# Snow-Induced PV Loss Modeling Using Production-Data Inferred PV System Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Sources and Acquisition

#### 2.1.1. Snow Cover

#### 2.2. Method

_{SL}over a certain period of time is given by the difference between the AC energy produced by the snow-impacted PV system, E

_{AC}, and the snow-free production, E

_{AC,sf}. Here, snow-free production is defined as the AC energy produced by the PV system when kept free of snow. However, none of the PV sites in this study have a mirrored set-up of two similar PV systems in which one is allowed to be covered with snow, and the other is kept snow-free. Therefore, the snow-free production is approximated by a modeled snow-free production E

^{*}

_{AC,sf,}and snow losses are given by

#### 2.2.1. Inference of Tilt and Azimuth

#### 2.2.2. Inference of Snow-Free Models

_{b}and S

_{d}are the shading factors associated with the direct (E

_{bn}and E

_{bh}) and diffuse (E

_{dh}) parts of the irradiance. The geometric factor R

_{b}denotes the ratio of beam radiation on the tilted surface to that on a horizontal surface and ρ is the albedo for the surrounding area. Now the shading factor for the direct component is a time-dependent function given by a weighted moving average as follows:

_{bk}and S

_{d}were estimated by minimizing the mean absolute error (again more robust to potential outliers than the squared error) between the modeled and measured hourly PV production.

#### 2.2.3. Snow Loss Approximations and Comparisons

_{AC}in Equation (1) with

_{day}and N

_{data}are the current month’s number of daytime hours and number of hours with AC yield data, respectively. Months not meeting the 90% availability criteria were ignored (values set to “NaN”).

^{2}°C). Then, for each hour where the condition in Equation (6) was met, the snow cover on the array as a whole was expressed in percent of the PV array row height and assumed to slide downwards by

- a PV array consisting of three portrait-oriented module rows;
- a PV array consisting of four landscape-oriented module rows.

## 3. Results

#### 3.1. Comparison of Snow Data

#### 3.2. Inference of PV Array Tilt and Azimuth

#### 3.3. Inference of Snow-Free Models

#### 3.4. Snow Loss Estimation

^{2}) equaled almost half the irradiation during December 2018, the month with the lowest irradiation (1726 Wh/m

^{2}). Even though this example is somewhat extreme, it clearly indicates that very few hours of snow loss in May are enough to contribute more than months such as November or December.

#### 3.5. Comparison with Existing Snow Loss Models

#### 3.6. Snow Loss and Panel Tilt Angle

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map of Sweden and surroundings showing site locations and data scope. Reference sites are marked with green dots (earlier study) and teal circles (current study), while survey and basic sites are marked with red dots.

**Figure 2.**An example of the dynamic evolution of the shading coefficient for the direct component throughout the year (thick blue line). Six fix shading coefficients (black dots), distributed symmetrically around the winter/summer solstice, were interpolated to any time of the year using the six corresponding Hann weighting functions (thin colored lines).

**Figure 3.**Histograms on the characteristics of the survey sites: (

**a**) the number of module rows in the PV array height and (

**b**) the orientation of the modules in the PV array.

**Figure 4.**Results from the estimation of panel geometries for different methods: (

**a**) tilt and (

**b**) azimuth.

**Figure 5.**Illustration of the performance of the inferred snow-free model: (

**a**) density plot for measured and modeled hourly snow-free production normalized with the installed effect and (

**b**) example of monthly modeled snow-free and actual observed production at one of the basic sites.

**Figure 6.**Boxplot for the mean of monthly contributions to the annual snow losses. Means are calculated for the sites with full winter season data (n = 257) and on average two winters (n = 542), with at least 90% data availability. Statistics presented in the plot are the median (green line), 50% (box), and 95% (whiskers) intervals as well as outliers (circles).

**Figure 7.**Results of normalized snow loss calculations in relation to the measured annual electricity yield for all quality-passed sites. The sites’ relative positions illustrate their geographical order by latitude: (

**a**) estimated snow losses for the months September–May and years 2014–2020 and (

**b**) annual mean yield for all years (June–May) with at least 90% data availability. Normalization for both graphs is by installed peak power (direct current at Standard Test Conditions—DC, STC).

