Simplified model to correct thermopile pyranometer solar radiation measurements for photovoltaic module yield estimation

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in Renewable Energy, Vol. 146, on February 2020, , available at: https://doi.org/10.1016/j.renene.2019.07.070 © 2019 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND 1 SIMPLIFIED MODEL TO CORRECT THERMOPILE PYRANOMETER SOLAR RADIATION 1 MEASUREMENTS FOR PHOTOVOLTAIC MODULE YIELD ESTIMATION 2 Lillo-Bravo I1, Larrañeta M2, Núñez-Ortega E1, González-Galván R1 3 1 Department of Energy Engineering, University of Seville. 4 2 Andalusian Association for Research and Industrial Cooperation (AICIA). 5 Corresponding author: 6 Miguel Larrañeta, Andalusian Association for Research and Industrial Cooperation, Camino de 7 los Descubrimientos, s/n. 41092, Seville, Spain. 8 Phone: (+34)954487237 9 E-mail: mlarr[email protected]s 10 Abstract 11 This article presents and evaluates the performance of a simplified model to generate 10-min 12 global horizontal synthetic solar radiation data that would correspond to the measurements of 13 a calibrated photovoltaic monocrystalline cell. The model, which only requires global horizontal 14 solar radiation data measured with a thermopile pyranometer as input, is based on the 15 characterization of the relation between the data measured with a thermopile pyranometer and 16 a calibrated cell as a function of the sky condition and the solar elevation. We have used an 17 extensive solar radiation database for the location of Seville (Spain) for the training of the model 18 that has been tested in Seville and Lancaster (USA), showing satisfactory results and suggesting 19 a global applicability with no local adaptation or calibration requirement. 20 The model shows the best results for high levels of solar radiation and solar elevations and 21 decreases its performance on days with high levels of diffuse irradiation and for very low solar 22 elevation angles. We obtain a daily RMSD between measured and synthetic data of 1.9% in 23 Seville and 5.2% in Lancaster. The frequency distribution of the synthetic datasets shows a KSI 24 of 3.7 W/m2 in Seville and 8.6 W/m2 in Lancaster. We also evaluate the ramp rates of measured 25 and synthetic sets through the KSI of the measured and synthetic ramp rates sets, obtaining 0.11 26 W/m2۰min in Seville and 0.20 W/m2۰min in Lancaster. 27 Keywords 28 Photovoltaic yield, calibrated solar cell, pyranometer. 29 1 Introduction 30 31 Photovoltaic (PV) modules have a practically identical spectral response, time, temperature 32 coefficient and angular response to a calibrated solar cell of the same technology and 33 encapsulation. For this reason, if radiation measurements are obtained by means of a calibrated 34 photovoltaic cell of the same technology and encapsulation as the photovoltaic modules of the 35 installation, most of the uncertainties in the determination of the electrical production of the 36 2 photovoltaic installation are already included. Meydbray et al., [1-2] compare the use of solar 37 reference cell and thermopile for PV applications, and point out that solar reference cells are 38 especially useful for precise characterization of the PV performance, allowing a better detection 39 of changes in PV system performance with time and shorter time assessment of the PV operating 40 efficiency. Table 1 summarises the comparison of the use of solar reference cell and thermopile 41 pyranometers for PV applications proposed by Meydbray et al [2]. 42 Table 1. Cell-pyranometer comparison made by Meydbray et al [2] for PV applications. 43 Reference Cell Thermopile pyranometer Spectral Response Can be made to closely match solar panel Broadband response needs to be corrected Angle of incidence Can be made to closely match solar panel Response to all angles Temperature response Temperature response is similar to PV system Are designed to minimize sensitivity to temperature. Not corrected for temperature Time response < milliseconds; matched to PV response Up to 30 seconds, can be problematic for measuring PV performance Other issues Emission to cold sky and transients in ambient temperature affects output International standards IEC 60904 ISO 9847, ISO 9845, ISO 9846 44 For PV applications, Dunn et al., [3] analysed the expanded uncertainties in the measurements 45 of the irradiance made