# Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}·day

^{−1}) and the second, after Athens, in global radiation levels (4.88 kWh·m

^{−2}·day

^{−1}) [5]. In this context and due to the sharp decrease in costs that it has experienced in recent years [6], the photovoltaic sector is key to this strategy [7]. It has therefore become the spearhead of the objectives of the Clean energy for all Europeans package [8] to generate a decarbonized energy model. Furthermore, it is a technology with a high capacity to generate employment [9]. However, the different changes that Spanish regulations have undergone regarding energy [10,11,12] have generated certain instability in the solar sector.

## 2. Materials and Methods

#### 2.1. Characterisation of the Power Produced by the Collectors

^{−2}) in the instant of time $t$,

^{−2}),

^{−1},

#### 2.2. Geometric Characterisation of the Collector Plane

#### 2.3. Characterisation of Obstructions

#### 2.4. Energy Balance

#### 2.5. Technical and Economic Valuation

- $NetPresentValue,NPV:$ accounts for the present value of all economic flows in the useful life of the project according to Equation (26), in which $F$ is the income of each annual period $j$ updated to the initial year (Equation (25)), $M$ is the maintenance expenses calculated in the initial year, ${C}_{0}$ is the initial investment $\left(j=0\right),$ $n$ is the useful life of the installation, $d$ is the discount rate or interest rate required of the investment, $i$ the inflation rate and $\mathsf{\Delta}p$ the rate of year-on-year increase in energy prices.$$NPV=-{C}_{0}+{\displaystyle \sum}_{j=1}^{n}F\frac{{\left(1+\mathsf{\Delta}p\right)}^{j}}{{\left(1+d\right)}^{j}}-{\displaystyle \sum}_{j=1}^{n}M\frac{{\left(1+i\right)}^{j}}{{\left(1+d\right)}^{j}}$$
- Period of return on investment or$Pay-back$: is an indicator that measures how long the total investment will be recovered at the present value. In this way, it serves to accurately reveal the date on which the initial investment will be covered. When the cash flows are the same every year, the calculation of the Payback will be given by Equation (27).$$Pay-back=\frac{{C}_{0}}{F}$$

## 3. Results

- Production, $P$:$$\begin{array}{l}P(\mathrm{kWh/year})=73.6+0.5419\xb7PP\xb7\mathrm{cos}\beta +\\ 340.02\mathrm{sin}\beta \mathrm{sin}\gamma +0.58\xb7PP-0.164\xb7\zeta \\ {R}^{2}=0.988& \\ \overline{{\epsilon}_{P}}=81.2\mathrm{kWh/a\xf1o}& \end{array}$$
- Compensation, $C$:$$\begin{array}{l}C(\mathrm{kWh/year})=-35.65-238.20\xb7\mathrm{cos}\beta +0.529\xb7PP\xb7\\ \mathrm{cos}\beta +289.9\mathrm{sin}\beta \mathrm{sin}\gamma +8.27\xb7{10}^{-5}\xb7P{P}^{2}\\ {R}^{2}=0.966& \\ \overline{{\epsilon}_{C}}=110.1\mathrm{kWh/a\xf1o}& \end{array}$$
- Self-Consumption, $A$:$$\begin{array}{l}A(\mathrm{kWh/year})=50.95+\left(325.02-0.0776\xb7PP\right)\xb7\mathrm{cos}\beta +\\ 0.6540\xb7PP-9.98\xb7{10}^{-5}\xb7P{P}^{2}-2.1478\xb7\zeta \\ {R}^{2}=0.963& \\ \overline{{\epsilon}_{A}}=35.2\mathrm{kWh/a\xf1o}& \end{array}$$

## 4. Conclusions

_{E}, p

_{C}, C

_{0,}i, d, Δp, and n. Although the values offered by the empirical model are approximate, its advantage lies in being able to carry out an analytical and quantitative study of the influence of the design parameters on the $NPV$ and $Pay-back$ indicators.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Vector magnitudes characteristic of the geometry of the collectors of a self-consumption PV plant.

