Intra-hour forecasts of solar power production using measurementsfrom a network of irradiance sensors

Vincent P.A. Lonij a,⇑, Adria E. Brooks a, Alexander D. Cronin a, Michael Leuthold b,Kevin Koch c

a University of Arizona, Department of Physics, 1118 E. 4th Street, Tucson, AZ 85716, United Statesb University of Arizona, Department Atmospheric Sciences, 1118 E. 4th Street, Tucson, AZ 85716, United States

c Technicians For Sustainability, 612 N. 7th Ave., Tucson, AZ 85705, United States

Received 26 March 2013; received in revised form 5 July 2013; accepted 3 August 2013Available online 31 August 2013

Communicated by: Associate Editor Jan Kleissl

Abstract

We report a new method to forecast power output from photovoltaic (PV) systems under cloudy skies that uses measurements fromground-based irradiance sensors as an input. This work describes an implementation of this forecasting method in the Tucson, AZ regionwhere we use 80 residential rooftop PV systems distributed over a 50 km ! 50 km area as irradiance sensors. We report RMS and meanbias errors for a one year period of operation and compare our results to the persistence model as well as forecasts from other authors.We also present a general framework to model station-pair correlations of intermittency due to clouds that reproduces the observationsin this work as well as those of other authors. Our framework is able to describe the RMS errors of velocimetry based forecasting meth-ods over three orders of magnitude in the forecast horizon (from 30 s to 6 h). Finally, we use this framework to recommend optimallocations of irradiance sensors in future implementations of our forecasting method.! 2013 Elsevier Ltd. All rights reserved.

Keywords: Irradiance; Variability; Cloud forecasting; Spatio-temporal correlation; Station-pair correlation

1. Introduction

Solar power generation at the utility-scale is a GrandChallenge (NAOE, 2008). A major problem is the intermit-tent output of solar power plants due to passing clouds andnighttime. Intermittency limits the adoption of solar powerby utility companies and industry because of potentiallyunpredictable grid instabilities that may result. Addition-ally, unpredicted fluctuations in solar power productionmay cause the utility to either overproduce electricity orpurchase additional electricity. Both scenarios are expen-

sive and can negate the benefits of using solar power(Gowrisankaran et al., 2011). Fluctuations and intermit-tency can be mitigated with energy storage, spinningreserves, or demand response. However, optimal manage-ment of these three methods requires accurate forecastsof PV power output on several timescales. Such forecastsare the subject of the research presented in this paper.

Forecasts with multiple forecast horizons are valuablefor utility operators and plant owners. For example, day-ahead forecasts are needed to determine an optimal energytrading strategy in the energy market. Hour-ahead andintra-hour forecasts are valuable for electric grid operatorsto better schedule spinning reserves and demand response.

Several methods exist to forecast PV power output,including numerical weather models and velocimetry of

0038-092X/$ - see front matter ! 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.solener.2013.08.002

⇑ Corresponding author.E-mail addresses: lonij@physics.arizona.edu (V.P.A. Lonij), cronin@

physics.arizona.edu (A.D. Cronin).

www.elsevier.com/locate/solener

Available online at www.sciencedirect.com

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Solar Energy 97 (2013) 58–66

clouds using satellite images or ground based measure-ments of clouds (SolarAnywhere, 2012; Jayadevan et al.,2012; Chow et al., 2011; Perez et al., 2010; Hamill andNerhkorn, 1993). However, these methods only outper-form the persistence model for horizons either <10 minor >60 min.

In Section 2, we present a novel method to forecast solarpower production using power measurements of a distrib-uted network of PV systems (Lonij et al., 2012c). We willshow how this method outperforms the persistence modelfor forecast horizons larger than 30 min. An advantage ofusing ground-based sensors is that PV power output canbe inferred directly from the output of other PV systemswithout independent estimates of the height, density, reflec-tivity, or spectral properties of clouds. This is unlike satel-lite based or Numerical Weather Prediction (NWP)methods, which have to use radiative-transfer models toconvert either satellite images or 3D cloud density data intoan estimate of surface-level POA irradiance (Muller et al.,2004; Perez et al., 2004). Note that we use power measure-ments of PV systems as a measure of irradiance, this elim-inates the need for a dedicated network of irradiancesensors and automatically accounts for the effect of temper-ature and angle of incidence on power forecasts.

