1. Introduction
Photovoltaic (PV) technologies are evolving rapidly, at both the cell level and the module level, resulting in significantly higher efficiencies and lower costs. In addition, PV systems are now being deployed in locations once thought to be unfavorable for solar-generated electricity. High-latitude locations, as well as other regions where snow prevails for many months of the year, are one example. Performance in these regions can be increased by installing PV modules at steeper tilt angles than are typical for middle latitudes [
1,
2]. Performance can also be increased by deploying bifacial modules, which receive irradiance from both the front and rear surfaces, resulting in higher electricity production and lower levelized cost of energy (LCOE) projections [
3,
4]. In high-albedo environments, for example, the amount of reflected light reaching the rear side of a module’s solar cells can increase the energy output by as much as 30 percent [
5,
6]. Given the energy gains and minimal price difference between bifacial and monofacial modules, bifacial modules now dominate the utility-scale market worldwide; moreover, they have driven the significant growth of utility-scale PVs in arctic regions, including Alaska [
7]. The International Technology Roadmap for Photovoltaic (ITRPV) notes that by 2030, bifacial PV cells are expected to account for 70% of the total world PV cell market [
8].
To accurately model the energy output of bifacial PV, rear-side plane-of-array (POA) irradiance data are needed [
9], and ground albedo is an important input for modeling the amount of irradiance incident on the rear side of a PV module [
10]. Accurate ground albedo measurements can also improve performance models for high-tilt-angle monofacial solar PV systems that are typically found at high latitudes [
11] where ground-reflected irradiance can contribute to the irradiance incident on the front of the modules. Existing models typically assume an albedo of 0.8 if snow is present, and 0.2 if no snow is present [
12,
13]. This can lead to inaccuracies because the albedo of snow can vary drastically, depending on the age and characteristics of the snow [
14], so the actual snow albedo at any point in time often lies between these two values. This is demonstrated in
Figure 1 below, which tracks the winter-time albedo in Fairbanks, Alaska; Flagstaff, Arizona; and Brookings, South Dakota. Between October 1st and April 30th, albedo is between the 0.2 and 0.8 thresholds 32% of the time, 36% of the time, and 68% of the time, respectively, supporting the need for an alternative to the current 0.8 and 0.2 snow/no-snow albedo assumptions often utilized.
This work presents the following:
A detailed description of our team’s multi-year analysis of time-series albedo data;
A method to model the albedo of snow as a function of temperature and time since the last snowfall;
A comparison of the performance of the developed model against existing common models for estimating ground albedo and remote sensing albedo measurements;
Evidence showing that improved snow albedo modeling can improve the accuracy of PV performance models.
Historical Approaches to Modeling the Changing Albedo of Snow
Historically, albedo models have been included within climate change models or considered as offshoots of snow melt models [
15,
16,
17]. As Dirmhirn and Eaton [
18] note, the daily variation in albedo from snow-covered surfaces is attributable to the varying contributions of angle-of-incidence-dependent specular reflection from the snow surface as well as from the metamorphism of the snow based on temperature and time. For the albedo model presented in this paper, we authors were not concerned about short-term sun-angle-dependent changes in albedo but instead focused on daily albedo changes linked to snow metamorphism. This premise is well documented in the literature: Rango and Martinec [
17], for example, note that older wet snow with a higher density has a lower albedo and a high liquid water content, and with each subsequent degree day becomes more melt-efficient. Anderson [
19] similarly demonstrates the negative relationship between snow density and albedo. Since snow albedo changes as snow metamorphoses, and snow density changes during the melt process, we surmised that some of the same inputs for snow melt models could also inform a snow albedo model.
Particularly in recent decades, satellite-based irradiance measurements have been used to construct historical time-series albedo datasets for regions with significant snow cover for at least part of the year. Calleja et al. [
16], for example, used data from the Moderate Resolution Imaging Spectroradiometer (MODIS) to examine the influence of incident radiation and ambient temperature on the snow albedo of a region of Antarctica. Jin and Simpson [
20] used data taken from the U.S. National Oceanic and Atmospheric Administration (NOAA) Advanced Very High Resolution Radiometer (AVHRR) and an established model for transforming top-of-atmosphere and reflected irradiance into values for albedo and implemented two methods to correct for non-isotropic snow cover. Stroeve and Nolin [
21] used data from the Multi-Angle Imaging Spectroradiometer (MISR) to generate snow albedo data over the Greenland ice sheet at a 1.1 km resolution. As discussed below, while albedo measurements from these sources can agree well with local ground measurements, their limited spatial resolution can potentially lead to inaccuracies in situations where albedo variation is highly dependent on local topography and micro-climates. Additionally, satellite imagery datasets tend to be published in batches at relatively infrequent time intervals, limiting their applicability to more-real-time modeling.
