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Solar irradiation from the energy production of
residential PV systems
C´edric Bertrand †∗, Caroline Housmans †, Jonathan Leloux ‡, and Michel
Journ´ee †
†Royal Meteorological Institute of Belgium, Brussels, Belgium.
‡Universidad Polit´ecnica de Madrid, Madrid, Spain
Abstract
Considering the dense network of residential photovoltaic (PV) systems imple-
mented in Belgium, the paper evaluates the opportunity of deriving global hor-
izontal solar irradiance from the electrical energy production registered at PV
systems. The study is based on one year (i.e. 2014) of hourly PV power output
collected at a representative sample of roughly 1500 residential PV installations.
Validation is based on ground-based measurements of solar radiation performed
within the network of radiometric stations operated by the Royal Meteorological
Institute of Belgium and the method’s performance is compared to the satellite-
based retrieval approach.
Our results indicate that inaccurate information about the PV systems ori-
entation and/or inclination can drastically reduce the method’s performance
and produce spatial artifacts in the distribution of the global solar irradiation
over Belgium. Another limitation is that there are certain sun positions (i.e.
low solar elevations) for which the method fails to produce a valid estimation.
Keywords: Photovoltaic system, solar radiation, decomposition model,
transposition model, remote sensing, ground measurements, crowdsourcing
∗Corresponding author. Tel.:+32 2 373 05 70
Email address: Cedric.Bertrand@meteo.be (C´edric Bertrand †)
Preprint submitted to Renewable Energy October 24, 2017
1. Introduction1
Appropriate information on solar resources is very important for a variety2
of technological areas, such as: agriculture, meteorology, forestry engineering,3
water resources and in particular in the designing and sizing of solar energy4
systems. Traditionally, solar radiation is observed by means of networks of me-5
teorological stations. However, costs for installation and maintenance of such6
networks are very high and national networks comprise only a few stations.7
Consequently the availability of solar radiation measurements has proven to be8
spatially and temporally inadequate for many applications.9
10
Over the last decades, satellite-based retrieval of solar radiation at ground11
level has proven to be valuable for delivering a global coverage of the global solar12
irradiance distribution at the Earth’s surface (e.g. [1], [2] [3], [4]). The recent13
deployment of solar photovoltaic (PV) systems offers a potential opportunity of14
providing additional solar information requiring the conversion of PV systems15
energy production to global solar irradiation (e.g. [5], [6]). As an illustration,16
in Belgium, the total installed PV capacity has increased dramatically in re-17
cent years from 102.6 MW in 2008 (26.55 MW in 2007) to 3423 MW at the18
end of 2016, according to data collected by local renewable energy association19
APERe (http://www.apere.org/), which has combined the figures released by20
the country’s three energy regulators Brugel (https://www.brugel.brussels/),21
VREG (http://www.vreg.be/)and CWaPE (http://www.cwape.be/). Of this22
capacity, 2451 MW (72%) are installed in the region of Flanders, while Wallo-23
nia and the Brussels Metropolitan Region have reached a cumulative capacity24
of 916 MW (27%) and 56 MW (2%), respectively. Note that each of Belgium’s25
three macro-regions has its own energy systems and its own policy for renew-26
able energies. In 2016, the country installed about 170 MW across 25.000 PV27
systems (2015: 100 MW). Systems with less than 10 kW capacity represented28
over 61% of the installed capacity. According to APERe, the newly installed PV29
power in Flanders is mostly represented by residential and commercial installa-30
2
tions, while in Wallonia around the half of the capacity installed last year comes31
from large-scale PV plants, a segment which has seen limited development in32
the region in previous years. In the year 2016, the Belgium produced 2.9 TWh33
of solar electricity that covered 3.7% of the country’s total electricity demand.34
35
However, using the energy production registered at PV systems as a solar36
irradiation sensor is not straightforward. It requires first to derive the solar37
irradiation from the energy production of the PV system (knowing that the38
power output of a PV system is not directly proportional to the solar irradiance39
that it receives). Second because modules are installed at a tilt angle close to40
local latitude to maximize array output (or at some minimum tilt to ensure41
self-cleaning by rain) this requires to convert the retrieved tilted global solar42
irradiance to horizontal. (Note that tilt and azimuth of residential PV systems43
often depend on the particular rooftop on which they are installed, rather than44
being designed for optimal performance). Towards this objective, operational45
data from a representative sample of Belgian residential PV installations have46
been considered to assess the performance of such an approach.47
2. Data48
2.1. Residential PV systems data49
This work is based on one year (i.e. 2014) of hourly PV power output50
collected at more than 2893 PV systems in Belgium installed from 2008 to51
2013. PV generation data was collected via the Rtone company website (Rtone,52
http://www.rtone.fr). The PV energy production data provided by Rtone was53
monitored using the commercial Rbee Solar monitoring product, which measures54
the energy production with a smart energy meter at a 10-min time interval. The55
information concerning the PV systems (i.e. metadata) were supplied by their56
owners. Each PV system is localized by its latitude and longitude, completed57
with the corresponding altitude. The PV generator is characterized by its ori-58
entation and tilt angles, its total surface, and its total peak power. Additional59
3
information about the PV module manufacturer and model, inverter manufac-60
turer and model, installer, year of installation, PV cell/module technology can61
also be provided.62
63
Metadata has been subjected to several checks in order to isolate and remove64
as much erroneous information as possible. The standard set of filters employed65
before analyses are the following:66
1. selection of single array systems since generation data cannot be decom-67
posed into constituent arrays,68
2. selection of systems located in Belgium,69
3. selection of systems with and orientation between -90oand +90ofrom70
south and a tilt from horizontal smaller than 60o.71
In some cases, system details were investigated manually to verify to a good72
degree of confidence whether the systems orientation, tilt or peak power are73
correct. Indeed, metadata are prone to errors on part of the owner, by entering74
incorrect system parameters but also in some cases due to installer’s conventions.75
As an example, tilt angle can be erroneously reported as 0osimply because 0 is76
used as default value in case of missing information. Another identified limita-77
tion is that many angles values are rounded as multiples of 5o.78
79
1470 systems have been selected and considered in this study after these80
preliminary checks (see Figure 1 for the location of the selected PV systems).81
It is worth pointing out that due to availability reasons, most of the data comes82
from Wallonia and Brussels. As indicated in Table 1, the vast majority of the83
