solar irradiance variability on small spatial and temporal...

Solar irradiance variability on small spatialand temporal scales

Gerald M. Lohmann

Erstgutachter Prof. Dr. Jürgen ParisiAbteilung Energie- und Halbleiterforschung, Institut für PhysikFakultät Mathematik und NaturwissenschaftenCarl von Ossietzky Universität Oldenburg

Zweitgutachter Prof. Dr. Jan KleisslDept. of Mechanical and Aerospace Engineering andUCSD Center for Energy ResearchUniversity of California, San Diego, USA

Drittgutachter Prof. Dr. Adam H. MonahanInstitute for Integrated Energy Systems andSchool of Earth and Ocean SciencesUniversity of Victoria, BC, Canada

Tag der Disputation 18. September 2017

It does not matter how slowly you goso long as you do not stop.

CONFUCIUS

Dedicated to my wife and children, born and unborn;To the salad days of green judgement and cold blood;And to the many great minds of old.

Contents

Kurzfassung viii

Abstract ix

1 Preface 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Irradiance normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Quantifying variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Spatial averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.4 Temporal averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Further research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 Local short-term variability in solar irradiance . . . . . . . . . . . . . . . . . . . . . 111.3.2 Simulating clear-sky index increment correlations . . . . . . . . . . . . . . . . . . . 121.3.3 Effects of temporal averaging on short-term variability . . . . . . . . . . . . . . . . 12

1.4 Available data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.1 Ground-based irradiance measurements . . . . . . . . . . . . . . . . . . . . . . . . 131.4.2 Satellite-derived irradiance and cloud motion vectors . . . . . . . . . . . . . . . . . 131.4.3 Sky imager photographs and ceilometer . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Local short-term variability in solar irradiance 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Measurement campaigns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Clear-sky index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Sky type variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Single-point statistics of clear-sky index . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Two-point correlations of clear-sky index . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Variability in clear-sky index increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Increment statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Two-point correlations of increments . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Variability in spatial averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Simulating clear-sky index increment correlations 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

v

3.2.1 Clear-sky index from pyranometer networks . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Sky imager photographs and ceilometer . . . . . . . . . . . . . . . . . . . . . . . . 363.2.3 MSG-based cloud index and motion vectors . . . . . . . . . . . . . . . . . . . . . . 373.2.4 Mixed sky subset selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Cloud edge fractal dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Cloud shadows and time series generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 Increment spatial correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.6 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Effects of temporal averaging on short-term variability 454.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Material and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Available irradiance datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.2 Data preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.3 Characterizing variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.1 Clear-sky index variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.2 Increment variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.3 Peculiarity in Varennes data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Conclusions 61

Bibliography 63

Acknowledgements 73

Curriculum vitae 75

List of publications 75

Erklärung gemäß Promotionsordnung 77

List of Figures

1.1 Diurnal cycle of irradiance and clear-sky index . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Probability density estimates of clear-sky index . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Kernel density estimates of clear-sky index increments . . . . . . . . . . . . . . . . . . . . 6

vi

1.4 An example of spatial averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Previously proposed models to estimate increment correlation structures . . . . . . . . . . . 91.6 An example of temporal averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Pyranometer coordinates and sensor pair distances for Jülich and Melpitz . . . . . . . . . . 192.2 Examples of different spatiotemporal variability in clear-sky index . . . . . . . . . . . . . . 202.3 Joint probability density function of clear-sky index mean and standard deviation . . . . . . 212.4 Two-point correlations of clear-sky index as functions of sensor pair distance . . . . . . . . 232.5 Statistics of clear-sky index increments for different time lags . . . . . . . . . . . . . . . . . 242.6 Two-point correlations of k∗ increments as functions of sensor pair distance . . . . . . . . . 252.7 Three representative realizations of randomly selected circles within a domain . . . . . . . . 262.8 Standard deviations of k∗ and its increments as functions of averaging area . . . . . . . . . . 27

3.1 Examples of available data sets utilized throughout chapter 3 . . . . . . . . . . . . . . . . . 373.2 Results from cloud edge box-counting analyses . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Example of sythetic fractal cloud shadows and corresponding k∗ time series . . . . . . . . . 403.4 Measured and modelled spatial autocorrelation structures of k∗ increments . . . . . . . . . . 413.5 Model sensitivity to parameter changes and input data variations . . . . . . . . . . . . . . . 42

4.1 Two examples of highly-variable clear-sky index time series . . . . . . . . . . . . . . . . . 504.2 Clear-sky indec distributions for mixed-sky conditions and two averaging times . . . . . . . 514.3 Structures of clear-sky index standard deviation as a function of averaging time . . . . . . . 524.4 As in Fig. 4.3, separately estimated for winter, spring, summer, and autumn . . . . . . . . . 534.5 Distributions of clear-sky index increments for different averaging times . . . . . . . . . . . 534.6 Structures of clear-sky index standard deviation as functions of increment time . . . . . . . 544.7 As in Fig. 4.6, separately estimated for winter, spring, summer, and autumn . . . . . . . . . 554.8 Distribution of Varennes-based single 900-s-block values of σ̂∆k

∗τ=0.1 s

T=0.1 s and an example of apeculiar time series of k∗ and its increments . . . . . . . . . . . . . . . . . . . . . . . . . . 56

List of Tables

2.1 Summary statistics used to visualize data spread throughout chapter 2 . . . . . . . . . . . . 212.2 Increment decorrelation length scales estimated from Fig. 2.6 . . . . . . . . . . . . . . . . . 26

3.1 Key variables used throughout chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Previous studies of short-term variability and the temporal resolutions considered . . . . . . 484.2 Details of the available clear-sky index data sets utilized throughout chapter 4 . . . . . . . . 49

vii

Kurzfassung

Für die erfolgreiche Netzintegration einer stetig steigenden Anzahl von Photovoltaikanlagen (PV) ist eswichtig, die Solarstrahlungsvariabilität auf kleinen räumlichen und zeitlichen Skalen zu charakterisieren.Da es allerdings zeitaufwändig und teuer ist, geeignete Datensätze zu sammeln, waren dieselben bislangnicht verfügbar. Frühere Untersuchungen blieben entweder auf begrenzte räumliche oder begrenzte zeitlicheAuflösungen beschränkt (oder beides). Mithilfe neu gewonnener Daten wird Strahlungsvariabilität im Rahmendieser Dissertation erstmals umfassend auf Skalen zwischen Dekametern und etwa 10 km sowie zwischen0,01 s und 15 min charakterisiert.

Zunächst werden dafür auf Basis gemessener 1 Hz Strahlungsdaten von bis zu 99 Pyranometern Felderdes Clear-Sky Index k∗ (i. e. auf wolkenfreie Zustände normierte Solarstrahlung) nebst entsprechenderInkremente (i. e. Änderungen innerhalb bestimmter Zeiträume) analysiert. Wolkenfreie, komplett bewölkte unddurchbrochen bewölkte Zeitfenster werden anhand eines einfachen Klassifikationsschemas mittels k∗ Statistikidentifiziert, wobei gezeigt wird, dass durchbrochene Bewölkungszustände für kurzfristige PV Fluktuationenam problematischsten sein können. Außerdem wird gezeigt, wie die räumliche Mittelung dieser Fluktuationensich von Bedeckungstyp zu Bedeckungstyp unterscheidet. Während räumliche Autokorrelationsstrukturen(i. e. die bestimmende Einflussgröße räumlicher Mittelung) der k∗ Inkremente sich für wolkenfreie undkomplett bewölkte Zeitfenster im Wesentlichen anhand eines früher aufgestellten Modells bestimmen lassen,gilt das nicht für durchbrochene Bewölkungssituationen.

Nächstfolgend wird ein fraktales Wolkenmodell vorgestellt, um vorgenannte räumliche Autokorrelati-onsstrukturen der Clear-Sky Index Inkremente für durchbrochene Bewölkung besser abzuschätzen. DiesesModell basiert auf satellitengestützen Wolkenbeobachtungen und daraus abgeleiteten Wolkenzugvektoren,sowie Schätzwerten der fraktalen Dimension von Wolkenkanten, die aus mehreren tausend Himmelsfotosund Satellitenbildern stammen (die ermittelten fraktalen Dimensionen liegen zwischen 1,4 und 1,6). DasModell synthetisiert kleinskalige fraktale Wolkenfelder und leitet aus ihrer virtuellen Bewegung über einenSatz Modellpyranometer entsprechende Clear-Sky Index Zeitserien und Inkrementkorrelationen ab. Im Allge-meinen bildet das Modell die aus Messdaten gewonnenen Korrelationsstrukturen sowohl für isotropische alsauch für nach Windrichtung getrennte Analysen gut ab, wobei sich keine starke Sensitivität gegenüber denFraktalmodellparametern ergibt.

