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Improved methodology to evaluate clear-sky direct normal irradiance with a multi-pyranometer array
Available online at www.sciencedirect.com
ScienceDirect Solar Energy 121 (2015) 123–130 www.elsevier.com/locate/solener
Improved methodology to evaluate clear-sky direct normal irradiance with a multi-pyranometer array Juan-Carlos Baltazar ⇑, Yifu Sun, Jeff Haberl Energy Systems Laboratory, Texas A&M Engineering Experiment Station, The Texas A&M University System, College Station, TX 77845, USA Received 12 January 2015; received in revised form 3 July 2015; accepted 9 July 2015 Available online 18 August 2015
Abstract An improved methodology to estimate the clear-sky direct normal irradiance based on an anisotropic clear-sky model using a Multi-Pyranometer Array (MPA) is analyzed in this paper. The MPA is a static platform with four sensors on surfaces at different tilt and azimuth angles, which measured solar irradiance can be used to estimate the normal incident component without the tracking devices that involve more detailed installation and maintenance. The array’s sensors are of the photovoltaic type, which require both the spectral and angle-of-incidence corrections. Based on the general expressions for the solar radiation on each of the MPA sloped surfaces and using a hybrid model that includes an anisotropic sky diffuse component, in this work an improved procedure is presented that estimate the normal direct solar irradiance by grouping two expressions of a couple of MPA sensors and analytically look for a satisfactory solution; one set for the horizontal and south sensors, and another for the east and west sensors. The solution for the east and west sensors is expanded to two new solutions by mirroring their readings according to the solar noon. A final switching scheme based on the expressions denominator size provides the satisfactory direct normal irradiance estimation. The analysis of the results for the analyzed days showed that the root mean square error (RMSE) decreased by around 14–43% in comparison with previous results reported in the literature. The proposed methodology is a promising new approach for MPA. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Multi-pyranometer array; Normal incident pyrheliometer; Anisotropic solar clear-sky model
1. Introduction Solar energy is a renewable and plentiful resource that has been widely used, every day more, in numerous applications; some of those using the total extend of the resource and others require the main components – as the direct normal solar irradiance. The measurement of the solar resource provides and validates the potential of any solar application. Global solar radiation can be measured by using a pyranometer based on thermopile or
⇑ Corresponding author. Tel.: +1 979 862 7175.
E-mail address: [email protected] (J.-C. Baltazar). http://dx.doi.org/10.1016/j.solener.2015.07.015 0038-092X/Ó 2015 Elsevier Ltd. All rights reserved.
photoelectric detectors. However, the measurements of the direct and diffuse components required of extra equipment, such as a solar tracker to follow the sun path and solar shading devices. In general, a precision Normal Incidence Pyrheliometer (NIP) is fixed on a tracker to obtain the direct normal component of the irradiance. The diffuse component is measured by shading a pyranometer by disks or spheres elements, either installed on the tracker or in a basic set with a shadow band. This procedure is accurate as long as the tracker can precisely follow the sun path. This equipment set is expensive and can be just justified in cases where there are solar applications that have favorable capital-benefit ratio. Also, the mechanical tracking system has the potential of failure
124
J.-C. Baltazar et al. / Solar Energy 121 (2015) 123–130
Nomenclature AMa h Ib,n Id IT Rb Rd Rr h b c
absolute air mass height above sea level (km) normal incident beam radiation (W/m2) diffuse component solar radiation (W/m2) total solar radiation (W/m2) beam component geometric factor diffuse component geometric factor reflective component geometric factor incidence angle of beam radiation (°) collector tilt angle (°) azimuth angle (°)
and maintenance issues, which could range from soiling clean up to realignment of the tracker. Therefore, looking for a less expensive, similarly robust and almost maintenance free devices to measure or estimate the direct and diffuse components of the solar irradiance, many ideas have been proposed. A Multi-Pyranometer Array (MPA) is one of them, which was originally developed approximately thirty years back. Ha¨ma¨la¨inen et al. (1985) developed an instrument, which included twenty-five solar radiation sensors mounted on a metal hemisphere. The sensors were average distributed and protected by a glass bulb. Since then, many others researchers or practitioners have further studied how to simplify the instruments and to improve the accuracy of the estimation procedure. Faiman and Zemel (1987) reduce the number of the sensors to four and discussed the practical application issues about using a MPA system and also described a calculation method (Faiman et al., 1992, 1993). The diffuse sky model that is used in the estimation of the solar radiation incident on any slope play a significant role in the calculation of the components of solar radiation. In general there is a non-isotropic character on the skylight and the radiation reflected from surroundings, so some of those effect should be considered. In their estimations, Faiman et al. used the diffuse isotropic sky model developed by Liu and Jordan (1961). In similar MPA study, Curtiss (1993) investigated the effect of different diffuse sky as well as ground reflectance models in the estimation of the components of the solar radiation and proposed a hybrid diffuse model by replacing the Hay’s (1979) isotropic diffuse term with the Temps and Coulson (1977) anisotropic diffuse equation. The solution of the MPA equation set requires an iterative procedure and the measurements of the ground reflectance. Munger and Haberl (2000) avoided the effects from the ground reflection by adding an artificial horizon to the MPA and rearrange the expressions to be solved explicitly but still there exist three non-conclusive separate solutions. Munger also showed that at any time at least a valid solution could exists and that the solution set can be filtered out by a