selected PV generators have a tilt angle between 20oand 50o, which generally84
corresponds to the slope of the roofs on which they are installed ([7]).85
86
2.2. Ground stations measurements87
The Royal Meteorological Institute of Belgium (RMI) has a long term ex-88
perience with ground-based measurements of solar radiation in Belgium (unin-89
4
terrupted 30 min average measurements in Uccle since 1951, in Oostende since90
1958, and in Saint-Hubert since 1959). Uccle is one of the 22 Regional Radiation91
Centres established within the WMO Regions. The incoming global horizontal92
solar irradiance is currently measured in 14 Automatic Weather Stations (AWS)93
in addition to the measurements performed in our main/reference station in Uc-94
cle (see Table 2).95
96
At the reference station in Uccle, measurements of the global horizontal so-97
lar irradiance (GHI) are performed by Kipp & Zonen CM11-secondary standard98
pyranometers. The CM11 pyranometer has a thermopile detector and presents99
a reduced thermal offset (i.e. about 2 Wm−2at 5 K/h temperature change).100
The pyranometer directional error (up to 80owith 1000 Wm−2beam) is less101
than 10 Wm−2and its spectral selectivity (300-1500 nm) is smaller than 2%.102
At the 14 RMI’s AWS, GHI observations for the year 2014, were made with a103
Kipp & Zonen CNR1 net radiometer. It consists of a pyranometer (model type104
CM3 complying with the ISO Second Class Specification) and a pyrgeometer105
(model type CG3) pair that faces upward and a complementary pair that faces106
downward. The pyranometers and pyrgeometers measure shortwave and far in-107
frared radiation, respectively. The CM3 pyranometer has a thermopile sensor108
and presents a limited thermal offset (i.e. ±4 Wm−2at 5 K/h temperature109
change). Its directional error (with 1000 Wm−2beam) is ±25 Wm−2and its110
spectral selectivity (350-1500 nm) is ±5%.111
112
All ground measurements are made with a 5-s time step and then integrated113
to bring them to a 10-min time step. The 10-min data have undergone a series114
of automated quality control procedures ([8]) prior to be visually inspected and115
scrutinized in depth by a human operator for more subtle errors. Because our116
radiometric station in Saint-Hubert (AWS 6476) was known to operate defi-117
ciently in 2014, all GHI measurements from this station were discarded. The118
geographical location of the remaining 13 ground measurement sites is provided119
in Figure 1 together with the selected residential PV systems. Note that be-120
5
cause the data quality control revealed that the CNR1 net radiometer installed121
in the Buzenol station (i.e. AWS 6484) has only performed well intermittently122
during the year 2014, GHI measurements from this station were not used for123
validation purpose.124
125
3. Methods126
3.1. Conversion of PV system energy production to tilted global solar irradiation127
The initial step of the approach consists in the derivation of global irradi-128
ance in plane of array, Gt, from the specific power output, P, of a PV system.129
Numerous models to calculate Pfrom Gtexist in the literature (e.g. [9], [10]).130
However, it is well known that the energy conversion efficiency of PV mod-131
ules depends on a number of different influences. Losses in PV systems can be132
separated in capture losses and system losses (e.g. [11], [12]). Capture losses133
are caused, e.g., by attenuation of the incoming light, temperature dependence,134
electrical mismatching, parasitic resistances in PV modules and imperfect max-135
imum power point tracking. System losses are caused, e.g., by wiring, inverter,136
and transformer conversion losses. All these effects cause the module efficiency137
to deviate from the efficiency measured under Standard Test Condition (STC),138
which defines the rated or nominal power of a given module.139
140
According to [13], the direct current (DC) power output of a PV generator141
can be properly described by:142
143
PDC =P?Gef f
G?.1 + κ(Tc−T?
c).a+bGeff
G?+cln Geff
G?. fDC (1)
where the symbol ?refers to STC [Irradiance: 1000 W m−2; Spectrum: AM144
1.5; and cell temperature: 25oC], PD C is the DC power output of the PV gen-145
erator (W), P?is the nameplate DC power of the PV generator (i.e. power at146
maximum-power point, in W), Geff is the effective global solar irradiance (W147
6
m−2) received by the PV generator (it takes into consideration the optical ef-148
fects related to the solar incidence angle), G?is the global solar irradiance under149
STC (W m−2), κis the coefficient of power variation due to cell temperature150
(%/oC) , Tcand T?
care respectively the cell temperatures under operating and151
STC conditions (oC), the three parameters a, b and c describe the efficiency152
dependence on irradiance, and fDC is a coefficient that lumps together all the153
additional system losses in DC (e.g. technology-related issues, soiling and shad-154
ing).155
156
The first term on the right-hand side of Eq. 1 goes a long way back ([14],157
[15]) and it considers that the PV module efficiency is affected by temperature,158
decreasing at a constant rate. Handling with this term just requires standard159
information: P?is the PV array rated power, which can be estimated as the160
product of the number of PV modules constituting the PV array multiplied161
by their nameplate STC power, and κis routinely measured in the context of162
worldwide extended accreditation procedures: IEC Standard-61215 (2005) and163
IEC Standard-61646 (2008) for crystalline silicon and thin film devices, respec-164
tively. P?and κvalues are always included in PV manufacturer’s data sheets165
or in more specific information as flash-reports.166
167
The second round bracket on the right-hand side of the Eq. 1 considers the168
efficiency dependence on irradiance. That was initially attempted by adding a169
base 10 logarithm ([14]) but it is better implemented by this empirical model170
proposed by [16] and [17] where a,band care empirical parameters. The effi-171
ciency increases with decreasing irradiance, due to series resistance effects, are172
represented by the term (b.Geff /G?), providing b≤0, while the efficiency173
decreases with decreasing irradiance, due to parallel resistance effects, are rep-174
resented by the term (cln Geff /G?), providing c≥0.175
176
The corresponding alternating current (AC) power output of the PV system177
from this DC power at the inverter entry is given by:178
7
179
PAC =PDC ηINV fAC (2)
where PAC is the AC power output of the PV generator, ηIN V is the yield of the180
inverter, and fAC is a coefficient that lumps together all the technology-related181
additional AC system losses.182
183
The energy produced during a period of time Tis finally given by:184
185
EAC =Zt=T
t=0
PAC dt (3)
To assess the technical quality of a particular PV system, energy performance186
indicators are obtained by comparing its actual production along a certain pe-187
riod of time with the production of a hypothetical reference system (of the same188
nominal power, installed at the same location, and oriented the same way). The189
Performance Ration (PR) which is the quotient of alternating current yield190
and the nominal yield of the generators direct current, is by far the most widely191
used performance indicator. It is defined mathematically as:192
193
P R =ηachieved
ηspec
=EAC /Gt
P?
N/G?(4)
where P?
Nis the nominal (or peak) DC power of the PV generator, understood194
as the product of the number of PV modules multiplied by the corresponding195
in-plane STC power. Because EAC ,P?