**Figure 7.**$NPV$ depending on peak power and degree of obstruction for optimal orientation and worst orientation.

**Figure 10.**(

**a**) Comparison of $NPV$ obtained with the proposed model (Equation (32)) versus that obtained by simulation; (

**b**) Comparison of $Pay-back$ obtained with the proposed model (Equation (33)) versus that obtained by simulation.

**Table 1.**Design variables and variation intervals considered for the systematic analysis of PV installations with self-consumption.

Variable | Inclination of Collectors $\left(\mathit{\beta}\right)$ | Azimuth of Collectors $\left(\mathit{\gamma}\right)$ | Power Peak $\left(\mathit{P}\mathit{P},\mathit{i}\mathit{n}\mathbf{W}\right)$ | Angle of De Obstruction $\left(\mathit{\zeta}\right)$ |
---|---|---|---|---|

Number of cases | 7 | 13 | 7 | 4 |

Minimum Value | 0° | −180° | 500 W | 0° |

Maximum Value | 90° | 180° | 3500 W | 30° |

Increase | 15° | 30° | 500 W | 10° |

Month | H (kWh/m^{2}) | k_{t} | Mean Temperature | Representative Julian Day |
---|---|---|---|---|

January | 2.06 | 0.46 | 9.1 | 17 |

February | 3.08 | 0.51 | 10.7 | 47 |

March | 3.93 | 0.50 | 13.5 | 75 |

April | 4.81 | 0.49 | 16.3 | 105 |

May | 5.28 | 0.48 | 19.4 | 135 |

June | 6.74 | 0.59 | 24.4 | 162 |

July | 7.14 | 0.64 | 27.9 | 198 |

August | 6.50 | 0.64 | 27.6 | 228 |

September | 5.00 | 0.58 | 24.3 | 258 |

October | 3.30 | 0.49 | 18.6 | 288 |

November | 2.29 | 0.46 | 13.6 | 318 |

December | 1.73 | 0.42 | 9.6 | 344 |

Parameters Considered | Value |
---|---|

Installation cost reflected in the power of the C_{U} itself (EUR/Wp) | 1.1 |

Overall return on investment and conversion losses: $\eta $ | 0.8 |

Price of energy purchased: ${p}_{E}$ (EUR/kWh) | 0.16 |

Compensation price for energy discharged to the grid: ${p}_{C}$ (EUR/kWh) | 0.04 |

Maintenance cost reflected in the power of the installation (EUR/Wp) | 0.02 |

Annual discount rate: $d\left(\%\right)$ | 3 |

Inflation rate: $i\left(\%\right)$ | 1 |

Year-on-year increase in energy Price: $\mathsf{\Delta}p\left(\%\right)$ | 2.5 |

Project useful life: $n$ (years) | 20 |

**Table 4.**Annual production (in kWh/year) of a 1000 Wp PV installation and an obstruction angle of 30° as a function of the azimuth and inclination of collectors.

Inclination(º) | |||||||||||||

90 | 466 | 496 | 553 | 618 | 671 | 701 | 703 | 700 | 670 | 616 | 552 | 495 | 466 |

75 | 542 | 584 | 659 | 740 | 810 | 856 | 869 | 855 | 808 | 738 | 657 | 584 | 542 |

60 | 664 | 693 | 772 | 859 | 937 | 991 | 1011 | 990 | 936 | 858 | 770 | 692 | 664 |

45 | 802 | 824 | 889 | 969 | 1044 | 1096 | 1117 | 1095 | 1042 | 968 | 887 | 823 | 802 |

30 | 942 | 958 | 1003 | 1062 | 1120 | 1162 | 1177 | 1161 | 1119 | 1061 | 1001 | 958 | 942 |

15 | 1067 | 1075 | 1098 | 1128 | 1159 | 1181 | 1189 | 1181 | 1158 | 1127 | 1097 | 1075 | 1067 |

0 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 | 1151 |

−180 | −150 | −120 | −90 | −60 | −30 | 0 | 30 | 60 | 90 | 120 | 150 | 180 | |

Azimuth(º) |

**Table 5.**Annual self-consumed energy (in kWh year) of a PV installation of 1000 Wp and an obstruction angle of 30° as a function of the azimuth and inclination of collectors.

