IEEE TRANSACTIONS ON POWER SYSTEMS 1 Cumulus Cloud...

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IEEE TRANSACTIONS ON POWER SYSTEMS 1

Cumulus Cloud Shadow Model for Analysisof Power Systems With Photovoltaics

Chengrui Cai, Student Member, IEEE, and Dionysios C. Aliprantis, Senior Member, IEEE

Abstract—Distributed photovoltaic (PV) power generationsystems are being rapidly deployed worldwide, causing technicalproblems such as reverse power flows and voltage rises in distri-bution feeders, and real and reactive power transients that affectthe operation of the bulk transmission system. To fully under-stand and address these problems, extensive computer simulationstudies are required. To this end, this paper sets forth a cloudshadow model that can be used to recreate the power generationof rooftop PV systems embedded in a distribution feeder, orthat of a utility-scale PV power plant, during days with cumulusclouds. Realistically shaped cumulus cloud shadows are modeledas fractals. The variation of the irradiance incident on each PVsystem in an area of interest is then obtained by considering themovement of the cloud shadow over the area. The synthesizedirradiance has satisfactory temporal and spatial characteristics.The proposed model is suitable for Monte Carlo simulations ofpower systems with high PV penetration.

Index Terms—Clouds, fractals, photovoltaic power systems.

I. INTRODUCTION

G RID-TIED photovoltaic (PV) power generation systemsare being rapidly deployed. The cumulative worldwide

installed PV generation capacity is expected to surpass 96.5 GWin 2013 [1], with a considerable amount installed at the distribu-tion level. However, distribution feeders are typically designedfor delivering electric energy to end-use customers, rather thanfor collecting it from distributed energy resources. Hence, a va-riety of technical issues related to PV system integration arises.To fully understand and address these problems, extensive

computer simulation studies are required. This is feasible todayusing specialized distribution feeder analysis software, such asCYMDIST [2], ETAP [3], GridLAB-D [4], or OpenDSS [5].These programs can represent distribution networks with highaccuracy, which is critical for—among other things—sheddinglight on the impacts of distributed generation resources. Oneshould account carefully for the variability of distributed powergeneration at appropriate temporal and spatial resolution, underall possible environmental conditions. Since the primary driverof PV power output is the solar irradiance (in ), high-fi-delity cloud shadow models are becoming increasingly impor-tant and timely for electric power system engineers, especiallyunder the premise that the exponential growth in PV capacity

Manuscript received December 14, 2012; revised June 04, 2013; acceptedAugust 08, 2013. This work was supported by the Electric Power ResearchCenter of Iowa State University. Paper no. TPWRS-01370-2012.C. Cai is with the Department of Electrical and Computer Engineering, Iowa

State University, Ames, IA 50011 USA (e-mail: [email protected]).D. C. Aliprantis is with the School of Electrical and Computer Engineering,

Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]).Digital Object Identifier 10.1109/TPWRS.2013.2278685

will continue to take place. The majority of prior PV integra-tion studies are based on single-point irradiance data obtainedfrom various sources (e.g., [6]). However, during cloudy days,using the same solar irradiance time series for calculating thepower output of hundreds or thousands of PV panels scatteredover an area can lead to significant error [7]–[10]. Therefore,results could be overly conservative, and the costs to mitigateany foreseeable issues might be over-estimated.In order to improve simulation fidelity, this paper sets forth

a model of the solar irradiance over a given area during timeswhen cumulus clouds are prevailing. The model is probabilistic,and intended for use in Monte Carlo simulations. It yields a rea-sonable representation of temporal variability (on a second-by-second basis) and spatial variability (down to a resolution of afew meters) without requiring extraordinary computational re-sources, since it does not rely on a physics-based cloud model.This is key in conducting any type of study that requires de-tailed knowledge of the power flow variation in a distributionfeeder over an extended time period of interest, e.g., over thecourse of several hours (the assumption of a series of quasisteady-state conditions is typical for the analysis of distributionfeeders) [11]–[14].The proposed model could be applied for the study of dy-

