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Solar Energy 112 (2015) 68–77

Short-term reforecasting of power output from a 48 MWe solar PVplant

Yinghao Chu, Bryan Urquhart, Seyyed M.I. Gohari, Hugo T.C. Pedro, Jan Kleissl,Carlos F.M. Coimbra ⇑

Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, Center for Renewable Resource Integration, University of

California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA

Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, Center for Energy Research, University of California, San Diego,

9500 Gilman Drive, La Jolla, CA 92093, USA

Received 31 March 2014; received in revised form 14 November 2014; accepted 17 November 2014

Communicated by: Associate Editor Frank Vignola

Abstract

A smart, real-time reforecast method is applied to the intra-hour prediction of power generated by a 48 MWe photovoltaic (PV) plant.This reforecasting method is developed based on artificial neural network (ANN) optimization schemes and is employed to improve theperformance of three baseline prediction models: (1) a physical deterministic model based on cloud tracking techniques; (2) an auto-regressive moving average (ARMA) model; and (3) a k-th Nearest Neighbor (kNN) model. Using the measured power data from thePV plant, the performance of all forecasts is assessed in terms of common error statistics (mean bias, mean absolute error and root meansquare error) and forecast skill over the reference persistence model. With the reforecasting method, the forecast skills of the three base-line models are significantly increased for time horizons of 5, 10, and 15 min. This study demonstrates the effectiveness of the optimizedreforecasting method in reducing learnable errors produced by a diverse set of forecast methodologies.� 2014 Elsevier Ltd. All rights reserved.

Keywords: Real-time reforecasting; Artificial neural networks; Genetic algorithm optimization; PV generation

1. Introduction

The importance of short-term solar forecasting systemsfor renewable integration has been discussed at length else-where (Lew et al., 2010; Inman et al., 2013). The variablenature of renewable power generation is an obstacle forachieving higher level of solar penetration into the powergrid. Uncertainty in solar power generation caused by

http://dx.doi.org/10.1016/j.solener.2014.11.017

0038-092X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Mechanical and AerospaceEngineering, Jacobs School of Engineering, Center for RenewableResource Integration, University of California, San Diego, 9500 GilmanDrive, La Jolla, CA 92093, USA.

E-mail address: ccoimbra@ucsd.edu (C.F.M. Coimbra).

atmospheric processes adversely affects the stability ofpower grid and increases the capital and operational costof reserves and ancillary generators. Smart generation con-trol based on accurate generation forecasts is essential forintegrating high level of cost-competitive solar power whilemaintaining a high level of grid stability (Hart et al., 2012;Inman et al., 2013). Motivated by the pressing need formore effective predictive ability for solar integration, differ-ent solar forecasting methodologies (physics-based, imag-ing, stochastic learning and regression models, etc.) havebeen developed for various temporal horizons rangingfrom minutes to several days (Kalogirou, 2001; Li et al.,2008; Bacher et al., 2009; Huang et al., 2010; Mellit and

Y. Chu et al. / Solar Energy 112 (2015) 68–77 69

Pavan, 2010; Hassanzadeh et al., 2010; Marquez andCoimbra, 2011; Pedro and Coimbra, 2012; Lave et al.,2012; Hart et al., 2012; Marquez and Coimbra, 2013a;Marquez et al., 2013; Inman et al., 2013; Quesada-Ruizet al., 2014).

The performance of individual forecasting method can befurther improved by real-time reforecasting, i.e., by adopt-ing stochastic tools based on the analysis of the forecastand error time series. Reforecasting is mostly used inweather and climate forecasts to diagnose systematic bias,recognize model deficiencies, statistically correct forecasterrors, and run data assimilation, thereby aiding in thecalibration of forecasts and improving forecast skills andreliability (Carter et al., 1989; Kalnay et al., 1996;Krishnamurti et al., 1999; Rajagopalan et al., 2002; Hamillet al., 2004; Hamill et al., 2006; Whitaker et al., 2006;Wilks and Hamill, 2007). In this work, the application ofre-forecasting advanced here is different than the one usedfor meteorological models. Reforecasting in meteorologyis used over long periods of historical data to fine tune theparameters of deterministic models, while here the proposedreforecasting is a method to statistically improve predictivemodel in real-time using optimized stochastic learning tech-niques. In this work, the reforcasting operates as adaptiveModel Output Statistics (MOS) enhancers for each of thebaseline forecasting models.

