## 1. Introduction

## 2. Method

- Position of the PV system (latitude, longitude, and altitude)
- Angle of the system (azimuth and tilt); all modules must have the same angle
- Time-zone, and whether a daylight-saving hour is added in summer
- Air temperature (constant or time-dependent)
- Wind speed (constant or time-dependent)
- PV system type (e.g., glass/polymer open rack)
- PV temperature coefficient (power change per temperature change)
- Albedo (ground reflection)
- Inverter efficiency (constant or power dependent)

## 3. Results

#### 3.1. Temperature Calculation

#### 3.2. Radiation Calculation

#### 3.3. Simulated PV System Power

_{p}, as best fit of the simulation to the measured system power. Then, a correction factor (measured GHI/simulated GHI) was applied to the simulated system power before determining the best fitting system peak power (Figure 4d–f). On average, the system peak power was found to be 12.15 kW without using a GHI correction and 11.4 kW, including measured GHI data. This means that the correction with measured GHI data reduced the obtained system peak power of that system by 6.9%. Using the GHI correction also minimized the scatter between the three days. However, this may overestimate the diffuse light at the PV system if surrounded by other buildings.

#### 3.4. Incident Angle Modifier

#### 3.5. Test of Different PV Systems

## 4. Discussion

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Jäger-Waldau, A. Snapshot of Photovoltaics—March 2021. EPJ Photovolt.
**2021**, 12, 2. [Google Scholar] [CrossRef] - International Energy Agency. Snapshot of Global Photovoltaic Markets: Report IEA PVPS T1-39:2021. Available online: https://iea-pvps.org/snapshot-reports/snapshot-2021/ (accessed on 24 June 2021).
- European Commission. Photovoltaic Geographical Information System (PVGIS). Available online: https://ec.europa.eu/jrc/en/pvgis (accessed on 24 June 2021).
- National Renewable Energy Laboratory (NREL). PVWatts Calculator. Available online: https://pvwatts.nrel.gov/ (accessed on 24 June 2021).
- Hay, J.E. Calculation of the Solar Radiation Incident on Inclined Surfaces. In Proceedings of the First Canadian Solar Radiation Data Workshop, Toronto, ON, Canada, 17–19 April 1978. [Google Scholar]
- Ineichen, P.; Perez, R. A New Airmass Independent Formulation for the Linke Turbidity Coefficient. Sol. Energy
**2002**, 73, 151–157. [Google Scholar] [CrossRef] - Atwater, M.A.; Ball, J.T. A Numerical Solar Radiation Model Based on Standard Meteorological Observations. Sol. Energy
**1978**, 21, 163–170. [Google Scholar] [CrossRef] - King, D.L.; Boyson, W.E.; Kratochvil, J.A. Photovoltaic Array Performance Model; Report SAND2004-3535; Photovoltaic System R&D Department, Sandia National Laboratories: Albuquerque, NM, USA, 2004. [Google Scholar]
- Saad Parvaiz, D.; Stefan, B.; Lukas, W.; Stefan, K. Photovoltaic Yield Prediction Using an Irradiance Forecast Model Based on Multiple Neural Networks. J. Mod. Power Syst. Clean Energy
**2018**, 6, 255–267. [Google Scholar] [CrossRef] - Holmgren, W.F.; Hansen, C.W.; Mikofski, M.A. Pvlib Python: A Python Package for Modeling Solar Energy Systems. J. Open Source Softw.
**2018**, 3, 884. [Google Scholar] [CrossRef] - Rinio, M. PVcheck Software. Available online: https://pvcheck.hotell.kau.se/ (accessed on 6 October 2021).
- Pvlib.Temperature.Sapm_cell—Pvlib Python 0.9.0+0.G518cc35.Dirty Documentation. Available online: https://pvlib-python.readthedocs.io/en/stable/generated/pvlib.temperature.sapm_cell.html (accessed on 8 October 2021).
- Linke, F. Transmissions-Koeffizient Und Trübungsfaktor. Beitrüge zur Physik der Atmosphére
**1922**, 10, 91–103. [Google Scholar] - Sveriges Meteorologiska Och Hydrologiska Institut GHI Data for Karlstad, Sweden. Available online: https://www.smhi.se/data/meteorologi/ladda-ner-meteorologiska-observationer/#param=globalIrradians,stations=all,stationid=93235 (accessed on 30 June 2021).
- Martin Chivelet, N.; Ruiz, J.M. Calculation of the PV Modules Angular Losses under Field Conditions by Means of an Analytical Model (Vol 70, Pg 25, 2001). Sol. Energy Mater. Sol. Cells
**2001**, 70, 25–38. [Google Scholar] [CrossRef] - Solar Energy Materails and Solar Cells. Available online: https://ur.booksc.eu/book/24106631/609401 (accessed on 4 October 2021).
- Reno, M.; Hansen, C.; Stein, J. Global Horizontal Irradiance Clear Sky Models: Implementation and Analysis; Sandia National Laboratories (SNL): Albuquerque, NM, USA; Livermore, CA, USA, 2012. [Google Scholar] [CrossRef]

**Figure 1.**Flow diagram of PVcheck version 0.1. Pascal program parts are shown in orange and green boxes. “Matching” of temperature data points means that temperatures are only taken for the same points in time where imported power data is available. This is normally done by interpolation from the imported temperature data file. Matching of wind speed and GHI is done analogously. Python program parts using pvlib are shown in blue boxes and are enframed. Module and inverter efficiency curves are parts of the parameters of the system that have to be loaded in the beginning.