in the plane of array throughout the course of a representative day with 46 a thermopile pyranometer and a PV reference device. They conclude that during most of the 47 day, uncertainties are in the order of ± 5% for a pyranometer, and ±2.4% for a calibrated cell, 48 both stated with 95% confidence intervals. So calibrated cells provide superior irradiance 49 measurements for PV power plant monitoring applications. Haeberlin et al., [4] compare the use 50 of a pyranometer and a calibrated silicon solar cell and reach similar conclusions: it is preferable 51 to use a calibrated Si-cell (of the same type as the cells in the PV array) as a reference device of 52 the irradiance measurements for PV plant monitoring than a pyranometer. 53 The PV module performance is generally evaluated by manufacturers under standard test 54 conditions (STC), which refer to 25°C module temperature, 1000 W/m2 of incident solar 55 irradiance on the PV module plane with 1.5 solar spectrum air mass (AM) as reference spectrum 56 (IEC 60904-3, 2016) [5]. Nevertheless, in real conditions the PV module temperature, solar 57 irradiance with incident angle and solar spectrum differ from STC. 58 3 Incident solar global irradiance that reaches the PV module is affected by the distribution of its 59 solar spectrum. This solar spectrum depends on cloud cover, AM, precipitable water (PW) and 60 aerosol optical depth (AOD) [6]. The short-circuit current is the main parameter affected by the 61 distribution of the solar spectrum, but in some technologiesefficiency, maximum power and fill 62 factor are also influenced. Polo et al, [7] analyse the use of reference modules as irradiance 63 sensor for monitoring and modelling rooftop PV systems for different PV technologies with 64 different spectral responses. If such affection is neglected and only the broadband irradiance is 65 used for PV-module performance modelling, then errors of up to 10% can be introduced in 66 extreme conditions [8]. 67 Therefore, a spectral correction shall be used in order to adjust the incident solar irradiance and, 68 hence, improve the PV module performance modelling [9]. This spectral correction depends on 69 the PV module technology. Polo et al., [10] give the spectral factor for seven photovoltaic 70 technologies and 124 sites. They found that the annual spectral factor for crystalline silicon 71 technologies is rather homogenous worldwide with maxima spectral losses and gains of ≈3% 72 and ≈1%, respectively. On the contrary, the spectral factor for thin film devices displays a higher 73 spatial variability. Nuñez et al., [11] proposed a spectral matching ratio for multijunction cells 74 within a concentrating photovoltaic module. The Sandia Array Performance Model applies a 75 fourth-order polynomial correction based on AM [12]. M. Lee et al., [13] included in their model 76 the influence of AM and PW in a similar function. Theristis et al., [14] developed a model 77 including AM, PW and AOD to improve concentrating photovoltaic system performance 78 modelling. Another option is to use direct measurements of spectral radiation, but the use of 79 spectroradiometers is not yet widespread and an adequate validation is required [15]. 80 Moreover, in many cases the PV module structure does not have two-axis tracking. 81 Consequently, the incident solar global irradiance that reaches the PV module is affected by the 82 solar angle of incidence (AOI). Angular losses in PV modules can introduce differences in the 83 short circuit current under STC of up to 3.5% under global normal irradiance conditions with 84 horizontal orientation compared with a device without losses due to angular effects [16]. King 85 et al., [17] suggest the use of a 5th order polynomial function to represent the angular optical 86 losses on the short circuit current. Martin et al., [18] formulated similar functions. Instead, given 87 the importance of these effects on the performance of the photovoltaic module, manufacturers 88 test their PV modules according to IEC standard 61853-2:2016 [19] where the effect of the angle 89 of incidence and the spectral responsivity on the output power of the photovoltaic module is 90 measured. The most used software for the design of photovoltaic installations has implemented 91 a model to take into account the angular losses. PVsyst software [20] has implemented the 92 model