In Section 3 we develop a framework to model station-pair correlations. We will show that our framework repro-duces the station pair correlations measured in this work aswell as correlations reported by several other authors. Wethen use our framework to predict the forecast accuracyof velocimetry based forecasts that span three orders ofmagnitude in the forecast horizon (from 30 s to 6 h).Finally, we discuss how to use this framework to devise astrategy to optimally place irradiance sensors for use inforecasting.

2. Forecasting of PV system output

The principal input to the forecasting algorithmdescribed here consists of measurements of PV power out-put from 80 residential rooftop systems distributed over a50 km by 50 km area are used to forecast PV power output.Details of the system specifications are given in Lonij et al.(2012b). Measurements are recorded at 15-min intervals,and each measurement represents the average AC powerover the previous 15 min. Data presented here are obtainedusing existing infrastructure. Each of the PV systems in thisstudy uses an inverter by SMA Solar Technology with adata communications card installed to record data. Dataare transmitted over the Internet using an SMA SunnyWebBox. Although the forecasts we report here are basedon historical data, a real-time forecasting system could beimplemented with a software-only modification to oursetup.

Fig. 1 shows power output for each of the 80 systems inthe Tucson area at three different times. The dark points(indicating low output, due to a cloud) on this day shift

from the southeast towards the northwest of the Tucsonvalley over the course of 1 h.

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Fig. 1. Measured PV output at three times separated by 30-min. Dark(red) circles indicate stations with low power output, light (yellow) circlesindicate stations with high power output. The background color is theinterpolated clearness index (K), white areas are cloudy, blue areas areclear. During the period shown, a cloud drifts from the south–east tot thenorth–west. A movie of this is available at (Lonij et al., 2012a). (Forinterpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

V.P.A. Lonij et al. / Solar Energy 97 (2013) 58–66 59

The ground sensor network presented here has an aver-age nearest neighbor spacing of about 3 km and providesmeasurements every 15 min. This results in better spatialand temporal resolution than currently availableoperational forecasts based on GOES satellite images(SolarAnywhere, 2012) (whose forecasts have 10 km resolu-tion and are updated approximately every hour).1

In our algorithm, once data are collected on a centralserver, PV output for each system is forecast as follows.First, a clear-sky expectation for the output of each systemis obtained, as described by Lonij et al. (2012b). Subse-quently, we correct for outages, system orientation, andpartial shade due to permanent obstacles (not clouds)(Lonij et al., 2012b). Finally, we infer that deviations fromthe clear-sky operation of the system are the effect ofclouds. Here we define the clearness index K at every loca-tion (x, y) and time (t) as

Kðx; y; tÞ ¼ POAðx; y; tÞPOAclearðx; y; tÞ

ð1Þ

where, POA(t) indicates the plane of array (POA) irradi-ance at time t and POAClear(t) indicates the modeledPOA irradiance in the absence of clouds. Since the outputof any particular PV system normalized by peak power(kW/kWpeak) is approximately proportional to POA irradi-ance, we can write

Ki ¼piðtÞ

pi;clearðtÞð2Þ

where pi(t) is the normalized power output (kW/kWpeak)for system i and pi,clear(t) is the normalized power thatwould be generated under a clear sky. Once K has beendetermined for every system at the present time t, then itis possible to forecast K at time t + dt, and at location(x, y) using

Kðx; y; t þ dtÞ ¼ Kðx& vxdt; y & vydt; tÞ ð3Þ

where vx and vy are the x and y components of the cloudvelocity.

Values of K at locations between the points where PVsystems are located are determined by interpolation as fol-lows. For each location (x, y), K is determined for the fourclosest PV systems {Ki}. We then compute K(x, y) = ME-DIAN ({Ki}). We use the median instead of, e.g., bilinearinterpolation, because cloud edges are relatively sharpcompared to the 3 km distance between measurement sta-tions. Therefore picking a representative nearby system toestimate cloud cover is preferable over averaging.

A significant challenge in our forecasts is to determinethe cloud edge velocity. Ground-based measurements ofwind are not an accurate measure of the velocity of clouds.For example, in Fig. 1, where the clouds progress towards

the northwest, ground-level wind measurements indicatewind from the northeast. Therefore we explored four waysof estimating cloud velocity:

1. Wind velocity from the National Oceanic andAtmospheric Administrations Rapid Update Cyclenumerical weather model (NOAA RUC NWP) atthe grid-point at latitude 32" and longitude &111"at an altitude of 700 mb.

