Presented at the 36th European PV Solar Energy Conference and Exhibition, 9-13 September 2018, Marseille, France

A COMPARISON OF TWO MODELS FOR THE SEPARATION OF DIRECT AND DIFFUSE IRRADIANCE
IN PLANE OF ARRAY
Dorian Esteban Guzman Razo, Sven Killinger, Björn Müller, Christof Wittwer
Fraunhofer ISE, Fraunhofer Institute for Solar Energy Systems ISE
Heidenhofstraße 2, 79110 Freiburg, Germany
Tel.: 0761 4588 5826, email: dorian.esteban.guzman.razo@ise.fraunhofer.de
ABSTRACT: For advanced automated operational monitoring of Photovoltaic (PV) systems, e.g. in order todistinguish losses due to self-shading from malfunctions, direct and diffuse irradiance at the module plane (plane ofarray, POA) are required separately. However, in practice, usually only measured values of global POA irradiance(GPOA) are recorded on site. There is thus a need for a model to divide GPOA into its direct (IPOA) and diffuse
components (DPOA). This paper compares and evaluates two models recently developed that perform such separation.
The aim of this work is to show the advantages and disadvantages that these two models offer and how they differ
from each other. In order to compare and evaluate them, two different POA data sets were used. The first data set isgenerated using SolarGIS satellite-based data for 16 different locations across Germany. The second data set contains
on-site measured DPOA and GPOA data from a PV power plant located in south-west Germany. Results from bothmodels show good matching with satellite-based data from the 16 different locations.
Keywords: Modelling, Photovoltaic, PV System, Solar irradiance models, Decomposition in plane of array

1

INTRODUCTION

Due to the increase of the global PV installedcapacity around the world, the challenges for an accurate
simulation of the overall system’s behavior under outdoor
conditions have increased as well. It is known that the PV
system’s energy output depends to a large extend on
GPOA and on its components: IPOA and DPOA.
Models for decomposing global horizontal solar
irradiance into its direct and diffuse components havebeen developed and evaluated for decades, e.g. [1–4].
These models are part of many calculation methods foryield forecasting and well validated for many regions of
the Earth. However, to the knowledge of the authors,similar models for irradiance decomposition in POA do
not exist up to now.
In practical terms, measurements of GPOA are
commonly available within medium and large-scale PV
installations which provide just enough data for the
calculation of plant’s key performance indicators (KPIs)
such as Performance Ratio (PR) and Availability. Yet, if
a more detailed and accurate PV system modelling isrequired in order to now-cast and forecast power
production, this information is not sufficient and adecomposition of GPOA into IPOA and DPOA is necessary.
1.1 Background and literature review
In early 2019, two different models for GPOA into IPOA
and DPOA separation have been introduced: [5,6]. Bothempirical approaches are similarly inspired by theconcept of the classic diffuse (K d ) and clearness index(K t ) fraction introduced by [7], described by Equation (1)and Equation (2).
𝐾𝑡 =
𝐾𝑑 =

𝐺𝐻𝐼
𝐺𝐸𝑋𝑇
𝐷𝐼𝐹
𝐺𝐻𝐼

(1)
(2)

Where GHI is the global horizontal irradiance, GEXT
is the extra-terrestrial irradiance and DIF is the diffuse
horizontal irradiance.
For convenience in the following sections of thiswork, the model in [5] will be referred as Mod1, themodel in [6] approach A will be referred as Mod2A andthe model in [6] approach B will be referred as Mod2B.

In the models compared here, both indexes aredefined on plane of array described by Equation (3) andEquation (4).
𝐾𝑡,𝑃𝑂𝐴 =𝐾𝑑,𝑃𝑂𝐴 =𝐺𝑃𝑂𝐴𝐺𝐸𝑋𝑇,𝑃𝑂𝐴
𝐷𝑃𝑂𝐴
𝐺𝑃𝑂𝐴

(3)
(4)