developed by Souka and Safat and adopted by the American Society of Heating, 93 Refrigeration, and Air Conditioning (ASHRAE) [17]. Plag F et al., [21] related the spectral and 94 angular effect showing that the spectral effect depends on the incident angle too. 95 An alternative to these methods to link the silicon photovoltaic calibrated cell measurement 96 with the measurements made by a thermopile pyranometer are specific simplified models to 97 relate the global solar radiation measured by both devices. J. J. Michalsky et al., [22] propose a 98 table of correction factors depending on the clearness index and the brightness index. It requires 99 the knowledge of diffuse and direct radiation from the site. King & Myers [23] correct the 100 response of the PV device as a function of the solar spectrum, AOI and temperature. This 101 4 corrected response allows us to obtain an improvement in the estimation of the total irradiance. 102 The method also requires diffuse and direct data among the calculation of functions related to 103 the absolute air mass, the AOI and a temperature coefficient. The main drawback in the 104 implementation of the described methods is the requirement of rarely available measurements 105 such as AOD, PW, diffuse and direct irradiance for the location under study. 106 The purpose of this work is the development of a simplified model for estimating the solar global 107 radiation that a calibrated monocrystalline silicon cell would measure using only thermopile 108 pyranometer measurements as input. This model could be used as a correction factor that would 109 allow us to characterize the PV module performance from thermopile pyranometer radiation 110 data. 111 Simplicity is the main advantage of this model because it depends only on the global irradiation 112 data measured by a thermopile pyranometer avoiding the need to use more complex variables 113 which are generally more difficult to obtain. The model has been developed for one location 114 (Seville) and tested in the same location (Seville) and a different one (Lancaster) without any 115 local adaptation showing satisfactory results. 116 The paper is presented as follows: Section 2 presents the database used for the training and 117 testing of the model. In Section 3 we explain step by step the development of the model. In 118 Section 4 we perform the result analysis of the model in two locations. Conclusions are then 119 made in Section 5. 120 2 Meteorological database 121 In this work, an extensive database is used for training the method proposed (Table 2). This 122 database is composed of 10-min averages values of GHI recorded during 42 consecutive months 123 from July 2012 to December 2015 for the location of Seville (Spain). The measurements were 124 taken with a sampling and storage time resolution of 5 s. A secondary standard pyranometer 125 Kipp & Zonen CMP21 and an Atersa calibrated monocrystalline silicon PV cell measured the GHI. 126 The devices used for the design and testing of the methodology are located at the 127 meteorological station of the Group of Thermodynamics and Renewable Energy of the 128 University of Seville and have been periodically calibrated, according to the recommendations 129 from the instrument manufacturers. Data used in this work have been subjected to quality- 130 control procedures [24] following the BSRN recommendations [25]. Only 1% of the data has 131 been filled for the entire database. Data recorded at sun altitude lower than 2º have not been 132 used in this study. 133 Table 2. Location selected for the method training. 134 Latitude (°N) Longitude (°W) Altitude (m) Köppen-Geiger Climate Period Seville 37.4 6.0 12 Csa 2012-2015 135 In addition, the model has been tested in two locations. We use data for one entire year for the 136 location of Seville and for the location of Lancaster, USA (Table 3). GHI data from Lancaster were 137 measured with a secondary standard pyranometer Kipp & Zonen CMP21 and an Atersa 138 calibrated monocrystalline silicon PV cell, and has been subjected to quality control procedures 139 5 [24] following the BSRN recommendations [25]. According to the Köppen-Geiger climate 140 classification system [26], Seville’s climate is classified as Mediterranean with hot summers and 141 Lancaster’s climate is classified as Mediterranean with mild summers. When performing the 142 quality control procedure of Lancaster database, we identify a period of 12 consecutive days 143 with irregularities in the measurements in October. This period is not computed in the 144 performance evaluation of the algorithm. 