# Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds
## 2 Materials and Methods
### Geometric Characterisation of the Collector Plane  
Figure 2 shows, in vector mode, the determining directions of the collector plane's behaviour regarding solar irradiance collection. To define these characteristic vectors, the terrestrial reference system is used, in which the Ox axis points west, the Oy axis points south, and the Oz axis coincides with the zenith direction, where \(\overset{\rightarrow}{i}\), \(\overset{\rightarrow}{j}\) and \(\overset{\rightarrow}{k}\) are the unit vectors associated with each of these axes, respectively. Thus, in this reference system, the solar vector, \(\overset{\rightarrow}{s}\), which is defined as a unit vector pointing to the Sun, is given by Equation (4) [39]  
\[\begin{array}{l}\overset{\rightarrow}{s}=s_{x}\overset{\rightarrow}{i}+s_ {y}\overset{\rightarrow}{j}+s_{z}\overset{\rightarrow}{k}=(\sin\Omega t\cos \delta)\overset{\rightarrow}{i}+\\ +(\cos\Omega t\cos\delta\sin\varphi-\sin\delta\cos\varphi)\overset{ \rightarrow}{j}+\\ +(\cos\Omega t\cos\delta\cos\varphi+\sin\delta\sin\varphi)\overset{ \rightarrow}{k},\end{array} \tag{4}\]  
where:  
\(\varphi\) is the latitude,  
\(\Omega\) is the speed of rotation of the Earth,  
\(\delta\) is the declination given by Equation (5) in which \(\Gamma\) is the daily angle given by Equation (6).  
\[\begin{array}{l}\delta(rad)=[0.006918\ -0.399912\cos(\Gamma)+\\ +0.070257\sin(\Gamma)\ -0.006758\cos(2\Gamma)+\\ +0.000907\sin(2\Gamma)\ -0.002697\cos(3\Gamma)+\\ +0.00148\sin(3\Gamma)],\end{array} \tag{5}\]  
\[\Gamma(rad)=\frac{2\pi\left(d_{j}-1\right)}{365} \tag{6}\]and \(t\) is the true solar time (in hours), whose relationship with GMT time (in hours) is given by Equation (7), in which \(L_{e}\) represents the longitude in degrees of the plant's geographical location and \(E_{i}\) (in minutes) is the so-called equation of time that is calculated using Equation (8).  
\[t=GMT-12+L_{e}/15+E_{i}/60 \tag{7}\]  
\[E_{i}=229.18\quad[0.000075+0.001868\cos(\Gamma)-0.032077\sin(\Gamma)\]  
\[-0.014615\cos(2\Gamma)-0.04089\sin(2\Gamma)] \tag{8}\]  
Furthermore, according to Figure 2, from (4), it is found that the solar height, \(\alpha\), that is, the angle that the solar rays form with the horizontal plane, can be determined by the Equation (9)  
\[\alpha=\arcsin(\cos\Omega t\ \cos\delta\cos\varphi+\sin\delta\sin\varphi), \tag{9}\]  
Finally, Figure 2 also represents the unit vector \(\overrightarrow{n}\) orthogonal to the plane of photovoltaic modules and representative of the spatial position of the collector, which depends on the inclination \(\beta\) and the azimuth \(\gamma\) of the collectors according to Equation (10). Thus, as seen in Figure 2, the angle between the vector \(\overrightarrow{n}\) and the solar vector \(\overrightarrow{s}\) is \(\theta\), while the angle between \(\overrightarrow{n}\) and \(\overrightarrow{k}\) coincides with the inclination of the collector, \(\beta\). Equations (11)-(13) determine the relationships between the angles and the vectors characteristic of the collector's position and that determine its radiative uptake.  
\[\overrightarrow{n}=(\sin\beta\sin\gamma)\overrightarrow{i}+(\sin\beta\cos \gamma)\overrightarrow{j}+(\cos\beta)\overrightarrow{k} \tag{10}\]  
\[\cos\theta=\overrightarrow{s}\cdot\overrightarrow{n}, \tag{11}\]  
\[\cos\theta_{z}=\overrightarrow{s}\cdot\overrightarrow{k} \tag{12}\]  
\[\cos\beta=\overrightarrow{k}\cdot\overrightarrow{n} \tag{13}\]