1.2 Outline and objective
The aim of this work is to compare both GPOA
separation models using two datasets. The first one
consists of GHI and DIF satellite-derived data provided
by SolarGIS, combined with a transposition model ([8])
and the second dataset consists of GPOA and DPOA
measured data.
The paper is structured in the following way: Section
2 contains the steps followed in order to compare the twomodels and describes, as well, the data used within thiswork. In Section 3, results from the comparison of thetwo models are provided. And finally, a brief summary ofthis work and avenues for future research are presented in
Section 4.
1.3 Models description
Mod1, on the one hand, is mainly based on the two
indexes above mentioned (Equations (3) and (4)).
Similar to the already known models for the horizontal
plane, in this case, however, the diffuse fraction in planeof array (K d,POA ) is defined as a function of the Clearness
Index (K t,POA ) and angle of incidence (AOI, in radians);see Equation (5).
𝐾𝑑,𝑃𝑂𝐴 = 𝑓(𝐾𝑡,𝑃𝑂𝐴 , 𝐴𝑂𝐼)

(5)

Mod2A and Mod2B, on the other hand, are describedin a general way by the diffuse fraction in plane of array
K d,POA as a piece-wise function of K t,POA , the sun’s
zenith angle (θz ), the PV system’s azimuth angle (α) and
the PV system’s tilt angle (β), with similar set of limits
for K d,POA to the model proposed by [9]. See Equation(6).
𝐾𝑑,𝑃𝑂𝐴 = 𝑓(𝐾𝑡,𝑃𝑂𝐴 , 𝜃𝑧 , 𝛼, 𝛽)(6)Presented at the 36th European PV Solar Energy Conference and Exhibition, 9-13 September 2018, Marseille, France

Once K d,POA has been calculated, it is possible toobtain DPOA using Equation (4), and thus to calculate
IPOA with Equation (7).
𝐺𝑃𝑂𝐴 = 𝐷𝑃𝑂𝐴 + 𝐼𝑃𝑂𝐴

(7)This work is focused on DPOA calculation from GPOA;
therefore, IPOA calculations are not presented.
In order to calculate K d,POA , Mod1 offers astraightforward third-degree-like equation defined asfollows:
𝐾𝑑 𝑃𝑜𝐴 (𝐾𝑡 𝑃𝑜𝐴 , 𝐴𝑂𝐼) = 𝐶1 𝐾𝑡 𝑃𝑜𝐴 + 𝐶2 𝐴𝑂𝐼
2+𝐶3 (𝐾𝑡 𝑃𝑜𝐴 ) + 𝐶4 𝐾𝑡 𝑃𝑜𝐴 𝐴𝑂𝐼 + 𝐶5 (𝐴𝑂𝐼)2
3

2+𝐶6 (𝐾𝑡 𝑃𝑜𝐴 ) + 𝐶7 (𝐾𝑡 𝑃𝑜𝐴 ) 𝐴𝑂𝐼+𝐶8 𝐾𝑡 𝑃𝑜𝐴 (𝐴𝑂𝐼)2 + 𝐶9 (𝐴𝑂𝐼)3 + 𝐶10(8)

All the constants are already optimized and definedfor any given case of K d,POA . Table I shows theconstants’ values presented in Equation (8).
Table I: Optimized values of Mod1 model’s constants.
Values for C1 to C10 to calculate any possible value of
K d,POA .

Constant
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10

Value
1.3052
0.9739-4.6871-1.8813-1.17492.73401.18
0.71270.444
0.7361Mod2A and Mod2B also propose two different
approaches in order to calculate the coefficientsai , bi , ci (i = 1,2,3), both as a function of β and α(fl (α, β)), as Equation (10), for approach A, and Equation(11), for approach B, indicate.
𝑓𝑙 (𝛼, 𝛽)𝐴 =Table II: Definition of sub-index i depending on K t,POA
values for Equation (9). When the K t,POA value isbetween 0 and 0.3, inclusive, i is equal to 1. When K t,POA
value is higher than 0.3 but lower than 0.78, i is equal to
2. When K t,POA value is 0.78 or higher, i is equal to 3.

Limits[0,0.3](0.3,0.78)[0.78, +∞)Index 𝒊123

∗ (𝜃𝑙 (𝛼) − 𝑚3,𝑙 ) + ∅𝑙 (𝛽)

(10)
(11)

Where the sub-index l is defined by the previous
definition of ai , bi , ci (i = 1,2,3). Equations (12) and (13)define functions ∅l (β) and θl (α), respectively.
∅𝑙 (𝛽) = 𝑑1,𝑙 ∗ 𝛽2 + 𝑑2,𝑙 ∗ 𝛽 + 𝑑3,𝑙(12)𝜃𝑙 (𝛼) = 𝑚1,𝑙 ∗ 𝛼 2 + 𝑚2,𝑙 ∗ 𝛼 + 𝑚3,𝑙(13)

