Horizontal-to-tilt irradiance conversion for high-latitude regions: a review and meta-analysis

This review focuses on the solar irradiance model chain for horizontal-to-tilted irradiance conversion at high latitudes. The main goals of the work are 1) to assess the extent to which the literature accounts for decomposition and transposition models specifically developed for high-latitude application; 2) to evaluate existing validation studies for these particular conditions; 3) to identify research gaps in the optimal solar irradiance model chain for high-latitude application (i.e., latitude ≥60°). In total, 112 publications are reviewed according to their publication year, country, climate, method, and keywords: 78 publications deal with decomposition models and 34 deal with transposition models. Only a few models (6) have been parameterized using data from Nordic countries. Here, we compare 57 decomposition models in terms of their performance in Nordic climate zones and analyze the geographical distribution of the data used to parametrize these models. By comparing the Normalized Root Mean Square Deviation coefficients for direct normal irradiation, the decomposition models Skartveit1 and Mondol1 are most effective on one-hour scale and Yang4 on one-minute scale. Recent studies on the empirical transposition models estimating the global tilted irradiation on vertical surfaces show the best performance for Perez4 and Muneer models. In addition, innovative methods such as artificial neural networks have been identified to further enhance the model chain. This review reveals that a validated model chain for estimating global tilted irradiation at high latitudes is missing from the literature. Moreover, there is a need for a universal validation protocol to ease the comparison of different studies.

Highlights

• In total, 47 one-hour and 10 one-minute decomposition models are parametrized for high latitudes.

• Skartveit1 and Yang4 are the most effective one-hour and one-minute models.

• Perez4 and Muneer can assess east-west vertical bifacial photovoltaics.

• Decomposition models can achieve sub-hour resolution, transposition models cannot.

• Validated model chains are lacking for east-west vertical surfaces at high latitudes.

1 Introduction

Achieving net-zero emissions by 2050 represents a milestone for the low-carbon built environment. However, reaching this target is hindered by continuously rising global energy consumption (International Energy Agency, 2021). In response, different governments have proposed increasing the share of energy production from renewables and enhancing the efficiency of new energy power plants (Gielen et al., 2019).

To achieve climate targets, monofacial photovoltaics (MPV) and bifacial photovoltaics (BPV) have been exploited in recent decades. While MPV produces electricity by collecting sunlight only from the front side (i.e., the backside is covered with an opaque sheet), BPV collects sunlight from both sides of the panel. Using two active sides (i.e., the front and the rear side) increases overall energy production because the backside of the panel can harvest sunlight scattered by the atmosphere and reflected by the ground or other surfaces behind the panel. When not integrated into the building envelope, MPV and BPV are generally south oriented (in the Northern Hemisphere) and tilted at an optimal angle, which maximizes annual energy production, depending on the location. Therefore, the peak in energy production is usually observed at around solar noon. However, there is a mismatch between typical solar production and the average consumption profile of a residential sector which peaks in the early morning (around 7–8a.m.) and late afternoon (around 5–6p.m.) (International Energy Agency, 2022). Such a temporal mismatch between solar power production and electricity load can be reduced by adopting east-west (E-W) vertically mounted BPV systems (VBPV) (Khan et al., 2017; Jouttijärvi et al., 2022), whose production peaks are specifically in the morning and in the afternoon or evening. Furthermore, the electricity spot price typically peaks during the evening, and therefore increasing the self-consumption of PV electricity should be prioritized. Increasing self-consumption and avoiding electricity use during peak price represent the main drivers for E-W VBPV.

E-W VBPV have the greatest potential at high latitudes (Jouttijärvi et al., 2022) based on the low average Sun elevation, wide azimuth range, and long periods of high ground albedo due to snowfalls (Guo et al., 2013; Khan et al., 2017). In these conditions, an E-W VBPV can outperform a south-facing photovoltaic (PV) system (Stein, 2018; Pike et al., 2021). Rodríguez-Gallegos et al., (2018) investigated the advantages and disadvantages of MPV and BPV installations from an economic perspective. Their analysis highlighted the importance of different parameters, such as latitude, weather data, albedo, and the applied model chain, when calculating global tilted irradiance (E_gt) (Perez et al., 1990a; Hofmann and Seckmeyer, 2017a). Knowing the E_gt is a key factor for determining the potential energy production of a new PV plant (Prasad et al., 2020; Ramadhan et al., 2021). To estimate the E_gt, the amount of direct normal irradiance (E_bn) and diffuse horizontal irradiance (E_dh) must be known. However, ground measurements of E_bn and E_dh are uncommon, and these quantities are usually estimated from global horizontal irradiance (E_gh) using empirical decomposition models (Hassan et al., 2021). Although the scientific community has improved and developed new decomposition models since they were first introduced in 1961, they continue to be strongly influenced by the selected location, as they are empirical models based on site-specific data (Gueymard and Ruiz-Arias, 2016; Li et al., 2017). The number of locations considered in the model parametrization has been increasing in the recent years by permitting to implement a group of quasi-universal models (i.e., decomposition models for global application) in contrast to local models (i.e., decomposition models for regional application). However, local models can outperform models presented as quasi-universal when applied to the location where the data originates (Starke et al., 2021).

In contrast to decomposition models, transposition models allow E_gt to be estimated based on solar irradiance (i.e., E_bn, E_dh), Sun geometry (i.e., solar zenith angle, solar azimuth angle), and surface geometry (i.e., tilt angle, azimuth angle). These models have used different assumptions about the spatial distribution of sky radiance over the hemisphere, ranging from isotropic (Liu and Jordan, 1961) to anisotropic (Perez et al., 1987a). Nowadays, the anisotropic assumption is used the most, as it accounts for the circumsolar and horizon brightening contribution beyond the isotropic sky radiance and the direct beam irradiance (Figure 1).

FIGURE 1

FIGURE 1. Different solar contributions to global tilted irradiance.

Within this framework, E-W VBPV systems in high-latitude application pose different challenges compared to south-facing PV systems. In particular, during the early and late hours of the day, when energy production peaks for E-W VBPV systems, the atmospheric thickness is high, which is a major source of error in many transposition models (Toledo et al., 2020). In addition, solar elevations less than five degrees are often omitted in the decomposition models because data quality check routines tend to filter them out as low-reliable datapoints (Utrillas and Martinez-Lozano, 1994; Hofmann and Seckmeyer, 2017b). This is only a minor drawback when modelling conventionally mounted PV systems. However, when modelling vertical E-W mounted systems at high latitudes, a significant part of the production is cut when solar elevations below five degrees are excluded (Figure 2). During 2021 in Trondheim (Norway), a direct solar irradiance of approximately 120 kWh/m² reached the ground throughout the year (around 15% of the total yearly direct normal irradiance) when the Sun elevation angle was lower than five degrees. Moreover, another source of error is related to the estimation of the horizon brightness component of diffuse irradiance, which has a greater influence on E-W VBPV than on south-facing MPV systems.

