“Look at the sky. Ask yourself: Has the sheep eaten the flower, Yes or no? And you will see how everything changes…”
Antoine de Saint Exupéry
I NDEX NOTE
Report Title Selection and performance quantification of the most appropriate Clear‐
Sky Model for the forecasting of solar radiation at the Reunion Island Curriculum International Master PM3E
Year 2013
Author Mickaël Edon
Company Réuniwatt
No. of Employees < 50
Address 14, rue de la Guadeloupe 97490 Sainte‐Clotilde France
Company Tutor Dr Sylvain Cros Function/Position R&D Engineer School Tutor Dr C. Mangwandi
Keywords Variable renewable energy, Photovoltaic production forecasting, Clear‐
Sky model, Atmospheric parameters, Solar irradiation, validation of ground measurement, Detection of clear‐sky moments.
Summary The thesis is focused on solar irradiance forecasting methods and more specifically on clear‐sky models (CSM). CSMs with many inputs are often the most accurate but their performance remains very sensitive to local climate conditions. Moreover, availability of some inputs is not always guaranteed at every location. The objective of the thesis is the selection and performance quantification of the most appropriate CSM for the forecasting of solar radiation at the Reunion Island.
Appendices I. Model input data summary
II. Model error (official and literature) III. Process organigram
IV.a Detail of results, table
IV.b Detail of results, correlograms
CONTENTS
CONTENTS
ACKNOWLEDGMENTS 4
EXECUTIVE SUMMARY 5
LIST OF FIGURES 6
LIST OF TABLES 6
1 INTRODUCTION 7
1.1 O N ENERGY AT A WORLD ’ S SCALE 7
1.2 E NERGY SYSTEM ON THE R ÉUNION ISLAND 7
1.3 A N ISLAND WITH GREAT POTENTIAL FOR S OLAR PHOTOVOLTAIC ENERGY 9
1.4 R EUNIWATT , AN INNOVATIVE COMPANY 10
1.5 S COPE : CHOOSING THE MOST APPROPRIATE CLEAR ‐ SKY MODEL 10
2 INTRODUCTION TO CLEAR‐SKY MODEL 11
2.1 C LEAR ‐S KY ATMOSPHERIC EFFECTS 11
2.2 O N THE IMPORTANCE OF USING AN APPROPRIATE C LEAR S KY M ODEL 13
2.3 C HOICE OF MODELS TO ANALYSE 14
2.4 M ODEL _1 15
2.5 M ODEL _2 15
2.6 M ODEL _3 E RREUR ! S IGNET NON DEFINI .
2.7 M ODEL _4 16
3 METHODOLOGY 16
3.1 G ROUND MEASUREMENTS 16
3.2 S ELECTION OF CLEAR ‐ SKY MOMENTS 17
3.3 SOURCE OF ATMOSPHERIC PARAMETERS 20
3.4 ASSESSING MODEL PERFORMANCE 23
3.5 D EALING WITH GROUND MEASUREMENT UNCERTAINTIES 24
4 MODEL PERFORMANCE 25
4.1 S UMMARY OF RESULTS 25
4.2 R ESULTS COMPARISON WITH THE LITERATURE 29
CONTENTS
4.3 S ELECTION OF THE MOST APPROPRIATE MODEL 31
5 CONCLUSION 33
BIBLIOGRAPHY 34
APPENDICES 37
I. M ODEL I NPUT DATA SUMMARY 38
II. M ODEL ERROR ( OFFICIAL AND LITERATURE ) 39
III. P ROCESS O RGANIGRAM 40
IV. A D ETAIL OF RESULTS , TABLE 41
IV. B D ETAIL OF RESULTS , CORRELOGRAMS 44
Acknowledgments
A CKNOWLEDGMENTS
This MSc. thesis has been carried out at Reuniwatt, at the Reunion Island, and supervised by Sylvain Cros, Earth Observation and Climate Business Intelligence Expert. I wish to express my sincere appreciation to Sylvain for his enthusiastic support and excellent supervision within both Clear‐Sky Models and computer science.
Particular thanks to Nicolas Schmutz, founder of Reuniwatt, for having welcomed me within this vibrant and ambitious company. I am also very much grateful to Nicolas for having inviting me to participate to the 1) International conference on solar forecasting in an insular context 2) “Journées Européennes du Solaire” 3) “Rencontre avec le monde économique dans le cadre du débat national sur la transition énergétique”. Besides, I will bring back home a very good memory of our lunches at La Jonque. I wish to Nicolas, and to the rest of Reuniwatt team, a complete success in the development of Soleka and future projects contributing to the energy autonomy of this lovely and unique island which is La Réunion.
I would also like to thank the all staff of Reuniwatt for making me feel welcome and contributed to a fruitful, creative and innovative atmosphere. Working here, with such enthusiastic colleagues, has been a really positive and enriching experience to me.
Finally, I would also like to take this opportunity to express my gratitude to the great ME3 (extended) family: for being the most diverse yet most integrated group of extraordinary talent I’ve ever had the chance to be part of.
Industry Thesis Advisor
Dr Sylvain Cros
Earth Observation and Climate Business Intelligence Expert
Reuniwatt
Academic Thesis Advisor Dr C. Mangwandi Queen’s University Belfast
United Kingdom
Cover page photo: Mickaël Edon
Executive summary
E XECUTIVE SUMMARY
As part of solar irradiance forecasting, there is a need for values of solar radiation under clear‐sky conditions. Uncertainty of solar irradiance under clear sky can affect significantly the forecast results accuracy. Clear‐sky models can compute this value, by using few or many inputs. Clear‐sky models (CSM) with many inputs are often the most accurate but their performance remains very sensitive to local climate conditions. Four models have been tested in this study:
‐ Model_1: A physical model requiring detailed inputs.
‐ Model_2: An empirical model
‐ Model_3: A physical model requiring simplified inputs.
‐ Model_4: A very new physical model using detailed atmospheric radiative transfer features.
The performance quantification has been established by using ground measurements in Sainte‐Marie (Réunion Island) covering 658 days, and during clear‐sky moments determined by the Perez et al. method. Atmospheric parameters were retrieved from the Aeronet database.
The study has revealed that, unlike what is often agreed in the literature, a model accuracy does not always depend on its number of inputs. Model_3, which require much less inputs than Model_2 and Model_1, have given more accurate results. Using more local atmospheric input, even on a daily basis, has not been effective for improving accuracy.