Nand Gtare given by the billing energy196
meter of the PV installation, the PV manufacturer and the integration of a197
solar irradiance signal, the PR value can be directly calculated. The difference198
between 1 and PR lumps together all imaginable energy losses (i.e. capture199
losses and system losses).200
201
For a given PV system and site, the P R value tends to be constant along202
the years, as much as the climatic conditions tend to repeat. When sub-year203
periods are considered, the P R dependence on unavoidable and time-dependent204
8
losses requires corresponding correction in order to properly qualify the tech-205
nical quality of a PV system. Based on Eqs. 1 to 3, we can reformulate Eq. 4 as:206
207
1
fG. fT. fAC . fP DC . fBOS
. P R = 1 (5)
in which the losses have been lumped into five main categories:208
1. fG: PV module’s yield in function of incident irradiance level,209
2. fT: PV module’s yield in function of cell’s temperature,210
3. fAC : yield of the conversion from DC to AC.211
4. fP DC : yield that represents the ratio of the real DC power and the rated212
DC power,213
5. fBOS : yield of the balance of system.214
Three of these five losses parameters can be expressed analytically. Based215
on Eq. 1, the efficiency dependence on irradiance is:216
217
fG=a+bGeff
G?+cln Geff
G?(6)
However, such a formulation of the fGparameter is useless here since the218
effective irradiance, Geff , is by definition unknown in our case. To overcome219
such a limitation, fGis split into its two main contributing factors:220
221
fG=firr . finc (7)
where firr represents the variation in the PV module efficiency with the level222
of the solar irradiance and finc the variation in the PV module efficiency as a223
function of the incidence angle of the solar irradiance, respectively. Then, ap-224
proximating the ratio Gt/G?by the Capacity Utilization Factor (CUF) defined225
as:226
227
CU F =EAC
T . P ?
N
(8)
with EAC the energy produced during a period of time T(see Eq. 3) and P?
N
228
the nominal (or peak) DC power of the PV system, firr can be estimated by:229
9
230
firr =a+b . C UF +cln(C U F ) (9)
In this equation, the three parameters a,band cvary according to the con-231
sidered PV module technology. Values representative of crystalline silicon cells232
technology (i.e., a=1, b=-0.01 and c=0.025) have been assumed for all PV mod-233
ules here. Finally, based on [18], finc can be expressed as follows:234
235
finc = 1 −1−exp(−(cos θi)/αr)
1−exp(−1/αr)(10)
where θiis the irradiance angle of incidence and αrthe angular loss coefficient,236
an empirical dimensionless parameter dependent on the PV module technology237
and the dirtiness level of the PV module. Typical αrvalues range from 0.16 to238
0.17 for commercial clean crystalline and amorphous silicon modules. In this239
work a value of 0.20 has been assumed for αrwhich is a typical value for crys-240
talline silicon PV modules presenting a moderate level of dirtiness.241
242
The second factor, fT, is defined as:243
244
fT= 1 + κ(Tc−T?
c) (11)
where the operating temperature of the solar cell, Tc, is calculated from the245
ambient temperature, Ta, using the following equation based on the Nominal246
Operation Cell Temperature (NOCT) defined as the temperature reached by247
the cells when the PV module is exposed to a solar irradiance of 800 W.m−2,248
an ambient temperature of 20oC, and a wind speed of 1 m.s−1(it is obtained249
from the manufacturer datasheets):250
251
Tc=Ta+NOC T −20
800 . Gt(12)
Similarly to Eq. 6 the CUF approximation is used to estimate Tcreformu-252
lating Eq. 12 as follows:253
254
Tc=Ta+(NOC T −20)/800.1000 . C UF (13)
10
The third factor, fAC , is computed from:255
256
fAC =PAC
P?
N. ηEU R
(14)
where, the so-called ”European efficiency” of the inverter, ηEU R, is given by the257
formula:258
259
ηEU R = 0.03 η5+ 0.06 η10 + 0.13 η20 + 0.1η30 + 0.48 η50 + 0.2η100 (15)
with η5,η10,η20 ,η30,η50 ,η100 the instantaneous power efficiency values at 5%,260
10%, 20%, 30%, 50% and 100% load.261
262
The fourth factor, fP DC , as well as the fBOS factor cannot not be directly263
estimated because the real energetic behavior of each PV system is a priori264
unknown. Lumping both factors together into a new losses factor, fP ERF , it265
follows from Eqs. 4 and 5 that266
267
fP ERF =1
fG. fT. fAC
.EAC /Gt
P?
N/G?(16)
where the fP ERF factor sums up all the performance losses that, on the first268
hand could be avoided and, on the other hand that cannot be modeled through269
a simple and general analytical expression. This factor can be estimated for270
each PV system from historical data of EAC and Gtusing the EAC data di-271
rectly from the energy meters and Gtdata obtained from the combination of272
clear-sky radiative model simulations and cloud cover information. It is worth273
pointing out that to reduce the uncertainties in its estimation, the fP ERF factor274
was determined on a monthly basis from clear sky situations.275
276
Similarly to [19], clear-sky situations were determined from the PV systems277
energy production time series using a modified version of the algorithm devel-278
oped by [20]. For each PV system, fP ERF , was calculated as being the ratio279
between the electrical energy produced by the PV system corrected by the three280
other losses factors (i.e. fG,fT, and fAC ) together with the quotient P?
N/G?and281
11
the calculated in-plane clear-sky irradiation received by the PV system during282
the considered month. Clear sky simulations were carried out by running the In-283
eichen and Perez clear-sky model [21] using monthly mean climatological Linke284
turbidity values from PVGIS/CMSAF (http://re.jrc.ec.europa.eu/pvgis/). Sim-285
ulated clear sky global horizontal irradiances were then transposed to tilted286
clear sky global irradiance using the ERB decomposition model ([22]; see Ap-287
pendices A.1) and the HAY transposition model ([23]; see Appendices B.1).288
289
Finally, once all losses factors are estimated, the derivation of the in-plane290
hourly global solar irradiance from the hourly PV system energy production is291
given by:292
293
Gt=1
fG.fT.fAC .fP ERF
.EAC
P?