2. A constant velocity for each hour that is numericallyoptimized (retrospectively) to minimize the RMSerror of our power forecast. Note that this is not atrue forecast since it uses future information. Weuse this method to explore how well our power fore-casts perform if cloud velocity were perfectly known.

3. A Kalman Filter (Welch and Bishop, 2006; May-beck, 1979) applied to the velocity determined bymethod 2. We use the Kalman Filter to extrapolatethe data obtained with method 2, 2 h ahead. Thismethod is therefore once again a true forecast. TheKalman Filter is applied separately to the x and ycomponents of the wind vector. We model windvelocity as vK(t + dt) = vK(t) + aKdt. The KalmanFilter returns maximum likelihood estimates forvK(t) and aK(t). For our forecast we then usev(t) = vK(t & dt) + aK(t & dt) *dt, where dt = 2 h.

4. The persistence model, which assumes that the clear-ness index at a future time t = t0 + dt is the same asthe clearness index at t0. This is equivalent to assum-ing the cloud velocity is equal to zero.

Table 1 shows RMS error using these three differentcloud velocity estimates for time horizons ranging from15 min to 90 min. Table 2 shows mean bias error (MBE)defined as MEAN (pforecast & pmeasured) (Chow et al., 2011;Perez et al., 2010). Both Tables 1 and 2 show results forcloudy days in the period May 1, 2011 through April 30,2012. Cloudy days are defined as days where Ki averagedover all systems and averaged over 24 h is less than 0.9.Fig. 2 shows the result of a forecast for one day in Augustof 2011.

For comparison, Tables 1 and 2 also list results from thepersistence model. For time horizons ranging from 30 min

Table 1RMS errors for power forecasts using forecast horizons ranging from15 min to 90 min. RMS values are for cloudy days only and in units ofnormalized PV system output power (kW/kWpeak). The results of differentmethods to determine cloud velocity are shown. Note that method 2. is nota true forecast since it uses future information.

Horizon NOAA Kalman Persistence Optimized(1.) (3.) (4.) (2.)

15 min 0.065 0.067 0.062 0.06130 min 0.082 0.082 0.084 0.05545 min 0.090 0.092 0.097 0.06960 min 0.098 0.100 0.105 0.07975 min 0.106 0.107 0.113 0.09290 min 0.112 0.114 0.120 0.102

1 Solar Anywhere recently started to offer forecasts with forecasthorizons less than one hour as well as 1 km ! 1 km spatial resolutions,however, at the time of writing no peer-reviewed validation of theseforecasts is available.

60 V.P.A. Lonij et al. / Solar Energy 97 (2013) 58–66

to 90 min, forecasts from our forecast provide smallerRMS errors and mean bias errors than the persistencemodel. This is significant because other forecasting meth-ods based on NWP models, satellite images (Perez et al.,2010) or all-sky images (Jayadevan et al., 2012; Chow etal., 2011) are currently unable to beat the persistence modelat these time-horizons for these (15 min) averagingintervals.

The best true forecast is obtained using cloud edgevelocity obtained using method 1 (wind velocity formNOAA RUC model). The improvement over the persis-tence model is modest (about 8% for 45 min ahead fore-casts), though this improvement is comparable to theimprovement obtained with satellite based forecastingmethods for a 1 h forecast horizon using 1 h averages(Perez et al., 2010).

Although method 2 is not a true forecast (because it usesa retrospectively optimized cloud velocity) it does give usan indication of how well the model can perform if cloudvelocity were better known. Method 2 results in an RMSerror that is 23% smaller than methods 1 and 3 at a forecasthorizon of 45 min. This suggests cloud velocity vectorsexist that will improve our forecast of PV system outputby 23%. Improved estimates of wind velocity may beobtained in the future from a regionally optimized WRF

model, ground based observations (e.g. using a camera),or by implementing a sensor network with higher temporalresolution (Bosch et al., 2013).

3. A model for spatio-temporal correlations of clouds

The success of velocimetry based forecasting methods isdependent on the spatio-temporal correlation properties ofclouds in the geographical region of interest. In this sec-tion, we develop a model that can simultaneously describecorrelations of power production as well as correlations forramp-rates, at multiple time scales and averaging intervals.We use our model to describe our measurements as well asmeasurements from several other authors (Hoff and Perez,2012; Perez et al., 2012; Perez et al., 2011; Mills and Wiser,2010; Hoff and Perez, 2010; Murata et al., 2007; Glasbey etal., 2001). We will use our model to describe RMS forecast-ing errors and to recommend optimal locations for irradi-ance sensors to implement the method described inSection 2.