Model_4 was built with less approximation on atmospheric radiation phenomena; and is the most accurate in this study. Model_3 is the most operational model, requiring the smallest number of atmospheric parameters and yet it shows very satisfactory performance.
Therefore, the most appropriate model to use, as part of solar radiation forecasting, would be Model_4. Model_3 would be the most appropriate one if there was a need for a clear‐sky model with small computation resources.
Index words: Variable renewable energy, Photovoltaic production forecasting, Clear‐Sky model, Atmospheric parameters, Solar irradiation, Validation of ground measurement, Detection of clear‐sky moments.
List of figures
L IST OF FIGURES
F
IGURE1‐1
E
NERGY SCHEMA OF THER
EUNIONI
SLAND... 8
F
IGURE1‐2
D
ISTRIBUTION BY POWER RANGE OF THE INSTALLEDPV
CAPACITY IN2010
(
DARK BLUE=
NUMBER OF INSTALLATIONS;
LIGHT BLUE=
INSTALLED CAPACITY) ... 9
F
IGURE2‐1
C
LEAR‐
SKY ATMOSPHERIC EFFECTS... 11
F
IGURE2‐2
R
ADIATION REDUCTION THROUGH ATMOSPHERIC EXTINCTION PROCESSES(
SOURCE:
C.
H
OYER‐K
LICK:
INTRODUCTION TO SOLAR RESSOURCE ASSESSMENT... 12
F
IGURE2‐3
A
IR MASS EFFECT... 13
F
IGURE2‐4
LOSSES ON INCIDENT IRRADIATION ON A TYPICALPV
CELL(
SOURCE:
TECHNIQUES DE L’
INGENIEUR) ... 13
F
IGURE2‐5
A
CCURATE AND INACCURATE MODELLED GHI... 14
F
IGURE2‐6
ACCURATE AND INACCURATE CLEAR‐
SKY INDEX... 14
F
IGURE3‐1
S
OLAR REFERENCE CELL USED FOR THE GROUND MEASUREMENTS... 16
F
IGURE3‐2
IRRADIATION MAP AT SAINT‐
MARIE,
IN[
WH/
M2],
FROM19
AUG2010
TO07
JUN2012 ... 17
F
IGURE3‐3
E
XTRA‐
TERRESTRIAL IRRADIATION,
MODELLED AND MEASURED IRRADIATION DURING ONE DAY... 18
F
IGURE3‐4
K
C,
KT AND KT' ... 18
F
IGURE3‐5
SOLAR ELEVATION ANGLE H ANDA
IR MASS(AM).
T
HE DASHED LINE SHOWS FIRST FILTER ON H>15° ... 18
F
IGURE3‐6
C
LEAR‐
SKY MOMENTS FOR H>15°
AND0,65<
KT'<=1 ... 19
F
IGURE3‐7
R
ELATIVE RMSE OF MODELLED/
MEASURED IRRADIATION VERSUS KT' ... 20
F
IGURE3‐8
W
ATER VAPOUR BETWEEN2007
AND2012
INS
AINT‐D
ENIS,
AERONET DATABASE... 22
F
IGURE3‐9
I
NTERPOLATEDAOD
AT700
NM BETWEEN2007
AND2012
INS
AINT‐D
ENIS,
A
ERONETD
ATABASE... 22
F
IGURE3‐10
MONTHLY AVERAGE OF THE LINKE TURBIDITY FACTOR AT THE REUNION ISLAND,
SOURCE:
CONFIDENTIAL... 22
F
IGURE4‐1
M
EASURED AND MODELLED IRRADIANCE OF ALL MODEL ON A RELATIVELY CLEAR‐
SKY DAY... 25
F
IGURE4‐2
C
ORRELOGRAM OF ALL MODELS(GHI
MINUTE)
FOR H>15°
AND0,65<K
T'<=1 ... 26
F
IGURE4‐3
M
ODEL RESULTS FOR MINUTE VALUES AND SOLAR ELEVATION ANGLE>
15°
AND0,65<
KT'<=1.
T
HE BLUE DASHED LINE INDICATES THE ASSUMED5,2%
UNCERTAINTY LIMITS.
T
HE RED DASHED LINE INDICATES THE ASSUMED2%
TYPE‐B
UNCERTAINTY LIMITS. ... 28
F
IGURE4‐4
M
ODEL RESULTS FOR HOURLY VALUES AND SOLAR ELEVATION ANGLE>
15
DEG AND0,65<
KT'<=1.
T
HE BLUE DASHED LINE INDICATES THE ASSUMED5,2%
UNCERTAINTY LIMITS.
T
HE RED DASHED LINE INDICATES THE ASSUMED2%
TYPE‐B
UNCERTAINTY LIMITS. ... 29
F
IGURE4‐5
A
CCURACY VERSUS OPERATIONALITY FOR EACH MODEL... 31
L IST OF TABLES T
ABLE3‐1
V
ERIFICATION OF THE APPLICABILITY OF THEÅ
NGSTRÖM’
SL
AW... 21
T
ABLE4‐1
R
ESULTS FORGHI
MODELLED PER MINUTE STEP.
F
ILTERS:
0,65<K
T’<=1
AND H>15°
ON AT LEAST,
ONE CONTINUOUS HOUR. ... 26
T
ABLE4‐2
R
ESULTS FORGHI
MODELLED PER HOURLY STEP.
F
ILTERS:
0,65<K
T’
<=
1
AND H>15°
AND AT LEAST ONE CONTINOUS HOUR. ... 27
T
ABLE4‐3
ORIGINAL MODEL OUTPUT AND MODIFICATION TO THE MODELS... 30
Introduction
1 I NTRODUCTION
1.1 O N ENERGY AT A WORLD ’ S SCALE
Our societies are facing a growing need for energy since the industrial revolution, and particularly since the Post‐World War II era. The world total primary energy consumption amounted 62’500 TWh in 1971 and 137’500 TWh in 2009 [1]. Energy production, Green House Gas (GHG) emissions and climate change are strongly correlated [2]. The energy sector accounted for 68% of the global GHG emissions in 2009. On current trends, the world is on track for a warming of +6°C by the end of the century. Because our world’s energy system is mostly based on finite fossil resources, the energy price increases along with the growing energy demand. The recent geopolitical events, such as the 2009 gas supply disruption to the EU, showed that the energy security is an increasing concern. From the climate/environmental, economic and energy security point of view, it is now obvious that it is vital to change the world’s energy system.