Snow melt models are common in the hydrology field and enable planners to predict water runoff and make water resource planning decisions. The need to understand and predict snow ablation extends from engineering applications to understanding the role of the Arctic in the global ecosystem [
14]. Two types of snow melt models are generally adopted: the data-intensive surface energy-balance models and the relatively simple degree-day models [
14,
17]. Energy-balance models are sophisticated and the inputs include net radiation, sensible and latent heat flux, conductive energy flux, and the energy required to heat the snowpack to 0 °C [
14]. Degree-day models (also referred to as the temperature index method) calculate snow melt depth by multiplying the number of degree days by the degree day ratio [
17], and the work presented here is inspired by degree-day models. Amaral et al. [
22] developed a similar albedo decay model using snow melt data collected in New Hampshire. However, Amaral et al. used a less-common snow density measurement as an input.
The melt-hour albedo model described herein presents a simple model for estimating the reduction in albedo of snow after a snowfall. The goal of this model is to maintain reasonable accuracy of the albedo prediction while minimizing the amount of input data and thus minimizing the barriers to its use. A more accurate estimate of snow surface albedo will enable PV performance modelers to improve the accuracy of their energy estimates. This improvement will be especially notable in high-latitude climates, which frequently utilize high PV tilt angles, as well as for bifacial PV modules which are becoming increasingly common in PV system deployments.
Within this paper, we first describe the data upon which the model is built, then describe how the model was developed from these data; we discuss the performance of the model, and then finally describe the impact of the model on improving the accuracy of PV system performance models.
4. Model Development
The development of the melt-hour albedo model is summarized in
Figure 4, which shows the data inputs, described in the previous section, and the process we used to develop the model from those data. After the filtering described previously, we combined the average daily albedo and daily snowfall data to visualize how snow albedo changes over time and specifically how albedo changes after a snowfall. We grouped the data into “snow events” (or, alternately, “events”), where a snow event is defined as a time period of three or more days that make up the interval between the end of one snowfall and the beginning of the next snowfall, as exemplified in
Figure 5 for Fairbanks, AK.
To observe the effect of time and temperature on albedo, we developed the melt-hour albedo methodology described above. We then incorporated the total number of melt hours into the daily albedo dataset so that each day was associated with the average daily albedo and the total daily sum of the melt hours. Then, events with melt hours equal to 0 were separated from events with melt hours greater than 0. To be more specific, during a snow event with a melt hour total of zero, the average hourly temperature would never have exceeded 0 °C.
Next, we applied an additional filtering step to remove events with maximum albedo values less than 0.5. Since freshly fallen snow typically has an albedo above 0.5, events with a maximum albedo less than 0.5 suggest either faulty data or incomplete snow coverage.
We then normalized all measured albedo values within each event so that the maximum normalized albedo within each event was equal to 1; this allows us to compare events and focus on the relative albedo change over time.
Figure 6 shows the normalized albedo versus the melt hours since the previous snowfall using the data from
Figure 5 to demonstrate how the melt-hour technique is useful for comparing snow events. This methodology is explained below and was used to develop the albedo change curves used in the model.
Once the data-cleaning process was completed, we plotted the normalized albedo data for all snow events with cumulative melt hour totals greater than zero for the six sites, as shown in
Figure 7. The data in
Figure 7 indicate two separate modes of albedo reduction: one in which the albedo declines quickly and levels off, and one in which the albedo declines slowly at first then rapidly declines with additional melt-hour accumulation.
By grouping the events according to the shape of their albedo reduction, we were able to determine that the presence of snow prior to a snow event had a high correlation with the shape of the event’s albedo loss. As shown in
Figure 8, in cases where the depth of snow on the ground exceeded 10 cm in all of the 3 days preceding the snow event, the normalized albedo of the snow decreases slightly over the first 200 melt hours, then quickly degrades to a minimum around 0.2.
However, in cases where the snow depth was less than 10 cm in any of the 3 days preceding a snow event, the normalized albedo quickly decays to a minimum value also around 0.2, as shown in
Figure 9. We have thus identified these two modes as the “exponential decay” mode, where there is little snow on the ground prior to a snowfall event, and the “slow decay” mode, where there is more snow on the ground prior to a snowfall event. The 10 cm snow depth was chosen because this snow depth appeared to be most effective in splitting the data into the two albedo change modes described above.
