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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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33

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

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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