In order to calculate coefficients ai , bi , ci (i = 1,2,3)
from Equation (9) with either approach A or B, firstcoefficients di,l and mi,l (i = 1,2,3) have to be defined
for l ∈ {1,2, … ,9}. Table III offers the optimum values for
those coefficients. K d,POA values higher than 1 are set to
1 and K d,POA lower than 0 are set to 0.
Table III: Optimized values for Mod2A and Mod2B
constants.
ApproachAandB
coefficients
di,l and mi,l (i = 1,2,3), needed to calculate any possible
value of ai , bi , ci (i = 1,2,3), defined by Equations (10),(11), (12) and (13).
𝒍
1
2
3
4
5
6
7
8
9
𝒍
1
2
3
4
5
6
7
8
9

(9)

Where the sub-index 𝑖, is chosen depending on the
𝐾𝑡,𝑃𝑂𝐴 value, as mentioned before, according to the limits
shown in Table II.

90°

𝑓𝑙 (𝛼, 𝛽)𝐵 = 𝜃𝑙 (𝛼) + ∅𝑙 (𝛽) − 𝑚3,𝑙Mod2A and Mod2B propose a general equation for
K d,POA calculation depending on the K t,POA value,considering a similar set of limits as the method proposedby [9]. This means that K d,POA can be calculated asfollows:
𝐾𝑑,𝑃𝑂𝐴 = 𝑎𝑖 + 𝑏𝑖 ∗ 𝐾𝑡,𝑃𝑂𝐴 + 𝑐𝑖 ∗ cos 𝜃𝑧𝛽(°)

𝒇𝒍 (𝜶, 𝜷)
𝑎1
𝑏1
𝑐1
𝑎2
𝑏2
𝑐2
𝑎3
𝑏3
𝑐3
𝒇𝒍 (𝜶, 𝜷)
𝑎1
𝑏1
𝑐1
𝑎2
𝑏2
𝑐2
𝑎3
𝑏3
𝑐3

𝒎𝟏,𝒍
-1.79E-5-4.5E-5
4.27E-5-2.72E-5
1.49E-5
3.17E-5-3.01E-5
0.68E-5
3.72E-5𝒅𝟏,𝒍0
5.21E-500-11.15E-5
6.55E-5
2.57E-5-9.19E-5
8.76E-5𝒎𝟐,𝒍-0.0001-0.000700.0002-0.00130.0007-0.00020.0008-0.0007𝒅𝟐,𝒍-0.0021-0.01110.0040-0.00690.0149-0.00030.00080.0075-0.0104𝒎𝟑,𝒍0.7635-0.5968
0.3956
0.7784-1.4297
0.76940.22650.5090-0.4251𝒅𝟑,𝒍0.9604-0.0191
0.03671.3824-1.87070.2692-0.04900.5763-0.1947

It is important to mention that, Mod2A and Mod2B
define the PV system’s azimuth angle (α) as 0° when issouth oriented, as -90° for the east and as 90° for thewest. Whereas in Mod1 south is defined as 180°, east as
90° and west as 270°.

2

METHODOLOGY AND DATA

Two different datasets were used for this work. The
first dataset consists of satellite-derived data provided bySolarGIS for 16 real PV systems distributed throughout
Germany (on site pyranometer measurements of GPOA are
not available for these sites). This dataset offers datafrom January 1994 until February 2018 with 15 minutes

Presented at the 36th European PV Solar Energy Conference and Exhibition, 9-13 September 2018, Marseille, France

time resolution. The second dataset provides on-site GPOA
and DPOA measured data from a SPN1 pyranometer
installed in south-west Germany, including dates from
22/03/2011 until 24/09/2015 with 5 minutes time
resolution. See Figure 1 and Table IV for details.

Figure 1: The 16 real PV systems and the PV system onsite measured data. The pointers in blue represent thegeographical location of the synthetic data generated forthis work. The red pointer represents the location of the
SPN1 pyranometer measured data.
Table IV: Azimuth and Tilt angles for each one of the 16
real PV systems and the PV system on-site measured.
Azimuth angles are defined based on the Mod1’s
standard. * On-site measurements.

System ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17*

System Azimuth
Angle [°]
177
213
190
215
182
203
172
152
162
191
156
195
174
160
208
180
237.5

System Tilt
Angle [°]
30
15
15
15
15
15
15
15
15
15
25
25
25
15
25
25
27

In order to generate DPOA data, SolarGIS data from
System ID 1 to 16 have been simulated using Perez

transposition method proposed by [8], included in PVlib
library ([10]). After that, 𝐾𝑡,𝑃𝑂𝐴 values from the 16 PV
systems have been calculated based on Equation (3).
Finally, based on Equations (8) and (9), K d,POA values
have been also calculated for Mod1, Mod2A and Mod2B.
Clearing for DPOA from Equation (4) allows one tocompare the DPOA simulated with the DPOA resulted fromthe two models.
K t,POA values for system ID 17 have been calculated
in a similar fashion, but in this case using on-sitemeasurements and Equation (3). Afterwards, K d,POA
values are calculated using the three models, based in
Equations (8) and (9). Thus, clearing again for DPOA from
Equation (4), including K d,POA values calculated
previously, it is possible to compare the three models’
resulting DPOA with DPOA on-site measured values.
The following filters have been applied for all thedatasets, system ID 1 to 17: DPOA and GPOA values lessthan 10 W/m2 (night-time) have been filtered out.
Whereas, in order to exclude incorrect measurements thatmight lead to inappropriate conclusions, exclusively for
System ID 17 additional filters have been included:
Incomplete days and measurement errors (e.g. negative
irradiance values) have been filtered out.
Finally, three common error indicators have been
selected with the intention of having a better insight ofthe models’ results: RSMD (Root Square Mean
Deviation), nRSMD (Normalized Root Square Mean
Deviation) and R2 (coefficient of correlation).
RSMD and nRSMD are used to measure thedifferences between K d,POA predicted (Ei ) and observed(Oi ) samples or populations. These indicators arefundamentally an accuracy index where higher
simulation deviations lead to higher RSMD and higher
nRSMD percentage values (RSMD is normalized by theaverage of the observed points). These two indicators
have been proposed before for this kind of experimentsby [11] and are given by Equations (14) and (15)respectively.
1

2
𝑅𝑆𝑀𝐷 = √ ∑𝑁
𝑖=1(𝐸𝑖 − 𝑂𝑖 )
𝑁

1

1

𝑂

𝑁

2
𝑛𝑅𝑆𝑀𝐷 = ̅ √ ∑𝑁
𝑖=1(𝐸𝑖 − 𝑂𝑖 )

(14)

(15)

RSMD and NRSMD have been calculated by K d,POAof both, calculated (Oi ) and simulated values (Ei ) of
Mod1, Mod2A and Mod2B.
In the case of R2, it is used to measure how well themodel replicated the observed values, in other words, thecorrelation between the observed values and the predicted
ones. This error measure has been included as well by[4], with the purpose of having an extra indicator of themodel’s performance and is given by the Equation (16).
∑𝑁 (𝑂𝑖 −𝐸𝑖 )2̅ 2
𝑖=1(𝑂𝑖 −𝑂𝑖 )𝑅2 = 1 − ∑𝑁𝑖=1(16)

R2 has been calculated with K d,POA simulated (ormeasured in the case of System ID 17) and K d,POA
resulted values from Mod1, Mod2A and Mod2B.
In the three equations, (14), (15) and (16), the lengthof the population defines N, E is defined by the estimated̅ is the mean value
value, O is the observed value and O
over all data points.

Presented at the 36th European PV Solar Energy Conference and Exhibition, 9-13 September 2018, Marseille, France

3

RESULTSAs a first impression, Figure 2 and Figure 3 show theplotted K d,POA results from Mod1, Mod2A and Mod2B
from System ID 9. The same plots have been generated
for the other datasets with similar results. Therefore, therest of the plots are not shown in this work.

Mod1

Mod1

Mod2A

Mod2A

Mod2B

Mod2B

Figure 2: K t,POA versus K d,POA classic plots for the threemodels, System ID 9 dataset. The green dots represent
the simulated SolarGIS data and orange dots represent
the results from Mod1, Mod2A and Mod2B.
In Figure 2, at the top, results from K t,POA versus
K d,POA Mod1 can be observed. In the middle of the
figure, results from K t,POA versus K d,POA Mod2A can beobserved. In the bottom of the figure, results from K t,POA
versus K d,POA Mod2B can be observed.