Accordingly, reforecasting is applied for 3 distinct intra-hour forecast horizons (5, 10 and 15 min ahead) of poweroutput for a photovoltaic power station in Boulder City,Nevada. The data used for model development and testingare discussed in Section 2. The three baseline forecastingmodels: a cloud tracking based deterministic model (Det),an Autoregressive and Moving Average model (ARMA),and a k-th Nearest Neighbor model (kNN) are describedin Section 3. Section 3 also covers the smart reforecastingmodel, the GA optimization, and the statistical metricsfor performance evaluation. Results and discussions arepresented in Section 4. The main conclusions of this workare summarized in Section 5.

Fig. 1. (a) Schematic image of the analyzed solar power plant operated by Semby the polygonal grayscale panel overlays. Each shade of gray is associated withthe deterministic forecast are also denoted, along with the distance between themCopper Mountain plant.

2. Data

Power output data is obtained from a 48 MW segment(approximately 1.3 km2) of the Sempra Generation CopperMountain solar power plant (114.993� W, 35.782� N,Fig. 1a). Cadmium telluride thin film panels are installedand fixed at an elevation angle of 25� with a due southazimuth. The generated power is collected by 96 inverters,and power output data is quality controlled by inspectingthe output from each individual inverter.

Occasionally, the output measurements from a smallsubset of inverters are unavailable (less than 4 on a singleday). As a result, the analyses and forecasts presented hereare based on the average of available measurements. FromNov 1st to Dec 5th, 2011, the inverter-average power outputare archived by a OSISoft PI Historian Server maintainedby Sempra and transmitted to a similar server at theUniversity of California, San Diego (UCSD). The samplinginterval of the power output is thirty seconds.

Two Total Sky Imagers (TSIs, Fig. 1b) were installed byUCSD at the Copper Mountain solar power plant in July2011 for automatic cloud observations. The TSI uses aspherical mirror to reflect the sky hemisphere into a down-ward pointing camera. Images are captured every 30 s at aneffective resolution of 420� 420 pixels. To reduce theintensity of reflected direct solar beam (i.e. the image ofthe sun itself), a strip of black rubber tape (a “shadow-band”) is affixed to the rotating mirror. The shadowbandimproves image quality and reduces potential sensor dam-age, but covers approximately 0.70 steradians of the skyhemisphere, which is about 14% of the image region usedin the deterministic forecasting model (<80� zenith angle).

The power and imaging data (26,638 points each) arefirst paired and then split for model estimation and evalu-ation. The first 15,000 data points are used to estimate theparameters of the Det, ARMA, and kNN models. Theremaining data are randomly divided into a learning subsetand a testing subset (70% and 30%, respectively). Thelearning set (approximately 8000 data points) is used for

pra Generation. The 48 MW subset of panels used in this work is indicatedone of the 96 inverters. The locations of the two sky imaging units used in. (b) The Total Sky Imager (TSI) mounted on an inverter enclosure at the

Fig. 2. Illustration showing the ray tracing procedure used to construct ageoreferenced mapping of the shadows. The grid size shown here isenlarged for illustration.

70 Y. Chu et al. / Solar Energy 112 (2015) 68–77

ANN training and optimization, and the testing set(approximately 3500 data points) is used to evaluate theperformance of the ANN-based reforecasts.

3. Methods

This section covers discussion of the three baseline fore-cast models, the smart reforecast models, the ANN trainingand optimization, the reference persistence model, and thestatistical metrics that are used to assess the forecasts.

3.1. Cloud tracking based deterministic model (Det)

The deterministic forecast (Chow et al., 2011 andUrquhart et al., 2013) predicts the power generation basedon the cloud tracking and shadow estimation.

Clouds are detected using the ratio of the red channel tothe blue channel in the digital sky images (Johnson et al.,1989; Johnson et al., 1991). The detection process providesa binary image of cloud location in the image coordinatesystem. To obtain a georeferenced mapping of the clouds,a pseudo-Cartesian transform (following Allmen andKegelmeyer (1997)) is applied with a modified scaling thatmaps cloud positions to a planar latitude-longitude grid atthe cloud altitude (Chow et al., 2011). This transformrequires geometric calibration of the imaging system suchthat each pixel has a known look angle (zenith and azimuthcoordinate pair). Cloud altitude was obtained from lidarbackscatter measurements taken at the KHND airport.The speed and direction of clouds were determined by ana-lyzing the change in cloud position between consecutiveimages. Changes in cloud position are detected using a nor-malized cross correlation procedure, the details of whichcan be found in Chow et al. (2011). The cloud velocity isthen used to advect the planar mapping of clouds forwardin time to generate a cloud position forecast. For this work,advected cloud position is computed at forecast horizons of5, 10 and 15 min for every captured image.