namic interactions of PV inverters with each other, or betweenthe inverters and an integrated Volt/VAr control system coordi-nating the actions of tap changing transformers, voltage regu-lators, and switched capacitor banks [15]–[18]. Such analysescan facilitate the design of advanced control schemes for mit-igating voltage rise [19], [20], minimizing distribution feederlosses [21], and reducing voltage fluctuations [22]. Moreover,they can be useful for estimating PV penetration limits [23],[24]. Finally, the proposedmodel could be applied, in lieu of realdata, for calculating the aggregate power output of large-scalecentralized or distributed PV systems, which is necessary fortransmission grid integration studies (investigating ramping is-sues, voltage stability, etc.) [25]–[27].A notable feature of the proposed model is its realistic rep-

resentation of the cloud shadow pattern. Ideally, one would useexperimental data from areal measurements of solar irradiance,but this requires the installation of a costly sensor network, andsuch data are not commonly available. An alternative is to makeuse of cloud images [28], but there is limited availability at thenecessary degree of temporal and spatial resolution.Practically the only remaining alternative is to devise numer-

ical models that generate random cloud shadow patterns on theground. This was the approach taken by Jewell et al., who gen-erated cloud patterns using information on the size distributionof clouds [29]–[34]. Garrett and Jeter have employed a similarmethod to synthesize cloud patterns based on statistical infor-mation [35], [36]. These methods generate a rigid cloud patternthat moves over a given area with constant speed, and simple

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2 IEEE TRANSACTIONS ON POWER SYSTEMS

geometries are assumed for the cloud pattern, such as rectan-gular or circular shapes. These models cannot reproduce the ir-regular shape of cloud shadows, which could be important forstudies of dynamic interactions between individual PV invertersand Volt/VAr control systems.Two notable recent modeling approaches are those developed

by Morf and Beaucage. Morf has proposed a series of modelsbased onMarkov chains to generate the solar irradiance for eachpoint on a grid over an area [37]–[39]. These models reproducestatistical properties of cloud cover (i.e., percentage of sky areacovered by clouds), but neglect the creation of realistic cloudshadow shapes, which could be important when modeling realdistribution feeders with geographical coordinates of installedPV panels. Overall, Morf’s approach places less emphasis onmeteorological parameters such as cloud type, cloud velocityor wind speed, and direction of cloud movement. Beaucage etal. have combined a mesoscale numerical weather prediction(NWP) model with a stochastic cloud generation model to sim-ulate the development of cloud patterns [40]. This model ispowerful but comes at high computational cost. Moreover, theinitialization of atmospheric physics parameters of the NWPmodel is rather complicated and requires a certain level of me-teorology expertise.Here, a representation of the cloud perimeter using fractals, as

originally proposed by Lovejoy [41], [42], and further studiedby Cahalan and Joseph [43] and others [44]–[46], is adopted.Fractal-based modeling can reproduce the naturally irregularshapes of the cumulus cloud shadow pattern, thereby yieldingmore realistic results than models based on simple geometricalshapes. In [47], Beyer et al. adopted the midpoint displacementalgorithm [48] to generate a cloud shadow pattern, which wasused to simulate the irradiance and validated by a statisticalanalysis.In this paper, we describe an improved cumulus cloud shadow

model with the following novel features: 1) Realistic irradiancewaveforms are generated, whereas in prior work, the synthe-sized irradiance typically varies between two fixed values forthe clear and shaded time periods. 2) The synthesis of a cloudshadow pattern of arbitrary time duration is possible, thus al-lowing longer-term simulation studies. 3) The model is not re-stricted to constant meteorological conditions, but can representvariable cloud velocity and cloud cover. 4) The algorithm issimple and can be easily implemented on a computer. 5) Param-eters can be tuned based on commonly available meteorologicaldata.The remainder of the paper is organized as follows: Section II

explains the basics of generating cloud shadows using fractals.Section III sets forth the proposed model for generating the solarirradiance over an area, and includes an illustrative example.Section IV concludes the paper.