Several authors have studied changes in irradiance (i.e.ramp-rates), rather than cloud derating, and havesuggested different empirical formulas to describe the spa-tio-temporal correlations of these ramp-rates (Perez et al.,2011; Mills and Wiser, 2010; Murata et al., 2007). Autocor-relation functions of ramp-rates in measured irradiancehave been studied for individual measurement stations byHoff and Perez (2012). In addition, correlation functionsfor ramp-rates have been studied as a function of stationpair distance (Mills and Wiser, 2010; Hoff and Perez,2010). Glasbey et al. (2001) proposed a formulation thatincluded both spatial and temporal dependence of the cor-relation, however, this formulation did not allow for thepossibility of forecasting because they did not include clouddrift. The effect of temporal and geographic smoothing ofirradiance variability has been studied using a waveletapproach (Lave and Kleissl, 2013; Lave et al., 2012a,b).

The formulation presented here is able to capture thequalitative features previously reported by several authorsincluding: monotonic decrease of the correlation functionfor ramp-rates as a function of station pair distance (Millsand Wiser, 2010; Murata et al., 2007), the zero-crossing ofthis correlation function when measured exclusively in thedirection of clouds velocity (Hoff and Perez, 2012), andfinally the forecasting accuracy of 3 different velocimetrybased forecasting methods, including the results of Section2 and work in Chow et al. (2011), Perez et al. (2010). Ourmodel accurately describes forecast accuracies over a rangeof forecast horizons that spans nearly 3 orders ofmagnitude (from 30 s to 6 h).

3.1. The framework

We define a measurement of clearness index at alocation ~x ¼ fx; yg at time t as Kð~x; tÞ. We assume thecovariance function between two such measurements canbe written as

Table 2Mean bias errors (!1000) for power forecasts using forecast horizonsranging from 15 min to 90 min. MBE values are for cloudy days only andin units of normalized PV system output power (kW/kWpeak). The resultsof different methods to determine cloud velocity are shown. Note thatmethod 2. is not a true forecast since it uses future information.

Horizon NOAA Kalman Persistence Optimized(1.) (3.) (4.) (2.)

15 min &2.80 &2.23 0.91 &0.2730 min &3.52 &3.14 2.08 &0.5645 min &2.58 &2.76 3.49 &0.1060 min &1.72 &1.91 5.09 1.0775 min 0.16 &0.60 6.84 2.8690 min 2.47 1.21 8.73 6.05

Fig. 2. Measurements and forecasts of PV performance using the networkof 80 PV systems. This 45-min ahead forecast is made using NCDCforecast for wind velocity. For the day shown in the figure, the RMS errorof the forecast is 0.07. The RMS error of the persistence model is 0.12.

V.P.A. Lonij et al. / Solar Energy 97 (2013) 58–66 61

Cov½Kð~x1; t1Þ;Kð~x2; t2Þ( ¼ f ðD~x;Dt; fsigÞ ð4Þ

where Dt = t1 & t2, D~x ¼ ~x1 & ~x2, and {si} is a set of param-eters for the specific functional form of f. From now on, wewill suppress {si} from the notation for the sake ofreadability.

From this formula we will now derive the correlationfunction for ramp-rates (the derivative _K ¼ dK=dt) as wellas the effect of time-averaging of either K or _K. We willuse the equation for the covariance between two stochasticvariables, that are themselves sums of multiple stochasticvariables:

CovX

i

X i;X

j

Y j

" #

¼X

i

X

j

CovðX i; Y iÞ ð5Þ

where {Xi} and {Yi} are sets of arbitrary stochastic vari-ables. Using Eq. (5), we can derive the correlation functionfor measurements averaged over a time !t defined asKð~x; tÞ ¼ ð!tÞ&1 R!t

0 Kð~x; t þ t0Þdt0. We find

Cov½Kð~x1;t1Þ;Kð~x2;t2Þ(¼ð!tÞ&2Z !t

0

Z !t

0

dtdt0f ðD~x;Dtþ t& t0Þ ð6Þ

Similarly we can derive the covariance function for _K byinserting the definition of the derivative,_Kð~x; tÞ ¼ limdt!0½Kðt þ dtÞ & KðtÞ(=dt, into Eq. (5); we find