145 We use 10-min averaged data because high-quality data for the testing of the method was 146 available in that time step, but the method can be used in any time resolution greater than 1- 147 min. 148 Table 3. Location selected for testing the method. 149 Latitude (°N) Longitude (°W) Altitude (m) Köppen-Geiger Climate Period Seville 37.4 6.0 12 Csa 2016 Lancaster 34.6 118.3 812 Csb 2016 150 3 Methodology 151 In this section we describe the methodology implemented for the synthetic generation of 152 reference PV cell solar radiation data (𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ) from pyranometer measurements (𝐺𝑝𝑦𝑟). The 153 model has been developed for 10-min time resolution and its only input is the 𝐺𝑝𝑦𝑟. The rest of 154 the used parameters are theoretically estimated from the geographic position of the location 155 under study (latitude and longitude) and the record time of the measurements. The method can 156 be described in a sequence of 5 steps. 157 3.1 Step 1: Data clustering 158 In the first step, we cluster the data from pyranometer and PV cell measurements as a function 159 of the clearness index 𝑘𝑡 and the solar elevation,𝛼. To characterize the type of day, other indices 160 such as the Perez clearness index (ε), Perraudeau's brightness (𝐼𝑁), sky ratio index (𝑆𝑅) [27] or 161 illuminance fluctuation frequency index [28] could be used, but they require the knowledge of 162 more variables such as diffuse and/or beam normal irradiance. Previous approaches have shown 163 that using only one radiometric variable, different sky conditions can be identified [29]. We 164 propose to cluster the data into 9 groups in terms of the clearness index, dividing the datasets 165 into intervals of 0.1 from 0 to 0.8. Values greater than 0.8 have been clustered into the same 166 group. For the solar elevation we chose 7 groups dividing the dataset into intervals of 5° from 167 20° to 40° and intervals of 10° from 40° to 60°. Data for solar elevations lower than 20° are 168 clustered in the same group as well as data for solar elevations greater than 60°. We group the 169 solar data into a total of 63 clusters. 170 3.2 Step 2: First regression 171 In the second step, we use a least squares procedure to perform a linear regression fit to the 172 cloud point obtained, when comparing the solar radiation values measured with the 173 pyranometer (𝐺𝑝𝑦𝑟) with the solar radiation values measured with the calibrated solar cell 174 (𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠), obtaining a function dependant on the solar elevation (α) and sky condition defined by 175 6 the clearness index (𝑘𝑡). See equation 1. In figure 1 we present an example of the cloud points 176 and the linear fit for the cluster interval corresponding to solar elevations lower than 30° and 177 greater than 25°, and clearness index lower than 0.6 and greater than 0.5. In table 4 we present 178 the first regression coefficient 𝑅𝐶 (α,kt) obtained for each cluster from the training dataset. 179 𝑅𝐶(α,𝑘𝑡)= 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠𝐺𝑝𝑦𝑟 ⁄ (1) 180 We can relate the first regression coefficient (RC) to the spectral and angular losses related to 181 the measurements of solar radiation with silicon monocrystalline cells in comparison to solar 182 radiation data measured with a pyranometer, since those losses show a strong dependency on 183 the solar elevation and sky condition. 184 185 186 Figure 1. Cloud points and first linear regression fit for solar elevations between 25° and 30° and 187 clearness indexes between 0.5 and 0.6 for the location of Seville. 188 Table 4. First regression coefficient (RC) for each cluster for the location of Seville. We shade the 189 result for the data exposed in figure 1. 