II. CLOUD SHADOW GENERATION USING FRACTALS

There exist ten principal types of clouds, viz., cirrus, cirrocu-mulus, cirrostratus, altocumulus, altostratus, nimbostratus, stra-tocumulus, stratus, cumulus, and cumulonimbus, which are cat-egorized according to height as high, middle, and low clouds[49]. To the authors’ knowledge, a universal algorithm that cangenerate all principal cloud types at high temporal and spa-tial resolution does not exist, due to the complexity of atmo-spheric physics. The proposed model represents the cloud typethat contributes the most to fluctuations of power output from

Fig. 1. Illustration of the midpoint displacement algorithm.

distributed PV systems, which is a primary concern of systemengineers. This is generally considered to be the low-altitudecumulus cloud because of its clearly defined edge and the deepshadow it creates [7], [9], [10], [31], [36], [47], [50], [51].The shapes of cumulus cloud shadow contours on the ground

can be modeled as fractals, which can be synthesized using themidpoint displacement algorithm [47], [48]. The required pa-rameters are the fractal dimension , the number of pixels alongthe edge of a square ground area , and the actual pixeldimension (in m; for example, means that one pixelrepresents an area of 49 , with the pixel at its center). Theground area containing the cloud shadow is .The fractal dimension is a key property, for which different

values have been proposed based on observations of cloud im-ages [43], [45], [47]. It has been found that two different di-mension values ( and ) are adequate. For example, [47]proposes values of and .To account for this in the midpoint displacement algorithm,changes from to at a certain stage, as suggested by [47].In our simulations, the values and areused. Of course, these parameters can be modified by the ana-lyst, if different types of low altitude cumulus clouds need to begenerated.A simple example showing the generation of a fractal surface

with 25 pixels is illustrated in Fig. 1. Themidpoint displacementalgorithm is recursive, and takes stages to com-plete. In each stage, center midpoint values are calculated basedon the four corner points of each square, and then values on themidpoints of the edges are calculated. The output of the algo-rithm is a three-dimensional fractal surface.An example of a square fractal surface with

pixels on each side is shown in Fig. 2(a). Note that the fractalsurface should not be confused with the actual cloud shape.Rather, it is an intermediate mathematical artifact that allowscloud shadows to be generated by intersecting the fractal surfacewith a horizontal plane of height . The pixels where the fractalsurface is below are shaded [black area in Fig. 2(b)]. The cloud


CAI AND ALIPRANTIS: CUMULUS CLOUD SHADOWMODEL FOR ANALYSIS OF POWER SYSTEMS WITH PHOTOVOLTAICS 3

Fig. 2. (a) 513-by-513 fractal surface. Solid lines represent cutting planes ofdifferent height. (b) Cloud shadow pattern obtained with .

Fig. 3. Relationship between the cutting surface height and the cloud cover.

cover is defined as the percentage of the ground area coveredby cloud shadow. Cloud shadow patterns with increasing cloudcover can be generated by raising the cutting plane from thebottom to the top of the fractal surface. The map corre-sponding to this fractal surface is shown in Fig. 3.Algorithm 1 describes the process to construct an elongated

fractal surface consisting of square frames. The main idea isto execute the “canonical” midpoint displacement algorithm fortimes, where is determined by the simulation duration and

cloud velocity, as explained in Section III-C. To ensure conti-nuity between frames, all points on the left edge of a new frameare assigned the same fractal value as the corresponding pointson the right edge of the previous frame. Some details worthwhileto note are: 1) The fractal dimension change from to oc-curs when the stage is , with and ,as suggested by [47]. 2) The parameter is arbitrarily assigned

and can be any positive number. Here, we are using .3) For the sake of brevity, the algorithm only describes in detailhow to obtain the midpoint of the upper edge; the other threeedges are found in a similar manner.

Algorithm 1Modified midpoint displacement algorithm

for to do

if equals 1 then

else

Make the first column of in stage equal to thelast column of in stage (stored in )

end if

for to do

if then

end if

for to do

for to , step 1 do

end for

end for



for to length of do

if equals 1 and is emptythen

else

end if

The midpoints on the other 3 edges arecalculated in a similar manner.

end for

end for

Assign to thecolumns of

end for

III. PROPOSED MODEL

This section describes subsequent steps that are taken once afractal surface has been obtained using the midpoint displace-ment algorithm. Solar irradiance data have been collected withan experimental station at Iowa State University (ISU) located at

. The experimental station con-sists of two PV panels with total rating of 270 Wp and max-imum power-point tracking capability. Various sensors aremon-itoring the system’s performance, including a LI-COR LI-200pyranometer that measures the global horizontal irradiance.