d / k q x h s se sw 0
declination (°) latitude (°) generic indicator for any tilted surface in the MPA ground reflectance hour angle (°) horizontal surface south facing tilted surface east of south facing tilted surface west of south facing tilted surface the values or parameters in mirror equations
switching scheme based on the value of the denominator of the MPA equations. Klima (2000) test several other switching schemes looking for an optimal method that could improve the accuracy of the MPA, thought a slightly improvement was reached he noted that any of his schemes could not find a valid solution for all the insolation conditions. In this study a new alternative solution procedure is proposed to improve the accuracy and the stability of the MPA expressions based on the general relationships for the solar radiation on each of the sloped surfaces of the MPA, and similarly to previous studies using a hybrid model that includes the diffuse component for anisotropic skies. The method analytically look for a satisfactory direct normal irradiance estimation based on two solutions expressions each one referred to a couple of MPA sensors, one set for the horizontal and south sensors, and another for the east and west sensors. The solution for the east and west sensors is expanded with their two mirror readings according to the solar noon. A final switching scheme based on the denominator size of the two solutions expressions delivers a satisfactory direct normal irradiance.
Fig. 1. Overall view of the solar test rig.
J.-C. Baltazar et al. / Solar Energy 121 (2015) 123–130
2. Measurement test rig setup The solar irradiance measurement sensors and the other supplementary devices are installed on the roof of Langford Architecture building at Texas A&M University, College Station campus. Fig. 1 shows the overall views of the test rig. The climate conditions in this region are hot and humid (climatic zone 2A). The test rig consists of several pyranometers for measuring global solar radiation, a couple of Normal Incidence Pyrheliometers (NIP) mounted on a solar tracker for measuring direct normal irradiance, two Black and White pyranometers (B&W) with focal shading disks to measure the solar diffuse component, and the Multi-Pyranometer Array (MPA). It also includes additional meteorological devices such as dry-bulb temperature, relative humidity, and wind speed and wind direction. The MPA consists of four pyranometers with photovoltaic sensor (Licor LI-200): one sensor mounted in a horizontal plane, and three others tilted 40° and distributed in azimuth angles of 60, 0, and 60° – toward southeast, south and southwest, respectively. A metallic band around the MPA is used as an artificial horizon to block the ground reflected radiation. Fig. 2 presents the MPA set built for this study.
To determine the accuracy of each of the individual sensor, these were compared for the whole month of October 2012, side by side with a track reference PSP. Fig. 3 shows the comparison between the four sensors against the PSP. All of the sensor in average have a good accuracy considering all the solar radiation conditions in the selected period. The spectral and the angle of incidence corrections (King and Myers, 1997) are determined by the correlation functions ((f1) and (f2), respectively, shown below 3
The MPA’s sensors are of photovoltaic-type that are not as accurate as the thermopile-type pyranometers like the Precision Spectral Pyranometers (PSP), due to the narrow band of light spectrum that they detect, but their cost correspond to a fraction of the PSP. King and Myers (1997) proposed methods to calibrate silicon-photodiode pyranometers to effects associated with solar spectrum, solar incidence angle and temperature. Other author have also, for pyranometers based on the same technology, have found error in the measurements of diffuse solar radiation or systematic deviations (Vignola, 1999, 2006). In this study the empirical expressions provided by King and Myers, which are location independent, for spectral and solar incidence angle calibration were applied to the MPA sensors.
2
f 1 ¼ 0:000263 ðAM a Þ 0:00632 ðAM a Þ þ 0:054 ðAM a Þ þ 0:932
ð1Þ
where AMa is the absolute air mass estimated by AM a ¼ e0:0001184h ½cosðhÞ þ 0:5057 ð96:080 hÞ1:634
1
ð1aÞ 3
f 2 ¼ 0:0000004504 h 0:00001357 h þ 0:0006074 h þ 1
2
ð2Þ
and h is the incidence angle in degrees and h is the height above sea level in km. And the calibrated tilted solar irradiance (Itcorr) is expressed as a function of the tilted one as following. I tcorr ¼ I t ðf 1 f 2 Þ
3. Sensors calibration
125
1
ð3Þ
Fig. 4 shows the percentage difference between the global horizontal solar irradiance measured for each MPA sensor after the spectral and incidence angle correction and the obtained from the PSP reference. 4. Improved methodology for the estimation of the clear-sky direct normal irrradiance In any surface the total incident solar radiation includes a direct beam, a diffuse and a ground reflection component. Considering that the ground reflection portion can be ignored by the integration of an artificial horizon in the MPA, the incident global solar radiation on each of its surfaces can be expressed by the following equations (4) to (7), for horizontal, toward southeast, south and toward the southwest surfaces, respectively.