FIGURE 2

FIGURE 2. Angular distribution of direct normal irradiance depending on sun elevation angle. Data refers to 2021 in Trondheim.

1.1 Aims and structure of the study

The ability to accurately predict PV energy production is based on implementing a solar irradiance model chain which comprises suitable decomposition and transposition models. Knowing which combination of decomposition and transposition models should be used to estimate E_gt in a new PV system is of primary importance. In fact, the accuracy of the model used for estimating the available solar energy is tied to uncertainty in the predicted energy production of the system. However, such models are usually created and validated for a specific geographical area (i.e., local, regional, global). Therefore, their accuracy can vary depending on where they are exploited.

Although a significant number of publications have validated decomposition and transposition models, the extent to which the literature covers the use of decomposition models in the Nordic geographical area is unclear. Moving from the extensive worldwide validation studies carried out by Ramadhan et al. (2021), and Yang (2022), the aim of this review is to explore decomposition models which can adequately estimate the E_dh and E_bn at high latitudes. In addition, transposition model validation studies are investigated to provide a complete theoretical framework for a model chain for E_gt estimation of E-W VBPV in Nordic setting. The paper is structured as follows: the materials and methods (Section 2) defines the literature search workflow, the statistical indicators, and the investigated location; the theory section (Section 3) outlines the theoretical framework behind the model chain for E_gt estimation; the results section (Section 4) provides an overview of decomposition and transposition models; the discussion section (Section 5) comments on the results and analyses the strengths, weaknesses, opportunities, and threats (SWOT), literature gaps, and study limitations. The review concludes by considering future developments and summarises the most important findings and the implications for future advancements in model development and application to high-latitude locations (Section 6).

2 Material and methods

2.1 Literature search

A systematic review approach was followed to investigate decomposition and transposition models for predicting E_gt on E-W VBPV in high-latitude locations. Relevant literature was identified using a keyword-based search on the Web of Science (WoS) and Scopus databases. Among the existing databases, these were selected as they return literature from highly reliable sources and provide wide geographic coverage. Identical search terms were used in both databases. Scientific journal articles that focused on the development and validation of methodologies for estimating E_gt were selected through a three-step screening process similar to (Manni and Nicolini, 2022). The first and the limited to selecting the best combination of search terms. Although using numerous search terms may be considered best, it can result in an unnecessarily high number of database hits which require longer and more extensive screening. In the third step, the eligibility of each database hit was “manually” assessed. The abstracts were screened to identify and exclude review articles and comparative studies which did not introduce new models.

Initially, all the possible combinations of three groups of terms were searched for in the titles, abstracts and keywords of the papers selected from both databases. The Boolean Operator “AND” was used to connect the three groups (A, B, and C) which included the following terms:

• Group A: “decomposition” and “transposition”;

• Group B: “*radia*“, and “solar”;

• Group C: “tilted”, “photovoltaic”, and “surface”.

Terms within the same group were mainly connected with the Boolean Operator “OR”. In particular, the query which was entered in both databases was: (“decomposition” OR “transposition”) AND (“*radia*” W/3 “solar”) AND (“tilted” OR “surface” OR “photovoltaic”). The operator W/3 (NEAR/3 in WoS) indicated that the search should only identify articles where the connected words appear within a range of 3 words in the selected search fields, while the asterisks were used to consider all the words including “radia” such as “irradiation”, “radiation”, “radiative”, and “irradiance”. The search was conducted in March 2022. The chosen search terms were found in more than 500 articles in Scopus and WoS. The order of the terms within the group and the order of the groups did not alter the returned literature: for example, (“decomposition” OR “transposition”) AND (“*radia*" W/3 “solar”) AND (“tilted” OR “surface” OR “photovoltaic”) and (“solar” W/3“*radia*”) AND (“photovoltaic” OR “tilted” OR “surface”) AND (“transposition” OR “decomposition”) returned the same publications. However, this search was deemed ineffective, as it returned a wide range of literature with topics beyond the scope of decomposition and transposition models. The search terms used in step two were refined with the help of the VOS Visualizer tool. This machine learning-based tool ranks the words contained in authors’ keywords from database hits and then applies clustering models to classify the most used words in groups. The resulting density visualization provided a quick overview of the main areas in the bibliometric network, highlighting the presence of journal articles about chemical photodegradation, photocatalysis, and x-ray diffraction. The density visualization in Figure 3 shows how the database hits belong to two different research areas that only share the “solar energy” term. Probably, using “decomposition” without “model” as well as including “photovoltaic” in the entered query led to such undesirable results.

FIGURE 3

FIGURE 3. Density visualization of words in the authors’ keywords from the journal articles selected during step one. The words included in the search were excluded from the figure.

In step two, the entered query was modified into: [(“decomposition” OR “transposition”) W/5 “model”] AND (“*radia*” W/3 “solar”). The density visualization map (Figure 4) highlights the exclusion of the unwanted scientific journal articles which appeared in step one. The chosen search terms were found in around 150 articles in Scopus and WoS. Finally, the screening process in the third step permitted the results to be further narrowed. Around 110 database hits were selected for a detailed review in this study. Among these, 78 research studies describe decomposition models, while 34 research studies focus on transposition models.

FIGURE 4

FIGURE 4. Density visualization of words in the authors’ keywords from the journal articles selected during step two. The words included in the search were excluded from the figure.

2.2 Climate zones

Studies related to Norway, Sweden and Finland were investigated in this work. According to the Köppen-Geiger climate classification (Köppen, 1900) updated by Peel et al. (2007), these countries mainly belong to subtypes of cold climate which begin with the letter D (Table 1). Alongside these, areas classified as subzones from polar climate (E) and mild temperate climate (C) are also present (Figure 5).

TABLE 1

TABLE 1. Description of Köppen-Geiger climate symbols and defining criteria for the cold climate.

FIGURE 5

FIGURE 5. Worldwide distribution of the Köppen-Geiger climate zones which are present in Norway, Sweden, and Finland (i.e., Cfb, Cfc, Dfb, Dfc, ET). The map was created by re-elaborating the original version in Beck et al. (2018).

Most Norwegian, Swedish, and Finnish regions belong to the “Dfc” sub-climate which corresponds to subarctic climates with the coldest month averaging below 0°C and up to 3 months averaging above 10°C. No significant precipitation difference is observed throughout the year. However, since only one weather station from the Dfc subzone is considered in the most relevant validation studies about decomposition models (Gueymard and Ruiz-Arias, 2016; Yang, 2022), the other climate zones present in Norway, Sweden, and Finland (i.e., Cfb, Cfc, Dfb, and ET) are also included in this study.

3 Theory

3.1 Global tilted irradiance calculation

This section outlines the theory behind decomposition and transposition models used to estimate E_gt. Regarding the taxonomy applied to decomposition and transposition models, these are named after the main author’s name with the addition of a number in case more models from the same author exist. In this article, the only exception to this is represented by the BRL decomposition model which is named after the initial of the three scientists who developed this: Boland, Ridley et al. (2010).

