# Mathematics-Driven Analysis of Offshore Green Hydrogen Stations
## 2 Materials and Methods
### PHASE II: Modeling of Solar Electricity Generation  
The sizing of the photovoltaic station required to power the electrolyzer and compressor used in the green hydrogen production process was carried out by the following: (i) calculating the relative position of the sun with respect to the selected location of the station, (ii) determining the solar radiation received perpendicularly on the surface of the photovoltaic panels, and (iii) assessing the panels' ability to convert solar radiation into electrical energy based on the selected technology per square meter of photovoltaic surface. 5.1 Modeling the Sun's Position Relative to the Location of the Photovoltaic Panels Powering the Green Hydrogen Station  
The first step is to model the set of trigonometric relationships used as the basis for calculating the absolute position of the sun relative to the final selected location of the \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Definition** \\ \hline Localization: & Localization of solar panels and green hydrogen station. \\ Time definition: & Total number of days in a year. \\ Time definition: & Specific day of the year. Day under study. \\  & Local time, referenced to 12 p.m. \\ Atmosphere and marine conditions: & Atmospheric state/quality. \\  & Coefficient of losses associated with ocean surface irregularities. \\  & Ambient temperature. \\ Photovoltaic panels: & Photovoltaic panel inclination and azimuth. \\  & Energy conversion efficiency. \\  & ALK type. \\  & PEM type. \\ Electrolyzer: & Hydrogen calorific power constant. \\  & ALK electrolyzer efficiency. \\  & PEM electrolyzer efficiency. \\  & Input pressure. \\  & Output pressure. \\  & Compression efficiency. \\ \hline \end{tabular}
\end{table} 
Table 1: Configuration parameters of the mathematics-based technical model. station [30; 31]. These relationships allow for the calculation of the Earth's declination angle (\(\alpha_{declination}\)), solar hour angle (\(\alpha_{solar}\)), solar elevation angle (\(\alpha_{solar\_elevation}\)), solar azimuth angle (\(\alpha_{solar\_azinuth}\)), and the day length (\(d_{length}\)). \(\alpha_{declination}\) is the tilt angle of the Earth's rotational axis and is modeled using [PERSON]'s equation [31]:  
\[\alpha_{declination}=23.45^{\circ}\times sin\left(\frac{2*\pi}{N_{days}}\times \big{(}\ N+C_{quinox}\ \big{)}\right) \tag{2}\]  
where \(N_{days}\) is the total number of days in the year, \(N\) is the day number within the interval [1, \(N_{days}\)] for which \(\alpha_{declination}\), is to be calculated, and \(C_{quinox}\) is the synchronization constant of \(\alpha_{declination}\) with the spring or autumn equinox (Equation (3)). In both cases, since it is a sinusoidal function, the result of the expression (Equation (2)) is synchronized with its zero crossings. \(\alpha_{declination}\) is a variable with a period of one year and values oscillating between \(\pm 23.45^{\circ}\)[32]. \[C_{\rm equinox}=\frac{-81}{284},\ \ spring\ equinox \tag{3}\]  
\(\alpha_{solar}\) is calculated relative to the selected location of the station. To perform this, the local time (\(h_{local}\)) was synchronized with the time of maximum solar elevation (12 p.m.), and the resulting value was multiplied by the number of degrees that the Earth rotates on its axis per hour. Considering that the Earth completes a full \(360^{\circ}\) rotation every 24 h, \(\alpha_{solar}\) is modeled as follows:  
\[\alpha_{solar}=\frac{360^{\circ}}{24}\times(h_{local}-12) \tag{4}\]  
where \(h_{local}\) represents the set of values within the interval [0, 24]. By itself, expression (Equation (4)) does not account for the fact that the apparent motion of the sun is not uniform and that the length of the solar day is not constant throughout the year [33]. The modeling of this variation was performed using the general equation of solar time, with results expressed in minutes (_min_) [31]:  
\[