190 1st Regression coefficient RC(α, kt) α < 25° 25° ≤ α < 30° 30° ≤ α < 35° 35° ≤ α < 40° 40° ≤ α < 50° 50° ≤ α < 60° α ≥ 60° kt ≤ 0.1 1.050 1.052 1.065 1.059 1.062 1.058 1.049 0.1 < kt ≤ 0.2 1.000 1.008 1.017 1.022 1.022 1.025 1.016 0.2 < kt ≤ 0.3 0.966 0.976 0.981 0.981 0.987 0.990 0.984 0.3 < kt ≤ 0.4 0.938 0.957 0.961 0.967 0.967 0.970 0.970 0.4 < kt ≤ 0.5 0.891 0.944 0.950 0.954 0.960 0.963 0.964 0.5 < kt ≤ 0.6 0.854 0.930 0.944 0.951 0.956 0.960 0.960 0.6 < kt ≤ 0.7 0.854 0.921 0.936 0.946 0.953 0.958 0.959 0.7 < kt ≤ 0.8 0.862 0.925 0.939 0.949 0.956 0.960 0.960 kt > 0.8 0.882 0.932 0.944 0.950 0.954 0.956 0.955 191 7 In figure 2 we present the first regression coefficient (RC) for all the sky conditions as a function 192 of the solar elevation selected intervals (a) and the first regression coefficient (RC) vs the solar 193 elevations as a function of the sky condition (b) 194 195 196 Figure 2. First regression coefficient (RC) versus all the selected intervals of solar elevations as a 197 function of the clearness indexes (a) for the location of Seville. First regression coefficient (RC) 198 versus the clearness indexes for all the selected intervals of solar elevations as a function of all 199 the selected intervals of solar elevations (b) for the location of Seville. 200 From table 4 and figure 2, we can observe that the first regression coefficient (RC) shows a 201 different tendency for solar elevations lower than 25° and clearness indexes greater than 0.4. 202 For 𝑘𝑡 values lower than 0.4 and solar elevations lower than 25°, we can observe in both figures 203 a, and b, that RC shows a change in its performance with respect to the rest of the 𝑘𝑡 and α 204 values. 205 This can be explained by the greater impact of the refraction effects for low solar elevations of 206 the solar radiation on the PV reference cell in comparison to the pyranometer. Reference cell 207 errors are particularly large at sunset and sunrise (up to 50 W/m2) according to the Hukseflux 208 Thermal Sensors report [30]. The greater the clearness index, the greater the direct component 209 amount on the total global radiation which in turn is strongly affected by the refraction effects 210 due to low solar elevations and horizon distortions. Moreover, the PV reference cells present a 211 greater angular dependence. 212 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 [0,25) [25,30) [30,35) [35,40) [40,50) [50,60) [60,90) First regression coefficient (RC) Solar elevations (º) kt ≤ 0.1 0.1 < kt ≤ 0.2 0.2 < kt ≤ 0.3 0.3 < kt ≤ 0.4 0.4 < kt ≤ 0.5 0.5 < kt ≤ 0.6 0.6 < kt ≤ 0.7 0.7 < kt ≤ 0.8 kt > 0.8 a) 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 First regression coefficient (RC) Clearnes index (-) α < 25°25°≤ α < 30°30°≤ α < 35° 35°≤ α < 40°40°≤ α < 50°50°≤ α < 60° α ≥ 60° b) 8 3.3 Step 3: Second regression 213 In the third step, we run again a least squares procedure but in this case we perform a second 214 order polynomic fit of the first regression coefficient (RC) as a function of the clearness index 215 (𝑘𝑡). The calculation is performed separately for each solar elevation selected interval following 216 equation 2. From this step, we obtain the three functions (𝑏(α),𝑐(α),𝑑(α)) dependant on the 217 solar elevation. In figure 3 we represent the polynomial fit of the first regression coefficient (RC) 218 as a function of the clearness index (𝑘𝑡) for each solar elevation interval. 219 𝑅𝐶(α,𝑘𝑡)=𝑏(α)·𝑘𝑡2+𝑐(α)·𝑘𝑡+𝑑(α) (2) 220 221 222 223 RC = 0.4658kt2- 0.6981kt+ 1.1234 R² = 0.9737 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0 0.2 0.4 0.6 0.8 1 First regression coeeficient (RC) kt 0º < α< 25º RC = 0.317kt2- 0.4612kt+ 1.0906 R² = 0.9946 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0 0.2 0.4 0.6 0.8 1 First regression coeeficient (RC) kt 25º≤ α< 30º RC = 0.3498kt2- 0.4877kt+ 1.1038 R² = 0.9863 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0 0.2 0.4 0.6 0.8 1 First regression coeeficient (RC) kt 30º≤ α< 35º RC = 0.3164kt2- 0.44kt+ 1.0952 R² = 0.9819 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0 0.2 0.4 0.6 0.8 1 First regression coeeficient (RC) kt 35º≤ α< 40º RC = 0.3122kt2- 0.4306kt+ 1.0961 R² = 0.9739 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0 0.2 0.4 0.6 0.8 1 First regression coeeficient (RC) kt 40º≤ α< 50º RC = 0.2865kt2- 0.399kt+ 1.091 R² = 0.974 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0 0.2 0.4 0.6 0.8 1 First regression coeeficient (RC) kt 50º≤ α< 60º 15 measured with the PV reference cell and 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ is the synthetically obtained PV reference cell 331 data. 332 In tables 6-7 we present the monthly and annual cumulative values of the measured and 333 synthetic time series for one year for the location of Seville and Lancaster