A. Solar Irradiance Characteristics

The global irradiance consists of a beam (also called direct)and a diffuse component [52]. The beam component is directlyand considerably affected by cloud shading, by a factor that de-pends on the thickness and type of cloud. The diffuse compo-nent is determined by numerous atmospheric factors, such asthe cloud cover and the cloud type. The proposed solar irradi-ance model treats these two components separately. The beamcomponent is determined by multiplying the maximum (clearsky) beam normal irradiance value with a factor related to theseverity of shading at each location, yielding relatively fast tran-sients. Our model is able to reproduce this type of behavior. Ex-perimental and simulated time-domain waveforms and statisticsare compared in a subsequent section. On the other hand, the dif-fuse component is assumed to maintain a user-defined constantvalue or a slowly-varying time profile, which is assumed to bethe same for the entire area.Our experimental setup has gathered many months’ worth of

global horizontal irradiance data, logged at 1-s intervals. An ex-ample is shown as the solid line in Fig. 4(a). Time segments fromdays with cumulus clouds are selected with the help of a sky

Fig. 4. (a) Measured global horizontal irradiance (solid) and the estimated dif-fuse horizontal irradiance (dotted). (b) Zenith angle. (c) Beam normal irradiance(solid) and digitization threshold (dotted). (d) Digitized shading condition.

camera installed on the ISU campus [53]. When the shading iscaused by an opaque cloud, the beam component drops consid-erably to a minimum. This usually happens repeatedly over thecourse of a few minutes, allowing the estimation of the diffusehorizontal irradiance level, plotted as the dotted line in Fig. 4(a).Hence, the beam horizontal irradiance is obtained by subtractingthe diffuse component from the global irradiance. The beamnormal irradiance is further obtained by dividing the beam hor-izontal irradiance by the cosine of the zenith angle [Fig. 4(b)][52].The beam normal irradiance [solid line in Fig. 4(c)] is further

digitized, in order to extract the duration and magnitude of eachshaded period. A shaded period is thought to occur whenever thebeam normal irradiance level drops below , whereis a constant and is the averaged beam normal irra-

diance level for all clear periods within the entire data segment.In this analysis, , and the threshold is plotted as thedotted line in Fig. 4(c). This process yields the digital shadingsequence shown in Fig. 4(d).Two quantities are used to capture the statistical properties of

the shading sequence on the beam normal irradiance, namely,the duration of a shaded period and the normalizedmag-nitude of the shaded period . The latter is defined by

, where is the aver-aged beam normal irradiance level for a particular shaded pe-riod. From the data, it can be observed that the majority of theshorter shaded periods have duration less than 200 s. Longershaded periods are due to large opaque and/orslowly moving cumulus clouds. The variation of iswithin 5% to 90% for short shaded periods, and within 10% to25% for longer shaded periods. Therefore, shaded periods withdurations less than 200 s are more interesting from a modelingstandpoint. Their statistics are plotted in Fig. 5.

B. Meteorological and Geographic Parameters

To generate the cloud shadow pattern for a given time span,the model requires two meteorological parameters as functions



Fig. 5. Statistics of the beam normal irradiance for shaded periods shorter than200 s, from experimental measurements.

of time, namely, the cloud cover and the cloud velocity.1 The specific waveforms used to generate the cloud

shapes in this study are plotted in Fig. 6. The wind direction isassumed to remain constant.The model also requires knowledge of the system’s geo-

graphic layout. Here, we perform a case study involving aresidential community with rooftop PV panels [56]. The exactlocation of each house is not available, but the coordinatesof all service transformers are included in the feeder data,indicated by circles in Fig. 7. In the simulation, the irradianceat each transformer also represents the irradiance received bythe PV panels at its immediate vicinity. The shading conditionof each point is obtained by virtually moving the generatedcloud shadow over the area under study. To account for thewind direction, the PV system layout is rotated accordingly,as shown in Fig. 8, which depicts the cloud shadow patternbetween 2:00 and 3:00 PM.