The total solar irradiance outside the Earth`s atmosphere (E₀) is almost constant at 1,361.1 W/m² (Gueymard, 2018). Absorption, scattering, and reflection phenomena occur within the atmosphere due to the presence of O₂, O₃, CO₂, and H₂O molecules (Calvin and WR, 2012). Therefore, the E_bn reaching the Earth’s surface is reduced by approximately 30% on a clear day and when the Sun is directly overhead (Calvin and WR, 2012), while E_dh is generated because of scattering events. Sunrays travel through the air mass (AM) following a path whose length is variable and depends on the position of the Sun. The AM length is the distance covered by sunray through the atmosphere and normalized by the shortest possible path.

As reported in Eq. 1, the sum of direct and diffuse irradiance incident to a horizontal plane is defined as E_gh.

$E_{gh}=E_{bn}\cos {\theta }_z+E_{dh}(1)$

where θ_z is the solar zenith angle (the angle between the direct line to the Sun and a vertical line).

The E_gh is used to calculate the clearness index (k_t) which is equal to the E_gh to E₀cosθ_z ratio. The diffuse fraction (k_d) is the share of the diffuse horizontal irradiance from the global horizontal irradiance (E_dh/E_gh). Due to the atmosphere inhomogeneity, the k_t and k_d indices vary with space and time.

The E_dh can be split into isotropic (E_d;iso), circumsolar (E_d;circ) and horizon brightening (E_d;hor) components. The E_d;iso is emanated uniformly from every point in the sky dome, while the E_d;circ originates from forward scattering rays near the path of sunrays. Finally, the E_d;hor consists of the brightness near the horizon and occurs when rays are scattered multiple times in the atmosphere in a band just above the horizon. Positive values for E_d;hor correspond to clear sky days (and they increase with the θ_z), while negative values are calculated for overcast days when the sky is not visible and the clouds near the horizon are darker than the rest of the hemisphere (Perez et al., 1990b). Cloud coverage influences irradiance scattering. E_gh can be reduced by more than 50% with a k_d close to one (heavy cloud cover day). However, some changes in the cloud pattern can increase E_gh above the clear sky conditions and the k_t index is consequently greater than one. Such a phenomenon is known as cloud enhancement (CE) (Inman et al., 2016; Gueymard., 2017a).

Determining the position of the Sun in relation to the investigated PV system is necessary to quantify the E_gt. The position of the Sun can be easily estimated for a specific combination of date, time, latitude, longitude, and elevation. Numerous algorithms determine the Sun position with high accuracy (Michalsky, 1988; Reda and Andreas, 2004; Blanc and Wald, 2012). Among these, the one proposed by Reda and Andreas (2004), which is used in the open-source pvlib-python tool (Holmgren et al., 2018) and National Renewable Energy Laboratory (2014), enables predicting the θ_z and solar azimuth angle (α_z) with an accuracy of 0.0003°. The θ_z and α_z as well as the tilt angle of the investigated PV surface (β) and its azimuth (α) allows calculating the angle of incidence of the sunrays (Φ) (Gueymard, 2022).

Following this, E_gt can be calculated with the following equation:

$E_{gt}={f_{b}E}_{bn}\cos \mathrm{\Phi }+f_d{E}_{dt}+E_{r;gr}(2)$

where E_dt is the calculated diffuse irradiance on the tilted surface and E_r;gr is the incident irradiance from ground reflections. The f_b and f_d factors account for the shading phenomena of direct and diffuse irradiance, respectively. To calculate E_gr, the View Factor (VF) from a specific ground area to the sky, the E_dh for that ground area, the ground albedo (ρ), and the VF from the tilted surface to the ground area must be known.

$E_{gr}={VF}_{srf\to gr}{\rho }_{gr}\left({VF}_{gr\to sky}{E}_{dh}+f_b{E}_{bn}\cos {\theta }_z\right)(3)$

The ground area is assumed as horizontal in Eq. 3.

3.2 Decomposition modeling

To estimate the solar irradiance collected by an oriented surface, E_dh and E_bn in the chosen location, the surface geometry configuration, and the characteristics of the surroundings (i.e., buildings, vegetation) in terms of horizon profile and albedo must be known. Quantifying E_dh and E_bn represents an issue due to the lack of measured data. Although it is feasible from a technological point of view, only E_gh is usually monitored for economic reasons (Gueymard, 2022). Therefore, implementing numerical models for decomposing E_gh into E_dh and E_bn are a topic largely debated by the scientific community.

Decomposition models enable E_dh and E_bn to be estimated by knowing E_gh. The first decomposition models to be developed exploited experimental data to identify empirical correlations between k_d and k_t. The reliability of these models was limited to climate conditions similar to the one used in the model parametrization. Therefore, more complex models have been developed which correlate k_d to more than one predictor. In general, such models calculate the k_d index through the Eq. 4 that was originally proposed by Yang and Boland (2019):

$k_d=\mu \left(X\right)+g°f\left(X\right)+\epsilon (4)$

where X is a vector of predictors, μ(X) describes a trend (constant or linear combination of X), g is the model function (typically exponential or logarithmic), f (X) is a linear function which can be a step function, and ε is the error estimation. Predictors are input parameters which can be based on measurements or deterministic calculations. The most popular predictors are the k_t, the θ_z, and the variability index (V). A decomposition model using only k_t as a predictor will probably result in less accurate results than a model using multiple predictors (Figure 6) (Gueymard and Ruiz-Arias, 2016).

FIGURE 6

FIGURE 6. Measured and modelled k_d vs. k_t at Trondheim (Norway). Measurements are retrieved from the NTNU-SINTEF SolarNet in Trondheim (Manni et al., 2023).

Several databases provide solar radiation datasets based on satellite monitoring or reanalysis models from Numerical Weather Prediction (NWP). Alongside these, Typical Meteorological Year (TMY) datasets are commonly used to evaluate building energy performance. Nonetheless, the use of solar irradiation data from statistic-based weather data files may result in the incorrect computing of diffuse fractions as well as in systematic error within the evaluation of the potential benefits. Databases from satellite observation are characterized by a specific spatial resolution and are not available for each time-period and for each latitude. Moreover, they present limitations regarding the assessed solar radiation components and land cover. The major issue related to the use of satellite images is distinguishing between snow and cloud coverage because they usually have the same pixel values and spatial distribution patterns. Conversely, reanalysis databases are created by running modern NWP models on previous data before correcting the outcomes with ground-measured meteorological data. The main advantage is that the datasets usually cover the whole Earth, although with a low accuracy and spatial resolution compared to satellite observations. Since large parts of Norway, Sweden, and Finland lack coverage of satellite-derived datasets, reanalysis databases have potential if they are more accurate than using empirical decomposition models with ground-measured E_gh. In this regard, the Copernicus Arctic Regional Reanalysis (CARRA) system, which is the first regional atmospheric reanalysis targeted for European parts of the Arctic areas, have been implemented in 2022 to specifically investigate the climate at high-latitude locations. Databases providing E_gh and another radiative component for the Nordic zone are listed in Table 2.