respectively with the 334 percentage differences in comparison to the reference PV cell measured data. We identified 335 continuous irregularities on twelve consecutive days in October in Lancaster. We have not 336 included this month in the performance evaluation of the model. We can observe that the 337 synthetically generated data show lower differences in the monthly cumulative values in 338 comparison to the monthly cumulative values of the reference PV cell-measured dataset varying 339 from -0.2% to 1.4% in Seville and from 0.3% to 3.3% in Lancaster, while the monthly cumulative 340 values of the pyranometer measurements show differences that range from 5.3% to 9.5% in 341 Seville and 4.4% to 11.1% in Lancaster. We can observe that the greater differences in the 342 pyranometer monthly data in comparison to the cell-measured monthly data are found from 343 September to March coinciding with the time of the year with the lower solar elevations. 344 Table 6. Monthly and annual cumulative values of the synthetic sets and their differences in 345 comparison to the measured reference cell data for the location of Seville. 346 347 Dataset Unit Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠 kWh/m2 62 87 154 159 195 229 229 210 165 114 77 70 1751 𝐺𝑝𝑦𝑟 kWh/m2 66 93 164 167 206 243 242 224 175 121 83 77 1861 𝐸𝑟𝑟𝑜𝑟𝑝𝑦𝑟 7.0 7.1 6.5 5.3 5.7 6.1 5.8 6.5 5.7 6.2 7.5 9.5 6.3 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ kWh/m2 62 87 154 160 197 232 231 212 165 114 76 70 1760 𝐸𝑟𝑟𝑜𝑟𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ -0.2 0.4 0.2 0.4 1.1 1.4 0.9 1.0 -0.4 -0.5 -0.7 -0.1 0.5 348 Table 7. Monthly and annual cumulative values of the synthetic sets and their differences in 349 comparison to the measured reference cell data for the location of Lancaster. 350 Dataset Unit Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year(*) 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠 kWh/m2 74 95 122 150 174 180 185 166 136 - 80 71 1432 𝐺𝑝𝑦𝑟 kWh/m2 80 102 130 159 181 190 194 176 149 - 89 78 1527 𝐸𝑟𝑟𝑜𝑟𝑝𝑦𝑟 7.8 8.0 6.1 5.6 4.4 5.6 4.7 6.4 9.4 - 11.1 9.3 6.6 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ kWh/m2 75 96 123 152 174 182 186 169 141 - 82 72 1452 𝐸𝑟𝑟𝑜𝑟𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ 0.8 1.9 1.0 0.9 0.3 1.4 0.4 1.6 3.7 - 3.3 1.3 1.4 (*)October not included in the total sum 351 The increased differences in Lancaster may be attributed to the good training of the model for 352 Seville (through the multi-step regression analysis) and the use of an empirical parameter 353 developed for Seville. 354 16 4.1 Mean evaluation 355 We evaluate the deviations between the modelled and measured PV reference cell data using 356 the Root Mean Squared Deviation (𝑅𝑀𝑆𝐷) calculated following equation 12. We also calculate 357 the RMSD of the pyranometer data in comparison to the calibrated cell-measured data in order 358 to show a base to compare (equation 13). 359 RMSD𝑐𝑒𝑙𝑙=√1 𝑁∑(𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠−𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ)2 𝑁 𝑖=1 (12) 360 RMSD𝑝𝑦𝑟=√1 𝑁∑(𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠−𝐺𝑝𝑦𝑟)2 𝑁 𝑖=1 (13) 361 where N is the number of data pairs, 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠 is the global horizontal irradiance measured with 362 the reference PV cell, 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ is the global horizontal irradiance synthetically obtained from 363 pyranometer measurements and 𝐺𝑝𝑦𝑟 is the solar radiation measured with the pyranometer. 364 The evaluation is performed in the 10-min and daily resolution. Only data for solar elevations 365 greater than 2º are considered in this analysis. In table 8 we present the RMSD𝑐𝑒𝑙𝑙 and in table 366 9 the RMSD𝑝𝑦𝑟 in the 10-min and daily resolution for the test datasets. We also present the 367 RMSD in % by dividing the RMSD by the average value (equations 14 and 15). 368 RMSD𝑐𝑒𝑙𝑙(%)=100∙(√1 𝑁∑(𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠−𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ)2 𝑁 𝑖=1 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠         ⁄) (14) 369 RMSD𝑝𝑦𝑟(%)=100∙(√1 𝑁∑(𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠−𝐺𝑝𝑦𝑟)2 𝑁 𝑖=1 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠         ⁄) (15) 370 371 For Seville, the average value (𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠         ) in the 10-min resolution is 391 W/m2 and in the daily 372 resolution is 4.7 kWh/m2. For Lancaster, the average value (𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠          ) in the 10-min resolution is 373 371 W/m2 and in the daily resolution is 3.8 kWh/m2. 