Inclination(º) | |||||||||||||

90 | 466 | 496 | 551 | 608 | 661 | 692 | 689 | 680 | 648 | 602 | 550 | 495 | 466 |

75 | 542 | 584 | 640 | 692 | 742 | 780 | 791 | 771 | 727 | 679 | 633 | 581 | 542 |

60 | 655 | 673 | 712 | 755 | 797 | 833 | 847 | 828 | 788 | 742 | 701 | 667 | 655 |

45 | 744 | 748 | 770 | 802 | 838 | 867 | 879 | 865 | 833 | 795 | 763 | 744 | 744 |

30 | 805 | 810 | 821 | 842 | 867 | 887 | 895 | 886 | 864 | 839 | 817 | 807 | 805 |

15 | 853 | 856 | 862 | 873 | 884 | 893 | 896 | 892 | 883 | 871 | 861 | 855 | 853 |

0 | 883 | 883 | 883 | 883 | 883 | 883 | 883 | 883 | 883 | 883 | 883 | 883 | 883 |

−180 | −150 | −120 | −90 | −60 | −30 | 0 | 30 | 60 | 90 | 120 | 150 | 180 | |

Azimuth(º) |

**Table 6.**Annual savings for the first year (in €/year) of a 1000 Wp PV installation and an obstruction angle of 30° as a function of the azimuth and inclination of collectors.

Inclination(º) | |||||||||||||

90 | 75 | 79 | 88 | 98 | 106 | 111 | 111 | 110 | 105 | 97 | 88 | 79 | 74.634 |

75 | 87 | 93 | 103 | 113 | 121 | 128 | 130 | 127 | 120 | 111 | 102 | 93 | 86.747 |

60 | 105 | 108 | 116 | 125 | 133 | 140 | 142 | 139 | 132 | 123 | 115 | 108 | 105.1 |

45 | 121 | 123 | 128 | 135 | 142 | 148 | 150 | 148 | 142 | 134 | 127 | 122 | 121.39 |

30 | 134 | 135 | 139 | 144 | 149 | 153 | 154 | 153 | 148 | 143 | 138 | 135 | 134.31 |

15 | 145 | 146 | 147 | 150 | 152 | 154 | 155 | 154 | 152 | 150 | 147 | 146 | 145.05 |

0 | 152 | 152 | 152 | 152 | 152 | 152 | 152 | 152 | 152 | 152 | 152 | 152 | 152.04 |

−180 | −150 | −120 | −90 | −60 | −30 | 0 | 30 | 60 | 90 | 120 | 150 | 180 | |

Azimuth(º) |

**Table 7.**Net Present Value $NPV$ of a PV installation of 1000 Wp and an obstruction angle of 30° as a function of the azimuth and inclination of collectors.

Inclination(º) | |||||||||||||

90 | -9 | 81 | 249 | 429 | 590 | 684 | 678 | 655 | 560 | 415 | 246 | 79 | -9 |

75 | 222 | 349 | 534 | 714 | 881 | 1003 | 1038 | 981 | 846 | 682 | 517 | 343 | 222 |

60 | 570 | 635 | 783 | 947 | 1104 | 1227 | 1273 | 1215 | 1081 | 918 | 758 | 621 | 570 |

45 | 880 | 906 | 1006 | 1140 | 1277 | 1384 | 1427 | 1378 | 1265 | 1123 | 988 | 895 | 880 |

