HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 47(1) pp. 57–63 (2019) hjic.mk.uni-pannon.hu DOI: 10.33927/hjic-2019-09 MODELING AND CALCULATION OF THE GLOBAL SOLAR IRRADIANCE ON SLOPES ROLAND BÁLINT *1,ATTILA FODOR1,ISTVÁN SZALKAI2,ZSÓFIA SZALKAI3, AND ATTILA MAGYAR1 1Department of Electrical Engineering and Information Systems, Faculty of Information Technology, University of Pannonia, Egyetem u. 10, Veszprém, 8200, HUNGARY 2Department of Mathematics, Faculty of Information Technology, University of Pannonia, Egyetem u. 10, Veszprém, 8200, HUNGARY 3Eötvös Loránd University, Egyetem tér 1-3, Budapest, 1053, HUNGARY The first step with regard to a simple model of a Photovoltaic Power Plant is developed in this paper based on astronomical and engineering principles. A solar irradiance model is presented in this paper that can be used to forecast the solar energy a surface on Earth is exposed to. The obtained model is verified against engineering expectations. The developed model can serve as a basis for forecasting the power of solar energy. Keywords: solar irradiance, modeling, solar geometry, inclined surface 1. Introduction well. The computational complexity of the model is also a key factor which should be kept to a minimum. As sustainable development is becoming highly impor- The organization of this paper is as follows: Section tant all over the world, the power production forecast 2 introduces the problem at hand and also presents the of Photovoltaic Power Plants (PVPPs) is becoming ever motivation of the paper. Afterwards, the elementary as- more necessary. The Sun is the greatest potential en- tronomical relationships are collected and the proposed ergy source available. The thermal energy it emits can be model is developed in Section3. The mathematical model transformed into electrical energy, or its irradiance can be is verified against basic engineering expectations in Sec- converted into electricity by solar panels. The amount of tion4. Finally, some concluding remarks are presented. energy produced by both of the solutions above is directly dependent on the solar irradiance. 2. Problem statement Several works have focused on the determination or The produced energy forecast of the renewable energy forecast of this energy and its dependence on the geomet- sources is one of the main problems of the Distributed rical parameters of the surface. A general model of the Energy Resource Management System. The operators of angle of incidence of solar irradiance with regard to the the electrical grids need precise mathematical models of azimuth is derived in Ref.[1] that is used for both fixed renewable power plants, e.g. PVPPs, wind farms or hy- and tracking surfaces. In Ref.[2], a simple spectral model droelectric power plants. The first step to develop a sim- is given that is applicable to inclined surfaces and clear ple model of a PVPP is presented in this paper based on skies. first astronomical then engineering principles. A very good comparative study was proposed in Refs. The energy production of a PVPP is a function of the [3] and [4] where different area-specific models were following: analysed with regard to the optimal tilt angle. Unfor- tunately, the Hungarian region is not covered by the • parameters of the PVPP, models used in this study. A semi-empirical method for short-term (hourly) solar irradiance forecasting was pro- • weather (cloud, temperature, wind), posed in Ref.[5] founded on the widely used Ångström- • quality of the atmosphere (dust, relative humidity), Prescott model. The aim of this work is to propose a model that is • activity of the Sun (11-year solar cycle). not only able to handle clear-sky conditions but clouds as The parameters of the PVPPs are the following: *Correspondence: [email protected] • position, 58 BÁLINT,FODOR,SZALKAI,SZALKAI, AND MAGYAR • orientation of the PV solar panels, 1:04 3 January Eq.1 Eq.2 – tilt angle, 1:02 – azimuth angle, 4 April 0 1 E • surface area (or the number of PV solar panels), 5 October 0:98 • nominal power of the PV solar panels, 0:96 4 July • efficiency of the solar inverter. 0 50 100 150 200 250 300 350 By using the first four parameters, the solar energy can be The day of the year calculated by the developed solar irradiance model which is presented in this paper. Figure 1: The calculated value of the reciprocal of the square of the Sun-Earth distance (E0) using Eqs.1 and 3. Astronomical relationships - 2 Sun-Earth geometries To estimate the solar irradiance from the Sun, some astro- Eqs.1 and2 are from Refs.