TABLE 2

TABLE 2. List of modelled databases which provide at least two irradiance components for a significant part of Norway, Sweden, and Finland. Metric conversion of 1° longitude is around 110 km, while 1° latitude can range between 63 km (at 55° latitude) and 36 km (at 71° latitude).

3.3 Transposition modeling

Once E_dh and E_bn have been determined, E_gt can be estimated by applying the transposition models in the next step of the model chain. Numerous transposition models have been implemented for quantifying E_gt on south-oriented MPV (Gueymard, 2009; Gueymard, 2022). Nonetheless, this calculation is still prone to significant errors when it comes to estimating E_gt characterized by a low angle of incidence or impinging on faces away from direct sunlight (i.e., rear of a BPV) (Riedel-Lyngskær et al., 2020). The Perez4 model (Perez et al., 1990b) is shown in Eq. 5 as an example:

$\left\{\begin{array}{c}{E}_{dt}=E_{dh}\left[\left(1-F_1\right)\left(\frac{1+\cos \beta }{2}\right)\right]+F_1\left(\frac{a}{b}\right)+F_2\sin \beta \\ a=\max \left(0,\cos \theta \right)\\ b=\max \left(\cos 85°,\cos {\theta }_z\right)\\ F_1=\mathit{max}\left[0,\left(f_{11}+f_{12}∆+\frac{\pi {\theta }_z}{180°}{f}_{13}\right)\right]\\ F_2=f_{21}+f_{21}∆+\frac{\pi {\theta }_z}{180°}{f}_{23}\\ ∆=\frac{E_{dh}AM}{E_0}\end{array}\right.(5)$

where f_nm varied according to sky conditions. However, empirical transposition models are often incorporated into software tools which perform VF or raytracing analyses.

Such tools usually assess the VF from the ground to the sky and from the tilted surface to the ground. The view factor describes the ratio of radiant energy emitted by a surface which is incident on another surface. This is calculated with Eq. 6:

${VF}_{1\to 2}=\frac{1}{A_1}{\int }_{A1}{\int }_{A2}\frac{\cos {\theta }_1\cos {\theta }_2}{\pi S^2}d{A}_{1}d{A}_2(6)$

When it comes to PV systems, the VF can be estimated by considering only two dimensions.

Raytracing assessment can be applied as an alternative to the VF evaluation to estimate shading. Raytracing is a method to mathematically determine the ray’s path within a tridimensional environment (Negi et al., 1986). The Monte Carlo simulation can be applied to track rays from the PV surface backwards to the light source as well (Hansen et al., 2016; Manni et al., 2020). If compared to the VF calculation, both raytracing assessment and the probabilistic approach require greater computational time (Berrian and Libal, 2020).

4 Results

4.1 Decomposition models in the Nordic climate zone

4.1.1 Bibliometric analysis

Figure 7 shows the number of articles published per year along with the time resolution adopted for the model outputs. More than half of the studies were published in the last two decades, even though the first study about decomposition modeling was published in 1961. A turning point in this research topic occurred in the year 2015. In fact, decomposition models characterized by sub-hour time resolution, which first appeared in 1988, became increasingly frequent between 2013 and 2015, when they finally outnumbered one-hour models.

FIGURE 7

FIGURE 7. Number and distribution of time resolution of publications by year. The black and red dashed lines indicate, respectively, the yearly percentage of the study adopting a sub-hour and an hour time resolution.

After 2015, all the decomposition models included in this review had sub-hour resolution. This represents a significant improvement because it allows some instantaneous events to be investigated such as the CE and albedo enhancement (AE) effects, which cannot be detected when aggregating data hourly.

The analysis of weather stations whose data were used for the model validations in the reviewed studies highlighted the scarcity of decomposition models that are specifically developed for high latitudes. In fact, around three quarters of these models were implemented for dry climates (B) and temperate climates (C), while the rest were divided into tropical climates (A), continental climates (D), and polar climates (E). Figure 8 highlights that the most investigated climate zones were the humid sub-tropical climates (Cfa), the oceanic climate (Cfb), and the Mediterranean hot summer climates (Csa). Moreover, the tropical savanna climates with dry-winter characteristics (Aw), the cold semi-arid climates (Bsk), the warm summer continental climates (Dfb), and the tundra climate (ET) were the most common from the other main climate zones. Data from the sub-artic climates (Dfc), which is most common in the territories of Norway, Sweden, and Finland, were used in the validation of only five decomposition models.

FIGURE 8

FIGURE 8. Number of publications by Köppen-Geiger main climate zones (A) and sub-climate zones (B). The red crosses represent the percentage of studies for each bin with respect to the totality.

The geographical distribution of the studies is reported in Figure 9. Among the investigated countries, up to six decomposition models were proposed for Norway, while none was found for Sweden or Finland.

FIGURE 9

FIGURE 9. Number of studies by countries.

Decomposition model inputs were chosen from a wide group of predictors. The number of predictors varied and ranged from one to seven in the reviewed studies. A unique predictor was used by 71 decomposition models (k_t in 70), while in 27 studies another predictor such as θ_z or V was added to k_t. The number of publications per quantity of predictors is reported in Table 3 along with the number of publications for each predictor.

TABLE 3

TABLE 3. Number of publications by number of predictors used as decomposition model input.

In total, around 20 different predictors were considered in the reviewed decomposition models. The analysis highlighted that the θ_z, the V, and the time of the day (t) are the three most used parameters after the clearness index (Figure 10).

FIGURE 10

FIGURE 10. Number of publications by used predictor.

A frequency analysis of the keywords associated with the reviewed scientific journal articles was carried out (Figure 11). On the one hand, the “diffuse solar radiation” and “clearness index” keywords were the most used and were found in 59 and 30 articles, respectively. On the other hand, the “clouds” and “statistical models” keywords were less frequent with only seven and six articles, respectively. The “other” category includes up to 26 different keywords that occurred less than five times.

FIGURE 11

FIGURE 11. Number of publications by authors’ keywords.

4.1.2 Meta-analysis

This section presents the decomposition models existing in the literature that were parameterized by considering solar radiation datasets from at least one of the climate zones characteristic of Norway, Sweden, and Finland (see Figure 5) (Tables 4, 5). Decomposition models which are characterized by an hourly time resolution are shown in the first part, while those having a one-minute time resolution are described in the second part.

TABLE 4

TABLE 4. Distribution of the climate zones used in the parametrization of the reviewed one-hour decomposition models. Multiple models presented in the same publication are clustered in the same row.

TABLE 5

TABLE 5. Distribution of the climate zones used in the parametrization of the reviewed one-minute decomposition models. Multiple models presented in the same publication are clustered in the same row.

Their performances were evaluated based on the extensive validation work carried out by Gueymard and Ruiz-Arias (2016) in the case of one-hour decomposition models, and on the novel study from Yang (2022) in the case of one-minute decomposition models. Moreover, the E_bn quantities predicted for locations classified as Cfb, Cfc, Dfb, Dfc, or ET in the Köppen Geiger climate map were considered to estimate the normalized Root Mean Square Deviation (nRMSD) values used to compare the models.