Fig. 2. MPA set used for this study.
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Fig. 3. Alignment for MPA sensors 1 through 4 with PSP sensor.
Fig. 4. Percentage difference between MPA sensors 1 through 4 after calibration and PSP.
J.-C. Baltazar et al. / Solar Energy 121 (2015) 123–130
127
Fig. 5. Block diagram of the proposed methodology.
Fig. 6. Multi-pyranometer four sensors records and measured NIP reference in the testing period.
I T ;h ¼ I b;n Rb;h þ I d;h
ð4Þ
I T ;se ¼ I b;n Rb;se þ I d;h Rd;se
ð5Þ
I T ;s ¼ I b;n Rb;s þ I d;h Rd;s
ð6Þ
I T ;sw ¼ I b;n Rb;sw þ I d;h Rd;sw
ð7Þ
where I represents the solar irradiance and the subscripts refers to the following: T, total; b, beam; n, normal; d, diffuse; h, horizontal; s, south; se, southeast; and sw, southwest. The beam and diffuse geometric factors Rb and Rd for each sensor can be estimated by expressions presented in Appendix A. Based on the equations (4)–(7) and two variables (Ib,n and Id,h), several expressions for calculating Ib,n can be found. In this study it was observed that the impact of the measurement error onto the estimation can be relatively reduced by the calibration and the transformation of the solution expressions. The improved proposed methodology is summarized next:
Grouping the equations for the MPA sensors, horizontal and south, through equations (4) and (6), the direct normal irradiance can be expressed as follows I bn;hs ¼
I T ;h Rd;s I T ;s Rb;h Rd;s Rb;s
ð8Þ
Similar equation for direct normal irradiance, using equations (5) and (7), can be derived using the measurement of the sensor toward southeast and its mirrored values, by Eq. (9). In the mirrored equation, the geometric factors, Rb,se0 and Rd,se0 , and the measurements are referred to the angles for the opposite position according to the solar noon. I bn;ew0 ¼
I T ;se Rd;sw0 I T ;sw0 Rd;se Rb;se Rd;sw0 Rb;sw0 Rd;se
ð9Þ
In a same way, other expression for the direct normal irradiance using the measurements of the sensor toward the southwest and its mirrored can be determined.
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Fig. 7. Measured and estimated direct normal irradiance.
I bn;we0 ¼
I T ;sw Rd;se0 I T ;se0 Rd;sw Rb;sw Rd;se0 Rb;se0 Rd;sw
ð10Þ
From these expressions, and to minimize the impact of the amplification effect inherent to the expression, another alternative solution is realized through a relationship of the sum of the numerators and denominators of both the equations (9) and (10) I bn ¼
I T ;se Rd;sw0 I T ;sw0 Rd;se þ I T ;sw Rd;se0 I T ;se0 Rd;sw Rb;sw Rd;sw0 Rb;sw0 Rd;se þ Rb;sw Rd;se0 Rb;se0 Rd;sw ð11Þ
The selection of the more suitable solution for a clear-sky would be the one that correspond to the maximum denominator of either Eq. (8) or (11). The whole process is outlined in the Fig. 5 below. 5. Results analysis Comparison of the measurements from NIP and B&W and estimations from Eqs. (4)–(11) were used to validate the proposed methodology, and evaluate their differences between the estimated and the actual solar irradiance measurements. Fig. 6 shows the MPA measured data and the
Fig. 8. Measured and estimated normal incident solar radiation for clear-sky days between (a) December 17–20, 2012, and (b) September 22–25, 2013.
J.-C. Baltazar et al. / Solar Energy 121 (2015) 123–130
129
RMSE, which is the root mean square error, and CVRMSE that is the coefficient of variation of the root mean square error. Table 2 shows the comparison of this indicators to other previously published. 6. Conclusion
Fig. 9. Measurement vs. estimation of direct normal irradiance.
Table 1 The comparison between MPA estimation of direct normal irradiance and the measured value (NIP) for clear-sky days in the indicated periods. Test dates
Difference (%) RMSE (W/m2) CV-RMSE (%)
November 14–24, 2012 December 17–20, 2012, September 22–25, 2013 All days
1.0 3.0 0.1 1.4
38 62 30 43
12 21 9 14
This study analyzed an improved methodology to estimate the direct normal irradiance for Multi-Pyranometer Arrays (MPA), which represent a cost fraction of a traditional tracking measurement devices. The photovoltaic sensors in the MPA for their narrow spectral response required both spectral and incidence angle corrections. The model for disaggregating beam and diffuse components is based on an anisotropic sky considerations and an improved methodology was illustrated based on the solutions that could be generated from the representative equations for a main pair of photovoltaic sensors in the MPA. The estimation of the direct normal irradiance for clear-sky days conditions have shown high reliability and stability. The total difference between the direct normal irradiance between the MPA estimated and the measured for the testing period clear days is 1.4%, and the CV-RMSE is around 14% (RMSE of 43 W/m2). The accuracy is better than similar studies reported in previous publications. The size of sample used is this study was small, and more tests for different seasons and period length need to be addressed to stress the reliability of the methodology. The proposed methodology is a promising new approach.
