# Mathematics-Driven Analysis of Offshore Green Hydrogen Stations
## 2 Materials and Methods
### PHASE II: Modeling of Solar Electricity Generation
#### 2.5.2 Modeling of Perpendicular Solar Radiation Received on the Surface of Photovoltaic Panels  
Once the relative position of the sun is determined, it is possible to model the expected solar radiation density (\(PV_{w}\)) received daily and perpendicularly on the surface of the photovoltaic panels (Equation (27)). The first step in this process is to model the solar radiation density received at the upper boundary of the Earth's atmosphere throughout each day of the year. The calculation of \(I_{solar}\) was performed using the following expression [30]:  
\[I_{solar}=1367\times\left(1+0.033\times cos\bigg{(}2\pi\times\frac{\rm N}{N_{ days}}\bigg{)}\right) \tag{14}\]  
where 1367 \(\rm W/m^{2}\) is known as the solar constant (\(\rm C_{solar}\)) and represents the density of received solar radiation. As it passes through the Earth's atmosphere, \(I_{solar}\) is attenuated (\(I_{attenuated}\)) due to its angle of incidence (\(90-~{}\alpha_{solar\_elevation}\)), the path followed by the solar rays from the upper atmosphere to the point on the Earth's surface where the solar panels are located--air mass (\(M_{air}\))--and the state/quality of the atmosphere itself (\(K_{\rm i}\)), where \(i\) depends on the different types of atmospheric absorption and scattering processes. [PERSON] and [PERSON] [34] defined air mass (\(M_{air}\)) as the distance that \(I_{solar}\) must travel from the upper boundary of the atmosphere to the terrestrial location of an observer--the location of the solar panels powering the green hydrogen station. According to [PERSON] and [PERSON], \(M_{air}\) can be modeled as follows:  
\[M_{air}=\frac{1}{cos\left(\frac{2\pi}{360}\times(90-\alpha_{solar\_elevation}) \right)+0.50572\times(96.07995-(90-\alpha_{solar\_elevation}))^{-1.6364}} \tag{15}\]  
[PERSON] [35; 36] modeled the attenuation that \(I_{solar}\) undergoes due to \(M_{air}\) (Equation (15)) and atmospheric absorption and scattering processes (\(\rm k_{\rm i}\)). Based on this, the solar radiation density (\(\frac{\rm w}{\rm m^{2}}\)) received on a specific day of the year (Equation (14)), at a specific location on Earth (Equation (15)), and under specific atmospheric conditions (Equation (16)), is referred to as \(I_{attenuated}\) and is modeled as follows:  
\[I_{attenuated}=\left|\left(I_{solar}\times e^{-M_{air}\cdot(K_{\rm a}+K_{\rm g}+K_ {\rm m02}+K_{\rm w}+K_{\rm o3}+K_{\rm o})}\right)\right| \tag{16}\]  
where \(k_{\rm i}\) represents [37]:  
* \(\rm k_{\rm a}\): Attenuation due to aerosols. * \(\rm k_{\rm g}\): Attenuation due to gases (carbon dioxide and oxygen). * \(\rm k_{\rm NO_{2}}\): Attenuation due to nitrogen dioxide. * \(\rm k_{\rm w}\): Attenuation due to water vapor. * \(\rm k_{\rm O_{3}}\): Attenuation due to ozone layer. In cases where the green hydrogen station is located offshore, the solar radiation density received on the surface of the photovoltaic panels is amplified due to the reflection of solar radiation on the marine surface. [PERSON] [33; 38] modeled the calculation of the solar rays' reflection angle (\(\alpha_{refraction}\)) based on the air reflection coefficient [39] (\(n_{air}\)), the water reflection coefficient (\(n_{water}\)), and the incidence angle of the solar rays (\(\alpha_{incident}\)) before impacting the marine surface:  
\[\alpha_{incident}=\frac{2\pi}{360}\times(90-\alpha_{solar\_elevation}) \tag{17}\]  
\[\alpha_{refraction}=arcsin\bigg{(}\frac{n_{air}}{n_{water}}\times sin(\alpha_{ incident})\bigg{)} \tag{18}\]  
[PERSON] [40; 41] modeled the calculation of the power reflected on the sea surface (\(I_{reflected}\)) as the average of the coefficients \(rs\) and \(rp\)[32] (\(k_{reflexion}\)), which in turn are calculated based on the coefficients \(n_{air}\), \(n_{water}\), and the angles \(\alpha_{incident}\) and \(\alpha_{refraction}\). [PERSON]'s law is modeled using the following expression:  
\[