Cov½ _Kð~x1; t1Þ; _Kð~x2; t2Þ( ¼ &@2f ðD~x;DtÞ

@Dt2ð7Þ

For numerical derivatives, such as those studied by Hoffand Perez (2012), Mills and Wiser (2010), Hoff and Perez(2010) and Murata et al. (2007), the data is also averaged.Therefore, we first apply Eq. (6) and then Eq. (7) to find

Cov½ _Kð~x1; t1Þ; _Kð~x2; t2Þ( ¼ &ð!tÞ&2Z !t

0

Z !t

0

dtdt0

! @2

@Dt2f ðD~x;Dt þ t & t0Þ ð8Þ

This can be rewritten as

Cov½ _Kð~x1; t1Þ; _Kð~x2; t2Þ( ¼ ð!tÞ&2½f ðD~x;Dt þ!tÞþ f ðD~x;Dt &!tÞ& 2f ðD~x;DtÞ( ð9Þ

If instead of covariance, we want to study correlation be-tween station pairs we may compute this using

Corr½X ; Y ( ¼ Cov½X ; Y (ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCov½X ;X (Cov½Y ; Y (

p ð10Þ

We will now validate this model with a specific expressionfor f.

3.2. Validation

We will validate the model of Section 3.1 using ourmeasurements described in Section 2. Based on our

measurement we will choose a specific functional formfor f ðD~x;DtÞ in Eq. (4).

Correlations between system pairs for Dt = 0 as a func-tion of north–south as well as east–west separation areshown in Fig. 3 (top). Fig. 3 (bottom) shows the station-pair correlation when Dt is chosen to maximize the correla-tion for each pair. Fig. 3 (bottom) shows an extended ovalregion of high correlations corresponding to D~x ) ~vcDt,where ~vc is the cloud velocity. This confirms that there isa drift component in the behavior of clouds. Fig. 4 showsa cross-section of Fig. 3 (top) along the east–west direction.

Based on Figs. 3 and 4 we conclude that for Dt = 0, thefunction f must be decreasing in jD~xj. For D~x ¼ 0, the func-tion must be decreasing in Dt. Finally, for a given Dt, the

Fig. 3. Top: Spatial correlation between station pairs for one day as afunction of separation in the east–west as well as north–south directions.Dark/red points in the center indicate high correlation, light/yellow pointsindicate low correlation. Bottom: Spatial correlation between station pariswith the relative time-shift between stations adjusted to maximizecorrelation. Correlations are significantly increased for stations that areseparated by a vector that is a particular multiple of the cloud velocity, i.e.,the central circular region of high correlation in the top figure has becomean extended oval. Correlations are calculated using 15 min averages ofnormalized power. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

62 V.P.A. Lonij et al. / Solar Energy 97 (2013) 58–66

covariance function reaches a maximum when D~x ¼~vcDt,where~vc is the cloud velocity. The corresponding functionfor ramp-rates must have a zero-crossing as a function ofdistance parallel to the wind direction, but has no zero-crossing if averaged over all directions (Hoff and Perez,2012), see Fig. 5.

There are two prevalent functional forms in the litera-ture to describe covariances of clouds. An exponentialform is used by Glasbey et al. (2001) and Mills and Wiser(2010). A function of the form 1=ð1þ x=!tÞ is used by Hoffand Perez (2012). The function proposed by Hoff and Perez(2012) does not yield the zero-crossing in Fig. 5, however,as we show below, an exponential function does producethis zero-crossing. We therefore choose and exponentialfunction:

f ðD~x;DtÞ ¼ D00 ¼ Ae&jD~x&~vcDtj=rx e&ðjDtj=rtÞq ð11Þ

where rx, rt, q, and A are parameters to be determinedfrom data. We define D01, D10, and D11 as the result ofEqs. (6), (7) and (9) respectively. The first exponential inEq. (11) describes a drifting cloud pattern with a correla-tion function that decays as a function of distance. The sec-ond exponential describes the changes of the cloud patternover time, i.e., the degree to which the cloud pattern at timet1 is not simply a spatially shifted version of the pattern attime t2. Although this simple functional form does not ex-actly match the average correlation shown in Fig. 4, con-sidering the large day-to-day variation in the data, weopt for a simpler function in this work that captures allthe qualitative features we require.