**Figure 8.**Boxplot for percentual snow losses per winter season for all 258 sites. The figure indicates mean (red star), median (green line), 50% (box), 95% (whiskers) intervals, and outliers (circles).

**Figure 9.**Scatterplots for estimated winter season snow losses (September-May for 2014–2020) from the inference method described in this article and two existing snow loss models (energy losses normalized by installed peak power (DC, STC)): (

**a**) Marion model [10], assuming arrays with unknown module mounting to have three rows of portrait-mounted modules. A single outlier point is left out and (

**b**) lin–temp model [7].

**Figure 10.**Snow loss during March 2018 as a function of panel tilt angle. Blue dots indicate losses for each of 161 individual sites as a fraction of the modeled snow-free production. The orange line shows the median value for each 10-degree interval.

**Table 1.**Overview of models for long-term prediction of snow losses, either directly (designated “Power” under Model Type) or through snow cover predictions (denoted as “Cover”).

Author (s) | Model Type | Number of PV Systems ^{1} | Number of Winters |
---|---|---|---|

Powers et al. [4] | Power; Empirical | 3; 1 location | 1 |

Townsend and Powers [6] | Power; Empirical | 4; 2 locations | 2 |

Andrews et al. [9] | Power; Empirical | 1 | 2 |

Hong et al. [15] | Power; Empirical | 70 | Up to 4 |

Zamo et al. [17] | Power; Empirical | 28 | 2 |

Shishavan et al. [18] | Power; Empirical | 2 | 3 |

Van Noord et al. [7] | Power; Empirical | 6; 3 locations | 2 |

Ross [11] | Cover; Principal | n/a | n/a |

Lorenz et al. [19] | Cover; Empirical | 14 (77) | 1 |

Marion et al.; Ryberg and Freeman [10,16] | Cover; Empirical | 6 (3); 2 locations | 2 (1) |

Bosman and Darling [20] | Cover; Empirical | (1) | 2 |

Rahmatmand [21] | Cover; Principal | n/a | n/a |

^{1}The number used in empirical model development. Where applicable, the number of systems used in model validation is listed in parenthesis.

Reference Sites | Survey Sites ^{1} | Basic Sites | |
---|---|---|---|

Number of sites | 5 | 102 | 296 |

Meta-data | |||

Peak power | X | X | X |

Location (lat, lon) | X | X | X |

Array tilt | X | X | |

Array azimuth | X | X | |

Module orientation | X | X | |

Number of module rows | X | X | |

Manual clearance of snow | X | X | |

Time series | |||

AC yield | X | X | X |

Snow-cover images | X |

^{1}Limited to site-owner-provided meta-data. Some variations in the meta-data-scope occur.

Basic Sites (Estimated Tilt and Azimuth) | Survey Sites (Estimated Tilt and Azimuth) | Survey Sites (Recorded Tilt and Azimuth) | |
---|---|---|---|

Number of sites | 239 | 19 | 19 |

Pearson correlation | 0.96 | 0.95 | 0.95 |

Bias (model—observation) | 0.12% | 0.022% | 0.020% |

Standard deviation | 6.1% | 6.9% | 6.9% |

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**MDPI and ACS Style**

van Noord, M.; Landelius, T.; Andersson, S.
Snow-Induced PV Loss Modeling Using Production-Data Inferred PV System Models. *Energies* **2021**, *14*, 1574.
https://doi.org/10.3390/en14061574

**AMA Style**

van Noord M, Landelius T, Andersson S.
Snow-Induced PV Loss Modeling Using Production-Data Inferred PV System Models. *Energies*. 2021; 14(6):1574.
https://doi.org/10.3390/en14061574

**Chicago/Turabian Style**

van Noord, Michiel, Tomas Landelius, and Sandra Andersson.
2021. "Snow-Induced PV Loss Modeling Using Production-Data Inferred PV System Models" *Energies* 14, no. 6: 1574.
https://doi.org/10.3390/en14061574