The forecast domain is defined by a grid overlaying theplant that is 4� 4 km in size with a resolution of 2.5 m perforecast cell (1600� 1600 cells), and each cell is resolved toa latitude, longitude, and altitude (the latter is obtainedfrom a digital elevation model). For each forecast cell, aray is traced along the vector to the sun, and the intersec-tion with the cloud map is determined (Fig. 2). If the inter-sected point is clear sky, then the ground location isdeemed clear, whereas if the intersection is cloudy, theground point is deemed shaded by cloud. Repeating theshadow mapping process for each forecast cell constructsa map of cloud shadows. This shadow map provides thepercentage of the plant that is shaded (only cells wherepanels exist are considered). Shadow maps are constructedfor each advected cloud map. Power output is assumed tobe an area-weighted average of a characteristic shaded andunshaded power output. The characteristic shaded andunshaded power output values are determined from a dis-tribution of historical power output observations. The

method to estimate power output from the shadow mapsis site specific, and additional details of the Det forecastcan be found in Urquhart et al. (2012) and Urquhartet al. (2013).

3.2. Autoregressive Moving Average model (ARMA)

ARMA is one of the most popular statistical tools fortime series analysis and forecast (Brockwell and Davis,2002; Box et al., 2011). The ARMA model considers thelagged past values and error terms, which is mathemati-cally defined as:

P t ¼Xp

i¼1

/iP t�i þXq

j¼1

hjet�j; ð1Þ

where P is the lagged past value, and e is the lagged pasterror. The parameters p and q and the coefficients / andh are determined by model identification tools in Matlab(Box et al., 2011). Hundreds of possible combinations ofparameters are tested and the selected one (p ¼ 22; q ¼ 4)achieves the lowest Root Mean Square Error (RMSE) onthe model estimation data set (Fig. 3).

3.3. k-Nearest Neighbor model (kNN)

The kNN model is one of the simplest machine learningmethods. This model classifies patterns based on the com-parison of a current pattern with training samples in thefeature space (Pedro and Coimbra, 2012). In forecast appli-cations, kNN identifies the elements among the trainingsamples that match “current” conditions most closely intherms of some predefined features: the neighbors. Theforecast value is then determined from the collections ofsubsequent values of the neighbors.

In this work, the features consist of the lagged past val-ues. The features space, created from the forecast model

5 10 15 20 25 301

2

3

4

5

6

7

8

9

10

p

q

0.158

0.16

0.162

0.164

0.166

0.168

Fig. 3. Map of the relative RMSE for different combination of param-eters. The red dot represents the combination (p = 22, q = 4) that achievesthe lowest rRMSE(0.1563). (For interpretation of the references to colourin this figure legend, the reader is referred to the web version of thisarticle.)

Y. Chu et al. / Solar Energy 112 (2015) 68–77 71

estimation data, were assembled in a matrix Aij. Each rowof Aij represents a feature vector for a specific time in themodel estimation data.

The nearest neighbor for a new data point at time t,characterized by the feature vector pj is compared withall the rows in the Aij in terms of Eucledian distances

ei ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

jðAij � pjÞ

2q

: ð2Þ

The distance values are sorted in ascending order, and thefirst k (set to be 12 in this work) matches are selected as thek nearest neighbors used to determine the forecast value:

P̂ tþDt ¼1

k

Xk

i¼1

P ti�matchþDt: ð3Þ

3.4. Artificial Neural Network (ANN)

ANN is a powerful classification and regression toolwidely used in the modeling and forecasting of solar irradi-ance (Bishop, 1994; Mellit and Kalogirou, 2008; Marquezand Coimbra, 2011; Chu et al., 2013). ANN consists ofinterconnected signal processing elements, which are calledneurons. In this work, feed forward ANNs are used, whichonly allow forward connections. Neurons are placed in lay-ers, and the layers between the input layer (the first) andthe output layer (the last) are called hidden layers. Neuronson one hidden layer calculate the sums of the weightedinputs received from the previous layer and add a bias orthreshold to the sums. After that, the sums are processedby the neurons’ activation function (Sigmoidal functionin this work) to generate the outputs. The outputs are thenused as the inputs for neurons on the following layer.

