# Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds
## 3 Results
In this way, Equations (28)-(30) show the dependency of the technical evaluation variables of the installation (production, compensation and self-consumption) with respect to characteristic parameters of the installation such as inclination \(\beta\) and orientation \(\gamma\) of the collectors, angle of obstruction \(\zeta\) and peak power \(PP\). It should be considered that it is not a statistical problem but rather an adjustment or approximation of a mathematical function with a simple expression to a complex mathematical function, therefore statistical methods are not applicable. To address the adjustment, the set of simple variables was expanded with composite variables obtained as products and quotients of the simple variables. The proposed functions (28)-(30) were selected as the most favourable from the set of fits to linear functions of composite variables. For each of the equations proposed as a quick approximation, the adjustment coefficients are shown as well as the mean of the absolute errors. \(\bullet\) Production, \(P\): \[ \begin{array}{ll}&P\left(\text{kWh}/\text{year}\right)=73.6+0.5419\cdot PP \cdot\cos\beta+\\ &340.02\sin\beta\sin\gamma+\ 0.58\cdot PP-0.164\cdot\zeta\\ &R^{2}=0.988\\ &\overline{\varepsilon_{P}}=81.2\cdot\text{kWh}/\text{ano}\end{array} \] (28) \(\bullet\) Compensation, \(C\): \[ \begin{array}{ll}&C\left(\text{kWh}/\text{year}\right)=-35.65-238.20\cdot \cos\beta+0.529\cdot PP\cdot\\ &\cos\beta+289.9\sin\beta\sin\gamma+\ 8.27\cdot 10^{-5}\cdot PP^{2}\\ &R^{2}=0.966\\ &\overline{\varepsilon_{C}}=110.1\cdot\text{kWh}/\text{ano}\end{array} \] (29) \(\bullet\) Self-Consumption, \(A\): \[ \begin{array}{ll}&A\left(\text{kWh}/\text{year}\right)=50.95+\ \left(325.02-0.0776\cdot PP \right)\cdot\cos\beta+\\ &0.6540\cdot PP-9.98\cdot 10^{-5}\cdot PP^{2}-2.1478\cdot\zeta\\ &R^{2}=0.963\\ &\overline{\varepsilon_{A}}=35.2\cdot\text{kWh}/\text{ano}\end{array} \] (30)  
Equations (29) and (30) constitute the basis of a simplified model since they allow evaluation of the savings \(F\) as a function of design variables by means of Equation (31), where \(p_{E}\) is the energy purchase price and \(p_{C}\) is the compensation price. \[F(PP,\beta,\gamma,\zeta,p_{E},p_{c})=p_{E}\ A(PP,\beta,\gamma,\zeta)+p_{c}C( PP,\beta,\gamma,\zeta) \tag{31}\]  
Therefore, Equation (31) provides, in a simple way, a first approximation of the possible savings associated to a PV installation for any orientation \((\gamma)\), tilt \((\beta)\) and degree of obstruction \((\zeta)\). In this way, this model allows an easy and immediate evaluation of the profitability and suitability of a domestic PV installation that must be configured according to a set of restrictions imposed by the environment in which it is located. Moreover, considering that the cost \(C_{0}\) has been considered proportional to the value of the peak power, the Equations (26) and (27) will be approximated by Equations (32) and (33), respectively, with a continuous dependence with respect to the variables of design. \[ \begin{array}{ll}&NPV_{model}=\ \ -C_{0}PP+F(PP,\beta,\gamma,\zeta,p_{E},p_{c}) \sum_{j=1}^{n}\frac{(1+\Delta p)^{j}}{(1+d)^{j}}-\\ &\sum_{j=1}^{n}M\frac{(1+i)^{j}}{(1+d)^{j}}\end{array}  \tag{32}\]\[Pay-back_{model}=\frac{C_{0}PP}{F(PP,\beta,\gamma,\zeta,p_{c},p_{c})} \tag{33}\]  
Finally, in Figure 10a the \(NPV\) values obtained by means of the model given by Equation (32) are represented against the simulated ones, obtaining a regression coefficient between both of \(R^{2}=0.9633\). Similarly, Figure 10b represents the modelled \(Pay-back\) values (Equation (33)) versus the simulated ones, with a regression coefficient between the two of \(R^{2}=0.9766\). Thus, although the model given by Equations (28)-(33) offers approximate values of \(NPV\) and \(Pay-back\), it notably facilitates the analysis of the qualitative influence of \(\gamma\), \(\beta\) and \(\zeta\) on these indicators.