One-hour models which might be suitable for high-latitude locations were mostly climate-specific models. In fact, only one fourth of them were found to consider more than one location during the parametrization process. In this regard, the Skartveit model family was established in (Skartveit and Olseth, 1987) based on data collected over 32 years in Bergen (Norway). Later, the models were improved by introducing the hourly variability index (V_hour) and the ground albedo predictor (Skartveit et al., 1998). The possibility of using ρ as an input parameter and exploiting climate data from Bergen make the Skartveit3 model suitable for high-latitude locations.

The Reindl model family was also parametrized with a solar radiation dataset from Norway, although it was used alongside datasets from the United States, Germany, Denmark, and Spain (Reindl et al., 1990). However, solar irradiation values from 1979 were used which cannot be considered as completely representative of the current scenario.

In addition, other decomposition models such as the ones from Hollands, Perez, Boland, and Ridley model families are evaluated in this section since they were developed for high-latitude climate zones. It is worth highlighting that the Yao model family (Yao et al., 2014) was included in the analysis even if it was parametrized with data from the Cfa climate zone. The reason is that these models were specifically developed for high albedo climate zones.

The analysis of the nRMSD index is reported in Table 6 and visualized in Figure 12. The Skartveit1 model was identified as the most effective model at high latitudes, although it was outperformed by Spencer in the E zone. In addition to this, Mondol1, which was parametrized with solar radiation data from Northern Ireland, exhibited high performance levels being the second in the global ranking and the third in the D zone ranking.

TABLE 6

TABLE 6. Error statistics about E_bn calculated for the high-latitude locations from Supplementary Material by Gueymard and Ruiz-Arias (2016). The best ten performing models are shown, with statistics averaged over the selected sites.

FIGURE 12

FIGURE 12. Error statistics about E_bn calculated for the high-latitude locations from Supplementary Material by Gueymard and Ruiz-Arias (2016). Values for each weather station are reported on the top, while on the bottom the average value is reported for each climate zone (dots) and globally (red line). Light green, orange, and dark blue dots correspond to a weather station in C, D, and E climate zone.

Regarding the analysis of one-minute models, those reviewed in the extensive validation study carried out by Yang (2022) and suitable for the high-latitude locations have been considered. Unlike one-hour models, one-minute models are usually developed to be globally applied. Therefore, datasets from different locations are exploited in their parameterization. Among these, the Engerer2 (Engerer, 2015) was considered to perform best, but recently the Yang4 was demonstrated to be able to replace it as the new quasi-universal model (Yang, 2022).

Alongside this, the one-minute model from Paulescu and Blaga (Paulescu and Blaga, 2019) was assessed since it proved to be as valid as the Engerer2 in all the locations considered in their study. The Paulescu was established for the Cfb climate and then validated against measured values from 35 locations, but none from the D climate zone. The authors argued that the irradiance (particularly the diffuse irradiance) showed odd features in these zones, as already stated by Gueymard, (2017a), Gueymard, (2017b).

In Yang and Boland (2019), two enhanced versions of Engerer2 (YANG1 and YANG2) were developed by adding the satellite derived diffuse fraction predictor (k_d,s). The k_d,s predictor consists of a one-minute interpolation between 30-min datapoints from satellite databases and is used to describe low frequency variability patterns. In fact, satellite observations can be exploited in the decomposition process by providing information about temperature and wind speed (Urraca et al., 2018), as well as about the ground irradiance which can be determined though specific methods (i.e., Heliosat-4) (Qu et al., 2017). However, Babar et al. (2020) demonstrated that satellite-based models are worse than empirical decomposition models despite the higher accuracy of their input datasets. Recently, Yang has introduced a new modeling strategy named temporal-resolution cascade (Yang, 2021). Such a strategy was incorporated into Yang3 (15-min model) and Yang4 (1-min model). Since Yang1, Yang2 and Yang3 are not as accessible nor as accurate as Yang4 (Yang, 2022), they were overlooked in the present performance assessment.

In 2019, Bright et al., (2019) updated the Engerer2 model by recalculating the parameters with datasets from 75 different stations worldwide and by removing inputs such as the Sun elevation angle and the clear sky irradiance (E_cs) (Engerer4). When datasets from the Australia region were used, the Root Mean Square Deviation was reduced by 17.9% from the original model. Conversely, in the cold climate zone, the authors reported an overall worsening in model performance.

In 2021, Starke et al. (2021) reviewed and updated their previous work (Starke18A and Starke18B) (Starke et al., 2018) by proposing new climate specific models based on BRL1M (Starke21). A new predictor, the hourly clearness index (k_t,hour), was added. A model for each climate zone and sub-zone defined in Köppen-Geiger climate classification was proposed and then verified against Engerer2, Skartveit3, Perez2, and BRL. The study revealed that the new models were the most accurate. It is worth highlighting that only the Starke family’s models developed for climate zones characteristic of the Norway, Sweden, and Finland regions were considered in this work.

In Figure 13, the comparative analysis on models’ performance levels in the high-latitude locations confirmed that Yang4 is the best one-minute model. Alongside this, the Starke family model can represent an effective solution particularly when applied to C and D zones (Table 7).

FIGURE 13

FIGURE 13. Error statistics about E_bn calculated for the high-latitude locations from Supplementary Material by Yang (2022). Values for each weather station are reported on the (A), while on the (B) the average value is reported for each climate zone (dots) and globally (red line). Light green, orange, and dark blue dots correspond to a weather station in C, D, E climate zone.

TABLE 7

TABLE 7. Error statistics about E_bn calculated for the high-latitude locations from Supplementary Material by Yang (2022). The best ten performing models are shown, with statistics averaged over the selected sites.

4.2 Transposition models

4.2.1 Bibliometric analysis

Figure 14 shows a timeline in which the reviewed studies about transposition modeling are distributed and clustered according to the model typology. Four model typologies were found in the literature: isotropic models, anisotropic models, artificial neural network (ANN) models, and probabilistic models. Isotropic models were the first model typology to be implemented in 1961; then, anisotropic models were developed (1977). ANN (e.g., multi-inputs convolutional neural networks, generalized regression neural networks, gradient-boosting frameworks for machine learning) and probabilistic models have been introduced recently in this research field (from 2013), but they are considered so promising that around 70% of the transposition models published since 2013 belong to these two groups. Conversely, the isotropic models have been investigated increasingly less over the last two decades. Nonetheless, ANN and probabilistic models have been presented in around 20% of the reviewed studies, while the isotropic and anisotropic models have occurred in 63% and 16% of the studies, respectively (Figure 15).

FIGURE 14

FIGURE 14. Number of publications and distribution of model typologies by year.

FIGURE 15

FIGURE 15. From the left: percentage distribution of model typologies, percentage distribution of sky conditions in the model validation, percent distribution of time resolution values. Percentage is calculated over all the reviewed transposition models.

