# Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds
## 2 Materials and Methods
### Energy Balance  
In PV plants with self-consumption, once the curve of photovoltaic power produced in the representative days of the year has been obtained in accordance with the methodology described, it is necessary to compare this curve to that of the demand required by the consumer to obtain an estimate of both the self-contained power submerged as the power taken from or injected into the grid. By way of example, Figure 4 shows in red the values of photovoltaic power generated by an installation, while the curve in blue shows the power demand.  
Figure 3: Obstruction profiles: (**a**) cylindrical chart representation; (**b**) geometric representation.  
According to Figure 4, in this work, the power balance for a PV installation with self-consumption is estimated according to Equations (15)-(17) in which the central instants of each of the six 10-min intervals in which each GMT hour is decomposed are considered and in which, for each instant \(x\) of the representative day of the month \(m\), \(P_{x,m}^{S}\) represents the generated and instantaneously self-consumed power, \(P_{x,m}^{PV}\) the solar power PV generated (Equation (1)), \(P_{x,m}^{D}\) the power demand required by the consumer, \(P_{x,m}^{E}\) the power imported from the grid and \(P_{x,m}^{I}\) the power injected into the network.  
\[P_{x,m}^{S}=\min\left(P_{x,m}^{PV};P_{x,m}^{D}\right), \tag{15}\]  
\[P_{x,m}^{E}=P_{x,m}^{D}-P_{x,m}^{PV}\]  
\[P_{x,m}^{I}=P_{x,m}^{PV}-P_{x,m}^{D}\]  
\[P_{x,m}^{E}=P_{x,m}^{PV}\]  
\[P_{x,m}^{I}=P_{x,m}^{PV}-P_{x,m}^{D}\]  
\[\left.\begin{array}{c}\text{(15)-(17)}\end{array}\right\}\text{if }P_{x,m}^{S}=P_{x,m}^{D}\]  
From Equations (15)-(17) it is possible to perform an hourly balance, integrating the power values obtained for each hour \(h\) of the representative days of each month \(m\) of the year, through Equations (18)-(20) in which the factor \(1/6\) represents the time in hours of which each instant considered is representative.  
\[P_{h,m}^{S}=\frac{1}{6}\sum_{x\in h}P_{x,m}^{S} \tag{18}\]  
\[P_{h,m}^{E}=\frac{1}{6}\sum_{x\in h}P_{x,m}^{E} \tag{19}\]  
\[P_{h,m}^{I}=\frac{1}{6}\sum_{x\in h}P_{x,m}^{I} \tag{20}\]  
Figure 4: Production-demand curve of a PV installation with self-consumption.  
Finally, the monthly balances of self-consumed energy \((S_{m})\), demanded from the network \((E_{m})\) and injected \((I_{m})\) in each month \(m\) can be evaluated with Equations (21)-(23) in which \(N_{m}\) is the number of days in month \(m\).  
\[S_{m}=N_{m}\sum_{h=1}^{24}P_{h,m}^{S} \tag{21}\]  
\[E_{m}=N_{m}\sum_{h=1}^{24}P_{h,m}^{E} \tag{22}\]  
\[I_{m}=N_{m}\sum_{h=1}^{24}P_{h,m}^{I} \tag{23}\]  
Thus, the energy saving in each month, \(m\), \(F_{m}\), can be obtained as the sum of two values, a direct saving due to self-consumption and a compensation term, according to Equation (24) in which \(p_{E}\) is the purchase price of the energy and \(p_{C}\) the compensation price  
\[F_{m}=p_{E}S_{m}+\min(p_{E}E_{m};p_{C}I_{m}), \tag{24}\]  
From (24), therefore, the annual consumption \(F\) will be estimated using Equation (25).  
\[F=\sum_{m=1}^{12}F_{m} \tag{25}\]