Despite these clear reasons for a change, investments in fossil‐fuel technologies are still greater than investments to best available sustainable energy technologies [1]. The potential of the various forms of renewable energies have not yet been fully appreciated. By using these technologies, together with changed behaviour, it is acknowledged that the long‐term increase of the mean global temperature can be limited to +2°C [2]. Using sustainable energy technologies would deliver benefits of enhanced energy security and sustainable economic development, while also reducing human impact on the environment.
The sun offers mankind virtually unlimited energy potential. Solar photovoltaic has been the fastest‐growing energy technology over the period 2000‐2011. Installed capacity was equal to 1,5 GW in 2000 and 65 GW in 2011 [3]. This energy’s potential is significant; according to optimistic scenarios [4], solar photovoltaic energy could account for 20% of the global electricity generation in 2060. By disregarding externalities associated with the use of fossil‐
fuel for energy production, solar photovoltaic is very close to being cost‐competitive, especially in sunny countries.
1.2 E NERGY SYSTEM ON THE R ÉUNION ISLAND
La Réunion is a French insular department situated in the Indian Ocean (21° South, 55° East).
It has an area of 2512 km
2and its population equals 840’000 inhabitants. Like most islands,
La Réunion is strongly dependant to fossil fuels (coal, refined gas and diesel) for its
transportation and energy sectors. In terms of primary energy consumption, this
dependency is about 90% (Figure 1‐1 [5]).
Introduction
FIGURE 1‐1 ENERGY SCHEMA OF THE REUNION ISLAND
La Réunion used to be energy autonomous some thirty years ago although the rapid population growth and greater use of air conditioning has led to a bigger use of coal in the electricity production as it accounts for 48,7% in 2012. The island is not interconnected with any surrounding islands. It is also located in a cyclonic area, which imposes more constraints on eg. solar and wind farm designs. In 2002, the regional administration set the objective of energy autonomy, all sectors included, at the horizon of 2030 [6]. This island possesses various exploited renewable energy sources for its electricity production. These are hydro (20,1%); sugar cane biomass (10%); solar photovoltaic, wind energy, biogas (3,7%). There are also projects to exploit wave energy, geothermal energy and ocean thermal energy.
During the last years, the grid operator has been facing a great increase of variable
renewable energy source in its capacity mix, mostly due to solar photovoltaic. As a
consequence, the regulatory limit of 30% of instantaneous production by variable energy
sources has been attained in 2011. Increasing the capacity of variable renewable energy in
the network decreases its stability. The management of the equilibrium
production/consumption is also more complex. Consequently, beyond this limit the grid
operator is authorised to disconnect any photovoltaic plant whose nominal power is greater
than 3 kilowatt‐peak (kWp). As seen on Figure 1‐2, it impacts a small number of installations
but most of the total capacity. The nominal power, in kilowatt‐peak, of a photovoltaic
module is its maximum power. In order to increase this share, the Energy Regulation
Commission (CRE) has published in 2012 a request for quotation making storage and solar
production forecasting compulsory for new solar plants.
Introduction
FIGURE 1‐2 DISTRIBUTION BY POWER RANGE OF THE INSTALLED PV CAPACITY IN 2010 (DARK BLUE = NUMBER OF INSTALLATIONS;
LIGHT BLUE = INSTALLED CAPACITY)
1.3 A N ISLAND WITH GREAT POTENTIAL FOR S OLAR PHOTOVOLTAIC ENERGY
Solar photovoltaic has a great role to play in order to meet the objective of energy autonomy. Its capacity has increased very sharply during the last years (42 MWp installed in 2009; 89 MWp in 2010 [5]). In 2011, the feed‐in tariff and tax incentive dropped so the regional administration decided to promote solar photovoltaic with an “energy cheque”[7].
In 2013, the installed photovoltaic capacity is 153 MW, 90% of which is in the industry sector. Because the space is limited on the island, there are projects where agriculture farm and photovoltaic farm are combined [8].
In order to increase the solar photovoltaic capacity, while keeping power grid stable and service reliable, forecasting and storage will be necessary compounds of future systems at La Réunion. A production prevision will be needed one day before in order to plan the needed production capacity mix. The prediction will be refined up to six hours before and a thirty minute prevision will allow anticipating steep ramp in the production capacity. Storage will be a necessary compound in order to smoothen the production variability of photovoltaic systems; it can also help to transfer energy during the peaks of consumption; and it can help the grid operator to maintain network stability [9].
Introduction
1.4 R EUNIWATT , AN INNOVATIVE COMPANY
This MSc. thesis is carried out with Reuniwatt. Reuniwatt is a young and innovative company, settled in Reunion Island, France, in the Indian Ocean. Reuniwatt’s development is based on three core activities:
Photovoltaic production forecasting
Deployment of networks of communicating smart sensors. Climatic data recording.
Climatic database management
Energy mix expertise
Being deeply involved with the university world and performing R&D on solar forecasting, Reuniwatt has developed, together with the Université de la Réunion, the project Soleka [10]. This project is also laureate 2013 of the “Concours National d’Aide à la Création d’Entreprise de Technologies Innovantes”, a national competition for companies with innovative technologies. Soleka is a tool intended to ease the insertion of variable energy in the energy mix. Soleka will allow forecasting the production of solar photovoltaic plant thirty minutes to one day ahead. This will help the electric grid operator to manage the balance production / consumption and to secure the electricity supply on the whole island. Soleka will also model the photovoltaic production in order for the production companies to size their storage system.
Reuniwatt is developing innovative solutions that will contribute to the energy autonomy of the Reunion Island; and accompany its customers in the realisation of their industrial investments that introduce cleaner processes.
1.5 S COPE : CHOOSING THE MOST APPROPRIATE CLEAR ‐ SKY MODEL
Solar irradiance forecasting methods can be described as the prediction of cloud behaviour over a specific area. This information is then combined with the value of solar radiation under clear‐sky conditions for the same area at the same forecast time. Uncertainty of solar irradiance under clear sky can affect significantly the forecast results accuracy.
Atmospheric radiative transfer models can compute this value. The inputs are varying from one model to another. The simplest models just take into account the solar elevation, while more detailed ones may include other inputs such as concentration of atmospheric components (aerosols, water vapor, ozone), elevation of the site, barometric pressure or ground albedo, in order to better model atmospheric transmittance.
