# Exploring the Benefits of Photovoltaic Non-Optimal Orientations in Buildings
## 2 Materials and Methods
### Case Study #1: Residential PVSCS
#### 2.2.4 Economic Balance  
The economic balance was calculated on an hourly basis using the prices of electricity published by the Spanish TSO Red Electrica de Espana in its webpage ESIOS [43] and the energy balances were calculated with Equations (1) and (2). The economic savings \(S\) are composed of two parts: the cost of self-consumed PV-produced energy \(A\) (not imported from the grid), calculated according to Equation (6); and the value of the surplus energy (PV-produced and not self-consumed) compensated with the grid \(C\). \[ \begin{split} A=\sum_{i=1}^{n}\Bigl{(}E^{i}_{PV}-E^{i}_{spl}\ \Bigr{)}\cdot Pr^{i}_{ret}\\ \text{being }i\text{ the }i-th\text{ hour of the month}\\ Pr^{i}_{ret}\text{ the retail price of electricity on the }i-th\text{ hour}\end{split}  \tag{6}\]  
The surplus compensation mechanism states that the surplus energy fed into the distribution grid is valuated at a price indexed to the wholesale market price (slightly lower), and that the final balance (the energy term in the monthly bill) cannot be negative. Thus, the value of compensated surplus energy can be expressed in Equation (7). \[ \begin{split} C=\sum_{i=1}^{n}E^{i}_{spl}\cdot Pr^{i}_{spl}\lor \sum_{i=1}^{n}\Bigl{(}E^{i}_{in}\cdot Pr^{i}_{ret}-E^{i}_{spl}\cdot Pr^{i}_{spl }\Bigr{)}\geq 0\\ C=\sum_{i=1}^{n}E^{i}_{in}\cdot Pr^{i}_{ret}\ \ \forall\sum_{i=1}^{n} \Bigl{(}E^{i}_{in}\cdot Pr^{i}_{ret}-E^{i}_{spl}\cdot Pr^{i}_{spl}\Bigr{)}<0 \\ \text{being }Pr^{i}_{spl}\text{ the price of surplus electricity on the }i-th\text{ hour}.\end{split}  \tag{7}\]  
So, the monthly economic savings are:  
\[S=A+C \tag{8}\]  
And the monthly cost of the electricity taken from the grid  
\[M=\sum_{i=1}^{n}E^{i}_{in}\cdot Pr^{i}_{ret} \tag{9}\]  
The electricity cost without PV would be the cost of the consumed energy valuated at retail price  
\[B=\sum_{i=1}^{n}E^{i}\cdot Pr^{i}_{ret} \tag{10}\]  
The annual balances for the reference PVSCS and old and new tariffs are calculated and presented in Table 2. Due to the changes in tariffs in June 2021, the hourly prices for the new tariff 2.0 TD before June 2021 and for the old tariff 20A after May 2021 were calculated accordingly to the tolls of both tariffs. For this consumer without PVSCS, the annual bill with the new tariff 2.0 TD is almost equal to the former tariff 2.0A. Regarding the bill with PVSCS installed, the new tariff 2.0 TD was 6.4% lower than the 2.0A, and the total savings were 12% higher for the new 2.0 TD tariff over the 2.0A. These higher economic savings for the PV self-consumed electricity were due to the highest electricity prices from 10 a.m. to 2 p.m. in this new tariff, just in a period of high PV production. These total savings are the cashflow that were then used for the calculation of the relevant economic indicators (TROI, NPV, IRR, LCOE) widely used in PVSCS financial analysis [45]. A linear degradation of PV production of 0.8% annually was assumed [46]. This translates to a reduction of 0.8% in the cashflow. The lifespan for the PVSCS is assumed to be 25 years, including the cost of a replacement of the inverter in the 13 th year. Considering the sizes of residential PVSCS installations, the operation costs would be zero and the maintenance cost were estimated to be 1% of the value of the installation paid on a yearly basis, in line with published results [47]. The discount rate was selected as 1% based in the economic indicators for Spain and the Euro zone. The study incorporates the time of return on investment (TROI) as the first economic measure. TROI is one of the most widely used methods for comparing the benefits of a program with the same costs per unit, per person or aggregated for the program as a whole. TROI was selected because it is a cost-benefit-oriented economic method [48]. Still, it is also used to calculate the return on investment (ROI), i.e., how much is produced by how much is invested. The cost of the systems under study is based in actual prices of this kind of installations.