Table 2 The comparison of this study and other publications. Publication
Difference (%)
RMSE (W/m2)
CV-RMSE (%)
This study Marion (1998) Maxwell et al. (1999) Klima (2000) Faiman et al. (1992) Curtiss (1993)
1.4 4.0 4.0 NA NA NA
43 76 76 54 50 114.9
14 14 14 NA NA NA
Appendix A. Solar energy expressions for the sensor of the multi-pyranometer array The total solar radiation incident upon the horizontal MPA sensor can be estimated directly form the following expression I T ;h ¼ I b;n Rb;h þ I d;h
reference NIP measurements for three selected complete clear-sky days. In Fig. 7 a comparison between the estimated 15-min direct normal irradiance and the corresponding measured value is presented for the same days. It is observed in Fig. 7 that the differences between the measured and estimated values of the direct normal irradiance still exist in certain conditions and time of the day, but the results are not diverging as in previous studies (Munger and Haberl, 2000; Klima, 2000) and follow the pattern expected for a clear-sky day. Similar results were found in different sample on different period as shown in Fig. 8a and b. Fig. 9 shows the comparison plot between estimation and measured for a group of clear days in all the different periods, and it is observed that the pattern is mostly close to the optimal diagonal line. Table 1 summarizes the estimation results, and statistical indicators, as the
ðA1Þ
Similar expression for the total solar radiation incident upon the surfaces of the MPA toward south, southeast and southwest are respectively presented next I T ;se ¼ I b;n Rb;se þ I d;h Rd;se þ I T ;h q Rr;se
ðA2Þ
I T ;s ¼ I b;n Rb;s þ I d;h Rd;s þ I T ;h q Rr;s
ðA3Þ
I T ;sw ¼ I b;n Rb;sw þ I d;h Rd;sw þ I T ;h q Rr;sw
ðA4Þ
The corresponding beam coefficient for each of the surfaces included in the MPA is Rb;k ¼ cosðhi;k Þ
ðA5Þ
where k indicates any surface in the MPA. The diffuse coefficients are calculated based on the Temps–Coulson (Temps and Coulson, 1977) model and are expressed for each tilted surfaces as
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Rd;se ¼
Rd;s ¼
1 þ cosðbse Þ 2 3 bse 1 þ sin 1 þ cos2 ðhi;se Þ sin3 ðhi;h Þ 2
1 þ cosðbs Þ 2 3 bs 1 þ sin 1 þ cos2 ðhi;s Þ sin3 ðhi;h Þ 2
Rd;sw ¼
1 þ cosðbsw Þ 2 3 bsw 1 þ sin 1 þ cos2 ðhi;sw Þ sin3 ðhi;h Þ 2
ðA6Þ
ðA7Þ
ðA8Þ
The incidence angles on the horizontal surface for any time interval are determined by cosðhi;h Þ ¼ cosð/Þ cosðdÞ cosðxÞ þ sinð/Þ sinðdÞ
ðA9Þ
and for the other MPA surfaces by the following expression cosðhi;k Þ ¼ sinðdÞ sinð/Þ cosðbk Þ sinðdÞ cosð/Þ sinðbk Þ cosðck Þ þ cosðdÞ cosð/Þ cosðbk Þ cosðxÞ þ cosðdÞ sinð/Þ sinðbk Þ cosðck Þ cosðxÞ þ cosðdÞ sinðbk Þ sinðck Þ sinðxÞ
ðA10Þ
References Curtiss, P., 1993. An analysis of methods for driving the constituent insolation components from multipyranometer array measurements. J. Sol.Energy Eng. 115. Faiman, D., Zemel, A., 1987. A method for monitoring insolation in remote regions. Sol. Energy 38 (5), 327–333.