Fig. 5 shows the correlation function for ramp-rates(D11) as a function of station-pair distance. There is a clearzero-crossing if D~x is parallel to the wind velocity, but not ifD~x is perpendicular or if D11 is averaged over all directions.This behavior is consistent with results from Hoff andPerez (2012) and Mills and Wiser (2010); Murata et al.,

2007. Furthermore, we can calculate the location of thezero-crossing as a function of averaging interval by neglect-ing the second exponential in Eq. (11), plugging this intoEq. (9) and setting the result equal to zero and solvingfor Dxk. This gives

Dx ¼ rx

2logð2 expð!tjvcj=rxÞ & 1Þ ) rx

logð2Þ2þ

!tjvcj2

ð12Þ

The approximately linear dependence on !t is consistentwith measurements by Perez et al. (2012).

3.3. Forecast errors

In this section we model irradiance forecast errors forvelocimetry based forecasting methods using the frame-work developed in Section 3.1. To this end, an expressionfor the Mean Square Error (MSE) is first derived in termsof the covariance between a forecast mf and a measurementms. By definition, MSE is given by

MSE ¼XN

i¼0

ðmf ;i & ms;iÞ2=N ð13Þ

where the subscript i refers to measurements at differenttimes. Given that average and variance of derating due toclouds is similar for measurements in the same climatic re-gion, it is reasonable to assume that the mean lf of the fore-cast and ls of the measurement are the same. We cantherefore insert lf–ls inside the parentheses of Eq. (13) toobtain

MSE )XN

i¼0

ðmf ;i & ms;i & ðlf & lsÞÞ2=N

¼ Varðmf & msÞ ð14Þ

We can rewrite the right hand side of this equation as

MSE ¼ Varðmf Þ þ VarðmsÞ & 2Covðmf ;msÞ ð15Þ

Fig. 4. A cross-section of Fig. 3 (top) in the east–west direction. The dotsindicate measurement from 3 h periods. One year of data is shown. Theblack circles are averages for each distance. The figure shows that, onaverage, the correlation decreases for larger D~x, but there is also a largespread. The model of Eq. (6) is shown as well. Correlations are calculatedusing 15 min averages of normalized power.

Fig. 5. Correlations between derivatives (ramp-rates) between differentstations are modeled. The model reproduces both the monotonicallydecreasing correlation as a function of distance when averaged over alldirections (Mills and Wiser, 2010; Murata et al., 2007), as well as the zero-crossing and subsequent minimum of the correlation if the stationseparation is parallel to the wind direction (Hoff and Perez, 2012).

V.P.A. Lonij et al. / Solar Energy 97 (2013) 58–66 63

Using D01 to compute the variance and covariance in Eq.(15) we can plot the predicted RMSE along with the ob-served RMSE. Fig. 6 shows the result of the model as wellas RMSE observed in different forecasting methods. Theparameters rt, A, and q in Eq. (11) are adjusted to matchthe RMS errors for the method presented in Section 2(the red circles in Fig. 6).

The RMS errors from other authors are scaled by a sin-gle factor to match with the model. This is justified becausethe other studies were conducted in different climate zonesand at different times (e.g. time of day, or time of year).The dependence of RMS errors on forecast horizon forall measurements shown in Fig. 6 is consistent with themodel.

Thus, we have demonstrated that we can predict RMSerrors for a given forecast horizon. In the next Sectionwe use this to determine the optimal locations for irradi-ance measurement stations.

3.4. Example: recommendation of sensor locations

In Section 2, the irradiance sensor (PV system) locationswere predetermined by existing hardware installations. Infuture implementations of our forecasting method it is pos-sible that no existing hardware is available, or so many sen-sors are already available that a subset must be chosen, inthese situations it is useful to determine optimal sensorlocations.

We now discuss optimal sensor placement in the casewhere forecasts are needed for a single large PV plant. Atrade-off is possible between the number of sensors in thenetwork (and therefore the cost), the accuracy of the fore-casts, and the maximum forecasting horizon. We assumethat wind can come from any direction with equal proba-bility. Therefore, we will design a point-symmetric sensorpattern, i.e., the PV system is located at the origin and

the sensors are located on concentric circles around thePV system.

First, we first determine the spacing between sensors,both in the radial and axial directions. At one instant(i.e. for a constant cloud velocity vector) the radial spacingcorresponds to spacing in the direction of cloud motionand axial spacing corresponds to spacing in the perpendic-ular direction.