Y ¼ fXN

j¼1

wjX j þ bj

!; ð4Þ

where X and Y are the input and output, respectively. w isthe weight, b is the bias, and f is the activation function.

The weight and bias values in each neuron are estimatedin a supervised training process with the learning data. Thetraining minimizes the error between the ANN outputs andthe targets by adjusting the weights and biases. This is aniterative process that stops once the discrepancy is lowerthan a preset value:

ðANN uðL;NÞ; I ; T Þ ����!TrainingANN p; ð5Þ

where ANN u represents the untrained feed-forward neuralnetwork with L hidden layers and N neurons per layer. I

represents the training inputs and T represents the trainingtargets. In this work, we employed a Bayesian regulariza-tion training process that uses the Levenberg–Marquardtoptimization. The resulting model ANNp was then usedto forecast power generation with new inputs In (not usedin the training phase):

P̂ ðt þ DtÞ ¼ ANN pðInðtÞÞ: ð6ÞCross validation (CVM) is used to prevent the overfit-

ting of ANN parameters to the training data. CVM is oftenused to determine whether a hypothesis model fits the dataof interest. Details of CVM implementation could be foundin (Efron and Gong, 1983; Geisser, 1993; Kohavi, 1995;Jain et al., 2000; Lendasse et al., 2003; Chu et al., 2013).In this work, the CVM divides the learning data into N

subsets Si ði ¼ 1; 2; . . . ;NÞ. Each subset Si contains inputsI i and target T i. N is set to 10 as suggested in the literature(Kohavi, 1995; McLachlan et al., 2004). One of the 10 sub-sets is used for validation, and the remaining 9 subsetsSj ðj ¼ 1; . . . ; 10jj – iÞ are used to train the ANN.

ANN uðL;NÞ; I i; T i

� � ����!TrainingANN i ð7Þ

The trained model ANN i is assessed on the validation sub-set Si and an RMSEi is obtained. This process is repeatedN times, each time subsequently using each subset as thevalidation set. Eventually, the CVM performance of avalidated model is defined as the average of all obtainedRMSEi. Higher values of average RMSE denote overfittedmodel that should be discarded. In this work, genetic algo-rithm is employed to identify the optimal model scheme(e.g. ANN structure) that has the smallest average valida-tion RMSE.

3.5. Reforecast model

The ANN-based smart reforecast model uses the predic-tions from a baseline forecast model and the measuredlagged generation values as inputs. There are ten possibleinputs available to the reforecast model. Seven of themare the time-lagged measured power generation values,with lags ranging from 0 to 30 min in steps of 5 min. Theother three are the power generation predictions from abaseline model for horizons of 5-, 10-, and 15-min.

72 Y. Chu et al. / Solar Energy 112 (2015) 68–77

GA is an efficient tool (Holland, 1992; Pai and Hong,2005; Mellit and Kalogirou, 2008; Marquez and Coimbra,2011; Pedro and Coimbra, 2012) that iteratively scanspotential solutions (individuals) of the search space to findthe optimal solution. GA optimization is initiated with ran-domly selected individuals (the population) whose fitness isevaluated according to an objective function, which is theaverage of CVM RMSEs in this work. Each individual, rep-resenting a possible scheme for ANN, is evaluated on thelearning data using CVM and the CVM performance isassigned as its fitness. Individuals with the best (lowest)fitness give highest forecast accuracy on the learning dataand are less likely to be affected by overfitting. Therefore,they are selected as parents for a new generation.

The majority of the new population is created by cross-ing-over the parents the remainder of the population is cre-ated via mutation. All the new individuals undergo thefitness calculation and this process repeats until the GAconverges (the difference between the average populationfitness and the fitness of the best individual is very small).The GA optimizations were submitted as parallel jobs to10 cores (2.39 GHz processor, 24 GB RAM). Each optimi-zation job takes about 1 day to complete.

After the GA optimization, the optimized ANN schemeis used to produce real time generation reforecast usingnew measured power and predictions from the baselinemodel. New forecasts are produced with minimal computa-tional time (less than 1 s). In this work, we develop andassess three ANN-based reforecast models for each of thethree baseline models: Det, ARMA, and kNN.