Faiman, D., Feuermann, D., Ibbetson, P., Zemel, A., 1992. A multipyranometer instrument for obtaining the solar beam and diffuse components, and the irradiance on inclined planes. Sol. Energy 48 (4), 253– 259. Faiman, D., Feuermann, D., Zemel, A.D., 1993. Site-independent algorithm for obtaining the direct beam insolation from a multipyranometer instrument. Sol. Energy 50 (1), 53–57. Ha¨ma¨la¨inen, M., Nurkkanen, P., Sla˚en, T., 1985. A multisensor pyranometer for determination of the direct component and angular direct component and angular distribution of solar radiation. Sol. Energy 35 (6), 511–525. Hay, J.E., 1979. Calculation of monthly mean solar radiation for horizontal and inclined surfaces. Sol. Energy 23 (4), 301–307. King, D.L., Myers, D.R., 1997. Silicon-Photodiode pyranometers: operational characteristics, historical experiences, and new calibration procedures. 26th IEEE Photovoltaic Specialists Conference, 1285– 1288. Klima, P.M., 2000. Improving the Reliability and Accuracy of a Multipyranometer Array Measuring Solar Radiation. Master thesis, Architecture Department, Texas A&M University. Liu, B., Jordan, R., 1961. Daily insolation on surfaces tilted towards equator. ASHRAE J. 3, 53–59. Marion, B., 1998. Multi-pyranometer array design and performance summary. Am. Solar Energy Soc. Annual Conf., 523–528 Maxwell, E.L., Wilcox, S.M., Cornwall, C., Marion, B., Alawaji, S. H., Mahfoodh, M., Amoudi, A.A., 1999. Progress Report for Annex II – Assessment of Solar Radiation Resources in Saudi Arabia 1993–1997. Technical Report NREL/TP-560-25374, National Renewable Energy Laboratory, Golden, CO. Munger, B.K., Haberl, J.S., 2000. Development of simplified calculations for a multipyrameter array for the measurement of direct and diffuse solar radiation. In: The Twelfth Symposium on Improving Building Systems in Hot and Humid Climates, San Antonio, TX. Temps, R., Coulson, L., 1977. Solar radiation incident upon slopes of different orientations. Sol. Energy 19 (2), 179–184. Vignola, F., 1999. Sollar cell based pyranometers: evaluation of diffuse responsivity. Am. Solar Energy Soc. Annual Conf., 260–265 Vignola, F., 2006. Removing systematic errors from rotating shadwband pyranometer Data. In: Proc. ASES Annual Conference, Denver, CO.
ScienceDirect Solar Energy 121 (2015) 123–130 www.elsevier.com/locate/solener
Improved methodology to evaluate clear-sky direct normal irradiance with a multi-pyranometer array Juan-Carlos Baltazar ⇑, Yifu Sun, Jeff Haberl Energy Systems Laboratory, Texas A&M Engineering Experiment Station, The Texas A&M University System, College Station, TX 77845, USA Received 12 January 2015; received in revised form 3 July 2015; accepted 9 July 2015 Available online 18 August 2015
Abstract An improved methodology to estimate the clear-sky direct normal irradiance based on an anisotropic clear-sky model using a Multi-Pyranometer Array (MPA) is analyzed in this paper. The MPA is a static platform with four sensors on surfaces at different tilt and azimuth angles, which measured solar irradiance can be used to estimate the normal incident component without the tracking devices that involve more detailed installation and maintenance. The array’s sensors are of the photovoltaic type, which require both the spectral and angle-of-incidence corrections. Based on the general expressions for the solar radiation on each of the MPA sloped surfaces and using a hybrid model that includes an anisotropic sky diffuse component, in this work an improved procedure is presented that estimate the normal direct solar irradiance by grouping two expressions of a couple of MPA sensors and analytically look for a satisfactory solution; one set for the horizontal and south sensors, and another for the east and west sensors. The solution for the east and west sensors is expanded to two new solutions by mirroring their readings according to the solar noon. A final switching scheme based on the expressions denominator size provides the satisfactory direct normal irradiance estimation. The analysis of the results for the analyzed days showed that the root mean square error (RMSE) decreased by around 14–43% in comparison with previous results reported in the literature. The proposed methodology is a promising new approach for MPA. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Multi-pyranometer array; Normal incident pyrheliometer; Anisotropic solar clear-sky model
1. Introduction Solar energy is a renewable and plentiful resource that has been widely used, every day more, in numerous applications; some of those using the total extend of the resource and others require the main components – as the direct normal solar irradiance. The measurement of the solar resource provides and validates the potential of any solar application. Global solar radiation can be measured by using a pyranometer based on thermopile or
⇑ Corresponding author. Tel.: +1 979 862 7175.
E-mail address: [email protected] (J.-C. Baltazar). http://dx.doi.org/10.1016/j.solener.2015.07.015 0038-092X/Ó 2015 Elsevier Ltd. All rights reserved.