374 Table 8. RMSD𝑐𝑒𝑙𝑙 in the 10-min and daily resolution for Seville and Lancaster test datasets. 375 Station (year) RMSD10-min RMSDdaily W/m2 % kWh/m2 % Seville (2016) 9.5 2.4 0.06 1.4 Lancaster (2016) 19.7 3.8 0.09 2.5 376 Table 9. RMSD𝑝𝑦𝑟 in the 10-min and daily resolution for Seville and Lancaster test datasets. 377 Station (year) RMSD10-min RMSDdaily W/m2 % kWh/m2 % Seville (2016) 29.6 7.5 0.33 6.9 Lancaster (2016) 41.7 8.1 0.29 7.7 378 17 The RMSD between measured data from the reference cell and pyranometer, mainly related to 379 spectral and angular losses, reach 8% on a daily basis. Moreover, when applying the correction 380 algorithm, the RMSD between measured data from the reference cell and synthetic data is 381 reduced to 2%. In figure 8 we present the daily 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠 vs the daily 𝐺𝑝𝑦𝑟 together, with the 382 daily 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ in Lancaster in the figure on the left and Seville in the figure on the right in a scatter 383 plot. We can observe how the BIAS is reduced significantly from values around 6% to values 384 lower than 2%. Reduction is even higher in Seville. 385 386 Figure 8. Scatter plot of the daily cumulative solar radiation values measured with the cell versus 387 the daily solar radiation values measured with the pyranometer and the synthetically generated 388 data for the location of Lancaster (a) and Seville (b). 389 4.2 Distribution evaluation 390 The distribution is evaluated since it is assumed that differences in these terms may lead to 391 differences in PV plant production. To evaluate the distribution, we calculate the KSI 392 (Kolmogorov-Smirnov test integral, equation 16) index defined as the integrated differences 393 between the CDFs of two datasets. It is a widely used index to compare cumulative distributions 394 of measured and synthetic solar radiation sets [32]. The unit of this index is the same for the 395 corresponding magnitude. We only analyse data for solar elevations greater than 2º. 396 𝐾𝑆𝐼=∫ 𝐷𝑛𝑑𝑥 𝑥𝑚𝑎𝑥 𝑥𝑚𝑖𝑛 , (16) 397 where, 𝑥𝑚𝑎𝑥 and 𝑥𝑚𝑖𝑛are the extreme values of the independent variable, and 𝐷𝑛 are the 398 differences between the CDFs of datasets evaluated in 100 points. The higher the KSI values, the 399 greater the differences in the CDFs of the evaluated datasets. We calculate the KSI as a 400 comparison of the CDFs of the solar radiation sets measured with the pyranometer to the solar 401 radiation sets measured with the PV reference cell in one hand, and the synthetically obtained 402 PV reference cell solar radiation sets to the solar radiation sets measured with the PV reference 403 cell in the other hand. This calculation is performed in order to quantify the impact of the 404 developed model when used as a correction factor for spectral and angular losses of silicon cells. 405 In Figure 9 we represent the CDFs of the measured and synthetic sets for Seville and Lancaster 406 2016. 407 18 408 Figure 9. CDFs of the measured and synthetic sets for Seville 2016 (a) and Lancaster 2016 (b). 409 𝐺𝑝𝑦𝑟 is the solar radiation measured with the pyranometer, 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠 is the solar radiation 410 measured with the PV reference cell and 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ is the synthetically obtained PV reference cell 411 data. 412 In Table 10 we present the KSI value (W/m2) of the evaluated datasets. The measurements with 413 pyranometer and reference PV cell present different CDFs. However, the synthetically generated 414 reference PV cell data present a significant improvement towards similarity to the measured PV 415 cell CDF. 416 Table 10. 𝐾𝑆𝐼 (W/m2) for Seville and Lancaster test datasets. 417 Parameter Station (year) 𝐺𝑝𝑦𝑟 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ KSI (W/m2) Seville (2016) 24.6 3.7 Lancaster (2016) 35.6 8.6 418 The similarity on the measured and synthetic reference PV cell data can be quantified with the 419 KSI. It is reduced by 85% for the location of Seville and by 75% in Lancaster. The model has been 420 trained with more than three years of measurements from the location of Seville, therefore its 421 performance is slightly better for Seville than for Lancaster. In any case, it is worth highlighting 422 that the model can be globally applied without any local adaptation. Obviously the errors are 423 lower for the location where the model has been trained. 