3.6. Testing

In this work, the persistence model is used as the testingbenchmark. The persistence model is the simplest forecastmodel, and it is very accurate for low variability periods.The persistence model assumes that the clear-sky index ofpower generation remains constant between t and t þ Dt:

P̂ ðt þ DtÞ ¼ P clrðt þ DtÞP clrðtÞ

� P ðtÞ; ð8Þ

where P clr is the modeled clear sky power output. Clear skyplane-of-array irradiance GIclr is modeled by computingglobal horizontal irradiance using the Kasten clear skymodel (corrected by Ineichen and Perez (2002)) and thentransposing it to the plane-of-array using the Muneertransposition model (Page, 2003). The clear sky power isthen computed as:

P clr ¼GIclr

GIstc� P stc; ð9Þ

where P stc is the rated capacity of the generator at standardtest conditions, and GIstc is the global irradiance at stan-dard test conditions (i.e. 1000 W/m2).

All the models are evaluated on the testing set defined inSection 2. The forecast performance is assessed and com-pared using 4 statistical metrics: Mean Biased Error (MBE)

MBE ¼ 1

m

Xm

t¼1

P̂ ðtÞ � PðtÞ� �

: ð10Þ

Mean Absolute Error (MAE)

MAE ¼ 1

m

Xm

t¼1

jP̂ ðtÞ � PðtÞj� �

: ð11Þ

Root Mean Square Error (RMSE)

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

m

Xm

t¼1ðP̂ðtÞ � P ðtÞÞ2

r; ð12Þ

and forecast skill (s), that measures the improvement of aforecast over the reference persistence model

s ¼ 1� RMSE

RMSEp; ð13Þ

where the subscript p denotes persistence.

4. Results and discussion

4.1. ANN model optimization

Three reforecast models, smart Det, smart ARMA, andsmart kNN, are optimized based on Det, ARMA, andkNN, respectively. The optimal set of input variables, hid-den layers, and the number of neurons per layer selected bythe majority of individuals in the converged population areshown in Table 1. These results show that the best smartreforecast models always consider the predictions fromforecast models they are based on. Table 1 also shows thatthe latest available values of generation measurements(variable number 1–3) are selected for all reforecast modelsregardless of forecast horizons. This is expected given thatthe last available generation measurements are highlyinformative for intra hour forecast. In terms of ANN archi-tecture, the optimal ANNs identified by GA always have asingle hidden layer with 7–10 neurons.

4.2. Smart reforecast

The smart reforecast models, as well as the three fore-cast models, are evaluated on the testing set that is indepen-dent of the learning process. Their performance metrics arepresented in Table 2. The results show that each smartreforecast model outperforms the persistence model,regardless of the forecast horizons. The RMSEs of allinvestigated models are illustrated in Fig. 4. The resultsshow that the optimized smart reforecast model correctsthe forecast bias and achieves higher forecast skill.

The most prominent overall enhancement is made bythe smart Det, which reduces the Det forecast RMSE byover 38% regardless of the forecast horizons (Fig. 4). Thisresult shows that the smart forecast model benefits mostfrom integrating the cloud cover information and laggedpower generation data, and outperforms the persistence

Table 1ANN inputs and parameters selected by the GA for different reforecast models and forecast horizons. Input indices represent ANN inputs: 1–7 are thetime-lagged measured power generation values and 8–10 are the forecasted power generation values from the forecast models. L represents the number ofhidden layers, and N represents the number of neurons per hidden layer.

Model 5 min forecast 10 min forecast 15 min forecast

Inputs’ indices L N Inputs’ indices L N Inputs’ indices L N

Smart Det 1 2 3 8 9 10 1 7 1 2 3 6 8 9 10 1 7 1 2 3 6 7 9 10 1 7Smart ARMA 1 2 3 4 6 8 9 1 8 1 2 3 6 7 8 9 10 1 10 1 2 3 5 6 8 9 10 1 10Smart kNN 1 2 3 4 5 6 8 9 1 7 1 2 4 5 9 10 1 10 1 2 3 4 7 8 9 10 1 8

Table 2Statistical metrics for testing results of methodologies for 5-, 10-, and 15-min horizons. The highlighted numbers identify the best performing method for agiven metric. Bold values indicate best performance.