photoelectric detectors. However, the measurements of the direct and diffuse components required of extra equipment, such as a solar tracker to follow the sun path and solar shading devices. In general, a precision Normal Incidence Pyrheliometer (NIP) is fixed on a tracker to obtain the direct normal component of the irradiance. The diffuse component is measured by shading a pyranometer by disks or spheres elements, either installed on the tracker or in a basic set with a shadow band. This procedure is accurate as long as the tracker can precisely follow the sun path. This equipment set is expensive and can be just justified in cases where there are solar applications that have favorable capital-benefit ratio. Also, the mechanical tracking system has the potential of failure
124
J.-C. Baltazar et al. / Solar Energy 121 (2015) 123–130
Nomenclature AMa h Ib,n Id IT Rb Rd Rr h b c
absolute air mass height above sea level (km) normal incident beam radiation (W/m2) diffuse component solar radiation (W/m2) total solar radiation (W/m2) beam component geometric factor diffuse component geometric factor reflective component geometric factor incidence angle of beam radiation (°) collector tilt angle (°) azimuth angle (°)
and maintenance issues, which could range from soiling clean up to realignment of the tracker. Therefore, looking for a less expensive, similarly robust and almost maintenance free devices to measure or estimate the direct and diffuse components of the solar irradiance, many ideas have been proposed. A Multi-Pyranometer Array (MPA) is one of them, which was originally developed approximately thirty years back. Ha¨ma¨la¨inen et al. (1985) developed an instrument, which included twenty-five solar radiation sensors mounted on a metal hemisphere. The sensors were average distributed and protected by a glass bulb. Since then, many others researchers or practitioners have further studied how to simplify the instruments and to improve the accuracy of the estimation procedure. Faiman and Zemel (1987) reduce the number of the sensors to four and discussed the practical application issues about using a MPA system and also described a calculation method (Faiman et al., 1992, 1993). The diffuse sky model that is used in the estimation of the solar radiation incident on any slope play a significant role in the calculation of the components of solar radiation. In general there is a non-isotropic character on the skylight and the radiation reflected from surroundings, so some of those effect should be considered. In their estimations, Faiman et al. used the diffuse isotropic sky model developed by Liu and Jordan (1961). In similar MPA study, Curtiss (1993) investigated the effect of different diffuse sky as well as ground reflectance models in the estimation of the components of the solar radiation and proposed a hybrid diffuse model by replacing the Hay’s (1979) isotropic diffuse term with the Temps and Coulson (1977) anisotropic diffuse equation. The solution of the MPA equation set requires an iterative procedure and the measurements of the ground reflectance. Munger and Haberl (2000) avoided the effects from the ground reflection by adding an artificial horizon to the MPA and rearrange the expressions to be solved explicitly but still there exist three non-conclusive separate solutions. Munger also showed that at any time at least a valid solution could exists and that the solution set can be filtered out by a
d / k q x h s se sw 0
declination (°) latitude (°) generic indicator for any tilted surface in the MPA ground reflectance hour angle (°) horizontal surface south facing tilted surface east of south facing tilted surface west of south facing tilted surface the values or parameters in mirror equations
switching scheme based on the value of the denominator of the MPA equations. Klima (2000) test several other switching schemes looking for an optimal method that could improve the accuracy of the MPA, thought a slightly improvement was reached he noted that any of his schemes could not find a valid solution for all the insolation conditions. In this study a new alternative solution procedure is proposed to improve the accuracy and the stability of the MPA expressions based on the general relationships for the solar radiation on each of the sloped surfaces of the MPA, and similarly to previous studies using a hybrid model that includes the diffuse component for anisotropic skies. The method analytically look for a satisfactory direct normal irradiance estimation based on two solutions expressions each one referred to a couple of MPA sensors, one set for the horizontal and south sensors, and another for the east and west sensors. The solution for the east and west sensors is expanded with their two mirror readings according to the solar noon. A final switching scheme based on the denominator size of the two solutions expressions delivers a satisfactory direct normal irradiance.
Fig. 1. Overall view of the solar test rig.
J.-C. Baltazar et al. / Solar Energy 121 (2015) 123–130
2. Measurement test rig setup The solar irradiance measurement sensors and the other supplementary devices are installed on the roof of Langford Architecture building at Texas A&M University, College Station campus. Fig. 1 shows the overall views of the test rig. The climate conditions in this region are hot and humid (climatic zone 2A). The test rig consists of several pyranometers for measuring global solar radiation, a couple of Normal Incidence Pyrheliometers (NIP) mounted on a solar tracker for measuring direct normal irradiance, two Black and White pyranometers (B&W) with focal shading disks to measure the solar diffuse component, and the Multi-Pyranometer Array (MPA). It also includes additional meteorological devices such as dry-bulb temperature, relative humidity, and wind speed and wind direction. The MPA consists of four pyranometers with photovoltaic sensor (Licor LI-200): one sensor mounted in a horizontal plane, and three others tilted 40° and distributed in azimuth angles of 60, 0, and 60° – toward southeast, south and southwest, respectively. A metallic band around the MPA is used as an artificial horizon to block the ground reflected radiation. Fig. 2 presents the MPA set built for this study.
To determine the accuracy of each of the individual sensor, these were compared for the whole month of October 2012, side by side with a track reference PSP. Fig. 3 shows the comparison between the four sensors against the PSP. All of the sensor in average have a good accuracy considering all the solar radiation conditions in the selected period. The spectral and the angle of incidence corrections (King and Myers, 1997) are determined by the correlation functions ((f1) and (f2), respectively, shown below 3
The MPA’s sensors are of photovoltaic-type that are not as accurate as the thermopile-type pyranometers like the Precision Spectral Pyranometers (PSP), due to the narrow band of light spectrum that they detect, but their cost correspond to a fraction of the PSP. King and Myers (1997) proposed methods to calibrate silicon-photodiode pyranometers to effects associated with solar spectrum, solar incidence angle and temperature. Other author have also, for pyranometers based on the same technology, have found error in the measurements of diffuse solar radiation or systematic deviations (Vignola, 1999, 2006). In this study the empirical expressions provided by King and Myers, which are location independent, for spectral and solar incidence angle calibration were applied to the MPA sensors.