N/G?(17)
where all losses factors except fP ERF are evaluated on a hourly basis using air294
temperature measurements performed within the RMI’s AWS spatially interpo-295
lated at each of the PV system locations in the computation of the operating296
solar cells temperature, Tc(see Eq. 13). fP E RF are determined monthly from297
the previous month EAC data.298
3.2. Tilt to horizontal global solar irradiance transposition299
The next step consists in the conversion of the retrieved in-plane global solar300
irradiance values from the PV systems energy outputs to global horizontal solar301
irradiance at each of the PV systems location. If many transposition models302
have been proposed in the literature (see [24] for a review) to convert solar ir-303
radiance on the horizontal plane, Gh, to that on a tilted plane, Gt, the inverse304
process (i.e. converting from tilted to horizontal) is only poorly discussed in305
literature (e.g. [25], [26], [27], [28], [5], [6], [29]). The difficulty relies on the fact306
that the procedure is analytically not invertible.307
308
Transposition models have the general form:309
12
310
Gt=Bt+Dt+Dg(18)
where the tilted global solar irradiance, Gt, is expressed as the sum of the in-311
plane direct irradiance, Bt, in-plane diffuse irradiance, Dt, and the irradiance312
due to the ground reflection, Dg. The direct component, Bt, is obtained from:313
314
Bt=Bncos θi=Bh
cos θi
cos θz
=Bhrb(19)
with Bnthe direct normal irradiance and Bhthe direct irradiance on a horizon-315
tal surface, respectively. θiis the incidence angle and θzthe solar zenith angle,316
respectively. Parameter rb= cos θi/cos θzis a factor that accounts for the di-317
rection of the beam radiation. The diffuse component, Dt, and the irradiance318
due to the ground reflection, Dg, can be modeled as follows:319
320
Dt=DhRd(20)
321
Dg=ρGhRr(21)
where Dhis the diffuse horizontal irradiance, Ghthe global horizontal irradi-322
ance (i.e., Gh=Dh+Bh), Rdthe diffuse transposition factor and ρthe ground323
albedo. The transposition factor for ground reflection, Rr, can be modeled un-324
der the isotropic assumption (e.g. [30]) as follows:325
326
Riso
r=1−cos β
2(22)
where, β, is the tilt angle of the inclined surface. See Figure 2 for angles defini-327
tion.328
329
Considering the effective global horizontal transmittance, Kt, the direct330
normal transmittance, Kn, and the diffuse horizontal transmittance, Kd(i.e.,331
Kd+Kn=Kt):332
13
333
Gh=KtIocos θz
Bn=KnIo
Dh=KdIocos θz
(23)
where Iois the extraterrestrial normal incident irradiance, Eq. 18 can be rewrit-334
ten as:335
336
Gt=KtIocos θi1−Kd
Kt+ cos θzKd
Kt
Rd+ρRr (24)
It comes from Eq. 24 that when only one tilted global solar irradiance mea-337
surement is considered, the conversion of Gtto Ghrequires the use of a decompo-338
sition model (i.e. model that separate direct and diffuse solar components from339
the global one) to estimate Kdfrom Ktin addition to a transposition model340
to solve the inverse transposition problem. Eq. 24 is solved by an iteration341
procedure, varying the target quantity Gh(through Kt) until the resulting G0
t
342
matches the input Gt(e.g. [26], [28], [5], [29]). Note that an alternative method343
to Eq. 24 based on the Olmo model ([31]) that presents the particularity of be-344
ing analytically invertible was proposed by [5]. But, if the overall performance345
of the inverted Olmo model was found comparable with the other approach,346
the results were slightly worse than those obtained by inverting the decompo-347
sition and transposition models in combination with an iterative solving process.348
349
When two (or more) tilted irradiances values (with different tilt angles350
and/or orientations) are involved in the inverse transposition process, only a351
transposition model is required. The idea that simultaneous readings of a multi-352
pyranometers system can be used to disangle the various components of solar353
radiation on inclined surfaces was originally proposed by [25] to solve in remote354
locations the periodic adjustment required by normal incidence and shadow-355
band pyranometers to ensure that their readings remain accurate when long-356
term data acquisition is in progress.357
358
14
Given ntilted pyranometers (with different inclinations and/or orientations),359
the inverse transposition problem can be represented in the matrix form (e.g.360
[27]):361
362
xTΛ x +B−C= 0 (25)
where Λ = {Ai}is a 2 ×n×2 third-order tensor, B={Bi}is a n×2 matrix,363
Cis a column vector with ngiven entries, and xa column vector with 2 variables:364
365
Ai=
0Ai
Ai0
∈ <2×n×2
B=
C1B1
C2B2
.
.
..
.
.
CnBn
∈ <n×2
C=
Gt1
Gt2
.
.
.
Gtn
∈ <n
xT=DhBh∈ <2
(26)
where the coefficients Ai,Biand Cidepend on the considered transposition366
model.367
368
The least square (hereafter referred to as LS) solution to Eq. 26 is given by:369
min P(x) = 1
2||xTΛ x +B−C||2:x∈ <2(27)
with ||.|| referring to the Euclidean norm. However, the LS is hard to solve and370
a standard technique to resolve Eq. 27 is to use a Newton type iteration method371
(e.g. [32]). As an alternative, Eq. 26 can also be solved by minimizing the errors372
(this approach is hereafter denoted to as EM - Errors Minimization). In this373
15
case, the solution is to minimize374
375
min (E(x) =
n
X
i=1
2
i(x) : x∈ <2)(28)
where, i(x)=(AiDhBh+BiBh+CiDh)−Gti, with i= 1, ..., ndenoting the376
tilted pyranometer.377
4. Results378
The Mean Bias Error (MBE) and the Root Mean Square Error (RMSE)379
statistical error indexes have been used to evaluate the prediction of the global380
horizontal solar irradiance from the energy production of residential PV systems.381
382
MBE =1
n
n
X
i=1
(ei)
383
RMSE =v
u
u
t
1
n
n
X
i=1
(e2
i)
where ei= (Gi,e −Gi,o) is the residual value, Gi,e are the estimated values and384
Gi,o represent the observed measurements. A positive MBE (resp. a negative385
MBE) means that the model tends to overestimate (resp. underestimate) the386
observed measurements.387
388
To obtain dimensionless statistical indicators we express MBE and RMSE389
as fractions of mean solar global irradiance during the respective time interval:390
391
MBE[%] = MBE
¯
M
392
RMSE[%] = RM SE
¯
M
where ¯
M=1
n
n
P
i=1
(Gi,o) is the measurements mean.393
394
16
Statistical error indexes were computed between in situ hourly irradiance395
measurements and the estimations computed from the hourly energy produc-396
tions of residential PV systems surrounding the measurement stations. An ini-397
tial radius of 5 km centered on the station was considered to select the residential398
PV systems for the validation purpose. When less than 4 PV installations were399
found within the delimited area, the radius was extended to 10 km. Table 3400
indicates for each of our measurement sites the number of neighboring PV in-401
stallations used for validation. No PV system was found in the vicinity of the402
Sint-Katelijn-Waver, Retie and Mont-Rigi stations (i.e. AWS 6439, AWS 6464403
and AWS 6494, respectively) and 3 others stations only have one surrounding404
residential PV system. At the opposite, the maximum number of installations405
surrounding a station is 37 for Ernage (i.e. AWS 6459).406
407
Based on our former evaluation of the inverse transposition problem ([29]),408
two different approaches have been considered to compute the global horizontal409
solar irradiance from the PV systems energy production. In the first approach,410
the tilt to horizontal conversion is performed independently at each PV installa-411
tions surrounding the validation site using Eq. 24 with the OLS decomposition412
model (see Appendices A.2) and the SKA transposition model (see Appen-413
dices B.2). The median value of the individual PV system estimates is consid-414
ered in the validation against AWS observations. This approach is referred to415