We acknowledge the presence of a middle decay population between the slow decay and exponential decay groups that is composed primarily of Flagstaff and Vermont data. Since these data appear in both decay shape datasets in
Figure 8 and
Figure 9, we believe that these snow events have snow depths in the prior 3 days that are close to the 10 cm division that would change their classification from one decay shape to another. The model in its present form accounts for the majority of observations.
We believe that this bifurcated albedo change is the result of snow melting to reveal patches of uncovered ground, which occurs more quickly when there is very little snow on the ground prior to a new snow event. The condition of low snow depth prior to a snowfall event is most common at the beginning of the snow season.
Based on the graph shown of the slow decay and exponential decay data in
Figure 7, equations of the best-fit lines for the slow decay and exponential decay normalized albedo are as follows:
where
is the slow decay normalized albedo;
is the exponential decay normalized albedo;
M is the accumulated melt hours since the most recent snowfall.
The exponential decay curve has a natural minimum of 0.2, which is approximately the albedo of the underlying surfaces at the test sites. For other sites, i.e., those with higher or lower ground albedo, the function can be altered to accommodate the higher or lower ground albedo by changing the first term in the sum.
These equations were used as the basis for the melt-hour albedo model presented here. As explained above, the only two required inputs are temperature and snow depth. Model accuracy, however, can be improved if actual data for a specific site are available to replace the default data provided below. The optional inputs, where default data can be replaced with measured data, include the following:
the albedo of the snow-free ground (default = 0.2);
the snow depth before the first time step of the dataset (default = 0 cm);
the minimum albedo present when snow is present (default = 0.4);
the albedo of fresh snow for the site (default = 0.8);
the minimum snow depth threshold used to indicate whether the modeled snow albedo or the ground albedo shall be used (default = 2.5 cm);
the change in daily snow depth needed to trigger a snow event that resets the melt-hour count (default = 0 cm).
The melt-hour model works sequentially over time-series data, as shown in
Figure 10. The model checks to see if the snow depth has increased since the last day of snow data. If the snow depth has increased, it is assumed that new snow has fallen and the melt-hour counter is set to 0. The model also determines if the snow event warrants albedo decay according to the exponential decay function or the slow decay function based on the snow depth values of the prior 3 days. If the total snow depth is less than 10 cm in any of the previous 3 days, then the exponential decay function is used; otherwise, the slow decay function is used. The model then begins stepping through the hourly time-series data and accumulating melt hours for any time period when the average hourly ambient temperature is greater than 0 °C. For each time step, the following tasks are executed:
The model-calculated albedo is given by Equation (
3):
where
is the albedo of the snow at a given time as calculated by the model;
is the normalized albedo, which may either be or depending on the snow conditions before the most recent snowfall;
is the albedo of freshly fallen snow for the site, an optional input to the model;
is the minimum albedo for the site, which is also an optional input to the model.
6. Conclusions
As bifacial PV systems proliferate across high-latitude locations, where snowfall predictably occurs, accurate albedo projections will increasingly factor into PV performance models. In this paper, we have presented a model for predicting snow albedo when time-series temperature and snow depth data are known. This model has several advantages:
It uses commonly available ambient temperature and snow depth data;
It allows users the ability to incorporate site-specific measured data to replace default values and improve model accuracy;
It is available in pvlib model repositories [
38];
It allows for real-time estimation of snow surface albedo.
However, the model is unable to account for:
Situations where snow depth may change independently of snowfall or temperature-driven melting, such as due to wind-blown snow or a rainfall event;
Snow albedo changes that are not caused by time and temperature, such as soiling of the snow by particulates, including ash, dust, debris, etc.
We acknowledge there are other models, such as the energy-balance model described earlier, that might better predict snow albedo, but we believe that the additional data requirements of those models render them impractical for many PV modeling applications.
We have compared the performance of the proposed melt-hour albedo model to that of other simple performance models, such as the binary model that applies a high albedo when snow is present and a low albedo when snow is absent, and conclude that the melt-hour albedo model better estimates the albedo of snow at a site. We have also found that the melt-hour albedo model does not perform as well as the NSRDB albedo estimates. While we acknowledge that albedo estimates based on remote sensing data, such as those from the NSRDB, do compare very favorably with measured albedo, these remote-sensing-based estimates are typically available after a significant time lag, which makes them unsuitable for near-real-time applications. In addition, the NSRDB is not currently available for locations north of 60° latitude, despite the rapid growth of solar PV in these regions.