2
f 1 ¼ 0:000263 ðAM a Þ 0:00632 ðAM a Þ þ 0:054 ðAM a Þ þ 0:932
ð1Þ
where AMa is the absolute air mass estimated by AM a ¼ e0:0001184h ½cosðhÞ þ 0:5057 ð96:080 hÞ1:634
1
ð1aÞ 3
f 2 ¼ 0:0000004504 h 0:00001357 h þ 0:0006074 h þ 1
2
ð2Þ
and h is the incidence angle in degrees and h is the height above sea level in km. And the calibrated tilted solar irradiance (Itcorr) is expressed as a function of the tilted one as following. I tcorr ¼ I t ðf 1 f 2 Þ
3. Sensors calibration
125
1
ð3Þ
Fig. 4 shows the percentage difference between the global horizontal solar irradiance measured for each MPA sensor after the spectral and incidence angle correction and the obtained from the PSP reference. 4. Improved methodology for the estimation of the clear-sky direct normal irrradiance In any surface the total incident solar radiation includes a direct beam, a diffuse and a ground reflection component. Considering that the ground reflection portion can be ignored by the integration of an artificial horizon in the MPA, the incident global solar radiation on each of its surfaces can be expressed by the following equations (4) to (7), for horizontal, toward southeast, south and toward the southwest surfaces, respectively.
Fig. 2. MPA set used for this study.
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Fig. 3. Alignment for MPA sensors 1 through 4 with PSP sensor.
Fig. 4. Percentage difference between MPA sensors 1 through 4 after calibration and PSP.
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Fig. 5. Block diagram of the proposed methodology.
Fig. 6. Multi-pyranometer four sensors records and measured NIP reference in the testing period.
I T ;h ¼ I b;n Rb;h þ I d;h
ð4Þ
I T ;se ¼ I b;n Rb;se þ I d;h Rd;se
ð5Þ
I T ;s ¼ I b;n Rb;s þ I d;h Rd;s
ð6Þ
I T ;sw ¼ I b;n Rb;sw þ I d;h Rd;sw
ð7Þ
where I represents the solar irradiance and the subscripts refers to the following: T, total; b, beam; n, normal; d, diffuse; h, horizontal; s, south; se, southeast; and sw, southwest. The beam and diffuse geometric factors Rb and Rd for each sensor can be estimated by expressions presented in Appendix A. Based on the equations (4)–(7) and two variables (Ib,n and Id,h), several expressions for calculating Ib,n can be found. In this study it was observed that the impact of the measurement error onto the estimation can be relatively reduced by the calibration and the transformation of the solution expressions. The improved proposed methodology is summarized next:
Grouping the equations for the MPA sensors, horizontal and south, through equations (4) and (6), the direct normal irradiance can be expressed as follows I bn;hs ¼
I T ;h Rd;s I T ;s Rb;h Rd;s Rb;s
ð8Þ
Similar equation for direct normal irradiance, using equations (5) and (7), can be derived using the measurement of the sensor toward southeast and its mirrored values, by Eq. (9). In the mirrored equation, the geometric factors, Rb,se0 and Rd,se0 , and the measurements are referred to the angles for the opposite position according to the solar noon. I bn;ew0 ¼
I T ;se Rd;sw0 I T ;sw0 Rd;se Rb;se Rd;sw0 Rb;sw0 Rd;se
ð9Þ
In a same way, other expression for the direct normal irradiance using the measurements of the sensor toward the southwest and its mirrored can be determined.
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Fig. 7. Measured and estimated direct normal irradiance.
I bn;we0 ¼
I T ;sw Rd;se0 I T ;se0 Rd;sw Rb;sw Rd;se0 Rb;se0 Rd;sw
ð10Þ
From these expressions, and to minimize the impact of the amplification effect inherent to the expression, another alternative solution is realized through a relationship of the sum of the numerators and denominators of both the equations (9) and (10) I bn ¼
I T ;se Rd;sw0 I T ;sw0 Rd;se þ I T ;sw Rd;se0 I T ;se0 Rd;sw Rb;sw Rd;sw0 Rb;sw0 Rd;se þ Rb;sw Rd;se0 Rb;se0 Rd;sw ð11Þ
The selection of the more suitable solution for a clear-sky would be the one that correspond to the maximum denominator of either Eq. (8) or (11). The whole process is outlined in the Fig. 5 below. 5. Results analysis Comparison of the measurements from NIP and B&W and estimations from Eqs. (4)–(11) were used to validate the proposed methodology, and evaluate their differences between the estimated and the actual solar irradiance measurements. Fig. 6 shows the MPA measured data and the
Fig. 8. Measured and estimated normal incident solar radiation for clear-sky days between (a) December 17–20, 2012, and (b) September 22–25, 2013.