as 1−PV-M hereafter. In the second approach all individual tilted global solar416
irradiance estimates are used simultaneously and the tilt to horizontal conver-417
sion is solved by EM (see Eq. 28) using the Powell’s quadratically convergent418
method ([33]) and the SKA transposition model (see Appendices B.2). Indeed419
the minimization carried out by using the Powell’s method has been found to420
systematically outperform the LS solution ([29]). It is a generic minimization421
method that allows to minimize a quadratic function of several variables with-422
out calculating derivatives. The key advantage of not requiring explicit solution423
of derivatives is the very fast execution time of the Powell method. In order to424
avoid the problem of linear dependence in the Powell’s algorithm, we adopted425
17
the modified Powell’s method given in [34] and implemented in [35]. This second426
approach is hereafter denoted to as X−PV-EM.427
428
Performance of the two approaches in the GHI estimation from PV systems429
AC power output was evaluated against in-situ observations and furthermore430
compared to the performance of GHI estimates retrieved from Meteosat Sec-431
ond Generation (MSG, [36]) satellite images as implemented on an operational432
basis at RMI. Description of the RMI’s MAGIC/Heliosat-2 algorithm used to433
retrieve the solar surface irradiance at the SEVIRI imager spatial sampling dis-434
tance above Belgium (e.g. about 6 km in the north–south direction and 3.3 km435
in the east–west direction) from MSG images is provided in Appendices C.436
437
While the MSG based retrieval method always provides GHI estimates dur-438
ing day time, there are certain sun positions for which the PV systems power439
output method fails to produce a valid estimation. Unsuccessful tilt to horizon-440
tal conversions are found for both the 1−PV-M and the X−PV-EM approaches441
at low solar elevation irrespective of the number of PV systems involved in the442
conversion process. Failure rates reported for the 1−PV-M and the X−PV-443
EM approaches at each validation sites are provided in Table 3 together with444
the total number of available hourly data points at each location for the year445
2014. Unsurprisingly the largest failure rates (up to nearly 40% in the case446
of the Melle station -AWS 6434-) are found at validation sites where only one447
PV installation is available. With more PV systems, the number of unsuccess-448
ful conversions after sunrise and before sunset is decreased. Table 3 tends to449
indicate that 1−PV-M starts to produce valid results at lower solar elevation450
conditions than X−PV-EM (i.e. an overall failure rate of about 12.4% is re-451
ported for 1−PV-M and of 19.6% for X−PV-EM, respectively) but it is largely452
relying on the angular configurations (i.e. tilt and azimuth angles) of the PV453
installations found within the group of PV systems.454
455
18
4.1. Hourly validation456
Table 4 compares hourly GHI estimates derived from PV production data457
with the 1−PV-M and X−PV-EM approaches as well as from MSG images with458
the corresponding ground measurements. To ensure that the comparisons are459
made between comparable data, special attention was given to the coherence of460
the data, the precision of the time acquisition, and the synchronization of the461
different data sets with the ground measurements. Because of inaccuracies in the462
orientations and/or inclinations of the PV installations provided by the PV sys-463
tems installers or owners, GHI computation from the energy production of only464
one installation can generate RMSE values as large as 189.34 W.m−2or 57.8%465
(i.e. at the Middelkerke validation site, AWS 6407). Increasing the number of466
PV installations involved in the estimation process smoothes the GHI estima-467
tion to some extent. This is particularly apparent for 1−PV-M which globally468
presents lower RMSE values than found for X−PV-EM (i.e. an overall RMSE469
value of 113.5 W.m−2or 41.4% is reported for 1−PV-M and of 121.9 W.m−2or470
44.4% for X−PV-EM, respectively). However, sensitivity experiments in which471
the number of PV installations involved in the GHI determination was varying472
revealed a larger variability in the resulting GHI estimations for 1−PV-M than473
found for X−PV-EM which produces a more stable solution.474
475
[5] reported a somewhat similar mean RMSE error of about 40% for GHI es-476
timates derived from 5 years (from 2010 through 2014) of five-minute resolution477
records of specific power of 45 PV systems in the region of Freiburg, Germany.478
In contrast, GHI retrieval from MSG images shows a better performance with479
an overall RMSE of 55.8 W.m−2or 20.3%. Moreover, while the satellite re-480
trieval tends to slightly overestimate the GHI values (i.e. MBE values ranging481
from 0.16 to 5.87%), there is no clear trend in the GHI computation from mea-482
sured AC PV output power. Positive and negative biases are reported for both483
1−PV-M and X−PV-EM approaches. Moreover it can appear that the sign of484
the bias even differs from one approach to the other (e.g. a negative MBE value485
of -9.08 W.m−2or -3.37% is reported at the Humain validation site (AWS 6472)486
19
for 1−PV-M approach while an overestimation of 7.86 W.m−2or 2.92% is found487
for X−PV-EM. In general, the magnitude of the bias is lower with X−PV-EM488
than with 1−PV-M (i.e. MBE values ranging from -5.6% to 5.5% and from489
-9.9% to 7.8%, respectively).490
491
Table 5 compares the performance of the 1−PV-M and X−PV-EM ap-492
proaches together with the MSG-based retrieval method in the hourly GHI com-493
putation as a function of the sky conditions. Note that error statistics were cal-494
culated from validation sites accounting for at least three residential PV instal-495
lations (i.e. the Middelkerke/AWS 6407, Melle/AWS 6434 and Stabroek/AWS496
6438 measurement stations were excluded). Clearly, the relative accuracy of497
the GHI estimates varies noticeably as the sky conditions moves from overcast498
to clear sky situations irrespective of the calculation method. With a reported499
RMSE value of roughly 112%, the 1−PV-M and X−PV-EM approaches fail to500
produce reliable estimations in overcast conditions (i.e. 0.0 ≤Kt<0.2) where501
the global radiation is mainly composed of diffuse radiation. With a reported502
RMSE in the order of 70% the satellite-based retrieval method also exhibits503
a poor performance in overcast conditions but shows a rapid performance im-504
provement as the sky becomes clearer and presents a minimum RMSE value of505
10.5% in partly clear conditions (i.e. 0.6 ≤Kt<0.8). Similarly, the accuracy506
of GHI estimations from measured AC PV output power increases as the sky507
conditions becomes clear but the magnitude of the errors is still at least twice508
the one found for the satellite retrieval method irrespective of the computation509
approaches. By contrast, while being still the best performing method in clear510
sky conditions the magnitude of the accuracy difference between the satellite511
and the PV systems methods is reduced (i.e. a RMSE of 19.4% for the satellite512
retrieval method vs. RMSE values of 25.4% and 24.3% for the 1−PV-M and513
X−PV-EM approaches, respectively). However, it is worth pointing out that514
the number of hourly data points present in the clear sky bin (i.e. 0.8 ≤Kt≤515
1.0) where the direct component largely domines is very low (i.e. 0.3% of the516
total data points) and all of them are for Ktvalues ≤0.85.517
20
518
4.2. Daily validation519
Computation of the statistical errors indexes on a daily basis is not as520
straightforward as for an hourly basis because as already mentioned both the521
1−PV-M and X−PV-EM approaches fail to produce valid GHI estimates at low522