Zuletzt wird die Ungenauigkeit der Abschätzung von Strahlungsvariabilität bei durchbrochener Bewölkungdurch zeitliche Mittelung anhand bodengebundener Messungen an sechs verschiedenen Orten charakterisiert.Auflösungsgrenzen zwischen 0,01 s and 1 s erlauben dabei eine Analyse zeitlicher Mittelungseffekte aufbeispiellos kleinen Skalen. Die Ergebnisse deuten auf eine Übergangszeitskala in der Größenordnung von 1 shin, dergestalt dass längere Mittelungszeiten zu einer erheblichen Unterschätzung der Strahlungsvariabilitätführen. Höher aufgelöste Messungen hingegen erhöhen den Aufwand von Datenmanagement und Qualitäts-sicherung, ohne eine merkliche Verbesserung in der Erfassung der Variabilität nach sich zu ziehen. Es gibtdabei keine Hinweise auf nennenswerte Unterschiede zwischen unterschiedlichen Orten oder Jahreszeiten.

viii

Abstract

Characterizing solar irradiance variability on small spatial and temporal scales is important for the successfulgrid integration of increasing numbers of photovoltaic (PV) power systems. However, suitable data sets aretime consuming and expensive to collect, and have not been available in the past. Consequentially, previousstudies have been either restricted to a limited spatial resolution, a limited temporal resolution, or both. Thisdoctoral thesis systematically characterizes variability on scales between tens of meters and about 10 km, aswell as between 0.01 s and 15 min, using new experimental data sets that have recently become available.

First, fields of clear-sky index k∗ (i.e., irradiance normalized to clear-sky conditions) and its increments(i.e., changes over specified intervals of time) are analyzed, using 1 Hz irradiance measurements from asmany as 99 pyranometers. Periods of overcast, clear, and mixed sky conditions are identified by meansof a simple classification scheme based on k∗ statistics, and it is demonstrated that mixed sky conditionsare the most potentially problematic in terms of short-term PV power fluctuations. Also, the character ofspatial averaging is shown to differ considerably between sky types. While spatial autocorrelation structures(i.e., the determining factor of averaging) of k∗ increments mostly resemble the predictions of a simplepreviously-proposed model for overcast and clear skies, this is not the case for mixed conditions.

Next, a fractal cloud model is presented in order to better estimate spatial autocorrelation structures ofclear-sky index increments under mixed sky conditions. This model relies on satellite-derived cloud imagesand cloud motion vectors, as well as box-counting analyses of cloud edges extracted from thousands offish-eye sky camera and satellite images (resulting in best-fit estimates of cloud edge fractal dimensionbetween 1.4 and 1.6). The model synthesizes small-scale fractal cloud fields and translates them across a setof model pyranometer locations to obtain corresponding clear-sky index time series and increment correlationstructures. In general, observed isotropic, along-wind, and across-wind structures are captured well by themodel, both in terms of overall value and shape, and the simulated correlations are not strongly sensitive tovariations in the fractal model parameters.

Finally, the changes in representation of mixed-sky temporal variability resulting from time averagingare characterized using surface irradiance measurements from six different locations. The original temporalresolution limits between 0.01 s and 1 s allow to study temporal averaging effects on variability in clear-skyindex and its increments on unprecedentedly small temporal scales. The results indicate that a temporal aver-aging time scale of around 1 s marks a transition in representing single-point irradiance variability, such thatlonger averages result in substantial underestimates of variability. Higher-resolution measurements increasethe complexity of data management and quality control without appreciably improving the representation ofvariability. The results do not show any substantial discrepancies between locations or seasons.

ix

CHAPTER 1

Preface

This publication-based doctoral thesis evolved from research undertaken at the University of Oldenburg inGermany, and the University of Victoria in British Columbia, Canada, between 2013 and 2017. The resultshave been compiled into three self-contained research articles, which form the basis of the dissertation andare included as-is in chapters 2, 3, and 4, respectively. This preceding chapter 1 provides an overview of thesubject and highlights the key research questions addressed throughout the thesis. Corresponding conclusionsare then drawn in chapter 5.

1.1 Introduction

The installed capacity of photovoltaic (PV) power has drastically increased in many regions of the worldduring the last decade, and it is expected to continue its recent fast-paced growth (e.g. 25 % in 2015 [1]). Fromwell over 200 GW in the beginning of 2016 [2], the world-wide PV capacity is estimated to multiply morethan tenfold until 2030, with capacity projections ranging between 3 TW and 10 TW [3]. As a consequence,the challenges associated with the inherent volatility of PV power production will considerably increaseas well [4, 5]. These challenges include the correct estimation of a PV system’s yield [6], the properdimensioning of energy storage [7], the balancing of generation and load [8], as well as the maintenance ofpower quality, such as voltage and frequency stability [9]. As PV power variability is primarily determined byweather-induced heterogeneity in solar irradiance fields [10], a comprehensive data-driven characterization ofirradiance variability is key to the planning and reliable operation of future power grids and their correspondingsubsystems.

Variability in both irradiance and irradiance increments (i.e., changes over specified intervals of time)are of interest in this context. On the one hand, variability in irradiance itself primarily affects the yield of aPV system and the dimensioning of energy storage. On the other hand, variability in irradiance incrementsimpacts the balancing of generation and load as well as the maintenance of power quality. Depending onthe dimensions of the power grid and PV capacity in question, relevant irradiance variability can span abroad range of spatio-temporal scales, from seconds and meters up to days and hundreds of kilometers [11].There is an ongoing need to understand the biases in representation of temporal variability resulting fromtemporally coarse-resolution observations [12, 13], as well as how spatial averaging (as would come fromhaving distributed PV over a region) mitigates variability [14, 15]. This is especially true for small sub-minuteand sub-kilometer scales, which have only begun to receive increased scientific attention during the last couple

1

2 Solar irradiance variability on small spatial and temporal scales

of years, and for which comprehensive data sets are time consuming and expensive to collect, and althoughneeded have not previously been available [10, 11, 16].

1.2 State of the art

There is a large body of literature touching on various aspects of irradiance variability and its underlyingprocesses. Recent topics of interest have included, for example,

• analyses of power spectra of PV systems and solar irradiance [13, 14, 17–22],

• comparisons of power fluctuations from specific PV plants with corresponding irradiance measurements[20, 23–25],

• characterizations of power variability as a function of PV plant size [25–28],

• developments of methods to infer irradiance and PV power estimates from images taken by fish-eye skycameras [29–34],

• estimations of spatial smoothing, correlation structures, and decorrelation length scales of irradianceand PV power, as well as their increments [10, 12, 16, 23, 35–41], and

• considerations of temporal averaging effects and differences in temporal variability of irradiance andPV power on time scales ranging from seconds to hours [9, 42–45].

On the one hand, early ground-based analyses characterized single-site irradiance on time scales ranging fromhours to months (e.g., [46, 47]), while later studies were often geared towards increasing temporal resolutionlimits between about 10 min and 1 s (and occasionally, even down to 0.01 s), but with strongly confined spatialcoverages. On the other hand, satellite-derived irradiance data were utilized for the analyses of large spatialscales on the order of tens of kilometers and more, but with restricted temporal resolution limits betweenabout 15 min and 1 h. Therefore, previous studies were either restricted to a limited spatial resolution, alimited temporal resolution, or both. For example, the few studies that were based on high-resolution 1 Hz PVpower observations only had measurements from a maximum of six PV systems at their disposal [20, 28].Though irradiance measurements with this high temporal resolution have on occasion been conducted withmore sensors, the spatial coverage has remained strongly confined (e.g., up to 45 sensors spread across∼ 2.5 km2 [26]). Some studies have used artificially generated data to overcome these restrictions, either bysimulating simple cloud shapes [23, 35], or by constructing virtual networks based on time shifted singlesensor measurements [16, 41]. However, these simulated data sets do not necessarily coincide with reality.

Despite the above-mentioned deficiencies in terms of spatial and temporal resolutions of previouslyavailable data sets, a strong knowledge base has been created concerning irradiance variability and correspond-ing methods of data analysis, which are relevant in connection with this thesis. These consist of commonprinciples of irradiance normalization, variability quantification with consideration of changes in irradiance,as well as means to characterize spatial and temporal averaging effects. The following subsections concicelyrecapitulate the essentials of these issues and summarize previous findings from the relevant literature in eachcase.

Chapter 1: Preface 3

1.2.1 Irradiance normalization

In general, the available global horizontal irradiance (GHI) at any given point on Earth’s surface is subject toinfluences from both astronomical and atmospheric processes. As for the former, the apparent movement ofthe Sun relative to Earth gives rise to diurnal and seasonal variations in GHI. These variations are accuratelypredictable and not large on short time scales of seconds or minutes. On the other hand, weather-relatedcontributions to irradiance variability are manifold and complex, and present on all time scales. For instance,growth, motion, and decay of clouds can affect the seasonal cycle in GHI (e.g., winter tends to be cloudierthan summer in mid-latitude low-lying land), and the rapid succession of sunlight exposure and cloud shadowin conditions dominated by fair-weather cumulus generates stochastic variability on short time scales ofseconds or minutes [48]. The presence of different kinds of cloud (e.g., layer vs. convective) at differentaltitudes and of different composition (e.g., high-albedo small cloud droplets vs. low-albedo large droplets inrain clouds) result in a complex set of influences on GHI over a broad range of time scales.