Figure 3: DPOA simulated values versus DPOA values
calculated with the models from System ID 9 arerepresented by accumulative grid. Lower accumulation ofvalues in a certain region is represented by dark-bluecolor while regions with 140,000+ values accumulated
are colored in yellow.
The red line in Figure 3 shows when correlation
equals to one. In the top of this figure, the plotted resultsof the Mod1 can be observed with 11 RMSD percentage
points, 15 nRMSD percentage-normalized points and0.88 R2. In the middle of this figure, the plotted results of
Mod2A can be observed with 14 RMSD percentage
points, 20 nRMSD percentage-normalized points and0.81 R2. In the lower part of this figure, the plotted resultsof Mod2B can be observed with 13 RMSD percentage
points, 20 nRMSD percentage-normalized points and0.82 R2.
Furthermore, deviation metrics were evaluatedseparately for each one of the 16 systems and themeasured system datasets. The condensed outcomes fromthe three models, Mod1, Mod2A and Mod2B are shownin Table V. System ID 17 has been excluded of thePresented at the 36th European PV Solar Energy Conference and Exhibition, 9-13 September 2018, Marseille, France

average calculation due to the low quality of the data
input.
Table V: Mod1, Mod2A and Mod2B results of the 16
systems and the measured system datasets. R2, nRMSD
and RMSD average, minimum and maximum values canbe observed.
* Not included in the calculation of average, minimum
and maximum.
Mod1

Mod2A

Mod2B

System
ID

R2

RMSD

R2

RMSD

R2

RMSD

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

0.86
0.85
0.88
0.86
0.89
0.87
0.90
0.88
0.88
0.89
0.85
0.86
0.87
0.89
0.84
0.86

0.12
0.11
0.11
0.12
0.10
0.11
0.10
0.11
0.11
0.10
0.12
0.11
0.11
0.11
0.12
0.11

0.80
0.80
0.81
0.81
0.83
0.82
0.83
0.83
0.81
0.83
0.82
0.80
0.81
0.84
0.80
0.81

0.14
0.14
0.14
0.14
0.14
0.13
0.13
0.13
0.14
0.13
0.14
0.14
0.14
0.13
0.14
0.14

0.80
0.80
0.81
0.81
0.83
0.82
0.83
0.83
0.82
0.83
0.82
0.80
0.81
0.84
0.79
0.81

0.14
0.14
0.14
0.14
0.14
0.14
0.13
0.13
0.13
0.14
0.14
0.14
0.14
0.13
0.14
0.14

Avg.
Max
Min

0.87
0.90
0.84

0.11
0.12
0.10

0.82
0.84
0.80

0.14
0.14
0.13

0.82
0.84
0.79

0.14
0.14
0.13

17*

0.75

0.17

0.60

0.22

0.57

0.22

Mod1

Mod2A

Mod2B

nRMSD

nRMSD

nRMSD

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

N
386329389808398272390095401778396173400671394811395631399891385501388206390770399051
380882391001

0.18
0.16
0.15
0.17
0.16
0.16
0.15
0.16
0.15
0.15
0.18
0.17
0.17
0.16
0.17
0.17

0.22
0.20
0.20
0.21
0.21
0.20
0.21
0.20
0.20
0.21
0.21
0.21
0.21
0.21
0.20
0.21

0.22
0.21
0.20
0.21
0.21
0.20
0.21
0.20
0.20
0.21
0.21
0.21
0.21
0.20
0.21
0.21

Avg.
Max
Min

393054401778380882

0.16
0.18
0.15

0.21
0.22
0.20

0.21
0.22
0.20

17*

183221

0.26

0.32

0.34

SystemID

Figure 4: Two days of System ID 16, July 2016. The top
plot shows a clear-sky-like day (19.07.2016) and thebottom plot shows a cloudy-like day (07.07.2016).
Table V shows the overall performance of the models
and the results of R2, nRMSD and RMSD calculation canbe observed. In the case of System ID 17, measured
values are taken into consideration just as a proof ofconcept for these two models. In comparison, system 1 to
16 with system 17, the models seem to perform slightly
worse. The reason may be attributed either to the fact
that, at least for model 1, the parameters were fitted using
satellite irradiance data, not measured data. Also,
measurement uncertainties for system 17 may be quite
high. Therefore, the results from System ID 17 are not tobe considered as conclusive results.
Finally, Figure 4 shows two different days from July
2016. On the one hand, the first plot is a clear-sky-likeday where also DPOA simulated data and results from
Mod1, Mod2A and Mod2B can be observed. It is notablethat both models (Mod1 and Mod2) over estimate the
DPOA simulated, mostly at over 600 W/m2 GPOA.
On the other hand, the second plot shows a cloudylike day where DPOA simulated data and results from
Mod1, Mod2A and Mod2B can be observed. In this case,both models (Mod1 and Mod2) show similar
performance, with slight under-estimation at the
beginning and the end of the day.
In both plots the blue dashed line represents the GPOA
simulated from SolarGIS, the orange dashed line
represents the DPOA simulated from SolarGIS, the green
dashed line represents the DPOA calculated with Mod1,
the red dashed line (in this example follows the Mod2B
results) represents the DPOA calculated with Mod2A andthe purple dashed line represents the DPOA calculatedwith Mod2B.

