The Sun

The sun is a sphere with a diameter of 1.39 x \(10^{9}\) m and on average its distance from the earth is 1.5 x \(10^{11}\) m. Seen from the earth it rotates on its axis every 4 weeks, however the rotation is not perceived equal from all the point of the earth but is about 27 days at the equator and 30 days on polar regions. The sun has an effective blackbody temperature¹ of 5777 K, but on its interior the temperature is much higher, then energy produced inside must be transferred out to the surface and then be radiated into space till the earth.

Sun-Earth Geometry.

However, in reality the sun does not work, as a blackbody radiator at a fixed temperature, rather the emitted solar radiation is the composite result of the several layers that emit and absorb radiation of various wavelengths. The final extraterrestrial solar radiation and its spectral distribution have now been measured by various methods in several experiments. The solar spectrum changes throughout the day and with location. Standard reference spectra are defined to allow the performance comparison of photovoltaic devices from different manufacturers and research laboratories. We can distinguish between three main standards:

• AM0 is the standard spectrum for space applications, it has an integrated power of 1366.1 \(W/m^2\)
• AM1.5 Global spectrum is designed for flat plate modules and has an integrated power of 1000 \(W/m^2\).
• AM1.5 Direct spectrum is defined for solar concentrator work, it includes the direct beam from the sun plus the circumsolar component in a disk 2.5 degrees around the sun. The direct plus circumsolar spectrum has an integrated power density of 900 \(W/m^2\).

The data for these three spectra are available for the download at the website². Assuming that the sun behaves as a blackbody, we could compare these spectra with the theoretical distribution obtained by the Planck’s law. Denoting as \(T = 5777 K\) the temperature of the sun, with \(R_s = 6.955 \cdot 10^8 \; m\) its radius and with \(R_{se} = 1.495 \cdot 10^{11} \; m\) the mean distance sun-earth, then the Planck’s Law allow to write the emittivity power of the sun as: \[\tag{2.1} E_{\lambda} = \biggl( \frac{R_{s}}{R_{se}} \biggl)^2 \frac{C_1}{\lambda^5 [exp(C_2/\lambda T)-1]}\] In equation (Eq. 2.1) \(\lambda\) is the wavelength expressed in micrometers, \(C_1\) and \(C_2\) are two contants given by \(C_1 = 3.742 \cdot 10^8 (W \mu^4)/m^2\) and \(C_2 = 1.439 \cdot 10^4 \mu K\). As consequence \(E_{\lambda}\) will be measured in \([W/(m^2 \mu m)]\). In Figure @ref(fig:solarspectra), the four spectra are compared: it is interesting to note the differences among the red and the blue and green curves, that represent the real and the adjusted spectral distribution of the sun, respectively. The air mass zero (AM0), or extraterrestrial spectrum used to generate the terrestrial reference spectra was developed by Gueymard [3] and is a synthesis of several air mass zero data sets. The comparison between AM0 and AM1 highlights by difference the effects of atmospheric attenuation on the direct component arriving on a surface orthogonal to the sun’s rays. These effects are particularly accentuated in the absorption bands of ozone (300-400 \(nm\)), water vapor (800-2000 \(nm\)) and carbon dioxide (above 2500 \(nm\)).

Spectal distribution of the sun under different standards. (Source: NREL)

Radiation Definitions

As the solar radiation goes through the atmosphere it suffers different processes of absorption, dispersion or scattering that result in lower levels of solar radiation being received at the Earth’s surface. These are due to the atmosphere components, such as ozone or \(CO_2\), and solid and liquid particles in suspension like aerosols or water vapour. However, the main source of attenuation is the cloud cover. In general, the processes of absorption and attenuation differently affect the wavelengths of solar radiation, therefore the spectral distribution of the solar radiation at ground level differs from the extraterrestrial one. We can distinguish between four quantities that in general are used to express solar irradiance:

• Extraterrestrial Radiation: it represents the amount of radiation that could be theoretically available if there were no atmosphere. Even if it could present some randomness, in general it is considered deterministic. Once the radiation enters in the atmosphere we could make two further distinctions in beam and diffuse radiation. The notations involved are several: \(G_{sc}\) is the solar constant, \(G_n\) is the value of \(G_{sc}\) corrected for the \(n\)-th day of the year. Moreover, \(G_{o,n}\) represents \(G_{n}\) computed for an horizontal surface. Finally \(H_{o,n}\) is the daily integral of \(G_{o,n}\). More details will be given below.
• Beam Radiation: it represents the solar radiation received from the sun without being scattered by the atmosphere. Consequently to its definition, the Beam radiation is maximum when the sky is clear³. In theory the Beam radiation can be zero: even if this could seem strange, in days when the sky is heavily covered by clouds, the instruments devoted to the measurement⁴ could register a value of zero. It is also denoted as Direct Normal Irradiance (DNI).
• Diffuse Radiation: it represents the solar radiation received from the sun after its direction has been changed and scattered by the atmosphere. In application, usually, this measure is considered with respect to an horizontal surface ( Diffuse Horizontal Irradiance or DHI).
• Global Solar Radiation: it represents the sum of the beam and the diffuse radiations. A common practice is to consider it with respect to an horizontal surface (Global Horizontal Irradiance or GHI).

Different definitions types of solar irradiance.

The Solar Constant \(G_{sc}\)

The solar constant \(G_{sc}\) is defined as the energy received from the sun per unit of time (typically a second), on a unit area of surface perpendicular to the direction of propagation of the radiation at the mean earth-sun distance outside the atmosphere. The most recent estimate from the Wold Radiation Center in Switzerland gives a mean value of \(\overline{G}_{sc} = 1367 \, W/m^2\). In order to prove this result, we have to consider the total energy irradiated from the sun per \(m^2\), given by: \(q_0 = 6.316 \cdot 10^7 W/m^2\). Denoting as \(R_s\) the radius of the sun and with \(R_{se}\) the mean earth-sun distance, the following simple equation with \(G_{sc}\) unknown can be solved: \[\tag{2.2} q_{0} 4 \pi R_{s}^2 = G_{sc} 4 \pi R_{se}^2 \, \, \Rightarrow \, \, \, G_{sc} = q_{0} \frac{R_{s}^2}{R_{se}^2} = 1367 \, W/m^2\] However, \(G_{sc}\) is actually not constant. For practical purposes it is possible to neglect the variations due to sunspots⁵ and to assume that the flow of energy radiated by the sun, \(q_0\), is constant. Consequently, \(G_{sc}\) will change according to the variation of the earth-sun distance. Counterintuitively the solar constant assumes its maximum value (around 1420 \(W/m^2\)) during the lasts days of December and its minimum (around 1320 \(W/m^2\)) during the summer in the late June. Spencer (1971), as cited by Iqbal [4], recovered an accurate equation (\(\underline{+} 0.01 \%\)) that can be used for the correction of \(G_{sc}\) during the year. Denoting as \(G_{n}\) the extraterrestrial radiation incident on the plane normal to the radiation on the \(n\)-th day of the year: \[\tag{2.3} G_{n} = G_{sc} \bigl( 1.000110 + 0.034221 cosB + 0.001280 sinB +0.000719 cos2B + 0.00077 sin2B \bigl)\] where \(B\) represents an angle in radians and can be computed as in (Eq. 2.4). \[\tag{2.4} B = (n-1) \frac{2 \pi}{365}\]

Values of the Solar Costant during the year in Bologna.

Equatorial coordinates

The position of the sun with respect to a point \(P\) on the Earth’s surface can be identified through an equatorial coordinate system. Each point on the Earth’s surface is identified by a pair of angular coordinates:

Relations between declination and latitude, and hour angle and longitude.