solar elevation conditions. In Table 6 RMSE and MBE indexes have been com-523
puted by assuming no incoming global horizontal solar irradiance in the com-524
putation of the daily global horizontal solar irradiation for data points where525
no valid hourly GHI estimates were obtained. Because unsuccessful GHI esti-526
mations from AC power output can be as large as 39.7% when only one PV527
installation is considered (see Table 3) the daily performance of the PV method528
can be very limited in some places in comparison to the satellite method. As529
an example, the RMSE of 1321.2 W.h.m−2or 42.9% and the MBE of -984.8530
W.h.m−2or -31.9% reported at the Middelkerke validation site (AWS 6407) in531
Table 6 for the PV system method reduce to a RMSE of 296.4 W.h.m−2or 9.6%532
and a MBE of 43 W.h.m−2or 1.4% for the MSG-based retrieval method at this533
validation site.534
535
Globally Table 6 indicate that the daily computation exhibits a better perfor-536
mance than hourly estimation irrespectively of the considered retrieval methods.537
An overall RMSE of 11% and a slight positive bias is reported for the satellite538
based method. 1−PV-M and X−PV-EM behave quite similarly in terms of539
RMSE (i.e. overall RMSE of 12%). Clearly the RMSE magnitude difference540
between the MSG-based and the PV systems power output methods is drasti-541
cally reduced when considering daily global solar irradiation quantities rather542
than hourly GHI values. However the RMSE spatial variation (i.e. from one543
validation site to another) is larger in the PV systems based method and, while544
a systematic positive bias is reported for the satellite retrieval method, the sign545
of the bias can vary from one validation site to another in the PV systems based546
method and even between the 1−PV-M and X−PV-EM approaches on a given547
21
site.548
549
4.3. Spatial validation550
Finally, to assess the spatial distribution of the solar surface irradiation551
computed from hourly PV systems power outputs, the 1470 residential PV in-552
stallations considered in our study (see Figure 1) were spatially aggregated into553
150 clusters (see black dots on panel C in Figure 3 for the clusters spatial dis-554
tribution) using the k-means algorithm ([37]). K-means clustering partitions555
a dataset into a small number of clusters by minimizing the distance between556
each data point and the center of the cluster it belongs to. A minimum of four557
PV installations by clusters was imposed except for two of them located in the558
vicinity of the Belgian coast because of the very low density of PV installations559
found in this area.560
561
Figure 3 presents an example of daily solar surface irradiation over the Bel-562
gian territory as computed from the interpolation of ground measurements using563
the ordinary kriging (OK) method (e.g. [38]), the MSG satellite derived estima-564
tion and the PV systems method using X−PV-EM. Clearly, due to the sparsity of565
the ground stations networks, interpolating ground data generates only a coarse566
distribution of the solar surface irradiation: large-scale variations of the solar567
irradiation (such as the south–east to north–west positive gradient) are identi-568
fied but local fluctuations remain unseen (Figure 3, panel a). Satellite-derived569
estimates, on the other hand, provide a global coverage and are therefore able570
to account for clouds induced small-scale variability in surface solar radiation571
(Figure 3, panel b). Regarding the PV systems method, the daily solar irradia-572
tion estimated at the PV clusters level were then interpolated by OK method to573
cover the entire Belgian territory (Figure 3, panel c). As we see, the south–east574
to north–west positive gradient is well apparent as well as some of the regional575
specificities. For instance, the Gaume region (area in the south-east of Belgium)576
located on the south side of the Ardenne (hilly mass) and that enjoys longer sun-577
22
shine time appears clearly on the mapping. In general, the PV systems method578
provides small-scale patterns partly supported by the MSG derived mapping.579
Some others appear as the signature of an erroneous estimation at the cluster580
level. Such artifacts are well apparent along the Ourthe valley in the Ardenne.581
582
5. Conclusions and perspective583
The lack of accuracy encountered in the information on the orientation584
and/or the inclination of the PV installations does not allow to retrieve reli-585
able solar irradiance data. It was found that the provided angles can typically586
bear inaccuracies up to 5-10o. However tilt angle and surfaces orientation have587
been found to have a large impact on the accuracy of the global horizontal solar588
irradiance calculation. Increasing the number of PV installations involved in589
the computation process allows smoothing the estimation to some extent. An-590
other limitation is that there are certain sun positions for which the method591
fails to produce a valid estimation. As an example, unsuccessful tilt to hori-592
zontal conversions occurs at low solar elevation irrespective of the number of593
PV systems involved in the conversion process. Validation results computed on594
an hourly basis provide a mean RMSE value of about 40% when considering595
a group of neighboring PV installations in the estimation process while values596
as large as 60% are reported when using a single installation. By comparison,597
satellite-based global horizontal irradiance estimation exhibits a better perfor-598
mance with an associated overall RMSE of about 20%. By contrast the overall599
performance of both methods does not differ significantly on a daily basis. How-600
ever, spatially the distribution of the solar surface irradiation computed from601
PV systems outputs presents small-scale pattern artifacts signature of erroneous602
estimations.603
604
Globally, the development of a procedure able to assess the true azimuth and605
tilt angle of a PV generator from the sole knowledge of its hourly energy output606
23
data would be largely benefit for the method. Towards this ob jective, it is worth607
mentioning that while the identification of a PV system’s location and orien-608
tation from performance data has received limited attention in the past, [39]609
have introduced a method for automatic detection of PV system configuration.610
And even more recently, [40] have proposed a method to estimate the location611
and orientation of distributed photovaltaic systems from their generation out-612
put data. The conversion of PV system energy production to tilted global solar613
irradiance can certainly be improved by taking into account the wind influence614
on the operating temperature of the solar cell as an example or by performing615
clear-sky radiative transfer computations with a more sophisticated model, etc.616
But such refinements will only have a limited influence on the final result since617
the main limitation is due to erroneous/invalid transposition of the titled global618
solar irradiance to the horizontal because of inaccurate information about the619
PV systems.620
621
Acknowledgments622
This study was supported by the Belgian Science Policy Office (BELSPO)623
through the Belgian Research Action through Interdisciplinary Networks (BRAIN-624
be) pioneer research project BR/314/PI/SPIDER Solar Irradiation from the625
energy production of residential PV systems .626
A. Decomposition models627
A.1. ERB model (Erbs et al., 1982):628
[22] developed a correlation between the hourly clearness index, Kt, and the629
corresponding diffuse fraction, Kdbased on 5 stations data. In each station,630
hourly values of direct and global irradiances on a horizontal surface were regis-631
tered. Diffuse irradiance was obtained as the difference of these quantities. The632