4

CONCLUSIONS AND FUTURE WORKIn this work, two models for POA irradianceseparation have been evaluated. Mod1 has a straightforward equation to calculate the diffuse index in POA
while Mod2 has two different approaches and it is apiece-wise like function defined by the clearness index in
POA value.

Presented at the 36th European PV Solar Energy Conference and Exhibition, 9-13 September 2018, Marseille, France

In general terms, for 17 different datasets containing
SolarGIS data and measured data, Mod1 has shown abetter performance than Mod2A, with 3 RSMD
percentage points, 5 nRSMD normalized percentage
points and 0.05 R2, and also than Mod2B, with 3 RSMD
percentage points, 5 nRSMD normalized percentage
points and 0.05 R2.
In future work, on the one hand, it is important tovalidate the models against high quality measurements of
DPOA at different sides. On the other hand, the sensitivityof the model accuracy to the overall modeling of shading
losses and energy yields has to be evaluated.

5

ACKNOWLEDGMENT

The authors acknowledge the financial support by
the Federal Ministry for Economic Affairs and Energy ofGermany (BMWi) in the project ALPRO (project number
0324054A).
6[1]

REFERENCES

D.G. Erbs, S.A. Klein, J.A. Duffie, Estimation of
the diffuse radiation fraction for hourly, daily andmonthly-average global radiation, Solar Energy 28(1982) 293–302.
[2] R. Perez, P. Ineichen, E.L. Maxwell, R.D. Seals, A.
Zelenka, Dynamic global-to-direct irradianceconversion models, ASHRAE Transactions 98(1992).
[3] C.A. Gueymard, J.A. Ruiz-Arias, Extensive
worldwide validation and climate sensitivity
analysis of direct irradiance predictions from 1-min
global irradiance, Solar Energy 128 (2016) 1–30.
[4] N.A. Engerer, Minute resolution estimates of the
diffuse fraction of global irradiance forsoutheastern Australia, Solar Energy 116 (2015)
215–237.
[5] D.E. Guzman Razo, B. Müller, C. Wittwer (Eds.),
Ein Modellansatz zur Bestimmung von Direkt- undDiffusanteil der Einstrahlung auf die PVModulebene, 34. PV-Symposium Bad Staffelstein,
2019.
[6] S. Halilovic, J.M. Bright, W. Herzberg, S.
Killinger, An analytical approach for estimating
the global horizontal from the global tilted
irradiance, Solar Energy 188 (2019) 1042–1053.
[7] B.Y.H. Liu, R.C. Jordan, The interrelationship andcharacteristic distribution of direct, diffuse andtotal solar radiation, Solar Energy 4 (1960) 1–19.
[8] R. Perez, P. Ineichen, R. Seals, J. Michalsky, R.
Stewart, Modeling daylight availability andirradiance components from direct and global
irradiance, Solar Energy 44 (1990) 271–289.
[9] D.T. Reindl, W.A. Beckman, J.A. Duffie, Diffuse
fraction correlations, Solar Energy 45 (1990) 1–7.
[10] W. F. Holmgren, C. W. Hansen, M. A. Mikofski,
pvlib python: A python package for modeling solar
energy systems, JOSS 3 (2018) 884.
[11] C. Gueymard, J.A. Ruiz-Arias, Performance of
Separation Models to Predict Direct Irradiance atHigh Frequency: Validation over Arid Areas, in: E.
Frank, P. Papillion (Eds.), Conference proceedings/ EuroSun 2014, International Conference on Solar
Energy and Buildings, Aix-les-Bains, France, 1619 September, International Solar Energy Society,Freiburg, 2015, pp. 1–10.