• Latitude \(\phi\): it is the angular location at the north or at the south of the equator, where by convention it is considered a positive sign in the north with respect to the equator and negative in the south. It varies in the range [-90°,90°]. In practice, it is the angular distance of \(P\) from the equator measured along the meridian passing through \(P\).
• Longitude \(\psi\): it is the angular distance, measured along the equator, of the meridian passing through \(P\) from the meridian of Greenwich \(G\). It is considered positive at the East of \(G\) and negative at West. It varies in the range [-180°,180°].
  Based on the definitions, latitude and longitude are equatorial coordinates: they refer exclusively to the equatorial plane and to the Earth’s axis⁶. By tracing the junction between the center of the Earth and that of the sun, it is possible to find the point of intersection S. In this context we define:
• Solar Declination \(\delta\): homolog of latitude, it is the angular position of the sun when it is situated on the local meridian with respect to the plane of the equator. As the latitude is positive in the North and negative in the South. It varies in the range [-23.50°,23.50°]. By definition it is equal for every point on the Earth.
• Hour Angle \(\omega\): homolog of longitude, it is the angular displacement of the sun at the East or West of the local meridian due to the rotation of the Earth on its axis. It is negative in the morning, zero at the solar noon and positive in the afternoon. It is defined as the difference between the longitude of \(P\) (fixed) and the longitude of \(S\) (variable). It is equal for all the points on the same meridian.

Terrestial Coordinates

The irradiance of a receiving surface depends on the cosine of the angle of incidence, i.e. the angle that the direction of the sunrays forms with the normal \(n\) at the surface itself. The variations of angle of incidence over time are the result of the rotational and revolutionary movements of the earth with respect to the sun. An alternative reppresentation of the equatorial coordinates is given by the terrestrial coordinates⁷. In solar engineering often this different system is used because it is more intuitive and easier to visualize. The geometric relationships between a plane of any particular orientation relative to the earth at any time and the incoming beam solar radiation,⁸ can be described in terms of several angles [6].

Position of the sun in terrestrial coordinates.

• Zenith Angle \(\theta_z\): it is the angle between the vertical line and the line to the sun, it represents the angle of incidence of the radiation on an horizontal surface.

• Solar Altitude Angle \(\alpha_s\): it is the angle between the horizontal line and the line of the sun and can be defined as the complement of the zenit angle.

• Solar Azimut Angle \(\gamma_s\): it is the angular displacement from the south of the projection of the radiation on the horizontal plane.

• Angle of Incidence \(\theta\): it is the angle formed by the Beam radiation on a surface and the normal to that surface. Considering an horizontal receiving surface the angle of incidence is actually equal to the zenit angle. If we consider an horizontal surface, the previous angles are the only ones needed for the computation of the incidence angle of the beam radiation. However it is possible to consider a more general relation that takes into account also the orientation and inclination of a surface three more angles can be defined:

• Slope \(\beta\): it is the angle between the plane of the surface and the horizontal. It varies in the range [0°, 180°].

• Angle of Incidence \(\theta\): it is the angle formed by the beam radiation on a surface and the normal to that surface.

• Surface Azimut Angle \(\gamma\): it represents the deviation of the projection on an horizontal plane of the normal to the surface from the local meridian. It is considered zero at the South, negative in the East and positive in the West.

Angular Relationships for an inclinated surface.