In order to distinguish the cloud-induced fluctuations from the slowly evolving, astronomically determinedapparent motion of the Sun, a GHI time series G at a particular location is typically normalized to either theextraterrestrial solar radiation Gextra (i.e., irradiance on Earth if there were no atmosphere), or the clear-skyradiation Gclear (i.e., irradiance on Earth with a cloud-free atmosphere). Knowing these, the clearness index1

k =G

Gextra(1.1)

and clear-sky index

k∗ =G

Gclear(1.2)

can be defined and estimated, respectively. The extraterrestrial solar radiation Gextra depends only on astro-nomical relationships, whereas a calculation of clear-sky radiation Gclear requires parameters of atmosphericconditions, such as typical water vapor concentration and aerosol load. As the exact characteristics of Gclearare model dependent, there is no single unique clear-sky index, and the use of any clear-sky model introducesnew uncertainties to the k∗ time series (which are absent from the original irradiance measurements).

In spite of these uncertainties, use of the clear-sky index is convenient because of the pronounced non-stationarity of the irradiance time series, and many authors have favored the clear-sky index over the clearnessindex because of the inherent consideration of basic non-weather-related atmospheric influences of the former[15]. Other sub-daily variations, especially those caused by changes in light scattering with solar angle α, areideally removed entirely from k∗ (although depending on how accurately the atmospheric conditions and theirvariability are actually estimated, such changes with α may still affect the clear-sky index to a minor degree).In practice, relatively low values of GHI occurring shortly after sunrise and just before sunset, coupled withpath prolongation and corresponding higher uncertainties in clear-sky irradiance calculations at these times,can result in unrealistic values of k∗ [27]. For this reason, data associated with low solar elevation angles(e.g., α < 15°) are typically removed from the record when estimating clear-sky index time series.

Figure 1.1 presents an example time series of measured global horizontal irradiance, simulated clear-skyirradiance [49], and estimated clear-sky index as per Eq. (1.2) in panels (a) and (b). Several typical features ofthe clear-sky index are evident in the time series. For example, the lowest clear-sky index values are typicallynot zero, because even the darkest of clouds do not attenuate all irradiance. Additionally, the upper limit can

1also called transmission coefficient [46]


200

600

GH

I [W

m−2

]

(a)0.

20.

61.

01.

4

k* [−

]

(b)

08:00 10:00 12:00 14:00 16:00 18:00

Time of day [UTC]

−0.

50.

00.

5

10 s

∆k τ

* [−

]

(c)

Measurement Clear−sky reference

Figure 1.1: (a) An example of a diurnal cycle of measured global horizontal irradiance (GHI) alongwith simulated clear-sky irradiance. (b) A corresponding clear-sky index estimate k∗ according toEq. (1.2). (c) Clear-sky index increments ∆k∗τ for time steps of τ = 10 s, calculated according toEq. (1.5). The data was measured with 1 Hz temporal resolution in Jülich on April 25, 2013 and hasbeen conditioned to exclude times of low solar elevation angles α < 15° after sunrise and beforesunset.

exceed 1, primarily due to short-term reflections from the sides of clouds (and also to a secondary degreedue to the limitations of clear-sky models). This phenomenon, known as cloud enhancement, can causesingle-point GHI to exceed its corresponding clear-sky irradiance value by more than 50 % on short timescales (on the order of minutes and shorter) under broken cloud conditions [13, 50–54].

On these short time scales, the probability density function of clear-sky index time series is typicallybimodal in nature, with the two peaks respectively corresponding to cloud-covered and cloud-free states[55–59]. Figure 1.2 correspondingly shows two estimates of the clear-sky index probability density functionbased on a histogram of the example time series shown in panel (b) of Fig. 1.1, as well as a kernel densityestimate (KDE) [60] of all available data from that sensor (about 4.5 · 106 s worth of 1 Hz data). While thehistogram does not represent a perfectly bi-modal distribution due to the limited number of data availablefrom the single-day example, the kernel density estimate clearly illustrates the two peaks corresponding tocloud-shadow coverage and sunlight exposure.


Clear−sky index [−]

Den

sity

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

2.5

HistogramKDE

Figure 1.2: Probability density estimates of clear-sky index k∗ based on the single-day example timeseries previously shown in Fig. 1.1 (histogram, black lines), as well as a considerably longer seriescontaining a total of about 4.5 · 106 s worth of 1 Hz measurements (kernel density estimate, red line).The KDE was derived using Gaussian kernels with a smoothing bandwidth (i.e., smoothing kernelstandard deviation) of 0.015.

1.2.2 Quantifying variability

The statistical moments associated with a given clear-sky index time series of length T , such as the samplearithmetic mean

k∗ =1

T

T∑

t=1

k∗(t), (1.3)

and the sample standard deviation

σk∗

=

√√√√ 1T − 1

T∑

t=1

(k∗(t) − k∗)2 (1.4)

can be used to characterize the time series’s variability to a degree. However, these measures are independentof the observations’ ordering in time and, thus, do not quantify how quickly the values of k∗ can change(i.e., randomly shuffling all clear-sky index data in time will affect neither the respecitve probability densityfunction nor the corresponding moments). Instead, characterizations of clear-sky index variability can begeared towards changes over specified intervals of time τ by deriving statistics of k∗ increments2

∆k∗τ (t) = k∗(t+ τ) − k∗(t), (1.5)

2also called ramp rates [27], step changes [41], or simply changes [16]


−1.0 −0.5 0.0 0.5 1.0

1e−

031e

−02

1e−

011e

+00

1e+

01

Clear−sky index increment [−]

Den

sity

τ = 5 sτ = 10 sτ = 60 s

Figure 1.3: Kernel density estimates of clear-sky index increments, based on about 4.5 · 106 s worthof 1 Hz data. The estimates were derived for increment time steps of 5 s, 10 s, and 60 s using Gaussiankernels with smoothing bandwidth of 0.0075.

which are a useful measure of intermittency [61]. The bottom panel (c) of the previously discussed Fig. 1.1shows an example of such a clear-sky index time series for a time step of τ = 10 s.

The probability density functions of single-point clear-sky index increments generally exhibit a narrowcentral peak (corresponding to a high probability of relatively small increments), surrounded by broad tails inwhich the probabilities slowly decrease. For increasing increment time steps, the numbers of high-magnitudechanges increase as well, leading to more pronounced tails of the corresponding distributions [19]. Thesegeneral characteristics are illustrated for τ = 5 s, τ = 10 s, and τ = 60 s by means of kernel density estimatesin Fig. 1.3, using the same data as for the KDE in Fig. 1.2, i.e., about 4.5 · 106 s worth of 1 Hz clear-sky indexdata.

Comparisons of averaged increment distributions of multiple pyranometers (or PV plants of differentcapacities) indicate that the numbers of high-magnitude changes and, thus, the width of the incrementdistribution are reduced for an increasing number of sensors (or larger PV plants) [25, 26, 28]. However,the presence of higher-than-normal (i.e., compared to a Gaussian distribution) probabilities of relativelyhigh-magnitude increments remains typical on all scales, with changes of up to tens of standard deviationsaway from the mean being recorded on a regular basis [38, 62].

Changes in increment variability are reflected by changes of the corresponding standard deviation σ∆k∗τ

(defined analogous to Eq. (1.4)), which offers a convenient way of characterizing typical excursions from themean. When considering averaged k∗ time series accross multiple pyranometers, the standard deviation ispractical to use because it will change as n−0.5 for a number of n uncorrelated locations, irrespective of the


0.2

0.6

1.0

1.4

k* [−

]

(a)(a)

08:00 10:00 12:00 14:00 16:00 18:00

Time of day [UTC]

−0.

50.

00.

5

1 m

in ∆

k τ* [

−]

(b)(b)

Single sensor Spatial average

Figure 1.4: An example of spatial averaging: (a) the same 1 s single-sensor clear-sky index k∗ timeseries as in Fig. 1.1 along with a corresponding spatially averaged 1 s clear-sky index computedover as many as 99 pyranometers dispersed over an area of about 80 km2; (b) 1 min clear-sky indexincrements ∆k∗τ derived according to Eq. (1.5) for both the single-sensor time series and the spatialaverage.

distribution of the data. For these reasons, the increment standard deviation has become a standard measure toquantify variability in irradiance, clear-sky index, and PV power output (e.g. [10, 11, 15, 63]). Note, however,that the standard deviation does not necessarily capture the size of extreme fluctuations appropriately, dueto the non-Gaussian character of k∗ increment statistics. For example, the widely used three-sigma rule ofthumb, according to which a range of ± 3σ∆k∗τ around the mean would cover 99.73 % of the values if ∆k∗τwere normally distributed [64], can be misleading when applied to k∗ fluctuations.