C. Cloud Shadow Pattern Generation

The canonical midpoint displacement algorithm has beenmodified to generate the cloud shadow pattern for a rectangular(i.e., not necessarily square) area, as described by Algorithm1. Consecutive square frames are generated and are stitchedtogether at their boundaries. To ensure continuity betweenframes, all points on the left edge of a new frame are assignedthe same fractal value as the corresponding points on the rightedge of the previous frame. The modified algorithm needsto know the number of square frames to generate, whichdepends on the time span of the simulation and the cloudvelocity. Let and denote the start and end times of the

1We lack the necessary instrumentation to measure these meteorological pa-rameters. Hence, for illustration purposes, we utilize data from the National Re-newable Energy Laboratory’s (NREL) Baseline Measurement System (BMS)[54]. We also assume that cloud velocity is equal to the wind speed atcloud height . The time resolution of the NREL data is at a 1-min. in-terval, so it is interpolated into 1-s data to be consistent with the simulation stepin the case study. The wind speed data is measured at a height of

, and has been modified to reflect the wind speed at a height of(a common height of cumulus clouds [49]) based on the power

law equation [55], with . Itshould be noted that these simplifying assumptions are not limitations of theproposed model itself, but rather of the available data for parameterizing themodel.

Fig. 6. Variation of: (a) cloud cover and (b) cloud velocity.

Fig. 7. Geographic layout of measurement points.

study, and let be a simulation time step, so thatand for appropriate integers and . The cloudshadow is moved to the left at times by an amountequal to , where . Then canbe determined by

(1)

In our case study, , and frames were necessary.In the canonical midpoint displacement method described in

Section II, a square fractal surface isgenerated first; then is intersected with a horizontal cuttingplane of height . The contour of the intersection generatesa static cloud shadow pattern for this square area associated witha given cloud cover . On the other hand, the proposed modelgenerates a rectangular, cloud shadowpattern for a variable cloud cover . This is achieved by inter-secting with a non-horizontal surface, which is storedas a matrix with identical rows but different elements ineach of its columns.Fig. 9 graphically demonstrates the technique used to ob-

tain the cutting surface. The light dashed lines represent theframes of the generated fractal surface . A new square

“horizon-defining” window is created every1 min, according to the cloud velocity, so that the center of areais always at a window’s center. In general, these windows

will not coincide with the frames generated by the midpointdisplacement algorithm, even though they have the same dimen-sion as each frame (only the first window coincides with the firstframe). The problem is to determine the appropriate value foreach column in the cutting matrix, so that the number of clouded



Fig. 8. (Top) Generated binary cloud shadow pattern for time period between 2:00 and 3:00 PM. The wind direction is SW. is the study area. (Bottom) Finalcloud shadow pattern, using a multi-layer rendering technique. The pixels of the hatched area on the right were not needed in this simulation.

Fig. 9. Windows and strips for the calculation of the cutting surface.

pixels within the window is equal to , at times,

Each consecutive window leads to the formation of a strip, asshown in Fig. 9. We define as the number of clouded pixelsin strip . In this illustration, the first window consists of threestrips, , , and . The value for all pixels in the firstwindow (i.e., the first frame) can be determined as discussed inSection II, given . The value for pixels belonging to thesecond window, which consists of , , and , is calculatednext. Since the clouded pixels in strips and have beenalready determined, we only need to determine the value forpixels in , such that .To find , the fractal surface corresponding to is intersectedby a horizontal surface of increasing height, until satisfiesthis equation. Similarly, for the third window, which consists of, , and , we find the value that yields the required .

This process is repeated until the simulation end time.2 In orderto eliminate discontinuities between strips, an interpolation isfurther performed. Instead of using the same value for entirestrips (i.e., ranges of columns in the matrix), the values ofthus obtained are only assigned to the middle column of each

strip, and the values for the remaining columns are obtained bylinear interpolation, as illustrated in Fig. 10. This process yieldsthe binary cloud shadow pattern of Fig. 8 (top), where the blackpixels represent the shadow.The final step is to represent the thickness of clouds. This ef-

fect has been often neglected in previous studies, thus resulting

2The rounding operations needed to obtain integer numbers of pixels andstrip-defining column ranges are not explicitly shown for expositional clarity.