There is a set of relations that connect the angle of incidence to the others that are time varying: \[\tag{2.10} cos \theta = T + U cos \omega + V sin \omega\] \[\tag{2.11} cos \theta = cos \, \theta_z cos \, \beta + sin \theta_z \, sin \beta \, cos \, (\gamma_s - \gamma)\] where \(T\), \(U\), \(V\), are three coefficients that can be computed as: \[\begin{cases} T = sin\delta \bigl( sin \phi \, cos \beta - cos \phi \, sin \beta \, cos \gamma \bigl) \\ U = cos\delta \bigl( cos \phi \, cos \beta - sin \phi \, sin \beta \, cos \gamma ) \\ V = cos\delta \, sin \beta \, sin \gamma \end{cases}\] The relations (Eq. 2.10), (Eq. 2.11) are equivalent and each one can be used to determine the \(cos \theta\), the only difference is in the required inputs. They can be applied for every oriented and inclined surface, however there are two special cases that are convenient to analyze. The first one is when we consider a surface oriented at south, in this case \(\gamma = 0\) and we are able to simplify the relation (Eq. 2.10) as: \[\tag{2.12} \cos \theta = sin \delta \, sin (\phi - \beta) + cos \delta \, cos (\phi - \beta) \, cos \omega\] The second special case is for an horizontal surface, in this case there is no inclination and the orientations are unconcerned, therefore \(\beta = 0\) implies that \(\gamma\) will no more enter in the formula: \[\tag{2.13} cos \theta = cos \theta_z = sin \delta \, sin \phi + cos \delta \, cos \phi \, cos \omega\] Moreover in the case (Eq. 2.13), the zenit angle \(\theta_z\) is equal to the incidence angle \(\theta\). In general this is not true, and \(\theta\) can be recovered given \(\theta_z\) and the other solar angles and applying the (Eq. 2.10) or (Eq. 2.11). As consequence of the relation (Eq. 2.13) there is the possibility to recover the hour angle at the sunrise and at the sunset, \(\omega_s\). In these two particular moments of the day we have that \(\theta_z = 0\), and applying the formulas of \(\omega_s\) for that day is given by: \[\tag{2.14} |\omega_s| = \underline{+} cos^{-1} \bigl( -tan \delta \; tan \phi \bigl)\] Considering an horizontal surface the sunset and sunrise hour angles are equal and opposite, in general this is not true for an oriented and inclined surface. Knowing the value of \(\omega_s\) allow to know if in a certain time of the day the sun will be visible or not in the sky, therefore we can establish also the maximum amount of sun hours for an horizontal surface which is obtained dividing the degree routes of the sun by the time spent (in minutes) to move one degree⁹. \[\tag{2.15} SunHours = \frac{2 \; |\omega_s |}{15}\]

Extraterrestrial Radiation

Considering the variation of the extraterrestrial radiation during the year, there are two sources of stochasticity. The first is given by a variation of the radiation emitted by the sun, in literature there are conflicting reports on periodic variations of intrinsic solar radiation. For engineering and modelling purposes this source of stochasticity is neglected. However the second source of variability is given by the different earth-sun distance during the year that leads to difference of \(\underline{+} 3.3 \%\). At any point in time, the solar radiation incident on a horizontal plane outside the atmosphere can be computed multiplying the cosine of the angle of incidence \(\theta_z\) by the solar costant as computed in (Eq. 2.2). \[\tag{2.14} G_{o,n} = G_{n} \, cos \theta_z = G_{n} \bigl(cos \delta \, cos \phi \, cos \omega + sin \delta \, sin \phi \bigl)\] This expression represents the extraterrestrial energy that arrives on an horizontal surface in any time between the sunrise and the sunset. A more useful expression in the application is the daily extraterrestrial horizontal irradiance, denoted as \(H_{o,n}\), that can be computed integrating the formula (Eq. 2.14) with respect to \(\omega\) as in (Eq. 2.15). \[\tag{2.15} H_{o,n} = G_{n} \frac{24 \times 3600}{\pi} \bigl(cos \delta \, cos \phi \, sin \omega_s + \frac{\pi \omega_s}{180} sin \delta \, sin \phi \bigl)\] If \(G_{sc}\) is expressed as \(W/m^2\), then \(H_{o,n}\) as computed in (Eq. 2.15) will be expressed in \(J/m^2/day\). Another very useful relation is given by the hourly extraterrestrial daily irradiance that can be computed as for a period between two hour angles, with \(\omega_2 > \omega_1\):
\[\tag{2.16} I_{o,n} = G_{n} \frac{12 \times 3600}{\pi} \biggl( cos \delta \, cos \phi \, \bigl( sin \omega_2 - sin \omega_1 \bigl) + \frac{\pi (\omega_2 -\omega_1)} {180} sin \delta \, sin \phi \biggl)\]

Global Horizontal Irradiance (black) and Extraterrestrial Horizontal Radiation (purple) for Bologna (Year 2019).

PVGIS Data and Solar Measurements

The empirical Beam and Diffuse radiation were obteined from the Photovoltaic Geographical Information System (PVGIS), that is a tool for citizens to evaluate their roof’s PV potential. It is a free database developed and maintained by the European Commission’s Joint Research Centre, it provides the information about solar radiation and PV system performance for any location in Europe. There is little doubt that the very best way to measure solar radiation is to use high-quality sensors on the ground such as measurements stations. But to be useful, these measurements should fulfill a number of conditions:

• Only high quality measurement sensors should be used.
• Measurements should be performed at least every hour.
• Sensors should be calibrated and cleaned regularly.
• Data should be available for a long time period.