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RMSE, which is the root mean square error, and CVRMSE that is the coefficient of variation of the root mean square error. Table 2 shows the comparison of this indicators to other previously published. 6. Conclusion
Fig. 9. Measurement vs. estimation of direct normal irradiance.
Table 1 The comparison between MPA estimation of direct normal irradiance and the measured value (NIP) for clear-sky days in the indicated periods. Test dates
Difference (%) RMSE (W/m2) CV-RMSE (%)
November 14–24, 2012 December 17–20, 2012, September 22–25, 2013 All days
1.0 3.0 0.1 1.4
38 62 30 43
12 21 9 14
This study analyzed an improved methodology to estimate the direct normal irradiance for Multi-Pyranometer Arrays (MPA), which represent a cost fraction of a traditional tracking measurement devices. The photovoltaic sensors in the MPA for their narrow spectral response required both spectral and incidence angle corrections. The model for disaggregating beam and diffuse components is based on an anisotropic sky considerations and an improved methodology was illustrated based on the solutions that could be generated from the representative equations for a main pair of photovoltaic sensors in the MPA. The estimation of the direct normal irradiance for clear-sky days conditions have shown high reliability and stability. The total difference between the direct normal irradiance between the MPA estimated and the measured for the testing period clear days is 1.4%, and the CV-RMSE is around 14% (RMSE of 43 W/m2). The accuracy is better than similar studies reported in previous publications. The size of sample used is this study was small, and more tests for different seasons and period length need to be addressed to stress the reliability of the methodology. The proposed methodology is a promising new approach.
Table 2 The comparison of this study and other publications. Publication
Difference (%)
RMSE (W/m2)
CV-RMSE (%)
This study Marion (1998) Maxwell et al. (1999) Klima (2000) Faiman et al. (1992) Curtiss (1993)
1.4 4.0 4.0 NA NA NA
43 76 76 54 50 114.9
14 14 14 NA NA NA
Appendix A. Solar energy expressions for the sensor of the multi-pyranometer array The total solar radiation incident upon the horizontal MPA sensor can be estimated directly form the following expression I T ;h ¼ I b;n Rb;h þ I d;h
reference NIP measurements for three selected complete clear-sky days. In Fig. 7 a comparison between the estimated 15-min direct normal irradiance and the corresponding measured value is presented for the same days. It is observed in Fig. 7 that the differences between the measured and estimated values of the direct normal irradiance still exist in certain conditions and time of the day, but the results are not diverging as in previous studies (Munger and Haberl, 2000; Klima, 2000) and follow the pattern expected for a clear-sky day. Similar results were found in different sample on different period as shown in Fig. 8a and b. Fig. 9 shows the comparison plot between estimation and measured for a group of clear days in all the different periods, and it is observed that the pattern is mostly close to the optimal diagonal line. Table 1 summarizes the estimation results, and statistical indicators, as the
ðA1Þ
Similar expression for the total solar radiation incident upon the surfaces of the MPA toward south, southeast and southwest are respectively presented next I T ;se ¼ I b;n Rb;se þ I d;h Rd;se þ I T ;h q Rr;se
ðA2Þ
I T ;s ¼ I b;n Rb;s þ I d;h Rd;s þ I T ;h q Rr;s
ðA3Þ
I T ;sw ¼ I b;n Rb;sw þ I d;h Rd;sw þ I T ;h q Rr;sw
ðA4Þ
The corresponding beam coefficient for each of the surfaces included in the MPA is Rb;k ¼ cosðhi;k Þ
ðA5Þ
where k indicates any surface in the MPA. The diffuse coefficients are calculated based on the Temps–Coulson (Temps and Coulson, 1977) model and are expressed for each tilted surfaces as
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Rd;se ¼
Rd;s ¼
1 þ cosðbse Þ 2 3 bse 1 þ sin 1 þ cos2 ðhi;se Þ sin3 ðhi;h Þ 2
1 þ cosðbs Þ 2 3 bs 1 þ sin 1 þ cos2 ðhi;s Þ sin3 ðhi;h Þ 2
Rd;sw ¼
1 þ cosðbsw Þ 2 3 bsw 1 þ sin 1 þ cos2 ðhi;sw Þ sin3 ðhi;h Þ 2
ðA6Þ
ðA7Þ
ðA8Þ
The incidence angles on the horizontal surface for any time interval are determined by cosðhi;h Þ ¼ cosð/Þ cosðdÞ cosðxÞ þ sinð/Þ sinðdÞ
ðA9Þ
and for the other MPA surfaces by the following expression cosðhi;k Þ ¼ sinðdÞ sinð/Þ cosðbk Þ sinðdÞ cosð/Þ sinðbk Þ cosðck Þ þ cosðdÞ cosð/Þ cosðbk Þ cosðxÞ þ cosðdÞ sinð/Þ sinðbk Þ cosðck Þ cosðxÞ þ cosðdÞ sinðbk Þ sinðck Þ sinðxÞ
ðA10Þ
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