1.2.3 Spatial averaging

There are substantial differences between the variability characteristics of single utility-scale PV plantscovering relatively small areas, and a large number of distributed systems with similar total capacities, butspanning relatively large areas [63]. In lieu of suitable PV power data, averages of clear-sky index incrementscan provide an estimate of the output variability of an ensemble of PV installations at multiple locations [10],and a corresponding example is shown in Fig. 1.4. Single-sensor clear-sky index estimates are contrasted withthe spatially averaged values of up to 99 synchronized pyranometers dispersed over an area of approximately80 km2 in panel (a), and 1 min increments are shown for both the single sensor and the spatial average inpanel (b). Pronounced spatial smoothing is evident in both panels.


As a characterization of the spatial structure of ∆k∗τ fields, it is useful to consider spatial two-pointcorrelation coefficients between locations i and j

ρ∆k∗τij =

T∑

t=1

(∆k∗τ,i(t) − ∆k∗τ,i)(∆k∗τ,j(t) − ∆k∗τ,j)√√√√

T∑

t=1

(∆k∗τ,i(t) − ∆k∗τ,i)2T∑

t=1

(∆k∗τ,j(t) − ∆k∗τ,j)2, (1.6)

which govern the process of spatial averaging3. In Eq. (1.6), ∆k∗τ,i(t) and ∆k∗τ,j(t) are the two individual

increment time series at the two locations i and j, while ∆k∗τ,i and ∆k∗τ,j are the corresponding arithmetic

means (computed as in Eq. (1.3)), and the quantity T denotes the number of data points in the two time series.These structures can be used, for example, to predict power variability of large and/or distributed PV

systems based on a single pyranometer, using a wavelet approach [24, 26, 39, 65]. Increment correlations(both in space and time) have been shown to depend on effective cloud speed [10, 63] and different dailyvariability categories [39, 41]. Moreover, correlation coefficients can depend on the orientation of a sensorpair relative to the direction of cloud movement [35, 37]. For along-wind correlations, the presence of negativepeaks in the increment correlation structures has been suggested [16, 23, 37, 41]. For all quantities andmethods considered, increment correlations at different locations have been shown to decrease with increasingdistance, with a smaller rate of decrease (longer decorrelation distances) for larger time increments.

Several models have been proposed to predict the behavior of increment correlation as a function ofdistance for specific temporal scales, either by using empirical fits to measured data [10, 23, 38, 39, 41, 66, 67]or by modeling simplified binary cloud shapes as unions of randomly positioned discs [35]. For example,hourly satellite-derived irradiance data from three different locations in the United States were used to analyzetwo-point correlations of k∗ increments as a function of pair distance, with samples being separated in therange of 10 km ≤ dij ≤ 300 km [10]. Taking increment time lags of 1 h ≤ τ ≤ 4 h and estimated relativecloud speed v into account, a model

ρ∆k∗τij = (1 +

dijτ · v )

−1 (1.7)

[10] was proposed that implies a linear relationship between pair distance dij and time lag τ for fixed valuesof the correlation coefficient. Similar implications were made by other proposed models, for example

ρ∆k∗τij = exp(

dij · ln(0.2)1.5 · τ · v ) (1.8)

[66] and

ρ∆k∗τij = exp(−

dij0.5 · τ · v ) (1.9)

[23, 24]. These Eq. (1.7), (1.8), and (1.9) were obtained as curve fits from different data with relatively coarsespatio-temporal resolutions and are compared to each other in Fig. 1.5. Although the exact shapes of thestructures vary, they are all based on an assumed linear relationship beteen pair distance and increment timestep. An initial verification of this relationship has also been suggested for smaller spatio-temporal scalesbelow 10 km and 15 min, based on virtual pyranometer networks (i.e., single sensor measurements shifted intime) with temporal resolutions of the single-point measurements being as low as 20 s [16].

3i.e., the standard deviation of the sum of two random variables x + y with correlation coefficient ρxy is σx+y =√σx2 + σy2 + 2ρxyσxσy


0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Distance / increment time step [km/h]

Cor

rela

tion

coef

fcie

nt [−

]

Eq. (1.7)Eq. (1.8)Eq. (1.9)

Figure 1.5: Previously proposed models to estimate the two-point correlation coefficient ρ∆k∗τ

ij ofclear-sky index increments based on relatively large-scale satellite-derived data. The models implya linear relationship between distance and increment time scale, and the structures are shown as afunction of the former over the latter for an assumed cloud speed of v = 5 ms−1.

While the the afore-mentioned empirical fits were always based on data sets of limited spatio-temporalresolutions, the simplified binary cloud shapes considered by the more theoretical model [35] did not accountfor the complexity encountered in real cloud shapes. Other methods used to simulate simple cloud shapesinclude, for example, randomly positioned and subsequently blurred squares (employed to document thespatial correlation structures’ strong dependence on cloud speed [23]), and fractal models [68, 69]. Thefractal approach has been applied to simulate irradiance and PV power time series for single points and spatialaverages, without considering model-derived spatial correlations of increments [68, 69]. As for appropriatecloud edge fractal dimensions, previous studies have suggested the presence of a sub-kilometer scale break inbox-counting dimension [68] and cloud area-perimeter relation [70, 71]. However, these studies were basedon relatively limited data sets (11 sky photographs [68] and single high-resolution Landsat satellite images[70, 71]). Moreover, other studies have presented evidence for the absence of such a scale break, arguing forunified scaling in atmospheric variability (e.g., [72–74]).

1.2.4 Temporal averaging

As indicated above, ground-based solar irradiance observations have often been averaged using a range ofdifferent temporal resolution limits (from hours to fractions of a second), and temporal averaging on timescales larger than minutes has been shown to introduce considerable smoothing to the clear-sky index, and to


0.2

0.6

1.0

1.4

k* [−

]

(a)

08:00 10:00 12:00 14:00 16:00 18:00

Time of day [UTC]

−1.

00.

01.

0

10 m

in ∆

k τ* [

−]

(b)

1 min averages 10 min averages

Figure 1.6: An example of temporal averaging: (a) Averages of 1 min and 10 min temporal reso-lution limits based on the 1 s single-sensor clear-sky index k∗ time series shown in Fig. 1.1. (b)Corresponding 10 min clear-sky index increments ∆k∗τ derived according to Eq. (1.5).

affect its probability distribution [55, 59]. Figure 1.6 presents examples of such temporal averaging, basedon the example period employed in the previous figures, using two different averaging periods of 1 min and10 min. Panel (a) shows the clear-sky index time series for these averages, while panel (b) illustrates theaveraging effect on 10 min increments in the clear-sky index. The misrepresentation of variability is evidentfor the longer averaging time in both panels.

In general, the larger the panel-covered area of a PV system, the less variable its power output is comparedto a single-point irradiance measurement [25], especially on short sub-minute time scales [20, 28]. Withregards to PV power applications, there is no consensus as to the proper temporal resolution of irradiancemeasurements needed to capture all relevant variability. When considering many inter-connected PV systemsin a relatively large area, e.g., all of Europe, spatial averaging appreciably reduces the necessary temporalresolution limit of data (the European Energy Exchange, for example, uses 15 minute time steps for electricitytrading). At the other end of scale, a single, relatively small 48 kWp PV plant has been shown to feature powerfluctuations of up to ± 50 % from one second to the next, and changes of more than 90 % within 20 s [28].Thus, high temporal resolution limits on the order of seconds and shorter may not be required to monitor largeutility-scale PV plants [75], while minute-averaged data may be too coarsely resolved [25]. When consideringsmaller roof top PV systems and/or partial shading (which can strongly reduce an inverter’s power output assoon as a few connected modules are shaded [76]), previous research as shown that the temporal resolutionlimit needed to capture irradiance variability on all time scales may, however, be as small as 0.1 s [13, 77] or0.4 s [12].


Those studies which determined the need for sub-second resolution limits were based on

1. using the 2nd temporal derivative of irradiance as a measure for instantaneous variation, and definingthe daily minimum (negative) value of this derivative as each day’s strongest instantaneous irradiancevariation, and finally calculating a hypothetical optimal averaging time based on an acceptable error of10 Wm−2 and an assumed parabolic shape of the variation for each of a few hundred days in spring andsummer [13];

2. analyzing the sample standard deviation of irradiance as a function of averaging time measured during7 hours on a single summer’s day [77]; and

3. separately studying two variability metrics as functions of averaging time for 7 selected days [12].

In addition to these methodical differences and relatively short data sets, each study was limited to a singlegeographic area, namely Southern Norway [13], Southern Finland [77], and Eastern Canada [12].

1.3 Further research questions

In the light of the above-mentioned findings and limitations of previous studies, the degree of smoothingintroduced to the aggregated power output variability of dispersed PV systems remains one of the majortopics in current solar variability research, predominantely on small sub-minute and sub-kilometer scales.Previous studies have indicated that positive correlation coefficients considerably influence the smoothing ofPV variability on these spatio-temporal scales [e.g., 16, 37], while, at the same time, the processes on theserelevant scales have only been characterized to a minor degree [11, 15].