**Figure 2.**Left column (

**a**–

**c**): Simulated and measured module temperatures on three different days (date in the upper left corner). Right column (

**d**–

**f**): Air temperature, wind speed, and average wind direction angle relative to the azimuth of the PV modules between 9 h and 15 h UTC on corresponding days.

**Figure 3.**Left column (

**a**–

**c**): Simulated and measured global horizontal irradiation (GHI) on three different days (date in the upper left corner). The measurement was done by a local weather station (Sveriges meteorologiska och hydrologiska institut, SMHI). Right column (

**d**–

**f**): Simulated and measured irradiation on corresponding days in the plane of the PV modules. The measurement was done using the above-named calibrated detector mounted directly near the PV modules. The simulations were done without shallow angle reflection correction (see Section 3.4).

**Figure 4.**Left column (

**a**–

**c**): Simulated and measured system power on three different days. The simulated curves using measured temperatures and wind speed, but without using measured GHI data, were fitted to the measured curves by adjusting the system peak power value to 12.2, 12.35, and 11.9 kW

_{p}. Right column (

**d**–

**f**): Simulated and measured system power on corresponding days, using temperature, wind speed, and measured GHI data by adjusting the system peak power value to 11.4 and 11.3 kW

_{p}. All power values were limited to 10 kW, which is the maximum inverter power. The simulations were done without shallow angle reflection correction (see Section 3.3).

**Table 1.**Peak power values determined with PVcheck for five different PV systems and three sunny days each. The calculations include a correction by the incident angle modifier for direct and diffuse light. The numbers are furthermore corrected for inverter efficiency and module lowlight behaviour to show the peak power of the modules. Electrical cable losses are neglected. Part (a) shows results obtained with the Ineichen clearsky model only (without GHI correction). Part (b) includes a correction factor (measured GHI/simulated clearsky GHI) for the simulated irradiation. The nominal peak power is the given module power at standard test conditions from the datasheets, multiplied by the number of modules. For each system, the maximum allowed degradation from the datasheets is given on the right.

PV System | Determined Peak Power of the PV System/kW | Nominal Peak Power Power/kW | Installation Date | Nominal Degradation/% | ||||||
---|---|---|---|---|---|---|---|---|---|---|

(a) without GHI Correction | (b) with GHI Correction | First Year | Next Years | |||||||

2021-04-16 | 2021-04-19 | 2021-06-04 | 2021-04-16 | 2021-04-19 | 2021-06-04 | |||||

A | 12.4 | 12.4 | 12.1 | 11.55 | 11.55 | 11.55 | 12.825 | 2018-08-01 | 3.00% | 0.60% |

B | 18.2 | 18.3 | 16.9 | 17 | 17 | 16.2 | 18.525 | 2019-12-19 | 2.00% | 0.54% |

C | 14.95 | 14.95 | 14.3 | 13.8 | 13.7 | 13.65 | 15.12 | 2019-08-03 | 2.00% | 0.54% |

D | 11.05 | 11.2 | 10.8 | 10.35 | 10.5 | 10.35 | 10.92 | 2020-02-12 | 2.50% | 0.65% |

E | 11.05 | 11.2 | 10.8 | 10.3 | 10.45 | 10.35 | 10.92 | 2020-02-12 | 2.50% | 0.65% |

**Table 2.**Determined peak power divided by the expected peak power of the five PV systems for three sunny days each. The expected peak power was taken as the nominal peak power that was reduced by the maximal nominal degradation over time from installation to the measurement; 100% means that the system runs as expected if the maximal allowed degradation has happened. Weather data was taken from the stations named in Section 2.

PV System | Determined Peak Power/Expected Peak Power | |||||
---|---|---|---|---|---|---|

Without GHI Correction | With GHI Correction | |||||

2021-04-16 | 2021-04-19 | 2021-06-04 | 2021-04-16 | 2021-04-19 | 2021-06-04 | |

A | 101% | 101% | 98% | 94% | 94% | 94% |

B | 100% | 101% | 93% | 94% | 94% | 89% |

C | 101% | 101% | 97% | 93% | 93% | 93% |

D | 104% | 105% | 102% | 97% | 99% | 97% |

E | 104% | 105% | 102% | 97% | 98% | 97% |

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