Clear‐sky models (CSM) with many inputs are often the most accurate but their performance
remains very sensitive to local climate conditions. Moreover, availability of some inputs are
not always guaranteed at every locations. The objective of the study is the selection and
performance quantification of the most appropriate CSM for the forecasting of solar
radiation at the Reunion Island.
Introduction to clear-sky model
The study is organized with the following tasks:
bibliography study identifying the most relevant and promising CSMs
CSM input data collection of the specific area.
Assessment of CSM performance by comparing their outputs to instrumental measurements.
2 I NTRODUCTION TO CLEAR ‐ SKY MODEL
2.1 C LEAR ‐S KY ATMOSPHERIC EFFECTS
The extra‐terrestrial solar radiation arriving to Earth is a relatively stable characteristic. Due to the elliptical shape of earth’s orbit, it varies by +/‐ 3% around its average value, the extra‐
terrestrial constant which is equal to 1,367 kW/m
2. This radiation arrives as a beam on top of earth’s atmosphere [11] and it is absorbed and scattered while passing through the different layers of the atmosphere [12]. Absorption means that the energy is taken up by matter while scattering means that radiation is deviated from straight propagation. As seen on Figure 2‐1, up to 30% of the incoming extra‐terrestrial solar radiation is lost by absorption and up to 25% of the total irradiation is scattered back to space [13].
FIGURE 2‐1 CLEAR‐SKY ATMOSPHERIC EFFECTS
Introduction to clear-sky model
The amount of radiation that reaches the ground is called the global radiation. It is composed of two components, the beam and diffuse compound (2‐1). The diffuse compound, a fraction of the total diffused extra‐terrestrial radiation, is between 5 and 22% of the global component [14].
(2‐1)
Absorption occurs from different components at different spectral range. As seen in Figure 2‐2, the ozone has two small absorption band near 290 nm and 600 nm. Water vapour absorbs strongly in the infrared part of the solar spectrum. Carbon dioxide is another strong absorber of infrared radiation.
There are two kinds of scattering effects: Rayleigh‐scattering and Mie‐scattering. The Rayleigh‐scattering is the scattering of electromagnetic radiation by particles which are much smaller than the wavelength of the radiation. The Mie‐scattering is the scattering by particles whose diameter is of about the same dimension as the wavelength or larger.
Rayleigh‐scattering is quite exclusively a function of Air Mass while the Mie‐scattering depends on local conditions. The Mie‐scattering has a strong forward pattern.
The atmospheric gases have a significant absorption band in the visible spectrum. These gases are mostly biatomic oxygen O
2and biatomic N
2, which accounts for 20,95% and 78,09% of the atmospheric gases respectively [15].
FIGURE 2‐2 RADIATION REDUCTION THROUGH ATMOSPHERIC EXTINCTION PROCESSES (SOURCE: C. HOYER‐KLICK: INTRODUCTION TO SOLAR RESSOURCE ASSESSMENT
The amount of scattering and absorbing depends on the length of the path through the atmosphere, which is expressed as the Air Mass (AM). It is simply the ratio of the actual path length to the path length when the sun is directly overhead. As seen in Figure 2‐3, the Air Mass can be calculated from the solar zenith angle.
A photovoltaic cell can only convert a theoretical maximum of 54% of the incident irradiation
[16]. As per Figure 2‐4, 18% of the incident energy in the infra‐red part is lost. The high
Introduction to clear-sky model
energy photons generate heat which leads to another loss on the ultra‐violet and visible band. This loss accounts for 28% of the total incident irradiation. Therefore, the useful incoming irradiance on a photovoltaic cell is varying due to scattering by aerosols and absorption by water vapour.
FIGURE 2‐3 AIR MASS EFFECT
FIGURE 2‐4 LOSSES ON INCIDENT IRRADIATION ON A TYPICAL PV CELL
(SOURCE: TECHNIQUES DE L’INGENIEUR)
2.2 O N THE IMPORTANCE OF USING AN APPROPRIATE C LEAR S KY M ODEL
Solar photovoltaic is the fastest growing source of renewable energy and the greatest project requires considerable financial investments [17]. The knowledge of clear‐sky irradiation reaching the ground is a key parameter in the field of solar radiation modelling and evaluation. Because equity investors evaluates projects based on the return on investment, magnitude of capital cost and perceived risk [18], the solar resource needs to be as accurately characterised as possible as it has a direct impact on the plant design and power output predictions. As an example, reducing the solar resource uncertainty by about 1% can allow the project to take on 1% more debt, and therefore reduce the capital cost of the owner. For a 50 MW project, valued at USD 125’000’000, it means that the loan can be increased by about USD 1’250’000 [18].
R&D is being done at the Réunion Island in order to develop “guaranteed PV systems”, which includes production forecasting and storage. Forecasting the solar resource means that clear‐sky irradiation and cloud coverage need to be estimated as precisely as possible. The clear‐sky index Kc (refer to 3.2 for its definition), which is proportional to the cloud coverage, is predicted by various means and used with a clear sky model in order to predict the incoming irradiation on e.g. a PV farm: the modelled irradiation is the product of the clear‐
sky index by the clear‐sky irradiation. The following Figure 2‐5 and Figure 2‐6 illustrate the importance of using an accurate model: an inaccuracy on the clear‐sky model has the same impact as over or under‐estimating the clear‐sky index. Therefore, the accuracy of the clear‐
sky model used has an impact on the accuracy of the predicted irradiation.
Introduction to clear-sky model
As part of a guaranteed PV system, the size of storage is directly proportional to the error in forecasting thus, reducing the error on clear‐sky irradiation has a positive impact on the storage size (internal communication).
FIGURE 2‐5 ACCURATE AND INACCURATE MODELLED GHI
FIGURE 2‐6 ACCURATE AND INACCURATE CLEAR‐SKY INDEX
Along with the model accuracy, its operability is equally important. If a model is to be industrially used, the high demand of atmospheric input parameter may be expensive and uneasy to use. Therefore, an “appropriate” model depends on its use and can be a compromise between both. The study aims at answering what is the most appropriate model to use for solar forecasting on the Réunion Island, from the accuracy and operability point of view.
2.3 C HOICE OF MODELS TO ANALYSE
There are numerous Clear‐Sky models [19]. The simplest ones aim at calculating the radiation that arrives on a specific location at a given time. Therefore the minimum parameters that are needed by a clear sky model are the solar elevation angle, the date and solar constant. More complexes models can include up to 8 atmospheric parameters, such as the aerosols and the water vapour [20]. Whilst the inputs needed vary from one model to another; the output are either irradiation (Wh/m
2) or irradiance (W/m
2). The GHI (Global Horizontal Irradiance) is given along with the beam (BHI) and diffuse (DHI) compound.