New radiation data sets of unprecedented spatio-temporal coverage have recently become available (cf.Sec. 1.4), which allow for the detailed characterization of small-scale irradiance variability on scales downto tens of meters and one hundredths of a second, respectively. These new high-resolution irradiance fieldmeasurements constitute a novel feature of this thesis and allow for a number of corresponding researchquestions to be attended to for the first time. In the following list, these questions are subdivided into therespective thesis chapters in which they are addressed.

1.3.1 Local short-term variability in solar irradiance

In chapter 2, field variability of clear-sky index and sub-minute k∗ increments is analyzed for distancesbetween tens of meters and about 10 km, using 1 Hz data recorded by as many as 99 pyranometers. Thefollowing research questions are addressed in this context:

1. Can different sky types be identified and linked to distinct short-term variability conditions?

2. How do these sky types affect small-scale variability, and what are the effects of spatial averaging?

3. To what degree are existing models suitable to simulate the sky types’ respective spatial structures ofincrement correlations?


1.3.2 Simulating clear-sky index increment correlations

In order to capture the influence of broken cloud fields on increment correlation structures, a fractal cloudmodel is proposed in chapter 3, entailing a characterization of the fractal properties of observed cloud edges.In the course of this analysis, the following specific questions will be attended to:

1. What are the fractal dimensions of cloud edges under mixed sky conditions?

2. Can previously suggested sub-kilometer scale breaks in cloud edge fractal dimension be observed usingboth ground-based and satellite observations?

3. Do synthetic clouds with fractal edges capture the structure of increment correlations, includingdifferences in along-wind and cross-wind directions?

1.3.3 Effects of temporal averaging on short-term variability

Chapter 4 is devoted to the systematic characterization of the effects of temporal averaging on the repre-sentation of short-term variability in solar radiation, with a specific focus on mixed sky conditions. Thecorresponding key research questions are:

1. Are clear-sky index variability and its increment variability affected differently?

2. What temporal resolution is necessary to capture relevant short-term variability?

3. Are there discrepancies between locations or seasons?

1.4 Available data sets

The data basis utilized throughout this thesis includes three different types of data, namely,

1. seven high-resolution multi-sensor irradiance data sets from four distinct regions on the NorthernHemisphere, measured near

(a) Jülich, Melpitz, and Oldenburg (Germany),

(b) Alderville and Varennes (Eastern Canada),

(c) Oahu (Hawaii), and

(d) Tucson (Southern USA),

2. cloud index maps and cloud motion vectors for the German locations, based on data from the SpinningEnhanced Visible and InfraRed Imager (SEVIRI) on board the Meteosat Second Generation (MSG)space craft, and

3. red-blue-ratio maps for cloud detection derived from photographs of a fish-eye sky imager and cloudbase height (CBH) estimations from a ceilometer, both co-located with the pyranometers near Jülich.

Among these data, the first-mentioned sets of ground-based irradiance measurements – and, especially, thedata from Jülich and Melpitz – are of particular importance because the likes of it have not been availableuntil recently.


1.4.1 Ground-based irradiance measurements

The data from Jülich (50.9° N, 6.4° E) and Melpitz (51.5° N, 12.9° E) originate from the HD(CP)2 Observa-tional Prototype Experiment (HOPE) [78]. The Jülich campaign took place from April 2 through July 24, 2013and featured a total of 99 EKO ML-020VM pyranometers deployed over an area of about 80 km2. The Melpitzcampaign lasted from September 3 until October 14, 2013, and saw deployment of 50 pyranometers, all ofwhich had previously been used in the Jülich campaign, on an area of about 4 km2. For both campaigns, thepyranometers were set to sample irradiance at 0.1 s intervals, which was averaged to 1 s during postprocessing[79]. With the exception of chapter 4, which utilizes the original 0.1 s samples from a five-sensor subset ofthe Jülich campaign, all available 1 s data from both Melpitz and Jülich are used throughout this thesis.

In Oldenburg (53.2° N, 8.2° E), three EKO ML-01 pyranometers have recorded 0.1 s irradiance samplessince April 28, 2015 as part of an in-house weather observation system of the Energy Meteorology Group inthe Energy and Semiconductor Laboratory of the Institute of Physics at the University of Oldenburg. Thesensors are horizontally mounted on a university building roof top with inter-sensor distances of about 15 m,and subject to bi-weekly maintenance, including verification of the horizontal orientation and cleaning of theglass dome. The quality of the measurements was verified for randomly selected periods using additionalmeasurements from a co-located well-established thermopile pyranometer. Data from May 1, 2015 throughDecember 31, 2016 are used in chapter 4 of the thesis.

The Canadian data from Alderville (44.2° N, -78.1° E) and Varennes (45.6° N, -73.4° E) were measuredusing LI-COR LI-200S pyranometers, with samples being taken every 0.001 s, and their averages evaluatedevery 0.01 s. Gagné et al. [12] provide detailed information about the locations, data aquisition units, and thelocal character of irradiance variability in Alderville and Varennes. In chapter 4, a subset of three sensorsfrom Alderville (providing data from January 2015 through March 2017), and two sensors from Varennes(with data from October 2015 through November 2016) are utilized. The inter-sensor distances are on theorder of 100 m each, and the two locations are about 400 km apart.

Near Kalaeloa Airport on Oahu (21.3° N, -158.1° E), the National Renewable Energy Laboratory (NREL)performed a measurement campaign using 17 LI-COR LI-200 pyranometers from March 2010 thoughNovember 2011, collecting irradiance data every 1 s [80]. This data set has previously been used and analyzedin several irradiance variability studies [e.g. 37, 40, 62, 81, 82]. Measurements from the six-pyranometersubset located on the premises of Kalaeloa Airport (each separated by a few 100 m) are used in chapter 4.

The available data set from Tucson (32.1° N, -110.8° E) was measured between April 5 and June 30, 2014with a temporal resolution limit of 1 s. While the corresponding measurement campaign featured differentphotodiode-based sensors, a subset of five identical Apogee SP-212 pyranometers is used in chapter 4, so thatall utilized Tucson data originate from the same type of sensor. The inter-sensor distances of the subset are onthe order of 100 m. Lorenzo et al. [83] have previously documented details of the entire data set, and used itto evaluate solar irradiance forecasts.

1.4.2 Satellite-derived irradiance and cloud motion vectors

Based on quarter-hourly SEVIRI measurements on board of the MSG satellite, cloud index estimates and cloudmotion vectors have been routinely processed and archived since 2004. The cloud index relates measuredreflectance to approximated surface albedo values and takes values between −0.2 (completely clear) and1.2 (completely cloudy). The data processing used to calculate cloud index images is presented in detail byHammer et al. [84]. Additionally, cloud motion vectors were estimated by matching similar cloud patterns


found in two subsequent cloud index images, as explained by Kühnert et al. [85]. In chapter 3, the originalcloud index product (with an irregular spatial resolution limit of about 1.2 × 2 km2 at the HOPE campaigns’locations) is used along with resampled images featuring a regular grid of 500 × 500 pixels with an edgelength of 2.5 km per pixel (centered over the Jülich and Melpitz campaign, respectively), and correspondingcloud motion vectors.

1.4.3 Sky imager photographs and ceilometer

During the daylight hours of the HOPE Jülich campaign, a sky imager (a digital CCD camera with fish-eyelens [86]) took hemispheric photographs every 15 s, realizing a resolution of 2592 × 1744 pixels and a 183°field of view. For the purpose of cloud detection, the color triplets (red, green, blue) of the original imagewere converted to values of red-blue-ratio r = red · blue−1, where low values indicate clear pixels and highvalues cloudy ones. Estimates of cloud base height were additionally available from a ceilometer located inthe immediate vicinity of the sky imager. The ceilometer’s original single-point measurements were recordedevery 20 s and then smoothed using a median filter [87] with a 600 s window. Using these smoothed CBHestimates, the red-blue-ratio images were undistorted and projected onto a Cartesian coordinate system, onwhich the values were resampled on a regular grid.

The images are considered in chapter 3, and they are essentially the same as those used by Schmidt et al.[34] (who also detail all of the necessary steps to convert the fish-eye sky images to geolocated maps ofred-blue-ratio values) with the exception of the following changes:

1. generating a higher-resolution final image of 500 × 500 pixels with an edge length of 10 m per pixel,

2. masking all areas where the required horizontal pixel size of 100 m2 is exceeded in the original image(either due to distorion by the fish-eye lens or high CBH values), and

3. masking a 25° angular distance around the position of the sun, if the mean pixel intensity (i.e., grayscale value in the range between 0 and 255) in this area indicates an unobstructed view of the sun. Thisdetermination is made if the gray scale exceeds a specified threshold (taken to be 180).