Fig. 10. Comparison of the cutting surface height before and after interpola-tion.

in simulated solar irradiance that drops or rises vertically, con-tradicting with experimental observations. To this end, a com-putationally efficient multi-layer rendering technique is devised.The objective is to obtain statistics of beam irradiance that arequalitatively similar to the experimental results (see Fig. 5).These statistics reveal that faster variations usually correspondto less severe shading (probably due to shading by the periph-eral parts of the clouds or by smaller clouds). On the otherhand, moderate to complete shading conditions correspond toslower variations (probably caused from the inner, thicker partsof larger clouds).The surface that has been determined by the previously

illustrated technique defines the external boundary of the cloudshadow. Afterwards, new cutting surfaces are formed by re-peatedly lowering the height of by a factor

, where and represent the highest and lowestelevation of the fractal surface . If layers are to be used,

extra cutting surfaces below the original one are formed.The cloud shadow pixels belonging to the th layer are assigneda random number using a uniform distribution , where

, , and . These valuesare representative of the cloud’s thickness, and are stored in an

shading matrix . In this illus-trative example, and . The obtained cloudshadow pattern is shown in Fig. 8 (bottom), and a magnifiedportion of this (indicated by a small box around 14:45) is dis-played in Fig. 11. The figures are in gray scale; darker pixelscorrespond to increased shading.

D. Synthesis of the Irradiance Time Series

Moving the cloud shadow pattern over the study area (underthe assumption that the study area has been rotated so that thecloud moves over it from “right” to “left”) is equivalent to sam-pling pixel values from the cloud shading sequence stored in

. For a given measurement point on the ground, the



Fig. 11. Magnified cloud shadow pattern.

shading level is obtained by sampling values from theappropriate row and column of , which are determinedby the geographic location of the point (affecting the row andcolumn offset) and the cloud velocity. Thanks to the rotation ofthe study area, the row index for a given point will be constant.The factor represents a cloud transparency level [seeFig. 12(a)].The global horizontal irradiance for point at time is

(2)

where [see Fig. 12(b)] is the maximum beam irradiancethat can be received by a horizontal surface under fully clearcondition, calculated using classical formulas [52]; and isthe diffuse horizontal irradiance. In this case study, has aconstant value (179 ) over the entire simulation period,but it could have been defined as time-varying. Fig. 12(c)shows the final synthesized solar irradiance time series for anarbitrarily selected measurement point, and Fig. 13 shows amagnified 15 min-long portion of the waveform to illustratethe similarity with experimental measurements [cf. Fig. 4(a)].All other measurement points exhibit similar patterns. Thespatially averaged solar irradiance over all measurement pointswithin the study area is depicted in Fig. 12(d). As expected, thevariability of the irradiance decreases considerably, comparedwith the variability of single-point measurements. Note thatFig. 12(d) shows the irradiance variation corresponding to asingle realization of a fractal cloud shadow, i.e., the one shownin Fig. 8. To estimate the sample mean of spatially averagedirradiance, 100 Monte Carlo simulations have been conducted,where each simulation uses a different fractal. It is interestingto observe that the spatially averaged irradiance of individualsimulations [Fig. 12(d)–(g)] can be substantially different fromthe sample mean [Fig. 12(h)], which resembles more closely thesmooth cloud cover variation of Fig. 6(a). The small decreasefrom 14:00 to 15:00 in Fig. 12(h) is due to the movement of thesun over that one hour. Statistics for the simulated irradiancefor all measurement points are shown in Fig. 14(a). A visualcomparison of this plot with Fig. 5 shows that the proposedmodel is able to qualitatively reproduce the statistical propertiesof the measured data.