MBE (kW) MAE (kW) RMSE (kW) S (%)

5 min 10 min 15 min 5 min 10 min 15 min 5 min 10 min 15 min 5 min 10 min 15 min

Persistence �1.1 �3 �3 21.9 30.6 33.7 41.8 52.7 57.6 0 0 0Det �31.2 �26 �19.5 48.1 45.7 43.9 71.6 68.9 68.5 �71.3 �30.7 �18.9Smart Det 0.1 �1.6 0.1 21.2 26.9 27.1 35.5 41.2 42.5 15.1 21.8 26.2

ARMA �3.2 �6.7 �7.5 21.6 29.4 32.7 39 48.5 53.7 6.7 8 6.8Smart ARMA 0.1 �1.5 1.2 20.7 26.8 29.3 36.4 44.1 46.4 12.9 16.3 19.4kNN �2.7 �7.4 �8.7 24.1 30.8 34.8 42.9 51.1 57.4 �2.6 3 0.3Smart kNN 0.1 �1 2.5 21.2 30.8 29.9 37.1 45.3 46.4 11.2 14 19.4

Forecasting Horizon (min)

RM

SE (k

W)

5 10 150

20

40

60

80

100Det. ARMA kNN Smart Det. Smart ARMA Smart kNN

Fig. 4. Plot of testing RMSEs for 5, 10, and 15 min horizons. The barswith lighter color represent testing RMSEs for target models, and the barswith darker color represent smart reforecast RMSEs. The dash linerepresents the persistence forecast RMSE for each horizon. (Forinterpretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

Y. Chu et al. / Solar Energy 112 (2015) 68–77 73

model with forecast skills ranging from 15.1 to 26.1%,depending on the forecast horizons.

ARMA yields the highest overall forecast skills (around7%) among the three forecast models. However, the RMSEreduction achieved by smart ARMA is the smallest amongthe three smart models (average decrease of 10%). Theforecast skill of smart ARMA is significantly lower thanthe forecast skill of smart Det.

The overall performance of the kNN forecast is parallelto the persistence forecast. The feature space of kNNemploys 15,000 data, which may not be enough to reflectall possible atmospheric conditions. The smart kNNachieves moderate RMSE reduction among the threesmart reforecast models. The forecast skill of smart kNNis inferior to both smart Det and smart ARMA. A larger

collection of feature space is expected to improve theperformance of both kNN and smart kNN models.

To quantitatively study how models perform undervaried meteorological conditions, the forecast performanceis assess as a function of variability V (Marquez andCoimbra, 2013b). The V is defined as the standard devia-tion of the step changes of the clear sky index kt:

V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

Xt

t�fhðkt � kt�1Þ2

r; ð14Þ

where fh is the forecast horizon, and each step is 30 s. Theadaptability of a model for different power output variabil-ity conditions can be evaluated by analyzing its perfor-mance against different V levels.

The plots of absolute forecast error jej versus V for allinvestigated models are plotted in Fig. 5. These figuresshow that the performances of most models except theDet are comparable during the low V periods (V < 0:01).During these periods, the power output is very stableresulting in low forecast error.

As expected, the forecast errors for all models increasewhen V increases. For both moderate V periods(0:015 < V < 0:035) and high V periods (V > 0:035), smartreforecasts achieve lower mean absolute forecast errors. Ingeneral, the Det model shows inferior performance to othermodels for all forecasts horizons. Several reasons canexplain this finding: (1) misidentification of clear andcloudy pixels within each image, especially near the sun,are caused by contamination of the clear sky library usedin the detection process (Chow et al., 2011); (2) cloudheight obtained from KHND may not reflect the heightof clouds over the Copper Mountain solar plant causinginaccurate georeferencing of clouds; (3) a planar treatment

Fig. 5. Average absolute forecast error (e) versus variability V for (a) 5-min-ahead, (b) 10-min-ahead, and (c) 15-min-ahead forecasts. Each mean absolutee is taken within the variability bin with a width of 0.005.

74 Y. Chu et al. / Solar Energy 112 (2015) 68–77

of the clouds ignores the fully 3D nature of clouds whichdegrades the performance of geometrically estimatingwhere clouds will cast shadows; (4) the shadowbandobstructs a large volume of the sky over the plant whereclouds may be causing line-of-sight obstructions betweensolar panels and the sun; (5) oversimplification of the rela-tionship between shadow fraction and average power out-put. A discussion of these issues can be found in Urquhartet al. (2013) and Gohari et al. (in press). Despite this, thereforcast based on Det, which considers the input variablesof both persistence and Det models, generates the mostconsistent and accurate forecast during different variabilityconditions. Moreover, it significantly outperforms othersmart reforecast models during the moderate V periods,particularly for the 10- and the 15-min horizons.