[6] and [7]. E0 denotes the nomical (geometrical) calculations to define the position reciprocal of the square of the Sun-Earth distance, r0 rep- of the Sun relative to the local horizontal surface need resents the mean Sun-Earth distance (1 AU) and r stands to be conducted. Many different equations are found in for the Sun-Earth distance on the dnth day of the year. the literature to calculate the same parameter. The size of Values of E0 using various equations are shown in Fig.1. these equations, in addition to the frequency of measure- This change will change the solar constant too. The ments, vary significantly. In this section some equations solar constant, I0, yields the following solar irradiance 2 are compared. value (in W/m ) at the edge of the Earth’s atmosphere. This constant value is 1367 W/m2 if the Earth is a dis- 3.1 Comparison of the available solar geome- tance of 1 AU (astronomical unit) from the Sun. This try models value is greater at perihelion (minimum Sun-Earth dis- tance ≈ 0:983 AU on 3 January) and less at aphelion Due to the elliptic orbit of the Earth around the Sun, some (maximum Sun-Earth distance ≈ 1:017 AU on 4 July). parameters exist with a 1-year cycle. These parameters Since the irradiance is proportional to the surface area, can be calculated with different equations and the dif- which is inversely proportional to the square of the dis- ference between values of these equations varies due to tance, the corrected solar irradiance can be calculated discrepancies in the description of the method. The cal- from culations regard a year as consisting of 365 days. Most In = I0E0: (3) of the results were obtained from Ref.[6] and references therein. Solar declination Relative Sun-Earth distance The declination of the Sun, δ, yields the latitude above which the path of the Sun follows. Therefore, the zenith The first parameter is the reciprocal of the square of the angle is zero at local noon. This value can be calculated Sun-Earth distance during the year. The elliptic orbit of from the following equations [6]: the Earth causes this to change in value. This relative dis- tance can be calculated from equations δ = 0:006918− 2 r0 E0 = = 1:000110+ d − 1 r − 0:399912 cos 2π n + d − 1 365 + 0:034221 cos 2π n + 365 d − 1 + 0:070257 sin 2π n − d − 1 365 + 0:001280 sin 2π n + 365 d − 1 − 0:006758 cos 4π n + d − 1 365 + 0:000719 cos 4π n + 365 d − 1 + 0:000907 sin 4π n − d − 1 365 + 0:000077 sin 4π n (1) 365 d − 1 − 0:002697 cos 6π n + and 365 ! 2 d − 1 180° r0 dn n (4) E0 = = 1 + 0:033 cos 2π (2) + 0:00148 sin 6π r 365 365 π Hungarian Journal of Industry and Chemistry MODELING AND CALCULATION OF THE GLOBAL SOLAR IRRADIANCE ON SLOPES 59 23:5° 20 Eq.8 Eq.9 21/22 June 10 Eq. 10 data series † 10° ] 20/21 March 22/23 September 21/22 December δ 0° min Eq.4 [ 0 Eq.5 t −10° E Eq.6 −10 Eq.7 −23:5° −20 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 The day of the year The day of the year Figure 2: The calculated value of the declination of the Figure 3: The values of the Equation of Time over a year Sun (δ) using Eqs.4–7 using Eqs.8–10 and a series for the year 2018 ! h 360 Et = 0:043 sin 2 4:8888 + 0:017214dn+ δ = arcsin 0:4 sin (dn − 82) (5) 365 + 0:033 sin(0:017214dn) − 2π δ = 23:45° sin (dn + 284) (6) − 0:03342 sin(0:017214dn)+ 365 i and + 0:2618tUTC − π 229:18 (9) and δ = arcsin 0:39779 sin 4:8888 + 0:017214dn+ dn Et = − 7:655 sin 2π + 365 + 0:033 sin(0:017214d ) (7) n d + 9:873 sin 4π n + 3:588 : (10) 365 The results of different equations that calculate the dec- lination of the Sun are shown in Fig.2. The Sun is Eqs.8 and9 yield Et in radians and the constant 229:18 above the equator on 20/21 March (vernal equinox) and converts this into minutes. This constant can be calcu- 22/23 September (autumnal equinox), moreover, the win- lated from ter solstice is on 21/22 December and the summer sol- 360° 360° 229:18 = 16 min (11) stice on 21/22 June. The difference between the equa- 2π 24 · 60 min tions in Fig.2 is minimal. The maximum change in the declination when calculated on a daily basis is less than Fig.3 shows the results according to Eqs.8–10 over a whole year, together with astronomical data for the year 0:5° (jδ(dn+1) − δ(dn)j). 2018. The largest difference was calculated by Eq. 10 when Equation of time compared with the other data.