Fig. 12. (a) Cloud transparency level. (b) Beam horizontal irradiance underclear sky condition. (c) Synthesized global horizontal irradiance time series. (d)Spatially averaged irradiance time series for the cloud shadow of Fig. 8. (e)–(g)Spatially averaged irradiance time series for three other fractals. (h) Samplemean of spatially averaged irradiance time series for 100 Monte Carlo simula-tions.

Fig. 13. Magnified global horizontal irradiance time series.

E. Sensitivity Analysis

The sensitivity of the model’s output with respect to changesin meteorological parameters is examined via four additionalcase studies. These are based on the original case, whose cloud



Fig. 14. Statistics of the simulated beam normal irradiance. (a) Base case. (b). (c) . (d) . (e) .

cover and cloud velocity waveforms are modified by 10% and5 m/s, respectively. The statistics are plotted in Fig. 14, from

which the following may be observed: 1) The cloud cover hasa significant impact on the number of shaded periods with rela-tively longer durations ( 100 s) and low irradiance. 2) Changesin cloud velocity tend to stretch the statistics along the hori-zontal axis, thus directly affecting the duration of shaded pe-riods.

F. Computational Requirements

The computing time required for completing the aforemen-tioned five sensitivity case studies is presented in Table I. Thestudies were run on a PC with an Intel i7 2.2-GHz CPU. Thetable lists the average times required for completing the threemain steps of the proposed algorithm, , , and ,

TABLE ICOMPUTING TIMES

TABLE IISTATISTICAL MEASURES OF EXPERIMENTAL DATA

for generating the fractal surface, generating the cutting surface,and applying the multi-layer rendering technique, respectively.The proposed model is computationally efficient, and suitablefor use in Monte Carlo simulations of distribution feeders.

G. Model Tuning

A practical method for tuning the parameters of the proposedmodel can be based on data that are commonly available frompublic weather stations, namely, hourly cloud cover, ground-level wind speed, and cloud height data. We use the weatherstation in Ames Municipal Airport, Iowa, USA [57]. The cloudvelocity is estimated using the power law equation

, using the ground-level wind speed andcloud height data. The exponent is the first tuned param-eter. The second tuning parameter is , which is used in themulti-layer rendering technique. The tuning aims to provide areasonably good match between the measured and simulatedstatistics of duration and normalized magnitude of shaded pe-riods, and , respectively.For analysis, the irradiance data measured by our experi-

mental station is divided into a high wind speed and a low windspeed group, based on average ground-level wind speed, usinga threshold of 8 m/s. Table II shows the mean , variance

, and skewness of the two groups of data. It can beobserved that both the mean and variance of decreasewith higher cloud velocity, as expected.First, the parameter is fixed to a value of 400, and only

is adjusted. All other parameters are kept constant, using previ-ously defined values. Then, Monte Carlo simulations are con-ducted using the data retrieved from the weather station for thesame time periods as the measured data. Fig. 15 shows the ef-fect of on the statistics of . It can be observed that asthe cloud velocity increases (i.e., for higher ), the model tendsto produce shaded periods with shorter duration. The tuningprocess shows that is a reasonably good value,leading to , , , and

, , . Fig. 16 further compares the cu-mulative distribution function of from simulations (with

) and measurement.Next, the tuning of takes place. Simulations are run with

and other parameters kept fixed. Fig. 17 shows howthe statistics of are affected by . Selectingleads to , , , and

, , . Fig. 18 shows



Fig. 15. Variation of statistical measures of with . Solid line: highwind speed. Dashed line: low wind speed.

Fig. 16. Comparison of the cumulative distribution function of fromsimulations (dashed line) and measurements (solid line). (Note that the shadingduration can exceed 200 s in some cases, so the CDF reaches a value slightlylower than 1 in this plot.)

a comparison of the cumulative distribution of fromsimulations (with ) and measurement.After tuning , the statistics of will be slightly af-

fected, so it might be necessary to iterate until a more satisfac-tory combination of and is found. The other parameters usedin the fractal generation (i.e., , , , and ) can also beadjusted based on captured cloud images, using the image pro-cessing method described in [47].

H. Model Summary

In summary, the proposed model proceeds as follows:1) Determine the geographic parameters of the area con-taining the PV, including the site’s altitude, latitude andlongitude, and coordinates of measurement points.