In this study, four models are selected according to three criteria: (i) they must be easily operational (ii) their quality needs to be recognized (iii) it must be already used in large‐scale processes. The four models chosen have all different approaches to modelling radiation.
They are the following:
Model_1: This model is well‐tried and easy to use; it is currently used by research
laboratories and Reuniwatt at the Réunion Island.
Introduction to clear-sky model
Model_2: This model has a good precision with the use of aerosol and water vapour parameters and is also widely used in the solar energy research community.
Model_3: This model has 3 simple inputs data and was used in the of a long term solar radiation database over Europe.
Model_4: This is a very new model. Atmospheric radiation computations are made with less approximation than the others models, and it is usable with an integrated aerosol and water database.
Input data for each model can be classified as: astronomical; geographical; surface data;
meteorological (column integrated); quantities related to atmospheric turbidity. A table summarizing the model input data is available in Appendices I.
2.4 M ODEL _1
The Model_1 is a simplified clear sky model for direct and diffuse insolation on horizontal surfaces. The Model_1 is built from three different codes. In each of these codes, a multi‐
layered atmospheric model is constructed with defined atmospheric parameters. Once the atmosphere is modelled, each code is used to calculate the radiative transfer at a given location and time. This model was written in an excel spreadsheet by its author, and translated into a Matlab script by Reuniwatt.
2.5 M ODEL _2
Model_2 was developed to design a new scheme converting meteorological stationary satellite images into solar irradiance map. In the Model_2, the irradiance is calculated based on radiative transfer models (RTM) and the Lambert‐Beer relation. The main input parameters are the atmospheric water vapour column [cm] and atmospheric aerosol optical depth [cm] [21]. The ozone content is taken constant at 340 Dobson units. The Model_2 is empirical; it is calculated from analytical expressions rather than RTM calculations. In this model, the aerosol optical depth (AOD) is taken for the whole broadband.
2.6 M ODEL _3
The Model_3 was created to build a solar energy atlas at continental scale. The Model_3 is composed by two sets of models, one giving the irradiance and one giving hourly irradiation.
Just like the Model_2, the Model_3 is also used to estimate solar radiation at ground level
from satellite images [14]. The model is based on the Rayleigh optical depth (or thickness)
parameterization and the Linke turbidity coefficient at air mass 2. The input parameter is the
Linke turbidity [22]. This describes the optical thickness of the atmosphere due to both the
absorption by the water vapour and the absorption and scattering by the aerosol particles
relative to a dry and clean atmosphere. It summarizes the turbidity of the atmosphere, and
hence the attenuation of the direct beam solar radiation [23]. The Model_3 model has been
Methodology
checked against the previous model that were used for similar usages and was found the most robust and accurate [14] [24].
2.7 M ODEL _4
Model_4 is a very new fast clear‐sky and fully physical model that replace empirical relations or simpler models used before [25] [26]. It uses the recent results on aerosol properties, and total column content in water vapour and ozone produced by a global mapping initiative of these parameters. The comparison with state‐of‐the‐art clear‐sky models showed that Model_4 surpasses them, especially regarding RMSE and correlation coefficient.
3 M ETHODOLOGY
3.1 G ROUND MEASUREMENTS
In order to quantify the performance of each model, irradiation data during clear‐sky periods is needed at the site where modelling of daily irradiation is performed. Measured and modelled irradiation can be correlated together only they cover the same time periods and location. In this study, data covering 658 days / starting the 19
thAugust 2010 / with a 1 minute resolution, are available for Sainte Marie, in the North of the island (Lat ‐20,8925;
Lon 55,5361). These irradiance data (in [W/m
2]) have been measured with solar reference cell (Figure 3‐1) and converted afterwards to irradiation values (in [Wh/m
2]). The solar reference cell has been calibrated with a thermopile pyranometer on the Reunion Island.
The following Figure 3‐2 shows the irradiation map of these data.
FIGURE 3‐1 SOLAR REFERENCE CELL USED FOR THE GROUND MEASUREMENTS
Methodology
FIGURE 3‐2 IRRADIATION MAP AT SAINT‐MARIE, IN [WH/M2], FROM 19 AUG 2010 TO 07 JUN 2012
3.2 S ELECTION OF CLEAR ‐ SKY MOMENTS
Identifying clear‐sky moments is not an easy task because there is no objective definition of a true clear‐sky moment. There are different methods used to identify clear‐sky moments [27]. These are based on cloud cover and/or sunshine fraction, while others are based on a combination of sky‐clarity indices, or the Linke turbidity. Keeping cloudless sky based on cloud cover is not a good method due to the wide scattering of clouds. Furthermore, these data are not available for this study. A combination of these methods will bring the most accurate results.
In this study, the detection of clear‐sky moments is based on modified clearness indices K
t’.
The clearness index K
tdoes not depend upon a clear‐sky model. It allows characterizing the insulation conditions at a given point of time when only the global component is known. It depends on the extra‐terrestrial irradiance G
0and the solar elevation angle h. The clearness index K
tis the ratio of incoming solar irradiation over the extra‐terrestrial irradiation; it is defined as per [28]. The clearness index is a measure of the transparency of the total atmosphere.
.
(3‐1)
The clearness index K
tdepends on the solar elevation angle and tends to be under‐estimated
for low solar elevation angle. In order to be used as a reliable sky condition descriptor, it can
be corrected with the formula from Perez et al. (1990) [28], which uses the Air Mass AM. The
modified clearness index, K
t’ is defined as:
Methodology
, . ,
, , ,
(3‐2)
The clear sky index k
cis a very usefull parameter for solar forecasting. It is used to represent the clearness of the sky, ie, how much irradiation reaches the ground at a certain time (GHI
site), compared how much it would be if the sky was clear (GHI
cs), it is therefore an indicator on cloud coverage:
(3‐3)
The following Figure 3‐3, Figure 3‐4 and Figure 3‐5 show the different irradiation, indexes, solar elevation angle and Air Mass during a typical day, with a clear‐sky in the morning and clouds forming in the afternoon.