When it comes to the simulated sky conditions, half of the reviewed transposition models were found to be validated in both clear and overcast sky conditions (Figure 15). Conversely, 18% of the models performed exclusively for overcast skies, while 16% of the models can operate only in case of a clear sky. Furthermore, most of the transposition models were found to provide hourly outputs (87%), and only three models could work with a sub-hour time resolution (8%). Contrary to decomposition modeling, there is a lack of trend in transposition modeling towards the reduction of the time resolution. Such a discrepancy might represent an issue and reduce the impacts of the decomposition model’s enhancement on the overall model chain. However, the hourly transposition models reviewed in (Gueymard, 2009) appear to be equally applicable to one-minute data. The tabular review (Table 8) provides information about the approach to modeling solar diffuse radiation (i.e., isotropic, circumsolar, horizontal bright). It is worth highlighting that recent ANN and probabilistic models based on machine-learning methods, can directly estimate the E_gt without considering the anisotropic sky dome.

TABLE 8

TABLE 8. Tabular review of the reviewed transposition models according to the typology, the evaluated diffuse radiation sources, and the time resolution. For the time resolution: M stands for “month”, h stands for “hour”, and m stands for “minute”.

4.2.2 Research evaluation

The review study about transposition models from Yang, (2016) highlighted that no universal model exists. However, the Perez4, Gueymard, Muneer, and Hay models showed high performance levels, locally. The analysis performed for locations in Switzerland (Loutzenhiser et al., 2007) and Poland (Nowak and Włodarczyk, 2009) revealed that Perez4 and Muneer perform the best for E-W vertical surfaces, with Perez4 performing slightly better. The nRMSD coefficients estimated for Perez and Muneer were around 32% and 34%, respectively (Nowak and Włodarczyk, 2009). This outcome was questioned in Toledo et al., (2020) where the Muneer model turned out to be more accurate than Perez4. The reason can be attributed to the location and datasets used to test the two models: data from North America, Germany, and Singapore was used in the first, whereas data from Spain was used in the second. Moreover, the Perez4 has been parameterized with data from nine locations in the United States of America and three locations in Europe, while Muneer is parameterized with data from five locations in Europe. Models are typically characterized by better error statistics when applied to a location where data exploited for the parameterization were measured. However, these climate conditions are different from Norway, Sweden, and Finland; therefore, it is not possible to express a preference with this information.

An innovative sky-radiance model was conceptualized in Yang (2016) and then implemented by Westb and rook, (2015), but the outcomes demonstrated that Perez4 was still superior for east- or west-facing vertical surfaces. Nonetheless, significant issues such as overestimating diffuse solar radiation were reported for the Perez4 model when assessing hours with high k_t due to CE or AE (Gueymard, 2017b). In fact, calculations during these hours of the day are critical for all the reviewed models, which showed systematic errors and an incapacity to simulate CE and AE effects. Therefore, exploiting empirical models which consider short time-enhancement effects are recommended. In particular, combining different transposition models as a function of the clearness index (i.e., Muneer for k_t<0.85 and Gueymard for 0.85 k_t) enables achieving a higher accuracy level (Gueymard, 2017a).

Appelbaum et al., (2019) proposed various enhancements to anisotropic radiance models for single rows of PV panels by neglecting sky obstructions. These corrections such as incorporating the sky VF (SVF) were applied to the KLUCHER model to account for adjacent rows and temporal deviations in E_d;circ and E_d;hor. The proposed approach is valuable for analyses performed in an urban environment characterized by an SVF lower than 0.9. The modified version of Klucher was found to be more accurate for the locations considered in the study. Similarly, Gu et al., (2020) established an optical model in which irradiance and VF are coupled for assessing single row irradiance.

Up to 26 parametric models (isotropic and anisotropic) were designed through Artificial Neural Networks (ANNs) and machine learning techniques in (de Simón-Martín et al., 2017). They developed a re-parameterized version of Perez4 based on a training dataset from Spain. Such a local version of Perez4 performed the best for East- and West-oriented facades. The outcomes prove that transposition models fitted with local data usually outperform models based on global data or data from other locations.

5 Discussion

5.1 Inconsistencies in the decomposition modeling

The literature on decomposition modelling is evolving rapidly with new innovative models being presented and more validation studies carried out. The methodology and standards used are well documented in most cases, but not consistent among the studies. New methods are frequently being developed, and an overview of recent development as well as historical works are presented in many recent review studies (Gueymard and Ruiz-Arias, 2016; Paulescu and Blaga, 2019; Starke et al., 2021; Gueymard, 2022; Yang, 2022). Up to six aspects such as the estimated value (i.e., E_bn, k_d), the statistic indicators, the benchmark dataset, the data time resolution, the quality control routine, and the method of input calculations were found to be inconsistent in the validation studies. Each of these aspects reduces the scientific validity of a comparative analysis between the models. Various qualitative and quantitative performance indicators were assessed in the reviewed validation studies, making it difficult to compare the outcomes. When it comes to the decomposition models, validation studies typically provide a graphical comparison between the estimated and the measured k_d values. Sometimes, the deviation of the estimated k_d from the measured quantity is also plotted against the k_t depending on the author’s preferences.

Conversely, almost all of the reviewed studies assessed quantitative performance indicators such as nMBD and nRMSD. However, features of the used datasets such as time interval, quality control, and model implementation are not homogeneous throughout the studies, making it difficult to compare them. A nonparametric statistical procedure, based on the sum of ranks, is presented in Stone, (1994) as a solution to the problem of ranking the overall performance of solar radiation estimation models at multiple locations. In addition, the use of large datasets is preferable since it reduces the influence of rare weather phenomena and improves the reliability of the validation process. Several studies used data collected by the Baseline Solar Radiation Network (BSRN) meteorological stations which were provided with a one-minute time step in the early 2000s with a high quality and time resolution (Gueymard and Ruiz-Arias, 2016). Nonetheless, the climate alteration due to solar cycles, human influence, and global climate change make it necessary to use only the most recent data. Authors of the reviewed studies considered data from a few years back and up to a few days before. Yang and Boland, (2019) exploited data from 2016 by providing the number of datapoints from each meteorological station, while Starke et al., (2021) stated that data after 2,000 was considered. Conversely, Bright et al., (2019) exploited datasets from the previous 5 years. To highlight inconsistency in the methodology, Table 9 illustrates the variance in inputs and outputs of three recent model validation studies, as an example.

TABLE 9

TABLE 9. List of weather stations, time period, model outputs, and statistical indicators used by Yang and Boland (2019), Starke et al. (2021), and Bright et al. (2019).