2) Specify the time period of the study, i.e., , , , as wellas year, month, and day.

Fig. 17. Variation of statistical measures of with . Solid line: highwind speed. Dashed line: low wind speed.

Fig. 18. Comparison of the cumulative distribution function of fromsimulations (dashed line) and measurements (solid line).

3) Calculate the clear beam horizontal irradiance andspecify the diffuse horizontal irradiance .

4) Specify the variation of cloud cover and cloud ve-locity , and a constant direction of cloud movement.3

(If not known, cloud velocity time series can be generatedusing ground-level wind speed measurements and a powerlaw equation with an exponent .)

5) Rotate the study area according to the cloud movementdirection.

6) Specify the parameters of the fractal cloud shadow model,i.e., , , , , , , and .

7) Calculate the number of frames using (1).8) Use the modified midpoint displacement algorithm to gen-erate the fractal surface .

3With a simple modification to the model, allowing for continuous rotationof the area under study, the cloud movement direction would not need to beconstant. It is kept constant here for the sake of simplicity.



9) Calculate the cutting surface according to the first halfof Section III-C.

10) Apply the multi-layer rendering technique described in thesecond half of Section III-C to obtain the shading matrix

.11) Move the synthesized cloud shadow pattern over the area

under study, and obtain the shading sequences for all mea-surement points .

12) Convert the shading sequence to an irradiance time seriesusing (2).

13) Convert irradiance to electric power output based on spec-ifications of PV systems, e.g., orientation of PV panels, tiltangle, efficiency, sun tracking method, etc. [58].

14) Go to step 8) and repeat for the required number of MonteCarlo simulations.

IV. CONCLUSION

Under an increasing penetration of distributed PV, high-fi-delity simulation models play a key role in system analysis. Inthis paper, a computationally inexpensive and relatively simplemodel was proposed for generating time series of solar irra-diance over an area of interest during time periods with cu-mulus clouds, which cause significant fluctuations in PV output.The main idea is to generate realistic cloud shadow patternsby modeling the clouds as fractals, which are moved over anarea of interest with embedded PV generation. Additional en-hancements to the model include representing the thickness ofclouds, yielding smooth and realistic irradiance transitions, andthe ability to conduct long-term (hour-long or more) studiesunder varying cloud cover and cloud velocity. The simulationresults were compared to experimental measurements.The model is meant to be used in future work as an add-on

module to distribution feeder simulation software such asGridLAB-D [4] or OpenDSS [5]. It will facilitate the studyof the dynamic behavior of distribution systems with highpenetration of PV when cumulus clouds are passing over, interms of variation of real and reactive power flows, voltagefluctuations, and actions of feeder and inverter control systems.

ACKNOWLEDGMENT

The authors would like to thank Prof. X. Wu in the Depart-ment of Geological and Atmospheric Sciences, Iowa State Uni-versity, Ames, IA, USA, for his expert advice, and the Iowa En-vironmental Mesonet, managed by Iowa State University De-partment of Agronomy, for providing the webcam data. The au-thors also would like to thank the journal editor and reviewersfor their valuable comments and suggestions for improving themanuscript.

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Chengrui Cai (S’10) received the B.S. degree in automation from Beijing In-stitute of Technology, China, in 2009. He is currently pursuing the Ph.D. degreein the Department of Electrical and Computer Engineering at Iowa State Uni-versity, Ames, IA, USA.His research interests include the modeling of distributed PV generation and

demand response.

Dionysios C. Aliprantis (SM’09) received the Diploma in electrical and com-puter engineering from the National Technical University of Athens, Greece, in1999, and the Ph.D. from Purdue University, West Lafayette, IN, USA, in 2003.He is currently an Associate Professor of Electrical and Computer

Engineering at Purdue University. His research interests are related to electro-mechanical energy conversion and the analysis of power systems. Morerecently his work has focused on technologies that enable the integration ofrenewable energy sources in the electric power system, and the electrificationof transportation.Prof. Aliprantis was a recipient of the NSF CAREER award in 2009.

He serves as an Associate Editor for the IEEE TRANSACTIONS ON ENERGYCONVERSION.

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