30 | 1126 | 1148 | 1207 | 1301 | 1401 | 1479 | 1509 | 1476 | 1395 | 1293 | 1198 | 1143 | 1126 |

15 | 1330 | 1342 | 1374 | 1421 | 1470 | 1508 | 1521 | 1506 | 1467 | 1418 | 1371 | 1340 | 1330 |

0 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 | 1463 |

−180 | −150 | −120 | −90 | −60 | −30 | 0 | 30 | 60 | 90 | 120 | 150 | 180 | |

Azimuth(º) |

**Table 8.**$Pay-back$ of a 1000 Wp PV installation and an obstruction angle of 30° as a function of the azimuth and inclination of collectors.

Inclination(º) | |||||||||||||

90 | 14.7 | 13.9 | 12.5 | 11.3 | 10.4 | 9.9 | 9.9 | 10.0 | 10.5 | 11.3 | 12.5 | 13.9 | 14.7 |

75 | 12.7 | 11.8 | 10.7 | 9.8 | 9.1 | 8.6 | 8.5 | 8.7 | 9.2 | 9.9 | 10.8 | 11.8 | 12.7 |

60 | 10.5 | 10.1 | 9.5 | 8.8 | 8.3 | 7.9 | 7.7 | 7.9 | 8.3 | 8.9 | 9.6 | 10.2 | 10.5 |

45 | 9.1 | 9.0 | 8.6 | 8.1 | 7.7 | 7.4 | 7.3 | 7.5 | 7.8 | 8.2 | 8.7 | 9.0 | 9.1 |

30 | 8.2 | 8.1 | 7.9 | 7.7 | 7.4 | 7.2 | 7.1 | 7.2 | 7.4 | 7.7 | 8.0 | 8.1 | 8.2 |

15 | 7.6 | 7.6 | 7.5 | 7.3 | 7.2 | 7.1 | 7.1 | 7.1 | 7.2 | 7.4 | 7.5 | 7.6 | 7.6 |

0 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 | 7.2 |

−180 | −150 | −120 | −90 | −60 | −30 | 0 | 30 | 60 | 90 | 120 | 150 | 180 | |

Azimuth(º) |

Optimal $NPV$ for the Best Orientation ($\gamma =0\xb0and\beta =15\xb0$) and Corresponding $PP$ | Minimum $NPV$ for the Worst Orientation ($\gamma =180\xb0and\beta =90\xb0$) and Corresponding $PP$ | |||
---|---|---|---|---|

Obstruction | $NPV$ (EUR) | Corresponding PP (Wp) | $NPV$ (EUR) | Corresponding PP (Wp) |

$\zeta $ = 0° | 1626.0 | 1500 | 48.5 | 1500 |

$\zeta $ = 10° | 1616.1 | 1500 | 14.8 | 1000 |

$\zeta $ = 20° | 1522.2 | 1500 | −4.38 | 500 |

$\zeta $ = 30° | 1319.1 | 1000 | −11.7 | 500 |

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**MDPI and ACS Style**

Varo-Martínez, M.; Fernández-Ahumada, L.M.; López-Luque, R.; Ramírez-Faz, J.
Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds. *Appl. Sci.* **2021**, *11*, 6517.
https://doi.org/10.3390/app11146517

**AMA Style**

Varo-Martínez M, Fernández-Ahumada LM, López-Luque R, Ramírez-Faz J.
Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds. *Applied Sciences*. 2021; 11(14):6517.
https://doi.org/10.3390/app11146517

**Chicago/Turabian Style**

Varo-Martínez, Marta, Luis Manuel Fernández-Ahumada, Rafael López-Luque, and José Ramírez-Faz.
2021. "Simulation of Self-Consumption Photovoltaic Installations: Profitability Thresholds" *Applied Sciences* 11, no. 14: 6517.
https://doi.org/10.3390/app11146517