencefference between the horizontal irradiance and the inclined irradiance is larger. Furthermore,Furthermore, wewe calculatedcalculated the the skill skill scores scores of of Reference Reference [19 [19]] on on dataset dataset B usingB using a 30-degree a 30-degree tilt angletilt angle and and 180-degree 180-degree orientation orientation as training as training data. data. We We denoted denote thed the model model trained trained by theby the 30-degree 30-degree tilt angle and 180-degree orientation as M . Since model M is trained using a 30-degree tilt angle and tilt angle and 180-degree orientation as fMf. Since model Mf f is trained using a 30-degree tilt angle and there is no no input input feature feature regarding regarding the the tilt tilt angle angle of of the the inclined inclined surface, surface, it exhibits it exhibits high high MAPE MAPE for 10 for- 10-degree and 15-degree tilt angle testing data. The MAPE of M on 10-degree and 15-degree tilt angle degree and 15-degree tilt angle testing data. The MAPE of Mf fon 10-degree and 15-degree tilt angle testing datadata isis muchmuch higher higher than than the the baseline. baseline. Recall Recall that that the the baseline baseline uses uses horizontal horizontal irradiance irradiance as the as estimationthe estimation result. result. Therefore, Therefore, the MAPEthe MAPE of baseline of baseline on 10-degree on 10-degree and 15-degreeand 15-degree tilt angle tilt angle testing testing data is relatively low. As a result, the skill scores of model M on these two tilt angles are negative. For data is relatively low. As a result, the skill scores of modelf Mf on these two tilt angles are negative. 20-degree and 30-degree tilt angle testing data, model M yields better results than the baseline since For 20-degree and 30-degree tilt angle testing data, modelf Mf yields better results than the baseline the testing tilt angles are closer to the training tilt angles. However, since model M is trained with data since the testing tilt angles are closer to the training tilt angles. However, since fmodel Mf is trained fromwith data only onefrom fixed only orientation, one fixed orientation, it still cannot it still yield cannot satisfying yield accuracy satisfying when accuracy the test when data the are collectedtest data usingare collected various using orientations. various Asorientations. a consequence, As a consequence, the skill scores the of s thekillproposed scores of methodthe proposed are still method much higher than those of model M on 20-degree and 30-degree tilt angle testing data. are still much higher than thosef of model Mf on 20-degree and 30-degree tilt angle testing data.

(a)

Figure 12. Cont.

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(b)

Figure 12. Skill scores on dataset B.

5. Conclusions 5. Conclusions In this work, artificial artificial neural networks were utilized to construct the transfer model to estimate solar irradiance irradiance on on tilted tilted surfaces. surfaces. Instead Instead of predicting of predicting tilted tilted irradiance irradiance directly, directly, the proposed the pr methodoposed methodestimated estimated the differences the differences between thebetween horizontal the horizontal irradiance andirradiance the irradiance and the on irradiance a tilted surface. on a tilted The surface.range of The the dirangefference of the between difference the horizontal between the and horizontal inclined irradiance and inclined was irradiance much smaller was thanmuch the smaller range thanof inclined the range irradiance. of inclined If we irradiance. estimated theIf we diff estimerence,ate itd wouldthe difference, prevent theit would estimated prevent inclined the irradianceestimated inclineddrift too irradiance far away from drift the too ground far away truth. from The the outputsground truth. of the The proposed outputs ANNs of the were proposed differential ANNs values. were Accordingdifferential to values. the experiments, According the to the design experiments, of differential the outputdesign can of differential help to enhance output the can estimation help to enhanceaccuracy. the The estimation experiments accuracy. also analyze The the experiments estimation accuracy also analyze using thediff erent estimation time distances accuracy between using differentthe training time and distances testing data. between Two the datasets training were and tested testing in data. the experiments. Two datasets First, were for tested the dataset in the withexperiments. a fixed orientation First, for the and dataset tilt angle, with the a fixed proposed orientation method and can tilt successfully angle, the estimateproposed inclined method solar can successfullyirradiance with estimate a lower inclined error compared solar irradiance to existing with methods.a lower error Second, compared for the todataset existing with methods. various θ Second,orientations for the and dataset tilt angles, with using various horizontal orientations irradiance and tilt and angles, inclined using irradiance horizontal of a irradiance tilt angle andas θ inclinedtraining irradiance data, the transfer of a tilt angl modele θ canas training infer the data, inclined the transfer irradiance model of anycan infer tilt angles the inclined between irradiance 0 and ofalthough any tilt those angles tilt between angles are0 and not θ included although in those the training tilt angles data. are Further, not included using inclined in the training irradiance data. at Furtherfour orthogonal, using inclined orientations irradiance as training at four data, orthogonal the transfer orientations model canas training estimate d theata, inclinedthe transfer irradiance model canof arbitrary estimate orientations.the inclined irradiance It was validated of arbitrary thatthe orientations. trained model It was can validated successfully that the estimate trained inclined model canirradiance successfully of arbitrary estimate orientations inclined irradiance and tilt angles of arbitrary in the testing orientations data with and satisfactory tilt angles accuracy. in the testing Authordata with Contributions: satisfactoryH.-Y.C. accuracy. is the corresponding author, who is responsible for literature review, algorithm Authordesign, manuscriptContributions: writing H.-Y.C. and is revision. the corresponding C.-C.Y. isresponsible author, who for is algorithmresponsible implementation for literature review, and experiment algorithm conduction. C.-C.C. is responsible of problem definition and correctness verification. K.-C.H. and M.-H.T. are designresponsible, manuscript for hardware writing installation, and revision calibration. C.-C.Y. andis responsible solar irradiance for algorithm data collection. implementation C.-L.L. is and responsible experiment for conduction.data cleaning, C. preprocessing-C.C. is responsible and error of problem checking. definition and correctness verification. K.-C.H. and M.-H.T. are responsible for hardware installation, calibration and solar irradiance data collection. C.-L.L. is responsible for Acknowledgments: This research was financially supported by Green Energy and Environment Research dataLaboratories, cleaning, Industrialpreprocessing Technology and error Research checking Institute. and Bureau of Energy, Ministry of Economic Affairs Acknowledgments:of Taiwan. This research was financially supported by Green Energy and Environment Research Laboratories,Conflicts of Interest: IndustrialThe Technology authors declare Research no conflict Institute of interest.and Bureau of Energy, Ministry of Economic Affairs of Taiwan.

Conflicts of Interest: The authors declare no conflict of interest.

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