FIGURE 3‐3 EXTRA‐TERRESTRIAL IRRADIATION, MODELLED AND MEASURED IRRADIATION DURING ONE DAY
FIGURE 3‐4 KC, KT AND KT'
FIGURE 3‐5 SOLAR ELEVATION ANGLE H AND AIR MASS (AM). THE DASHED LINE SHOWS FIRST FILTER ON H>15°
Methodology
In order to keep only moments with a clear‐sky, K
t’ values are first filtered for h > 0°. When the sun is below the horizon, the measured GHI may not be equal to 0 but, because AM is calculated with h, it would not be possible to calculate K
t’. Furthermore, K
t’ values for h comprised between 0 and 15° are not kept in order to eliminate all risks of errors caused by shading of obstacles just above the horizon. This has the consequence of systematically excluding the very data point that are usually modelled with the lowest accuracy and therefore, the performance assessment results are on the optimistic side [20] [19]. Then, K
t’ values are filtered for 0,65 < K
t’ ≤ 1 in order to only keep clear sky conditions, as per the Perez et al. method [28]. These limits are arbitrary but coherent with other classifications. At last, only measurements fulfilling these conditions and covering at least one continuous hour are kept.
After following this methodology, 17,3% of the period covered by the 658 days filtered for h
> 15°, are kept and are considered to have clear‐sky conditions. These moments can be seen in red in Figure 3‐6. The incoming irradiation doesn’t have a single value for a given solar elevation angle because it depends on atmospheric parameters, which have a seasonally pattern, as explained in 3.3.
FIGURE 3‐6 CLEAR‐SKY MOMENTS FOR H>15° AND 0,65<KT'<=1
However, there are researchers that oppose the use of clear‐cut K
tvalue for defining clear‐
sky moments because each researcher tend use his/her own values depending on the location and the month of the year [29]. To confirm the pertinent usage of the Perez et al.
method, the RMSE of modelled and measured data was plotted for various K
t’ (Figure 3‐7).
Refer to 0 for explanation on the RMSE index definition. The preliminary results with the Model_4 and a threshold of 0,65 < K
t’ ≤1 are very good. This gives confirmation that the clear‐sky data kept in this study are of good quality and prove the validity of the Perez et al.
method for the clear‐sky measured data.
Methodology
FIGURE 3‐7 RELATIVE RMSE OF MODELLED/MEASURED IRRADIATION VERSUS KT'
3.3 SOURCE OF ATMOSPHERIC PARAMETERS
Irradiance is affected by astronomical, geographical and atmospheric parameters. The first two classes can be obtained easily; they have a big impact on the predicted irradiance but can be known very accurately. On the contrary, atmospheric parameters have large effects.
The Aerosol Optical Depth (AOD) (or thickness) has the greatest impact, and then come Water Vapour (WV). The challenge in getting accurate parameters is that they changes rapidly over both time and space, and are difficult to predict [20].
The aerosol optical depth or optical thickness (τ) is defined as the integrated extinction coefficient over a vertical column of unit cross section. In other words, it is the degree to which aerosols prevent the transmission of light by absorption or scattering of light;
therefore it has no unit. The water vapour is defined as the amount of water which would be obtained if all the water vapour in a specified column of the atmosphere were condensed to liquid; therefore its unit is in cm
3/cm
2or g/cm
2[30]. The Linke turbidity basically describes the attenuation of the solar radiation in terms of a clean and dry atmosphere, and allows for an accurate and efficient way to describe the solar radiation during a cloud free day [31]. It is defined as the number of clean and dry atmospheres that would be necessary to produce the same attenuation of the extra‐terrestrial solar radiation that is produced by the real atmosphere [32].
Atmospheric parameters are obtained thanks to the AErosol RObotic NETwork (AERONET)
database. The AERONET programme is a federation of ground‐based remote sensing aerosol
network established by the NASA, and PHOTONS (Univ. of Lille 1, CNES, and CNRS‐INSU)
[33]. Atmospheric measurements that cover the studied period are retrieved from their
database. They were done in Saint‐Denis, which is just a few kilometres away where the
irradiance measurements were made, thanks to a multiwavelength sunphotometer. 31’224
measurements were made between the May 2007 and October 2012. The downloaded
parameters are of level 2 in terms of quality assurance criteria [34].
Methodology
The Linke turbidity is available online. Monthly climatic values are given for a specific latitude and longitude.
The data available for Saint‐Denis do not include the AOD at 700 nm, which is needed by the Model_2 simplified. This AOD is evaluated thanks to the AOD at 500 and 550 nm [21], thanks to the Ångström’s Law:
( 3‐4)
Where, Τ
λis the AOD at the wavelength λ, β is the Ångström turbidity coefficient. It represents the aerosol vertical quantity in the atmosphere, α is the wavelength exponent. It is linked to the size of particles and varies between 0 (for big particles) and 4 (for aerosol with the size of molecules).
Part of the AOD at 380 and 500 nm, which are needed are also missing. Consequently, these data are inter/extrapolated with the same method. A verification is made when possible to verify the accuracy of this method with the Aeronet data. The following Table 3‐1 validates the inter/extrapolation made to fill holes in the database: the results found are sufficiently good in comparison with the models’ sensitivity to these atmospheric parameters variation.
The Model_1 and Model_2 are fed with daily, monthly average and climatic monthly average atmospheric data. The monthly values are calculated from the daily ones if at least the data 75% of the month. The climatic monthly averages are found by averaging all the data available between 2007 and 2012 for a given month.
TABLE 3‐1 VERIFICATION OF THE APPLICABILITY OF THE ÅNGSTRÖM’S LAW
The Box and Whiskers plots (Figure 3‐8 and Figure 3‐9) show the variation of the water vapour and the AOD at 700 nm between 2007 and 2012. The water vapour shows a clear seasonal pattern, with higher value during the austral summers. The AOD at 700 nm is relatively constant through the years, with very small peaks during the austral springs.