Validation studies also differ in the models exploited to predict E_bn and the k_d. Eq. 1 was applied by some authors to validate the models for calculating E_bn (Gueymard and Ruiz-Arias, 2016), while some others exploited such equations to validate only k_d (Starke et al., 2021) or both E_bn and k_d (Paulescu and Blaga, 2019). Furthermore, the quality control routine which was applied to filter impossible or unlikely data from the datasets is not always the same. Despite the guidelines identified in (Sengupta et al., 2017), there is not one universally accepted way to perform quality control on irradiance data. In this regard, Paulescu and Blaga, (2019) replicated the dataset [D FIT 1 in (Paulescu and Blaga, 2019)] used by Gueymard and Ruiz-Arias, (2016), to validate Paulescu against the Engerer2. Also, the same quality control filters were applied to make outcomes comparable. Nonetheless, negative solar irradiance values were obtained in (Paulescu and Blaga, 2019) at the end of the quality control since Gueymard and Ruiz-Arias, (2016) manually modified data by changing all k_t values greater than one to one. Finally, the calculation of model inputs differs. When estimating the k_t index, the E₀ is required, but some authors did not provide the E₀ values (Yang and Boland, 2019), some used an equation based on the day of the year (Bright et al., 2019), and some applied a constant of either 1,366.1 W/m² or 1,361.1 W/m² (Gueymard and Ruiz-Arias, 2016). Such an inconsistency might represent an issue when using a model to estimate the k_d based on k_t values which are calculated with a different method from the one used for its training. Moreover, many models used the clear sky clearness index (k_tcs) as a predictor (Engerer, 2015; Yang and Boland, 2019), which requires the use of a clear sky decomposition model. The accuracy of such all-weather models depends on the chosen clear sky model. Gueymard and Ruiz-Arias, (2016) used the Perez–Ineichen clear sky model, while Starke et al., (2021) applied Solis clear sky model. Data sources were also different: the first used Solar radiation Data (SoDa), while CAMS Reanalysis was exploited in the second. An important step towards identifying a shared methodology to validate and compare different models was done by Yang, (2022). Yang introduced a universal benchmarking dataset that permits testing the generalization ability of decomposition models. This will enable the resolving of inconsistencies about the meteorological station, the data time interval, and the quality control routine by proposing a database with data from up to 126 sites, over a 5-year period from 2016 to 2020 which result from a very stringent quality control sequence.

5.2 Inconsistencies of the transposition modeling

As observed for decomposition models, the existing literature about transposition models is characterized by some inconsistencies in the applied methodologies. In fact, there is a lack of shared protocols for the implementation of transposition models as well as in their validation. However, some methodologies such as the Taylor diagrams (Taylor, 2001) exist in the Literature that might represent a potential and effective solution to harmonize the presentation of the results from different studies and make simpler the comparison of the calculated statistical indicators. Alongside those aspects which were found to be inconsistent in decomposition modeling (i.e., the statistic indicators, the data time interval, the quality control routine, and the method of input calculations), others related to the experimental apparatus used for the validation were found to be inconsistent in transposition modeling. Validation studies differ for the geometry configuration of the tilted surface. The tilt angle as well as the orientation changed depending on the experimental setup implemented by the authors. The validation study carried out by Nassar et al., (2020) on 24 models considered only horizontal and 30° south-facing planes, while Mahachi and Rix, (2016) focused exclusively on the geometry configuration of the investigated PV plant. A wider range of orientations was assessed in Toledo et al., (2020) where vertical surfaces facing the four main cardinal points were evaluated. Similarly, Mubarak et al., (2017) presented an articulated apparatus coupling pyranometers and crystalline silicon PV devices with individual temperature sensors to monitor the E_gt at various orientations (i.e., north, south, east, west, south-East, south-west) and tilt angles (from 0° to 70° with a 10° angular step). Furthermore, the experimental facility exploited in the validation of transposition models changes among the studies. Different equipment such as pyranometers or crystalline silicon PV devices were used to monitor E_gt, and uncertainties from measurements were often neglected. In this regard, applying the quality standards defined by the World Meteorological Organization and the National Renewable Energy Laboratory (Sengupta et al., 2021) for the measuring instruments could enhance the accuracy of the datasets and ease the comparison of results from different studies.

5.3 SWOT analysis

Even though the decomposition models require many inputs, the k_t is commonly used since it can be measured. Alongside this, other parameters are used such as θ_z, V, and AM which are deterministic and can be easily calculated. In particular, the θ_z and the AM can be calculated from dataset timestamps, while the V is typically determined from a moving window of the k_t values. As observed in Engerer2, decomposition models are usually comprised of a single function which makes them easy to implement in the Python environment. In this regard, the pvlib-python consists of a library of decomposition model implementations and supporting functions which can be used to calculate model parameters such as apparent solar time (AST), θ_z, and AM. This Python library is continuously developed and maintained and can be edited, so that new models can be implemented. Moreover, the k_t can be used to combine different decomposition models by either choosing the output of a single model depending on the k_t, or by summing the weighted outputs of multiple models, where the weight of each model depends again on the k_t. Different approaches exist, and combining models is still a developing topic. Exploiting a combination of decomposition models which are defined through ANNs and machine learning-based models depending on the k_t and time of year is also a solution. Model parameters can be re-fitted for local datasets to produce location tailored models. Major issues currently concern data availability and data quality-control routines. These have yet to be sufficiently discussed in the literature although this would improve the comparability of outcomes from different validation studies. Recent models such as the one presented in Starke et al., (2021) use data from the last two decades to calculate model parameters. Using as much data as possible for creating a model typically increases its accuracy, but since weather patterns and global temperature are changing, measurements from the early 2000s may not be representative of current climates. Climate change is also a threat to old models, as their accuracy is assumed to decrease for the climate they are intended to be used in, which means new studies will be required to assess their performance (Table 10). In fact, modifications of the atmosphere composition can alter the scattering phenomena of sunrays or cloud properties, by reducing the model capability to simulate such events. When it comes to a quality control routine, there are several methods which are documented in the literature for solar irradiance applications (Muneer et al., 2007; Lee et al., 2013). Nonetheless, all of them undergo the same limitations (Urraca et al., 2020). Solar irradiance can significantly change with cloud cover; therefore, all these methods cannot detect low magnitude measuring errors such as shadows, calibration drifts, soiling, and snow accumulation. The six-step quality control procedure which was proposed in Yang et al., (2018) for the BSRN data might represent best practice in this topic. The study concluded that measurements of E_gh, E_bn, and E_dh are needed to ensure reasonably good quality control. Otherwise, satellite-derived irradiance should be used to contrast the on-site irradiance measurements.

TABLE 10

TABLE 10. SWOT analysis of decomposition modeling.

Empirical transposition models are usually less user-friendly than decomposition models. The pvlib-python tool includes the most common transposition models by allowing model chains to be built in Python. It is worth highlighting that transposition models such as Perez4 struggle with the hours of the day characterized by low solar elevation angles, making uncertainties high when the power production of the E-W VBPV system peaks. This issue is not addressed in the literature, as almost all the validation studies exclude the hours of low solar elevation (Table 11).

TABLE 11

TABLE 11. SWOT analysis of transposition modeling.

As mentioned in the above paragraph for decomposition models, numerous approaches exist that enable combining different transposition models to increase the model accuracy. Furthermore, machine learning-based methods can contribute to ANN-created transposition models and have shown improvements over empirical transposition models due to lower computation times (faster routines and faster hardware).