These conservative modifications of the processing procedure ensure high-resolution red-blue-ratio maps andminimize cloud shape deformation as well as misclassification of bright but clear pixels as cloudy around theposition of the sun.

CHAPTER 2

Local short-term variability in solar irradianceResearch article published in Atmospheric Chemistry and Physics

Citation

G. M. Lohmann, A. H. Monahan, and D. Heinemann. Local short-term variability in solar irradiance.Atmospheric Chemistry and Physics, 16(10):6365–6379, May 2016. ISSN 1680-7324. doi:10.5194/acp-16-6365-2016.

Author contributions

GML conceived the study, wrote the computer code, analyzed the data, interpreted the results, and wrote themanuscript; AHM contributed to the final design of the study, gave conceptual advice, interpreted the results,and thoroughly edited the manuscript; DH gave conceptual advice; All authors critically revised the finalpaper.

15

http://dx.doi.org/10.5194/acp-16-6365-2016http://dx.doi.org/10.5194/acp-16-6365-2016

Atmos. Chem. Phys., 16, 6365–6379, 2016www.atmos-chem-phys.net/16/6365/2016/doi:10.5194/acp-16-6365-2016© Author(s) 2016. CC Attribution 3.0 License.

Local short-term variability in solar irradianceGerald M. Lohmann1, Adam H. Monahan2, and Detlev Heinemann11Energy Meteorology Group, Institute of Physics, Oldenburg University, Oldenburg, Germany2School of Earth and Ocean Sciences, University of Victoria, Victoria, BC, Canada

Correspondence to: Gerald M. Lohmann ([email protected])

Received: 1 January 2016 – Published in Atmos. Chem. Phys. Discuss.: 19 January 2016Revised: 20 April 2016 – Accepted: 3 May 2016 – Published: 25 May 2016

Abstract. Characterizing spatiotemporal irradiance variabil-ity is important for the successful grid integration of in-creasing numbers of photovoltaic (PV) power systems. Us-ing 1 Hz data recorded by as many as 99 pyranometers duringthe HD(CP)2 Observational Prototype Experiment (HOPE),we analyze field variability of clear-sky index k∗ (i.e., irra-diance normalized to clear-sky conditions) and sub-minutek∗ increments (i.e., changes over specified intervals of time)for distances between tens of meters and about 10 km. Bymeans of a simple classification scheme based on k∗ statis-tics, we identify overcast, clear, and mixed sky conditions,and demonstrate that the last of these is the most potentiallyproblematic in terms of short-term PV power fluctuations.Under mixed conditions, the probability of relatively strongk∗ increments of ±0.5 is approximately twice as high com-pared to increment statistics computed without conditioningby sky type. Additionally, spatial autocorrelation structuresof k∗ increment fields differ considerably between sky types.While the profiles for overcast and clear skies mostly resem-ble the predictions of a simple model published by Hoff andPerez (2012), this is not the case for mixed conditions. Asa proxy for the smoothing effects of distributed PV, we fi-nally show that spatial averaging mitigates variability in k∗

less effectively than variability in k∗ increments, for a spatialsensor density of 2 km−2.

1 Introduction

The number of photovoltaic (PV) power systems has dras-tically increased in many regions of the world during thelast decade, reaching a total nominal capacity of more than178 GW installed worldwide at the end of 2014. The futureglobal PV capacity is expected to continually increase fur-

ther, with predictions for 2019 ranging from 396 to 540 GW(SPE, 2015). In consequence, the challenges associated withthe inherent volatility of PV power production and its funda-mental cause, weather-induced heterogeneity in solar irradi-ance fields, will considerably increase as well (Stetz et al.,2015). Variability in both irradiance and irradiance incre-ments (changes over specified intervals of time) are of inter-est in this context. On the one hand, variability in irradianceitself primarily affects the yield of a PV system and the di-mensioning of battery storage. On the other hand, variabilityin irradiance increments impacts the balancing of generationand load, as well as the maintenance of power quality suchas voltage and frequency stability. Depending on the dimen-sions of the power grid and the PV capacity in question, rel-evant variability in irradiance and its increments can span abroad range of spatiotemporal scales, from seconds and me-ters up to days and hundreds of kilometers. There is a need tounderstand the biases in representation of temporal variabil-ity resulting from temporally coarse-resolution observations(Yordanov et al., 2013b), as well as how spatial averaging (aswould come from having distributed PV over a region) mit-igates variability (Hoff and Perez, 2010). Characterizing thespatiotemporal volatility of irradiance fields and their incre-ments is key to the planning and reliable operation of futurepower grids and their corresponding subsystems.

Recent studies of PV-related variability have analyzedpower spectra of PV systems and solar irradiance (Califet al., 2013; Curtright and Apt, 2008; Klima and Apt, 2015;Lave and Kleissl, 2010; Marcos et al., 2011a; Tabar et al.,2014; Yordanov et al., 2013b), compared power fluctua-tions from specific PV plants with corresponding irradi-ance measurements (Lave and Kleissl, 2013; Lave et al.,2013; Marcos et al., 2011a; van Haaren et al., 2014), andcharacterized power variability as a function of PV plant

Published by Copernicus Publications on behalf of the European Geosciences Union.

Chapter 2: Local short-term variability in solar irradiance 17

6366 G. M. Lohmann et al.: Local short-term variability in solar irradiance

size (Dyreson et al., 2014; Lave et al., 2012; Marcos et al.,2011b; van Haaren et al., 2014). Furthermore, spatial auto-correlation structures and decorrelation length scales of in-crements in irradiance and clear-sky index (i.e., irradiancenormalized to clear-sky conditions), and PV power outputhave also been studied for a range of spatial scales and in-crement values (Arias-Castro et al., 2014; Elsinga and vanSark, 2014; Hinkelman, 2013; Hoff and Perez, 2012; Laveand Kleissl, 2013; Mills, 2010; Perez et al., 2012; Perpiñánet al., 2013). For all quantities and methods considered, in-crement correlations at different locations have been shownto decrease with increasing distance, with a smaller rate ofdecrease (longer decorrelation distances) for larger time in-crements.

While satellite-derived irradiance data are convenient forthe analysis of large spatiotemporal scales, comprehensivedata sets for local short-term variability are time consumingand expensive to collect. They are needed but have not pre-viously been available (Hoff and Perez, 2012; Perez et al.,2012). Therefore, previous studies are either restricted to alimited spatial resolution, a limited temporal resolution, orboth. For example, the few studies that have been basedon high-resolution 1 Hz PV power observations only hadmeasurements from a maximum of six PV systems (Marcoset al., 2011a, b) at their disposal. Though irradiance measure-ments with this high temporal resolution have on occasionbeen conducted with more sensors, the spatial coverage re-mains strongly confined (e.g., up to 45 sensors spread across∼ 2.5 km2; Dyreson et al., 2014). Some studies have usedartificially generated data to overcome these restrictions, ei-ther by simulating simple cloud shapes (Arias-Castro et al.,2014; Lave and Kleissl, 2013), or by constructing virtualnetworks based on time-shifted single-sensor measurements(Perez et al., 2012). However, these simulated data sets donot necessarily coincide with reality.

To fill the gap in understanding of small-scale spatial andtemporal variability in irradiance, we use an extensive exper-imental data set of global horizontal irradiance (GHI) fieldsamples from two measurement campaigns to characterizesub-minute variability of clear-sky index for distances be-tween tens of meters and about 10 km. A high temporal res-olution of 1 Hz and the use of up to 99 synchronized sili-con photodiode pyranometers yields a robust basis for theanalyses (these data are described in detail in Sect. 2). Basedon this data set with its unprecedentedly fine resolution inboth space and time, we study single-point statistics and two-point correlation coefficients of clear-sky index, and developa simple classification scheme to identify overcast, clear, andmixed skies (Sect. 3). Conditioned on these three sky types,we then analyze the probability distributions of sub-minuteincrements in clear-sky index for single sensors and largespatial averages of about 80 km2, as well as their correspond-ing spatial autocorrelation structures (Sect. 4). Finally, wespatially average randomly selected sensors from the data set,covering different area sizes but maintaining a fixed spatial

density, as a proxy for the smoothing effects of distributedPV power production (Sect. 5). Discussions and conclusionsfollow in Sects. 6 and 7.

2 Data

2.1 Measurement campaigns

The data sets on which this study’s analyses are basedoriginate from two extensive measurement campaigns per-formed during the HD(CP)2 Observational Prototype Exper-iment (HOPE) using a set of autonomous silicon photodiodepyranometers. These instruments measure the downwellingshortwave radiation at the Earth’s surface in the range be-tween 0.3 and 1.1 µm. Although this wavelength band doesnot span the entire solar irradiance spectrum, it correspondswell with the relevant bandwidths of typical semiconduc-tor materials used for photovoltaic applications (Pérez-Lópezet al., 2007). In fact, the pyranometers themselves may essen-tially be thought of as tiny PV systems, reduced in space toa single point. Equipped with a battery power supply for upto 10 days, they store their data onsite with a temporal reso-lution of 10 Hz (averaged to 1 Hz during postprocessing). AGPS-based clocked control is used to ensure synchronizationbetween sensors, and for proper positioning data.