−0.4 −0.2 0 0.2 0.40

1

10

100

1000

Error (MW)

Freq

uenc

y

PersistenceDetSmart Det

−0.4 −0.20

1

10

100

1000

Erro

Fig. 6. Plots of error (e) distributi

−0.4 −0.2 0 0.2 0.40

1

10

100

1000

Error (MW)

Freq

uenc

y

PersistenceDetSmart Det

−0.4 −0.20

1

10

100

1000

Erro

Fig. 7. Plots of error (e) distributio

The performance of ARMA and kNN models are notsignificantly superior to the persistence forecast for lowand moderate V periods (sometimes it is even worse thanthe persistence forecast). But both achieve significantimprovement during the high V periods. Unlike the smartdeterministic model, their forecast errors are not consis-tently reduced in the moderate V periods.

To further understand the models performance, theirerror (e) distributions for the three investigated horizonsare studied in Figs. 6–8, respectively. The standarddeviation, Pearson’s skewness, and kurtosis for each distri-bution are calculated and listed in Table 3. Standard devia-tion is used to quantify the level of dispersion for thedistributions. Skewness (defined as (mean-mode)/Standarddeviation) is used to quantify the level of forecast bias.

0 0.2 0.4r (MW)

PersistenceARMASmart ARMA

−0.4 −0.2 0 0.2 0.40

1

10

100

1000

Error (MW)

PersistencekNNSmart kNN

ons for 5 min ahead forecasts.

0 0.2 0.4r (MW)

PersistenceARMASmart ARMA

−0.4 −0.2 0 0.2 0.40

1

10

100

1000

Error (MW)

PersistencekNNSmart kNN

ns for 10 min ahead forecasts.

−0.4 −0.2 0 0.2 0.40

1

10

100

1000

Error (MW)

Freq

uenc

y

PersistenceDetSmart Det

−0.4 −0.2 0 0.2 0.40

1

10

100

1000

Error (MW)

PersistenceARMASmart ARMA

−0.4 −0.2 0 0.2 0.40

1

10

100

1000

Error (MW)

PersistencekNNSmart kNN

Fig. 8. Plots of error (e) distributions for 15 min ahead forecasts.

Table 3Statistical results of the error distribution analysis.

Standard Deviation (MW) Skewness Kurtosis

5 min 10 min 15 min 5 min 10 min 15 min 5 min 10 min 15 min

Persistence 0.0418 0.0527 0.0575 �0.0257 �0.0564 �0.0516 12.3055 8.3966 8.3158Det 0.0645 0.0638 0.0657 �0.4834 �0.407 �0.2976 5.7441 5.6172 6.1051Smart Det 0.0355 0.0412 0.0425 �0.0037 �0.0387 0.0032 10.6125 7.2039 6.6759ARMA 0.0389 0.0481 0.0531 �0.0818 �0.1389 �0.1417 11.4599 8.6004 8.2282Smart ARMA 0.0364 0.0441 0.0464 0.0019 �0.0348 0.025 10.0702 8.1636 6.7361kNN 0.0428 0.0506 0.0567 �0.0632 �0.1461 �0.1532 8.8997 7.3065 7.4041Smart kNN 0.0371 0.0453 0.0464 0.003 �0.0224 0.0536 9.0757 7.565 6.2277

Y. Chu et al. / Solar Energy 112 (2015) 68–77 75

For example, a distribution with a longer or fatter right tailcompared to the left tail yields a positive skewness. Kurtosisis used to quantify the “peakness” of the distributions andheaviness of their tails, and is defined as the distributions’fourth central moment divided by the fourth power of theirstandard deviation. For example, a high kurtosis corre-sponds to a distribution with a sharper peak and longer,heavier tails.

The negative skewnesses (-0.026 to �0.0516) of the per-sistence error distributions indicates that they lean slightlyto the left side for all the three forecast horizons. Theyshow moderate level of deviation but the high kurtosis indi-cating a sharp central peak. This is the result of excellentperformance of the persistence forecast for low V periods,which comprises more than 50% of the testing period. Nev-ertheless, performance of persistence forecast degrades as V

Fig. 9. Sample time series of 10-min horzion power generation forecasts and aband smart ARMA, and (c) kNN and smart kNN.

increases, so these distributions also show fat tails, result-ing in relatively high standard deviation.