424 4.3 Autocorrelation evaluation 425 There is a relation between consecutive values that should be maintained when generating 426 synthetic solar radiation data. To that end, we evaluate the autocorrelation qualitatively by 427 calculating and representing the ramp rates (RRs), defined as the difference between successive 428 data points over 10 minutes (equation 17). 429 430 𝑅𝑅=(𝐺𝐻𝐼𝑘−𝐺𝐻𝐼𝑘−1)∆𝑡 ⁄ (17) 431 where Δt refers to an interval of 10 minutes. The units will be given in W/m2∙min. We calculate 432 the absolute RR values for the annual datasets taking into account only daytime observations. 433 We qualitatively evaluate the RRs of each dataset through the CDFs (Figure 10) and 434 19 quantitatively through the Kolmogorov Smirnov Index of the CDFs of the Ramp rates (KSIRR) 435 (Table 11). 436 Figure 10. CDFs of the RRs of the measured and synthetic sets for Seville 2016 (a) and Lancaster 437 2016 (b). 𝐺𝑝𝑦𝑟 is the solar radiation measured with the pyranometer, 𝐺𝑐𝑒𝑙𝑙 𝑚𝑒𝑎𝑠 is the solar 438 radiation measured with the PV reference cell and 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ is the synthetically obtained PV 439 reference cell data. 440 441 Table 11. 𝐾𝑆𝐼𝑅𝑅 (W/m2۰min) for Seville and Lancaster test datasets. 442 Parameter Station (year) 𝐺𝑝𝑦𝑟 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ KSIRR (W/m2۰min) Seville (2016) 0.22 0.11 Lancaster (2016) 0.41 0.20 443 From the RRs evaluation we can observe a reduction in the KSIRR of around 50% in both locations 444 from 𝐺𝑝𝑦𝑟 to 𝐺𝑐𝑒𝑙𝑙 𝑠𝑦𝑛𝑡ℎ . The model, which has been trained with data measured in Seville, presents 445 therefore a better performance in terms of autocorrelation for the location of Seville than for 446 the location of Lancaster, but we can highlight its global applicability without any local 447 adaptation. Nevertheless, more locations should be evaluated to assess the performance of the 448 model in a wider range of climates. 449 5 Conclusions 450 In this paper, we present a simple model to obtain reference cell 10-min solar radiation data 451 from thermopile pyranometer measured data series with reasonable accuracy. Our model does 452 not require any other input data besides the pyranometer measurements and can be globally 453 applied with no local adaptation or calibration. We have trained the model in one location 454 (Seville, Spain) and applied it in two locations, Seville and Lancaster (USA), obtaining satisfactory 455 results. The model is developed from an initial cluster of the training data and several 456 regressions, taking into account the main parameters affecting the angular and spectral losses; 457 the solar elevation and the clearness index. At the time of using it, only thermopile pyranometer 458 data and a few geometrical calculations are required. This way, the model shows great simplicity 459 facilitating its applicability. 460 20 The model shows better performance on sunny days with high solar elevations, when most of 461 the energy is obtained. We have evaluated the mean, distribution and autocorrelation of the 462 synthetic sets in comparison to measured data with a reference cell using measured data with 463 a pyranometer as input. Daily RMSD is reduced from 7-8% to 1-2% when evaluating a complete 464 annual set. The frequency distribution of the synthetic sets is also improved. It has been 465 quantified in an 80% reduction of the KSI. The autocorrelation is quantified through the KSI of 466 the RRs obtaining also a reduction of 50%. The model shows better performance for the location 467 of Seville, where it has been trained, since solar radiation depends on local phenomena. Future 468 works will focus on the improvement of the model by using data from more different climates 469 for its training, and tests will focus on the evaluation of its global applicability. 470 Acknowledgments 471 The authors are grateful to Quintas Energy, S.A for providing the data from Lancaster. 472 References 473 [1] J. Meydbray, K. Emery. 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