The first field campaign with these instruments took placenear Jülich, Germany (50.9◦ N, 6.4◦ E), from 2 April to24 July 2013. It featured a total of 99 pyranometers de-ployed over an area of about 80 km2. The second was con-ducted near Melpitz, Germany (51.5◦ N, 12.9◦ E), and lastedfrom 3 September until 14 October 2013. During this time,50 pyranometers, all of which had previously been used inthe Jülich campaign, were deployed over an area of about4 km2. During the measurement campaigns each instrumentwas subject to regular weekly maintenance. This mainte-nance included data transfer, battery replacement, thoroughcleaning of the glass dome, and re-leveling of the mountingplatform (if necessary). As part of this process, the states ofcleanliness and orientation were recorded, in order to facili-tate identification of periods of bad data. For each observa-tion of tilt or fouling, that week’s worth of data was flaggedaccordingly, even though the specific problem did not neces-sarily last for the entire preceding week.

The geometry of the pyranometer locations for the Melpitzand Jülich campaigns, as well as a histogram of all sensorpair distances dij is presented in Fig. 1. The sensor layoutof the Melpitz campaign, with many sensors concentratedin the center and fewer towards the edge of the domain, ismore structured than that of the Jülich campaign. This differ-ence is due to the much larger spatial domain in the Jülichcase, which entailed external restrictions on the instrumentlocations, such as road access, setup permission, and agricul-tural land use. Consequently, most of the very short sensorpair distances (dij < 1 km), and some intermediate distances

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Figure 1. Panels (a, b) show maps of the coordinates (UTM projec-tion, grid zone 32U) of all pyranometers deployed during the twofield campaigns in (a) Melpitz and (b) Jülich. Panel (c) displaysa histogram of the combined station pair distances from both datasets.

(1 km< dij < 2 km) are attributed to the Melpitz campaign,while sensor pair counts with dij > 3 km are all associatedwith the Jülich campaign (Fig. 1c).

Taking into account the final data sets’ high temporal res-olution of 1 Hz, along with the corresponding dense spa-tial coverages, the two field campaigns provide the basis forunique analyses of irradiance variability, particularly regard-ing potential PV power fluctuations. At the same time, thelimited durations of the campaigns result in a data set thatextends from mid-spring to mid-autumn, and may not be rep-resentative of other times of the year. Schmidt et al. (2016)use data from the Jülich campaign for a performance evalua-tion of sky-imager-based solar irradiance forecasts, and Mad-havan et al. (2016) present a more detailed discussion of thecampaign and the instrumentation. To the best of the authors’knowledge, no other PV-related studies based on comparablydense and high-frequency irradiance sensor networks havebeen published to date.

2.2 Clear-sky index

The available global horizontal irradiance (GHI) at any givenpoint on the Earth’s surface is subject to influences from bothastronomical and atmospheric processes. As for the former,the apparent movement of the Sun relative to Earth gives riseto diurnal and seasonal variations in GHI. These variationsare accurately predictable and not large on short timescalesof seconds or minutes. On the other hand, weather-relatedcontributions to irradiance variability are manifold and com-plex, and present on all timescales. For instance, the growth,

motion, and decay of clouds can affect the seasonal cycle inGHI (e.g., winter tends to be cloudier than summer in mid-latitude low-lying land), and the rapid succession of sunlightexposure and cloud shadow in conditions dominated by fair-weather cumulus generates stochastic variability on shorttimescales (seconds–minutes) (Woyte et al., 2007). The pres-ence of different kinds of cloud (e.g., layer vs. convective) atdifferent altitudes and of different composition (e.g., high-albedo small cloud droplets vs. low-albedo large droplets inrain clouds) result in a complex set of influences on GHI overa broad range of timescales.

In order to distinguish the cloud-induced fluctuations fromthe slowly evolving, astronomically determined apparentmotion of the Sun, a GHI time series G at a particular lo-cation may be related to either the extraterrestrial solar radi-ation Gextra (i.e., irradiance on Earth if there were no atmo-sphere), or the clear-sky radiation Gclear (i.e., irradiance onEarth with a cloud-free atmosphere). Knowing these, we candefine the clearness index

k =G

Gextra, (1)

and clear-sky index

k∗ =G

Gclear, (2)

respectively. The extraterrestrial solar radiation Gextra de-pends only on astronomical relationships, whereas a calcu-lation of clear-sky radiation Gclear requires parameters of at-mospheric conditions, such as typical water vapor concentra-tion and aerosol load. As the exact characteristics of Gclearare model dependent, there is no single unique clear-sky in-dex. The use of any clear-sky model thus introduces new un-certainties to the k∗ time series which are absent from theoriginal irradiance measurements. In spite of these uncertain-ties, use of the clear-sky index is convenient because of thepronounced non-stationarity of the irradiance time series.

While all atmospheric influences on GHI variability are in-cluded in k, variations in the clear-sky index are dominatedby changes in cloud cover. Other sub-daily variations, espe-cially those caused by changes in light scattering with solarangle α, are ideally removed entirely from k∗ (although de-pending on how accurately the atmospheric conditions andtheir variability are actually estimated, such changes withα may still affect the clear-sky index to a minor degree).With k∗ thus being better suited to remove trends in GHIvariability, the focus of this analysis will be clear-sky irra-diance time series computed for the respective locations ofboth field campaigns, using the clear-sky model describedby Fontoynont et al. (1998). A limitation of this model is thatit is based on climatological means and does not account forall variations in scattering or absorption properties. More-over, relatively low values of GHI occurring shortly aftersunrise and just before sunset, coupled with path prolonga-tion and corresponding higher uncertainties in clear-sky ir-

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radiance calculations at these times, can result in unrealisticvalues of k∗ (Lave et al., 2012). In consequence, we onlyconsider data associated with α > 15◦ throughout the study.While the resulting clear-sky index remains an approximatemodel-based quantity, rather than a direct measurement, itallows us to focus directly on weather-related variations insurface irradiance.

The lowest values of k∗ are typically not zero, becauseeven the darkest of clouds do not attenuate all irradiance.Additionally, the upper limit can exceed 1, primarily due toshort-term reflections from the sides of clouds (and also toa secondary degree due to the limitations of the clear-skymodel). Under broken cloud conditions this phenomenon,known as cloud enhancement, can cause single-point GHI toexceed its corresponding clear-sky irradiance value by morethan 50 % on short timescales (Luoma et al., 2012; Piacentiniet al., 2011; Yordanov et al., 2013a).

To characterize the modulation of k∗ variability by the pre-vailing sky type (e.g., overcast vs. clear sky), we divide thetime series at each sensor into non-overlapping 15 min win-dows. This sub-hourly timescale is short enough that it is typ-ically dominated by a single sky type, but long enough thatthere is enough variability to make statistical analyses mean-ingful. We will use differences in the statistics of k∗ withinthese 15 min windows to define different sky type categories.

To illustrate the wide range of cloud influences on k∗

statistics in these 15 min windows, Fig. 2 presents three dis-tinct examples of spatiotemporal variability in k∗. These rep-resentative subsets have been manually selected from a poolof random windows sampled from the entire duration of theJülich campaign. Each panel includes summary statistics forall sensors in the domain for the period (represented as boxplots), as well as the variability of a single randomly selectedsensor. The box plots each consist of a lower “whisker”wlow,the first quartile Q1, the median Q2, the third quartile Q3,and an upper “whisker” wup (summarized in Table 1). Fol-lowing common practice when presenting box plots, wlow(wup) is defined as the lowest (highest) data point that stillfalls within the range of Q1− 1.5 · IQR and Q3+ 1.5 · IQR,with IQR denoting the interquartile range IQR=Q3−Q1(Devore, 2015). Any data below wlow or above wup are con-sidered outliers.

The 15 min window in Fig. 2a features very little spatialvariability and a continually low range of k∗ values, corre-sponding to a time of overcast conditions during which afairly homogeneous cloud layer spanned the entire domain.In contrast, the majority of sensors in Fig. 2b show a con-tinually high range of k∗ with little spatial variability, withthe exception of some pronounced rapid and short-lived de-creases in k∗. Clear-sky conditions dominated the domainat this time, with occasional short-duration shadows cast onsingle sensors (although not the single example sensor). Fi-nally, the data shown in Fig. 2c display considerable vari-ability throughout the domain at all times, with a consistentIQR value of ∼ 0.5. The trace of the example sensor clearly

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Figure 2. Examples of different spatiotemporal variability in clear-sky index k∗ for three distinct cases of sky types: (a) mostly over-cast, (b) mostly clear, and (c) mixed. These representative subsetshave been manually selected from the Jülich campaign and span15 min each. The time series of randomly selected sensors (redcurves) are contrasted with summary statistics of field variability,represented as box plots (Table 1).

illustrates the predominant condition of mixed skies in thiscase, with an alternation between cloud coverage and clear-sky exposure. The characteristic differences in the temporalaverage and variability of k∗ evident in these example datasets indicate that a natural classification scheme for sky typewithin 15 min can be developed in terms of the statistics ofk∗.