proposed correlation combines a linear regression for 0 < Kt≤0.22, a fourth633
24
degree polynom for 0.22 < Kt≤0.8 and a constant value for Kt>0.8634
Kd=
1−0.09 KtKt≤0.22
0.9511 −0.1604 Kt+ 4.388 K2
t−16.638 K3
t+ 12.336 K4
t0.22 < Kt≤0.8
0.165 Kt>0.8
(29)
A.2. OLS model (Skartveit and Olseth, 1987)635
[41] estimated the direct normal irradiance, Bn, from the global horizontal636
irradiance, Gh, and the solar elevation angle, γ, for Bergen (Norway, 60.4oN)637
with the following equation based on hourly records of global and diffuse hori-638
zontal irradiances with mean solar elevation larger than 10oduring 1965-1979:639
Bn=Gh(1 −Ψ)
sin γ(30)
where Ψ is a function of the clearness index, Kt. The model was validated with640
data collected in 12 stations worldwide. The function Ψ reads as:641
642
Ψ =
1 for Kt< c1
1−(1 −d1)hd2c1/2
3+ (1 −d2)c2
3ifor c1≤Kt≤1.09c2
1−1.09c2
1−Υ
Ktfor Kt>1.09c2
(31)
where:643
c1= 0.2644
c2= 0.87 −0.56 −0.06 γ
645
c3= 0.51 + sin π(c4
d3−0.5)
646
c4=Kt−c1
647
d1= 0.15 + 0.43 e−0.06 γ
648
d2= 0.27649
d3=c2−c1
650
Υ=1−(1 −d1)d2cc1/2
3+ (1 −d2)cc2
3
651
cc3= 0.51 + sin π(cc4
d3−0.5)
652
cc4= 1.09 c2−c1
653
25
B. Transposition models654
Each model develops the diffuse transposition factor (i.e., the ratio of diffuse655
radiation on a tilted surface to that of a horizontal), Rd, according to specific656
assumptions.657
B.1. HAY model (Hay, 1979):658
In the HAY model ([23]), diffuse radiation from the sky is composed of an659
isotropic component and a circumsolar one. Horizon brightening is not taken660
into account. An anisotropy index, FHay , is used to quantify a portion of the661
diffuse radiation treated as circumsolar with the remaining portion of diffuse662
radiation assumed to be isotropic, i.e.,663
RHAY
d=FHAY rb+ (1 −FH AY )1 + cos β
2(32)
where FHAY =Bh/(Iocos θz) is the Hay’s sky-clarity factor and rb= cos θi/cos θz
664
the beam radiation conversion factor. θzis the solar zenith angle, θiis the in-665
cidence angle of the beam radiation on the tilted surface, Bhis the direct hori-666
zontal solar irradiance and Iois the extraterrestrial normal incident irradiance.667
B.2. SKA model (Skartveit and Olseth, 1986):668
Solar radiation measurements indicate that a significant part of sky diffuse669
radiation under overcast sky conditions comes from the sky region around the670
zenith. This effect vanishes when cloud cover disappears. [42] modified the HAY671
model (Eq. 32) in order to account for this effect,672
RSK A
d=FHAY rb+Zcos β+ (1 −FH AY −Z)1 + cos β
2(33)
where Z= max(0,0.3−2FHAY ) is the Skartveit-Olseth’s correction factor. If673
FHAY ≥0.15, then Z= 0 and the model reduces to the HAY model.674
C. Description of the RMIs MAGIC/Heliosat-2 algorithm675
Information on GHI over Belgium is retrieved from MSG data by an adapta-676
tion of the well-known Heliosat-2 method ([43]) initially developed for Meteosat677
26
First generation satellites. The principle of the method is that a difference678
in global radiation perceived by the sensor aboard a satellite is only due to a679
change in the apparent albedo, which is itself due to an increase of the radiation680
emitted by the atmosphere towards the sensor (i.e. [44], [45]). A key parameter681
is the cloud index (also denoted as effective cloud albedo), n, determined by the682
magnitude of change between what is observed by the sensor and what should683
be observed under a very clear sky. To evaluate the all-sky GHI, a clear-sky684
model is coupled with the retrieved cloud index which acts as a proxy for cloud685
transmittance. Inputs to the Heliosat-2 method are not the visible satellite im-686
ages in digital counts as in the original version of the method ([44]) but images687
of radiances/reflectances:688
nt(i, j) = ρt(i, j)−ρt
cs(i, j )
ρt
max(i, j )−ρt
cs(i, j )(34)
where nt(i, j) is the cloud index at time tfor the satellite image pixel (i, j );689
ρt(i, j) is the reflectance or apparent albedo observed by the sensor at time t;690
ρt
max(i, j ) is the apparent albedo of the brightest cloud at time t;ρt
cs(i, j ) is the691
apparent ground albedo under clear-sky condition at time t.692
693
The MAGIC/Heliosat-2 method implemented at RMI ([46]) is applied to694
the visible narrow-bands of the Spinning Enhanced Visible and Infrared Imager695
(SEVIRI) on board of the MSG platform. ρt
cs(i, j ) is derived from the fifth696
percentile of the ρt(i, j) distribution related to the last 60 days ([47]) while697
ρt
max(i, j ) is estimated from theoretical radiative transfer model calculation (see698
[46]). GHI estimates are computed for each pixel and MSG time slot by the699
combination of the satellite cloud index, n, and the GHI in clear sky condition700
calculated by the Mesoscale Atmospheric Global Irradiance Code (MAGIC)701
clear-sky model ([48]). Daily global solar radiations are computed by trapezoidal702
integration over the diurnal cycle of the retrieved GHI t(i, j ). The retrieval703
process runs over a spatial domain ranging from 48.0oN to 54.0oN and from704
2.0oE to 7.5oE within the MSG field–of–view. In this domain, the SEVIRI705
spatial sampling distance degrades to about 6 km in the north-south direction706
27
and 3.3 km in the east-west direction.707
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List of Figures846
1 Location of the selected 1470 residential PV systems within the847
Belgian territory. Red dots indicate the location of the RMI’s848
ground radiometric stations considered in this study . . . . . . . 35849
2 Definition of angles used as coordinates for an element of sky850
radiation to an inclined plane of tilt β............... 36851
3 Comparison between the daily spatial distribution of surface so-852
lar global irradiation (in W.h.m−2) over Belgium as computed853
(a) by the interpolating ground measurements, (b) by the MSG854
satellite retrieval method and, (c) by the PV systems power out-855
put method. Illustrations are for the 20 June, 2014. Black dots856
in panel (a) indicate the location of the RMI’s in-situ measure-857
ments sites. Black dots in panel (c) indicate the location of the858
150 clusters of PV systems . . . . . . . . . . . . . . . . . . . . . . 37859
34
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Figure 1: Location of the selected 1470 residential PV systems within the Belgian territory.
Red dots indicate the location of the RMI’s ground radiometric stations considered in this
study
35
Figure 2: Definition of angles used as coordinates for an element of sky radiation to an inclined
plane of tilt β
36
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Figure 3: Comparison between the daily spatial distribution of surface solar global irradiation
(in W.h.m−2) over Belgium as computed (a) by the interpolating ground measurements, (b)
by the MSG satellite retrieval method and, (c) by the PV systems power output method.
Illustrations are for the 20 June, 2014. Black dots in panel (a) indicate the location of the
RMI’s in-situ measurements sites. Black dots in panel (c) indicate the location of the 150
clusters of PV systems
37
Table 1: Distribution of the number of PV systems as a function of the orientation and tilt
angle.
Tilt <–East Deviation from South (o) West –>
angle (o) -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90
00 0 0 0 0 1 0 0 0 3 0 0 0 0 0 0 0 0 0
10 4 0 0 0 0 6 0 2 2 7 0 2 0 0 2 0 0 0 0
20 4 2 1 2 0 19 10 12 10 20 8 5 9 0 18 2 4 3 0
30 13 4 14 4 0 50 24 24 18 71 9 21 14 0 47 7 6 6 0
40 67 8 21 13 0 188 38 36 30 223 16 32 23 0 147 24 12 15 0
50 4 2 2 2 0 12 3 6 4 16 2 3 1 0 17 5 6 1 0
60 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
80 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
38
Table 2: Location of the ground stations involved in the RMI solar radiation monitoring
network in which the global horizontal radiation is recorded.
Station Station Lat. Lon. Alt.