The error distribution of Det forecasts also shows a neg-ative skewness with highest absolute value among all distri-butions (<�0.3), which agrees with the highest magnitudeof the testing MBE (<�20 kW). This result suggests thatthe Det is more likely to underestimate the power genera-tions, resulting in overestimating of ramp rates, as evi-denced in Fig. 9a. The lowest kurtosis and higheststandard deviation for the Det forecasts indicates a highRMSE. The error distributions of smart Det achieves a sub-stantially lower magnitude of skewness (<0.01) than the Detfor all forecast horizons, indicating that the smart Det suc-cessfully identifies and reduces the systematic forecast bias.Because the smart forecast estimates a more reasonableramp rate, it reduces the occurrence of high absolute value

solute forecast errors: (a) deterministic and smart deterministic, (b) ARMA

76 Y. Chu et al. / Solar Energy 112 (2015) 68–77

errors (shown in Fig. 9a). These together produce an errordistribution with shorter tails, rounder shoulders, and amoderate kurtosis, the lowest standard deviation and low-est RMSE.

The ARMA forecasts have lower value of skewness thanthe persistence model, indicating that it underestimates thepower generation more often. The error distributions havea relatively low standard deviation and relatively high kur-tosis, which represents a narrowed distribution with lessoccurrence of high absolute value error. This is due tothe nature of the ARMA model that only considers thelag time series in the past 30 min. Thus, it estimates theramp only as a more sophisticated persistence model. Anexample is presented in Fig. 9b, where the ARMA forecasttime series follows the overall power generation profile butmisses most of the large ramps. The smart ARMA demon-strates similar characteristics resulting in the lowestimprovement among the three smart models. SmartARMA identifies and reduces the systematic bias and pro-duces a low skewness. However, smart ARMA does notimprove the accuracy of ramp rate estimation as signifi-cantly as the smart Det (Fig. 9a and b). These togetheryield a right shift of the error distributions, which reducesthe occurrence of negative errors with the side effect of aslightly increase in occurrence of positive errors (shownin Figs. 6–8).

Finally, the kNN method is similar to the zero skill ofthe persistence forecast. The error distribution of thekNN forecasts has relatively high standard deviationsand a large absolute magnitude of skewness. It has the low-est kurtosis and highest standard deviations among allsmart forecasting models (see Fig. 9c). Negative skewnessis observed for all target models, which can be explainedby the fact that solar power gradually decreases duringthe period under study: the period (early November) usedfor the forecast model estimation has higher average clearsky irradiance than the testing period (late Novemberand early December). Therefore, the parameters of themodels determined on the estimation set tend to overesti-mate the power generation in the testing set. This tendencyshows that the seasonal factors need to be readjusted for all3 power generation forecast models. Smart kNN improvesthe performance of its target model in the same way assmart ARMA.

5. Conclusions

Real-time reforecasts are developed to enhance theintra-hour generation forecasts for an operational 48MWe PV plant. The smart reforecasting methods, whichare based on GA-optimized ANNs, reforecast the poweroutputs from the solar plant for each of the three models:a cloud tracking based deterministic model, an auto-regres-sive moving average model, and a k-th nearest neighbormodel. Assessment on the independent data shows thatthe smart reforecasts add the following benefits to theforecasts:

1. Identify the bias of the baseline forecasting models andsubstantially reduce the mean bias error.

2. Statistically predict more accurate ramp rates.3. Reduce the absolute forecasting error for both moderate

and high variability periods.4. Reduce the occurrence of large magnitude errors and

therefore significantly narrow the error distribution.5. Improve the overall forecasting skill for multiple fore-

cast horizons (5, 10 and 15 min).

These results demonstrate the potential of smart real-time reforecasting for short-term solar integration and mit-igating solar variability due to cloud cover. This work isparticularly relevant to inverter control, regulation, fastdemand control and real-time dispatch.

Acknowledgements

The authors gratefully acknowledge the partial financialsupport given by the California Public Utilities Commis-sion (CPUC) under the California Solar Initiative (CSI)Program, and the partial support by the National ScienceFoundation (NSF) EECS (EPAS) Award No. 1201986,which is managed by Dr. Paul Werbos.

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