To make forecasts, we use Eq. (3), which assumes thatthere is a station located at~x ¼ &~vcdt. If there is no stationat that location, we can use the average of two nearbystations:

msð~xÞ ¼ msð~xiÞ=2þ msð~xjÞ=2 ð16Þ

where ~xi and ~xj indicate the locations of the two nearestirradiance sensors. If we plug this expression for msð~xÞ intoEq. (15) and use Eq. (5) to expand the covariance and var-iance terms we find

MSE ¼ 3

2Varðmsð~xiÞÞ þ

1

2Covðmsð~xiÞ;msð~xjÞÞ

& 2½Covðmf ;msð~xiÞÞ=2þ Covðmf ;msð~xjÞÞ=2( ð17Þ

We can use Eq. (17) to calculate the spacing between sta-tions if we assume that xi and xj are equidistant from x(the worst case scenario) and set the RMSE to be no morethan a given threshold (e.g. 3% higher than the minimumRMSE). Using the expression for covariance in Eq. (6)and the expression for f in Eq. (11), Eq. (17) can be solvednumerically to yield the maximum distance between xi andxj. We can follow this procedure for station separationseither perpendicular or parallel to the direction of cloudpropagation.

Finally, we have to choose and averaging time ð!tÞ; weassume that for larger forecast horizons it is acceptable

Fig. 6. RMS error as a function of forecast horizon for three differentexperiments. Triangles represent forecasts from Chow et al. (2011),squares represent forecasts from Perez et al. (2010), and circles representthe present study. Data from Chow et al. (2011) and Perez et al. (2010) arecorrected for averaging interval using Eq. (6) and multiplied by 0.1 and0.45 respectively. RMS error is also modeled using Eq. (15).

Fig. 7. Station locations calculated by numerically evaluating Eq. (17).The equivalent forecasting horizons, assuming a cloud velocity of 40 km/hare also shown.

64 V.P.A. Lonij et al. / Solar Energy 97 (2013) 58–66

to use larger averaging intervals. This reflects the fact thatsmall clouds are less significant when they are further away.Here we choose !t ¼ Dt=4. Fig. 7 shows the results for theparameters shown in Table 3.

4. Conclusion

We presented results of an irradiance forecastingmethod that uses measurements from a network of residen-tial PV systems. Forecasts using 15-min interval measure-ments from a network of distributed PV systemsoutperform the persistence model for forecast horizonsranging from 30 min up to 90 min. This is an improvementover forecasts based on satellite images, which are notavailable at 15 min intervals for forecasts horizons rangingfrom 30 min up to 90 min.

We observed that the RMS error of our forecasts formay be improved by 23% (for forecast horizon of 45 min)if we can find better ways to determine cloud velocity. Thisshows the overall capability of our model. Determiningcloud velocity from measured PV data is challenging forour data set because the geographical area spanned byour dataset is small relative to the time resolution of ourmeasurements. Simply stated, this is because it takes onlya few time steps (1 h or so) for clouds to transit acrossour entire 50-km network.

Using wind velocities obtained from numerical weathermodels gives improved results. However, because cloudedge velocity is not always the same as wind velocity, thereare still significant errors. In future work we will thereforeexplore other techniques to determine cloud velocity,including analysis of cloud images from satellites and fromground-based cameras. We will also explore if we can use anumerical weather models to predict on which days veloc-imetry based forecasts will perform well and on which daysvelocimetry based forecasts produce large errors.

Finally, we developed a framework to describe spatio-temporal correlations of cloud deratings. This frameworkis able to reproduce results from several different authorswho studied different geographical regions. The model isable to describe both correlations in power output as wellas correlations of ramp-rates.

We applied our framework to model forecasting errorsand were able to predict RMS forecasting errors over threeorders of magnitude in the forecasting horizon (from 30 sto 6 h). We also used our framework to determine optimalirradiance sensor locations for future implementations ofour forecasting method.

Role of the funding sources

This work was funded by Tucson Electric Power (TEP),the University of Arizona, and the Arizona Research Insti-tute for Solar Energy. TEP suggested that forecasts of PVpower production could be useful if possible. The fundingsources had no role in study design; in the collection, anal-ysis and interpretation of data; in the writing of the report;and in the decision to submit the article for publication.

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Table 3Parameters used in Eq. (11).

Parameter Value

A 0.024rx 20 kmrt 6 hq 0.65

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