Nowadays the number of ground-based radiation measurements that fulfill these criteria is low and the stations often are very spaced making the connession between spaces even more difficult. For these reasons the use of satellite data to estimate the solar radiation at the earth surface is become a popular methods. Moreover the sources used are mostly geostationary meterological satellites, such as the METEOSAT¹⁰. The disadvantage of using satellite data is that the solar radiation at ground level must be calculated using a number of fairly complicated mathematical algorithms which do not use just satellite data, but also data on atmospheric water vapour, aerosols (dust, particles) and ozone. Some conditions can cause the calculations to lose accuracy, e.g. snow which can be mistaken for clouds, dust storms which can be difficult to detect in the satellite images. Nevertheless, the accuracy of satellite-based solar radiation data is now generally very good. For this reason, most of the solar radiation used in PVGIS are based on the satellite algorithms.

Descriptive analysis

The main focus of this thesis will be on the city of Bologna, however in order to eveluate the performance of the models proposed in Chapters 3 and 4, we have decided to consider a larger sample of cities that should present different characteristics in terms of solar irradiance. Excluding Bologna, the other cities choosen are:

• Amsterdam, Berlino, Oslo and Parigi. These cities were chosen to evaluate the model on latitutes in which the amount of solar irradiance should be lower and more variable.
• Catania, Rimini, Roma. We have chosen Catania as representative city of the south, where we expect to have, in mean, the maximum amount of solar irradiance among the sample. Rimini, as Catania, is a city located on the sea, it can be used as proxy to see if being near sea lead to a different dynamic of irradiance with respect to Bologna.

In Figure @ref(fig:BeamDiffuse) there are shown the empirical time series (2019) for Bologna of the Beam Horizontal Irradiance and the Diffuse Horizontal Irradiance, that are the two parts that composed the Global Horizontal Irradiance. We note how the behaviour of the Beam irradiance is different from the Diffuse one. The Beam component often presents values that are very small or zero¹¹. On contrary the Diffuse component, that is the irradiance received by the whole sky, is never zero, this ensures that the GHI can assume small values during the year but cannot be zero.

Data from PVGIS of the Beam Horizontal Irradiance and Diffuse Horizontal Radiation for Bologna (Year 2019).

In Table @ref(tab:SummaryStatistics) the descriptive statistics of the Beam and the Diffuse components are reported. Rimini and Bologna present a mean for the diffuse component higher in Bologna, however as expected the Beam Irradiance, in mean, is greater in Rimini. The standard deviation of Rimini is similar to those of Catania and Rome, however these two last cities present a mean value that is much greater:for Catania we have a value of 3.37 \(kWh/m^2/day\), Rome 2.89 \(kWh/m^2/day\) and Rimini 2.5 \(kWh/m^2/day\). Considering the kurtosis we have an opposite behaviour considering the north European cities and the ones in Italy: for the first group the kurtosis of the Beam radiation is greater with respect to the diffuse one, the opposite happens for the cities in Italy, that in general, present larger deviation from the normality assumption.

The descriptive statistics for almost all the cities, shows that the normality assumption could be violated. In Chapter 3, we will propose a stochastic differential equation that could be used to describe the dynamics of the GHI, however the normality assumption is often rejected by the formal test, expecially for the Beam component. As we can see from Table @ref(tab:SummaryStatistics), Italian cities have a higher average for both components than northern European cities. The standard deviations instead have a different behavior: they are greater in North Europe than in Italy considering the diffuse component while the opposite happens considering the Beam. Moreover in the North the distributions are less asymmetrical considering the Diffuse component. Finally, the kurtosis of many locations deviates from the value three, benchmark for the normal distribution. In general, a value greater than three denotes that the curve is more pointed than the normal one, while a value below three denote that it is more flat than the normal case.