5.4 Literature gaps

The main literature gaps that were found in this review study are highlighted in this section. Firstly, the difference in temporal resolution of data between decomposition and transposition models is addressed. The former can provide one-minute estimations and validations, while the latter usually deals with hourly predictions which is also the same time resolution used when assessing PV energy production. Indeed, PV simulations tools aim to estimate the economic prospects of different system configurations, and if the system is contracted with a set price per kWh electricity produced, the hourly or sub hourly mismatch between production and load profiles is not a factor in the calculations. For this study, however, the penetration potential of E-W VBPV systems is investigated at a sub-hour temporal resolution for more accurate results enabling user to account for CE and AE effects.

Secondly, the numerical models implemented for assessing BPV systems mainly focuses on south-oriented cells with a tilt angle that can range between 30° and 45°. In such optimally exposed configurations, the diffuse fraction of solar radiation and the solar radiation reflected by the ground play a secondary role. Conversely, these factors are relevant in E-W VBPV, particularly when simulating solar irradiation with low Sun elevation angles (i.e., significant contribution from horizontal brightening). Being the reviewed models mostly empirical, their performances can reasonably worsen when applied to conditions different from the one considered for the model parametrization. Therefore, the use of models particularly accurate in the estimation of the horizontal brightening as well as with the ground albedo as input parameter should be recommended for E-W VBPV systems. Among the reviewed models, the transposition model from Olmo et al., (1999) is the only having the ground albedo as an input. Despite of this, whether higher or lower accuracy is expected is difficult to say a priori, and it has to be demonstrated through a comparison of different combinations of decomposition and transposition models.

Thirdly, there is an evident gap with high-latitude conditions. Decomposition models are mostly parametrized with datasets from climate zones which are not representative of the territories of Norway, Sweden, and Finland. In particular, only one weather station was considered from the Dfc climate in Gueymard and Ruiz-Arias, (2016), and none in Yang, (2022) by making it difficult to univocally identify an accurate model chain to be exploited in these countries. In this regard, additional efforts are needed to collect data from these countries as well as to perform experimental validations which can contribute to identify room for improvement of decomposition and transposition models. For example, a decomposition model developed for mostly clear sky locations will probably fail in accurately estimating k_d in locations mostly characterized by overcast sky conditions. Similarly, a transposition model which cannot estimate properly the scattering of sunrays near the horizon will luckily provide estimations of E_gt with low accuracy in a location where the Sun is low in the sky for most of the time.

5.5 Limitations of the study

The evaluation of decomposition models based on outcomes from Gueymard and Ruiz-Arias, (2016) and Yang, (2022) focused on a performance average over all stations in the same climate zone of Norway, Sweden, and Finland, and performance was not assessed in a specific station from the Dfc zone. A more detailed investigation may have revealed the potential of additional decomposition models for high-latitude locations.

The data provided as Supplementary Material from Gueymard and Ruiz-Arias, (2016) and Yang, (2022) usually do not include daily or monthly or seasonal error statistics since the results are averaged over the full time period of the dataset. A one-minute dataset for each station including irradiance measurements and model estimations is provided by Starke et al., (2021). However, such results are only available for the models presented in the publication and it is not possible to conduct an inter-comparison of seasonal model performance. Following this, it has not been possible to assess how the performance of decomposition or transposition models varies with clearness index k_t, which is required for a discussion on how multiple models can be combined to reduce the overall uncertainty of the decomposition step in the modeling chain.

Finally, ensemble model output statistics such as the one described in Yang and Gueymard, (2020) were excluded from the comparative analysis presented in this work, although they seem to be promising. The authors preferred to limit the field of investigation to individual decomposition models and their match with transposition models within the model chain. The potential for combining various decomposition models to assess solar radiation in high-latitude locations is worth exploring in specific validation work. Note also that the ensemble model output statistics from Yang and Gueymard, (2020) was found to outperform some reliable stand-alone models such as Yang2 and Engerer2.

6 Conclusion

This systematic review investigated the solar irradiance model chain for horizontal-to-tilted irradiance conversion at high-latitude locations. Although a considerable number of publications exist on this topic, it is a scarcity of examples in the literature about the use of decomposition models at these latitudes. Moving from Gueymard and Ruiz-Arias, (2016) and Yang, (2022), a list of decomposition models suitable for high-latitude locations was provided. Moreover, validation studies concerning transposition models were reviewed based on the aim to identify transposition models which can be used in the model chain for estimating E_gt on E-W VBPV.

This work demonstrates that validation studies lack in terms of their inter-comparability and relevance to E-W VBPV in Norway, Sweden, and Finland. The major issues are the variety in the applied methodologies in recent decomposition model validation studies, the lack of validation studies for empirical transposition models for E-W VBPV, and the absence of one-minute transposition models. While the newest decomposition models (i.e., Yang4, Engerer4, Starke21) show promising results, more data and climate-specific validation of these models are required. The development of a common and shared validation protocol should be prioritized to ease the inter-comparison of the numerous decomposition models. Such a gap has been partially covered by the recent work from Yang, (2022). Regarding transposition models, the Perez4 was found to locally perform best for E-W oriented surfaces and vertically mounted surfaces, but it is preferable to combine different transposition models depending on the k_t.

The main findings from the present review can be summarized as follows:

• Skartveit1 and Mondol1 are the most effective one-hour model at high latitudes, although they are outperformed by Spencer in the E zone;

• Yang4 is the best one-minute model followed by the Starke family model which represents an effective solution particularly when applied to C and D zones;

• Perez4 and Muneer transposition models perform best for East- and West-facing vertical surfaces;

• Hour time resolution is mostly adopted in transposition models, while decomposition models can already perform one-minute analyses;

• A validated model chain for E_gt estimation on E-W VBPV in Norway, Sweden, and Finland was not found in the literature.

Therefore, the model chain resulting from the matching of Yang4 and Perez4 models could be recommended for performing one-minute analysis of E-W VBPV at high-latitude locations. Nonetheless, further insight into the best model choice for optical modelling is necessary. This includes:

• Validating the best performing decomposition models (see Table 5) with data from Norway, Sweden, and Finland by assessing decomposition model performance both on a seasonal level, and as a function of k_t;

• Evaluating how well the model validation with other D-type climate zone data works for Dfc-zone;

• Validating transposition models with irradiance measurements from an E-W VBPV system;

• Assessing the potential of novel methods within the field such as ANN and probabilistic models.

Author contributions

MM, MDS, and GL contributed to conception and design of the study. MM and JT organized the database. MM and JT performed the statistical analysis. MM and JT wrote the first draft of the manuscript. SJ and KM wrote sections of the manuscript. All authors contributed to the article and approved the submitted version.

Acknowledgments

The authors acknowledge the financial support from the Norwegian Research Council (research project FRIPRO-FRINATEK No. 324243 HELIOS), the Finnish Cultural Foundation, and Emil Aaltonen Foundation.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Nomenclature