Code Name (oN) (oE) (m)
6407 MIDDELKERKE 51.198 2.869 3.0
6414 BEITEM 50.905 3.123 25.0
6434 MELLE 50.976 3.825 15.0
6438 STABROEK 51.326 4.365 5.0
6439 SINT-KATELIJNE-WAVER 51.076 4.526 10.0
6447 UCCLE 50.798 4.359 101.0
6455 DOURBES 50.096 4.596 233.0
6459 ERNAGE 50.583 4.691 157.0
6464 RETIE 51.222 5.028 21.0
6472 HUMAIN 50.194 5.257 296.0
6476 SAINT-HUBERT 50.040 5.405 557.0
6477 DIEPENBEEK 50.916 5.451 39.0
6484 BUZENOL 49.621 5.589 324.0
6494 MONT RIGI 50.512 6.075 673.0
39
Table 3: Unsuccessful conversion (in %) reported for the PV systems 1−PV-M and X−PV-EM
approaches, respectively. Also provided is the total number of hourly data points available at
each validation sites and the number of PV installations found in the vicinity of the measure-
ment stations. ?indicates the PV installations located within a radius of 10 km surrounding
the validation site.
AWS PV systems Hourly PV Method
code number values 1−PV-M X−PV-EM
6407 1 4184 36.74% /
6414 5 4529 12.89% 17.40%
6434 1 4477 39.74% /
6438 1?4666 22.29% /
6439 0 / / /
6447 12?4698 16.69% 18.09%
6455 4?4346 13.71% 15.55%
6459 37?4573 13.21% 21.21%
6464 0 / / /
6472 5 4583 17.72% 23.04%
6477 3?4469 14.03% 22.04%
6484 10 / / /
6494 0 / / /
40
Table 4: Comparison between global horizontal solar irradiance produced by the PV systems
power output method for both the 1−PV-M and X−PV-EM approaches and retrieved from
the MSG satellite images with the corresponding ground measurements. The RMSE and MBE
error statistics (in W.m−2and %) are calculated on a hourly basis over the full year 2014.
?indicates the PV installations located within a radius of 10 km surrounding the validation
site.
AWS PV system MSG
code Nbr 1−PV-M approach X−PV-EM approach
RMSE MBE RMSE MBE RMSE MBE
6407 1 189.34 (57.76%) -53.30 (-16.26%) / / / / 69.55 (21.22%) 0.54 (0.16%)
6414 5 116.83 (41.81%) -24.06 (-8.61%) 113.81 (40.73%) -9.00 (-3.22%) 56.68 (20.28%) 8.68 (3.11%)
6434 1 116.94 (39.11%) -10.48 (-3.50%) / / / / 62.58 (20.93%) 11.61 (3.88%)
6438 1?128.05 (46.26%) -32.51 (-11.75%) / / / / 62.80 (22.69%) 13.86 (5.01%)
6447 12?106.14 (39.23%) 1.85 (0.68%) 107.76 (39.83%) -9.03 (-3.34%) 53.13 (19.64%) 7.67 (2.84%)
6455 4?114.88 (43.17%) 20.86 (7.84%) 120.87 (45.42%) 14.63 (5.50%) 59.52 (22.37%) 15.63 (5.87%)
6459 37?112.23 (39.66%) -28.03 (-9.90%) 129.64 (45.81%) -15.84 (-5.60%) 57.00 (20.14%) 1.01 (0.36%)
6472 5 120.55 (44.76%) -9.08 (-3.37%) 129.87 (48.22%) 7.86 (2.92%) 52.95 (19.66%) 11.26 (4.18%)
6477 3?110.13 (39.70%) -17.95 (-6.47%) 129.15 (46.55%) -3.56 (-1.28%) 55.05 (19.84%) 13.32 (4.80%)
41
Table 5: Performance in term of RMSE of the PV systems power output method for both the
1−PV-M and X−PV-EM approaches and the MSG retrieval method in the global horizontal
solar irradiance estimation as a function of the sky condition. Absolute (in W.m−2) and
relative (in %) RMSE are computed from 21716 hourly 2014 data points. Validation sites
accounting for less than 3 residential PV installations were discarded for the errors indices
computation (i.e. AWS 6407, AWS 6434 and AWS 6438).
Sky Method Data
Condition 1−PV-M X−PV-EM MSG Number
0.0 ≤Kt<0.2 72.01 (112.18%) 71.82 (111.88%) 44.20 (68.86%) 5258 (24.21%)
0.2 ≤Kt<0.4 107.68 (60.29%) 115.29 (64.55%) 55.32 (30.98%) 6136 (28.26%)
0.4 ≤Kt<0.6 125.15 (37.81%) 139.87 (42.26%) 61.41 (18.55%) 5458 (25.13%)
0.6 ≤Kt<0.8 139.14 (25.07%) 147.53 (26.59%) 58.06 (10.46%) 4798 (22.09%)
0.8 ≤Kt≤1.0 209.51 (25.36%) 201.06 (24.34%) 160.21 (19.39%) 66 (0.31%)
All Sky 113.49 (41.37%) 121.87 (44.43%) 55.75 (20.32%) 21716 (100.0%)
42
Table 6: Comparison between daily cumulated surface solar irradiation produced by the PV
systems power output method (for both the 1−PV-M and X−PV-EM approaches) and re-
trieved from the MSG satellite images with the corresponding daily ground measurements.
The RMSE and MBE error statistics (in W.h.m−2and %) are calculated on a daily basis over
the full year 2014.?indicates the PV installations located within a radius of 10 km surrounding
the validation site.
AWS PV systems MSG
code Nbr 1−PV-M approach X−PV-EM approach
RMSE MBE RMSE MBE RMSE MBE
6407 1 1321.23 (42.85%) -984.80 (-31.94%) / / / / 296.38 (9.61%) 43.03 (1.40%)
6414 5 308.40 (14.25%) -175.35 (-8.10%) 304.64 (14.08%) -106.57 (-4.92%) 268.53 (12.41%) 116.29 (5.37%)
6434 1 534.81 (16.39%) -257.06 (-7.88%) / / / / 337.75 (10.35%) 150.23 (4.60%)
6438 1?640.95 (22.04%) -434.83 (-14.95%) / / / / 322.61 (11.09%) 154.50 (5.31%)
6447 12?250.26 (8.48%) 5.98 (0.20%) 242.89 (8.23%) -115.75 (-3.92%) 232.35 (7.87%) 94.28 (3.19%)
6455 4?348.20 (19.69%) 168.84 (9.55%) 306.30 (17.32%) 116.07 (6.56%) 260.73 (14.74%) 143.62 (8.12%)
6459 37?225.21 (10.48%) -137.49 (-6.40%) 180.90 (8.42%) -64.94 (-3.02%) 231.97 (10.79%) 40.94 (1.90%)
6472 5 151.70 (9.48%) -13.18 (-0.82%) 226.74 (14.18%) 43.76 (2.74%) 185.28 (11.58%) 112.68 (7.04%)
6477 3?266.96 (10.56%) -29.97 (-1.19%) 259.35 (10.26%) 100.60 (3.98%) 271.98 (10.76%) 161.13 (6.37%)
43