Number of data 18121
Correlation coefficient 0,997
Variance of the difference of the 2 time series 1,4E‐05
mean difference 0,002
Relative mean error (mean difference / mean AOD500) 3,35%
Relative variance (variance / mean AOD500) 0,02%
Number of data 31224
Correlation coefficient 0,995
Variance of the difference of the 2 time series 3,3E‐05
mean difference 0,001
Relative mean error (mean difference / mean AOD380) 0,89%
Relative variance (variance / mean AOD380) 0,04%
Interpolation at AOD500
Extrapolation at AOD380
Methodology
FIGURE 3‐8 WATER VAPOUR BETWEEN 2007 AND 2012 IN SAINT‐DENIS, AERONET DATABASE
FIGURE 3‐9 INTERPOLATED AOD AT 700NM BETWEEN 2007 AND 2012 IN SAINT‐DENIS, AERONET DATABASE
The Linke turbidity is retrieved with on a monthly step. Figure 3‐10 shows the variation of this factor throughout the year:
FIGURE 3‐10 MONTHLY AVERAGE OF THE LINKE TURBIDITY FACTOR AT THE REUNION ISLAND, SOURCE: CONFIDENTIAL
Methodology
The concentration of water vapour in the atmosphere has a strong seasonal pattern, with a peak during the austral summer. This is due to a higher average irradiation which causes more evaporation. The variation of concentration of aerosol in the atmosphere is due to several factors [35]. These are of anthropogenic sources such as agricultural, industrial, air‐
transportation, etc. and of natural sources such as dust storms, sea salt particles, etc. The peak of aerosol concentration at the Réunion Island is due to slash‐and‐burn in Madagascar during the dry season [36].
3.4 ASSESSING MODEL PERFORMANCE
Accuracy indicators are needed in order to compare two times series against each other, such as measured and modelled values. Various indicators are used in the literature [20]
[19]; the most common being the Mean Bias Error (MBE) and the Root Mean Square Error (RMSE).
They are defined as:
∑
(3‐5)
And,
∑
(3‐6)
Where:
n is the number of couples of eg. measured and predicted values,
is the mean value of these measured and predicted values,
is the difference between the measured and the predicted value,
The RMSE of a pairwise difference of two time series can serve as a measure on how far on average the error is from zero. The MBE is the mean difference between the two time series.
The relative RMSE and MBE, in percentage, give a value of the correlation between two time series which is easier to compare with other models.
The correlation coefficient is another useful accuracy indicator. It is a statistical measure of how well the model approximates the real data points. It is defined as:
∑ ∑
∑ ∑ ∑ ∑
(3‐7)
Where, ̅ is the average of the first time series, and is the average of the second one.
Correlograms are also used to visually check the correlation quality between two time series.
A correlogram is a scatter plot with each pair of the time series plotted on X and Y. A higher
Methodology
RMSE value will give a thicker line while a higher MBE will give the line a positive or negative shift.
The more accurate a model is, the smaller the RMSE and MBE are. A MBE can be corrected unlike the RMSE, which is a value intrinsic to the model performance and its input parameters. A “good” value for a RMSE or a MBE depends on different aspects. First the time step chosen for the modelling is very important, modelling irradiation every minute will bring a higher accuracy than modelling it every hour, which involve averaging steps. The filters used have an impact too. The quality of the measured irradiation and the chosen clear‐sky moments is also very important since it is meant to be a reference for the model. In this study, a more stringent filtering on K
t’ allows increasing the quality of the clear‐sky moments kept. The choice of filters on e.g. the solar elevation angle can have a positive impact, as stated in 3.2.
There are different thresholds for defining “good” RMSE and MBE values in the literature.
(Badescu, 2011) defines a good calibration with global hourly irradiation if ‐5%<MBE<5% and RMSE<15%. (Gueymard, 2011) ranks models with a MBE equal to ±2% and RMSE equal to
±5% as best ones. These thresholds are valid for irradiance value and solar elevation angle greater than 15°.
3.5 D EALING WITH GROUND MEASUREMENT UNCERTAINTIES
Assessing a model performance would ideally be done against a reference source of measurement which is error‐free. In reality, both modelled and measured data series are uncertain and must still be compared together. This is why some researchers prefer to use the terms of “difference” with the MBD and RMSD indicators rather than MBE and RMSE, as chosen in this study. Not being able to appreciate these uncertainties can lead to false conclusion. For example, if the mean difference between a model A and the reference measurements are ‐1% whereas it is 3% for model B, it is tempting to conclude that model A is more accurate than model B. It is likely to be exact although the opposite conclusion would have been reached if it has been established that the measurements themselves had a systematic difference of ‐2% compared to the value of the measurand [20].
The measurements used in this study are obtained from a solar reference cell. This sensor is a pertinent tool for measuring irradiance on a solar PV farm since both exhibit a matching response which varies with climatic and astronomic parameters [18]. A solar reference cell measures a spectrally corrected irradiance (i.e. usable fuel for PV systems as per Figure 2‐4).
The cell used in this study, of the solar reference cell gives a broadband irradiance value [37]
which is therefore directly comparable with broadband models. The total measurement
uncertainty is the combination of both the basic calibration and field measurement
uncertainties. As said in 3.2, the solar reference cell has been calibrated for the local light
spectrum with a thermopile pyranometer. The pyranometer has a total expanded
uncertainty (95% confidence interval) of about 4,3% on broadband resource data [38]. The
Model performance
uncertainty of the solar reference cell is known to be below 3% [37]. Therefore, it is now possible to calculate the total uncertainty with the composed uncertainty formula [39]:
2
∑
2( 3‐8)
Thus, the total uncertainty for field measurement is equal to 5,2%. Part of this total uncertainty consists of a systematic type‐B uncertainty. It represents a bias that cannot be corrected in general. In this study, it is assumed to be equal to 2% [20].
4 M ODEL PERFORMANCE
4.1 S UMMARY OF RESULTS
All models were tested with various atmospheric parameters as explained in 3.3, and over different time step, one minute and one hour. Figure 4‐1 illustrates on a relatively clear‐sky day, the 4
thof September 2010, the difference in modelling the GHI by the different models.
Model_3 appears to be the least accurate during that day although it is not when looking at it over many clear‐sky moments. A great number of measurements reduces the influence of anomalies.
FIGURE 4‐1 MEASURED AND MODELLED IRRADIANCE OF ALL MODEL ON A RELATIVELY CLEAR‐SKY DAY
The following Figure 4‐2 is a correlogram showing all models correlations on one scatter:
Model performance
FIGURE 4‐2 CORRELOGRAM OF ALL MODELS (GHI MINUTE) FOR H>15° AND 0,65<KT'<=1
It shows that Model_4 performs very well in this study; it has the thinnest plot. The increasing plot thickness with the GHI is due to the fact that errors are greater on greater values. Model_3 is also performing well, with a small over‐estimation of the irradiation for higher irradiation values. The Model_2 seems to have the thickest plot although it performs better than Model_1 for lower irradiation values. A detail of the correlograms is given in the appendices. Model_4 is used differently from other models. It has input data which are different from other model, which may explain its superiority.