3 Sky type variability

3.1 Single-point statistics of clear-sky index

In order to assess the character of irradiance variability con-ditioned on sky type, we group subsets of similar sky con-ditions by means of two simple statistics. Specifically, wecompute the sample arithmetic mean

k∗i =1N

N∑

t=1k∗i (t), (3)




Table 1. Summary statistics used to visualize data spread throughout this study.

Name Symbol Definition

First quartile Q1 75 % of the data >Q1, and 25 % Q2, and 50 % Q3, and 75 % Q1− 1.5 · IQRUpper whisker wup highest value


The resulting distributions of spatial autocorrelation func-tions ρk

∗

ij are shown as functions of sensor pair distance dij inFig. 4 for each of the grid boxes from Fig. 3. The same boxplot statistics listed in Table 1 are computed for 10 logarith-mically spaced bins of dij . A pair of sensors is only includedin these calculations if its members share the same sky typefor at least 60 min over the observational period. The numberof pairs used to derive the box plot information is given ineach panel.

Pairs of sensors with very high σ k∗

i and very low k∗

i (pan-els A3 through A5), are found to be virtually non-existent,although these ranges are occupied by individual sensors(cf. Fig. 3). Similarly, combinations with relatively high σ k

∗

i

and moderate or large k∗i (panels B4/5 and E4/5) also lack ahigh number of available sensor pairs. The remaining well-sampled grid boxes all show spatial autocorrelation functionsρk∗

ij that decrease with increasing dij , as expected. However,the rates of decrease vary considerably across the differentgrid boxes. The differences in autocorrelation structure be-tween two adjacent grid boxes (e.g., A1 and B1) are gener-ally small, but become more pronounced when comparingthose farther apart (e.g., A1 and D4).

For further analyses, and consistent with the manually se-lected exemplary periods previously shown in Fig. 2, we con-sider a classification of three distinct sky types based on thegrid boxes:

1. overcast (A1 and B1),

2. clear (D1 and E1), and

3. mixed (A3 through E5).

This classification is based upon the subjective identificationof different grid boxes in Figs. 3 and 4 with characteristic sta-tistical properties. Although some data corresponding to in-termediate sky conditions (panels A2 through E2, as well asC1) are neglected using this classification scheme, the struc-ture of ρk

∗

ij is appreciably distinct for all three identified skytypes.

Under overcast conditions, correlation coefficients remainρk∗

ij ≈ 1 for dij.1 km, while clear conditions deviate fromρk∗

ij ≈ 1 even for relatively small separations dij.0.05 km.Correlation values ρk

∗

ij ≈ 1 appear under mixed sky condi-tions only for very small dij.0.05 km. With increasing dis-tances, the three sky types’ spatial autocorrelation structuresalso differ in their rates of decay. For example, the character-istic distance to reach ρk

∗

ij ≈ 0.5 is about 10 km for clear-skyand overcast conditions, while it is an order of magnitudesmaller (∼ 1 km) for the mixed sky type.

The k∗ autocorrelation structures within the different skytypes are consistent with the associated cloud patterns. Theresults for overcast and clear-sky conditions both suggestfairly large and homogeneous structures (cf. 10 km to reachρk∗

ij ≈ 0.5), corresponding to large stratus-type cloud layers

in the former case, and homogeneously clear skies (with in-frequent and localized shadowing of the sensors) in the latter.During times classified as mixed, the structure of ρk

∗

ij indi-cates that heterogeneous cloud fields dominate, with muchsmaller length scales than those under overcast conditions.The decay length scale of correlations under mixed skiescorresponds well with typical cloud length scales .2 km ofcumulus-type clouds (Neggers et al., 2003).

4 Variability in clear-sky index increments

The previously discussed properties of the observed k∗ fieldsare independent of their ordering in time, i.e., randomly shuf-fling all sensor-pair data in time (within the 15 min win-dows) will result in the same spatial autocorrelation struc-tures. While this overall variability is of some interest (e.g.,when considering long-term yield of PV systems), it does notcharacterize how rapidly k∗ fields can change. A useful mea-sure of intermittency in the clear-sky index is the statistics ofk∗ increments

1k∗τ (t)= k∗(t + τ)− k∗(t) (6)

for different time lags τ .

4.1 Increment statistics

PDFs of 1k∗τ estimated from data from the Jülich campaignfor three short-term time lags τ = (1,10,60) s are presentedin Fig. 5. The results are first conditioned on the sky typeclassification (first three columns) and then shown again forall sky types together (rightmost column). As a first illustra-tion of the effect of spatial averaging on fluctuation statistics,the PDF of area-averaged increments is also included in eachpanel.

All PDFs are characterized by a narrow central peak – cor-responding to a high probability of very small increments –surrounded by broad tails in which the PDF decreases slowly.With increasing τ , these tails become flatter and the proba-bility of very large excursions increases. While the PDFs ofsingle-sensor observations all exhibit such tails, these fea-tures are much less prominent for the spatially averaged k∗

increment distributions. This result is consistent with previ-ous findings of e.g., Lave and Kleissl (2010), Marcos et al.(2011b), Dyreson et al. (2014), and van Haaren et al. (2014).Comparing the distributions of increments from multiplepyranometers and PV plants of different capacities for var-ious temporal resolutions, these previous studies all showedhigh magnitude changes with increasing time lag, and fewerhigh magnitude changes with increasing PV plant size (ornumbers of sensors).

Under overcast conditions, the central peaks of the distri-butions are generally prominent and the tails are not particu-larly pronounced. The PDFs of clear skies also have a strongcentral peak but display broad flat tails (higher than for over-




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rela

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coef

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ij[-

]Median IQR wup, wlow Outliers

Figure 4. Spatial two-point correlation coefficients ρk∗

ijof clear-sky index k∗ as functions of sensor pair distance dij for each of the grid

boxes from Fig. 3, based on data from both field campaigns. The summary statistics (cf. Table 1) are based on 10 logarithmically scaled binsof dij , and include only those sensor pairs whose members simultaneously correspond to the same grid box for at least 60 min. The totalnumber of pairs used to derive the statistics is given in each panel. Grid boxes to be subsequently grouped as similar sky types are indicatedby colored boxes.

cast conditions), representing the rare large excursions evi-dent in Fig. 2. The central peak is wider under mixed con-ditions, and the flanks are flatter than for clear skies. Undermixed conditions, probabilities of large excursions are en-hanced by the rapid changes associated with passing cloudedges. This distribution of increments is consistent with theindividual time series shown in Fig. 2. The statistics obtainedwhen all sky types are taken together feature the same shapesin the tails as those of the mixed sky type, because the ex-treme fluctuations in k∗ are most common under mixed con-ditions.

A measure of the extent of the tails of the PDF is theprobability P(1k∗τ =±0.5) of a single sensor to fluctuateby ±0.5. The values for this quantity, quoted in each panelof Fig. 5, take different orders of magnitude among the dif-ferent sky types. These probabilities increase from overcastto clear and then to mixed sky conditions, while the val-

ues associated with the overall statistics of all sky types arelocated somewhere in between the last two classifications.Compared to the statistics of all sky types, and independentof τ , P(1k∗τ =±0.5) is more than twice as high under con-ditions classified as mixed. Thus, for applications such as themaintenance of grid stability, where worst case scenarios (interms of strong short-term PV fluctuations) are of interest, theconditioning of 1k∗τ statistics on different sky types demon-strates the strong dependence on specific sky conditions ofthe likelihood of severe fluctuations occurring shortly one af-ter another.

While changes in increment variability are reflected bychanges of σ1k

∗τ (dashed lines in Fig. 5), the standard de-

viation is not an appropriate measure of the size of extremefluctuations due to the non-Gaussian character of k∗ incre-ment statistics. In consequence, the widely used three-sigmarule of thumb, according to which a range of 1k∗τ ± 3σ

1k∗τ

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Den

sity

[-]

Single sensors Spatial average ∆k∗τ ± σ∆k∗τ (single sensors)

Figure 5. Statistics of clear-sky index increments 1k∗τ for different time lags τ , based on data from the Jülich campaign. The first threecolumns display distributions conditioned on different sky types, while the rightmost column presents the combined statistics for all skytypes. The estimated probability density functions of all single sensors (solid black lines) are supplemented with the range of ±1 standarddeviation σ1k

∗τ around their means 1k∗τ (dashed black lines), and contrasted to the much narrower distributions of the spatially smoothed

average of all pyranometers (solid gray lines). The respective average probabilities P(1k∗τ =±0.5) of single sensors to fluctuate by ±0.5are quoted in each panel.

would cover 99.7

solar irradiance variability on small spatial and temporal...

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