# 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
\begin{array}{c}rs=\left(\frac{n_{air}\times cos(\alpha_{incident})-n_{ auto}\times cos\big{(}\alpha_{refraction}\big{)}}{n_{air}\times cos(\alpha_{incident}\big{)}+n_{auto}\times cos\big{(}\alpha_{ rafraction}\big{)}}\right)^{2}\\ rp=\bigg{(}\frac{n_{autor}\times cos(\alpha_{incident})-n_{ air}\times cos\big{(}\alpha_{refraction}\big{)}}{n_{autor}\times cos\big{(}\alpha_{incident}\big{)}+n_{air}\times cos\big{(}\alpha_{ rafraction}\big{)}}\bigg{)}^{2}\\ k_{reflectation}=\frac{rs+rp}{2}\end{array}  \tag{19}\]In this way, the reflected solar radiation density is calculated by combining the results of expressions (Equations (16) and (19)):  
\[I_{reflected}=I_{attenuated}\times k_{reflexion} \tag{20}\]  
In any case, the previous expression (Equation (20)) assumes that the marine surface is completely flat, meaning that it has no irregularities that could affect the way solar rays are reflected. This assumption may be quite bold in marine environments, where the marine surface is subject to currents, winds, waves, or even the passage of ships that can create surface undulations, leading to a greater or lesser reduction in the actual value of \(I_{reflected}\). This situation can be modeled by incorporating a loss coefficient (\(K_{marine}\)), which attenuates the original value of \(I_{reflected}\). The values of \(K_{marine}\) range within the interval [0, 1], where \"0\" represents the ideal condition of a perfectly flat marine surface, and \"1\" represents a completely irregular surface. \[I_{reflected}=I_{reflected}\times(1-K_{marine}) \tag{21}\]  
At this point, the total solar radiation density (\(I_{total}\)) received on the surface of the panels is equal to the sum of the solar radiation densities: \(I_{attenuated}\) (Equation (16)), \(I_{reflected}\) (Equation (21)), and diffuse radiation (\(I_{diffused}\)). Throughout this paper, the value of \(I_{diffused}\) is zero, as this type of radiation depends on uncontrollable atmospheric conditions that are highly variable and difficult to model. \[I_{total}=I_{attenuated}+I_{reflected}+I_{diffused} \tag{22}\]  
The sum of the solar radiation densities shown in expression (Equation (22)) is only possible after calculating their geometric contributions relative to the normal of the photovoltaic panel surface, both for direct solar radiation density (\(K_{direct}\)) and reflected radiation density (\(K_{reflected}\)). This condition is necessary due to the following: (i) the different incidence angles at which each type of radiation impacts the surface of the panels and (ii) the photovoltaic panels' conversion capacity, which is specified based on the total solar radiation density received perpendicularly on their surface (\(I_{n\_total}\)), the surface area of the photovoltaic panels in square meters (\(M_{panels}\)), and the technological efficiency coefficient (\(K_{panels}\)). This last parameter is specified in the datasheets provided by the photovoltaic panel manufacturer. Expression (Equation (23)) presents the geometric conversion factor necessary to normalize the direct solar radiation density (Equation (16)) impacting the panel surface (\(K_{direct}\)). This normalization depends on \(\alpha_{solar\_elevation}\) (Equation (8)), \(\alpha_{solar\_ azimuth}\) (Equation (9)), the panel inclination angle relative to the ground (\(\beta_{inclination}\)), and the panel azimuth angle (\(\beta_{ azimuth}\)) relative to the north or south, depending on the hemisphere in which the installation is located [30]. \[ \begin{split} K_{direct}=sin(\alpha_{solar\_elevation})\ *\ sin(\beta_{inclination})+cos(\alpha_{solar\_elevation})*cos(\beta_{inclination})\\ *cos(\alpha_{solar\_ azimuth}-\beta_{ azimuth})\end{split}