The following Table 4‐1 gives the results for a GHI modelled every minute, and with filters on 0,65<K
t’≤1 and h>15°. The Table 4‐2 is the same as the previous table except that the simulations are made on an hourly basis. The details of results are available in the Appendices.
TABLE 4‐1 RESULTS FOR GHI MODELLED PER MINUTE STEP. FILTERS: 0,65<KT’<=1 AND H>15° ON AT LEAST, ONE CONTINUOUS HOUR.
Model Mean value at site [Wh/m2]
MBE [Wh/m2]
Relative MBE [%]
RMSE [Wh/m2]
Relative RMSE [%]
Correlation coefficient
Number of measurements
Model4 12.30 ‐0.14 ‐1.13% 0.46 3.78% 0.991 67386
Model2_cte 12.30 ‐0.26 ‐2.12% 0.70 5.69% 0.982 67386
Model2_Daily 12.32 ‐0.15 ‐1.25% 0.67 5.42% 0.981 50919
Model2_Monthly 12.30 ‐0.19 ‐1.52% 0.63 5.15% 0.984 67282
Model2_MonthlyClim 12.30 ‐0.22 ‐1.82% 0.65 5.32% 0.984 67386
Model1_cte 12.30 ‐0.56 ‐4.58% 0.77 6.29% 0.988 67386
Model1_Daily 12.32 ‐0.47 ‐3.85% 0.68 5.55% 0.989 50919
Model1_monthly 12.30 ‐0.50 ‐4.09% 0.70 5.68% 0.990 67282
Model1_monthlyClim 12.30 ‐0.55 ‐4.45% 0.74 5.98% 0.989 67386
Model3 12.30 ‐0.19 ‐1.52% 0.53 4.31% 0.990 67386
Model performance
Table 4‐1 shows the number of measurements is in the same range for all models thus their results can be directly compared. All the simulations present a correlation coefficient above 0,98; which can be considered as a very satisfactory value. Model_4 is the most accurate model; it has the lowest relative MBE, the lowest relative RMSE and the highest correlation coefficient. Model_3 comes just behind with values of relative MBE and RMSE close to Model_4. The Model_2 and Model_1, which are used with daily, monthly averaged and climatic monthly averaged atmospheric parameters are coming behind with a relative RMSE above 5%. Using daily parameters does not give Model_1 and Model_2 a significant advantage compared to e.g. Model_3, which uses monthly averaged atmospheric parameters.
It is worth noting that the relative RMSE of Model_2 is not reduced by using daily parameters compared to monthly parameters. Model_1 exhibits contrary trends. This could be due to the fact that the Model_2 is an empirical model and is more sensitive to noise in its input parameters. As explained in 2.4, Model_1 is a radiative transfer model, based on physical equations. The use of varying atmospheric parameters, coming from the Aeronet database, does improve the model accuracy so much compared to the use of constant parameters.
All the models have a negative bias in these simulations, meaning they all over estimate the irradiation.
TABLE 4‐2 RESULTS FOR GHI MODELLED PER HOURLY STEP. FILTERS: 0,65<KT’ <= 1 AND H>15° AND AT LEAST ONE CONTINOUS HOUR.
All models have been tested on an hourly basis for several reasons, one being the calculation time required. Table 4‐2 shows that Model_4 is still the best performing model in terms of relative RMSE and correlation coefficient. Model_3 has a relative RMSE 1% greater although its relative MBE is a little less than 2% better. The Model_2 Simplified, which was slightly better than Model_1 on a minute basis, is now less accurate on an hourly basis. This leads to the fact that ranking of models may depend upon the time step chosen.
All models are less accurate on an hourly basis. It is worth noting that the number of measurements has fallen drastically from the minute step simulations to the hourly ones.
The sampling size (called “number of measurements” in the above table) is an important factor when calculating a model accuracy and when comparing two, or more, models together. As part of a model performance quantification, values which are less accurately
Model Mean value at site [Wh/m2]
MBE [Wh/m2]
Relative MBE [%]
RMSE [Wh/m2]
Relative RMSE [%]
Correlation coefficient
Number of measurements
Model4 752 ‐34.79 ‐4.62% 62.26 8.28% 0.965 2621
Model2_Daily 744 ‐39.95 ‐5.37% 84.95 11.41% 0.925 1875
Model2_Monthly 752 ‐43.40 ‐5.77% 84.72 11.26% 0.929 2609
Model2_MonthlyClim 752 ‐45.71 ‐6.08% 86.34 11.48% 0.928 2621
Model1_Daily 744 ‐54.18 ‐7.28% 77.42 10.40% 0.960 1875
Model1_monthly 752 ‐57.28 ‐7.62% 79.61 10.58% 0.960 2609
Model1_monthlyClim 752 ‐60.02 ‐7.98% 81.68 10.86% 0.960 2621
Model3 752 ‐21.38 ‐2.84% 70.52 9.37% 0.957 2621
Model performance
modelled have a greater impact on the results accuracy if the sampling is small. This effect has been verified when filtering data on more stringent clear‐sky conditions: keeping
“clearer” clear‐sky moments and thus reducing drastically then number of measurements has a negative impact on the model accuracy results.
Another reason why the hourly simulations give less accurate results than the minute ones is because the clear sky moments are calculated using irradiation of site measurements with a minute step which are, on average, considered as a clear‐sky. There could be the case of one hour considered having clear‐sky conditions with short moments with a significant cloud coverage, or longer moments with a very light cloud coverage. On average, the irradiation over one hour can be great enough so that this hour is considered as a clear‐sky. This source of error can potentially lower the quality of the overall clear‐sky moments. When increasing the lower threshold on the clearness index k
t’ to 0,7; all models show better results on both relative MBE, RMSE and on the correlation coefficient. By doing so, the number of measurements does not decrease a lot; consequently there is no negative impact on the results.
The following Figure 4‐3 and Figure 4‐4 shows on a diagram the relative MBE and RMSE as presented in Table 4‐1 and Table 4‐2.
FIGURE 4‐3 MODEL RESULTS FOR MINUTE VALUES AND SOLAR ELEVATION ANGLE > 15° AND 0,65<KT'<=1. THE BLUE DASHED LINE INDICATES THE ASSUMED 5,2% UNCERTAINTY LIMITS. THE RED DASHED LINE INDICATES THE ASSUMED 2% TYPE‐B UNCERTAINTY
LIMITS.
