Analysis of Solar Generation and Weather Data in Smart Grid with Simultaneous Inference of Nonlinear Time Series

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1 Te First International Worksop on Smart Cities and Urban Informatics 215 Analysis of Solar Generation and Weater Data in Smart Grid wit Simultaneous Inference of Nonlinear Time Series Yu Wang, Guanqun Cao, Siwen Mao, and R. M. Nelms Department of Electrical and Computer Engineering, Auburn University, Auburn, AL Department of Matematics and Statistics, Auburn University, Auburn, AL Abstract Smart Grid is an important component of Smart City, were more renewable power generation and better energy management is required. Forecast on renewable power generation, from sources suc as solar and wind, is crucial for better energy management. However, te current forecast metods lack a compreensive understanding of te natural processes, and are tus limited in precise prediction. In tis paper, we introduce simultaneous inference to analyze te solar generation and weater data for better predictions. We first introduce a local linear model for nonlinear time series, and present te construction of te simultaneous confidence bands (SCB) of te time-varying coefficients, wic provide more information on te dynamic properties of te model. We ten use te simultaneous inference for solar intensity prediction using a real trace, were te superior performance of te proposed sceme is demonstrated over existing approaces. I. INTRODUCTION In recent years, as te development of te modern tecnologies in informatics, communication, control and computing, our living environment is becoming smart. Smart Home and Smart City ave gradually become part of our lives, and are no longer merely future concepts for te public. An important component of Smart City is te Smart Grid (SG), wic is regarded as te next generation power grid to create a widely distributed energy generation and delivery network. Te SG features te incorporation of power generation from renewable energy sources, especially solar and wind, wic meanwile requires a better energy management system in te SG [1]. Energy management in te SG as been studied in many previous works [1] [3]. It is indicated in [3] tat ig efficient power management cannot be realized witout a better forecast on te grid load and renewable power generation in te SG. Te problem of grid load forecasting as been studied by many researcers wit different tecniques suc as state space models [4], artificial neural networks and support vector macine (SVM) [5], and nonparametric functional time series analysis [3]. Te prediction on te renewable energy generation in te SG as also attracted great interest. Predictions on solar and wind generation can be found in [6] and [7] using SVM regression and JPDF forecast respectively. Because of te weater dependence nature of te forecasting problem, statistical metods can be found in almost every related literature. On te oter and, te wide range of applications elps te improvement of te statistical teory on nonparametric analysis [8] [1], and nonstationary time-series analysis [11], [12]. Recently, te autors of [13], [14] propose te metod of constructing a simultaneous confidence band (SCB) for time-varying coefficients. Tese researc advances improve te understanding of time-series, and provide better tecniques in renewable energy generation prediction. Te previous work on predicting solar power generation provides acceptable results using SVM regression [6]. However, by simply trying different SVM kernels after some basic data processing statistically, it lacks a deep analysis of te solar power generation and weater data, and tus is limited in precise predictions of oter data set. For example, te ceck of assumptions is missing on independence of variables and errors. Furtermore, te renewable energy generation is a function of weater variables, and is a stocastic process on nonlinear time series. Terefore, te relationsip between te weater variables and te power generation sould be analyzed over a long time for a compreensive understanding of teir dynamic relations. For example, a coefficient varying by time overall may stay constant for sort periods. Tese drawbacks will limit te applications for predictions in oter cases. Terefore, a metod tat can capture te dynamic property of te process would be igly desirable for better predictions on solar power generation in different scenarios. Motivated by tis observation, we introduce simultaneous inference of nonlinear time series proposed in [13] for understanding compreensively te deep and dynamic relationsip between renewable power generation process and te weater variable processes. Te simultaneous inference is based on te SCB of time-varying coefficients in te local linear model, wic we use for nonlinear time series analysis. It is based on te assumption of nonstationary processes for te error and weater variables, wic matces te case of our problem were many weater variables are sown to ave an obvious seasonal pattern, meaning te observations are not stationary assumed by many oter forecast tecniques. And te SCB sows te confidence bands of te coefficients over any lengt of time, wic can be used to test if te coefficients are truly time-varying or not. Tis elps us to refine te model by omitting variables wic tat are not significant. Te main contribution of tis paper is te introduction of /15/$ IEEE 6

2 Te First International Worksop on Smart Cities and Urban Informatics 215 te local linear model and SCB for analyzing te solar power generation as a nonlinear time series. By cecking te dynamic properties of te coefficients from its SCB, we are able to acieve a more compreensive understanding of te model, and based on tis, we can furter refine te model and use it for predicting te renewable energy generation. As an example, we apply it in predictions on daily solar power generation. Tis metod as a wide range of applications, wic is not limited to analyze and predict te solar energy generation. It can also be used for predictions in different time scales, from minutes to monts, depending on different purposes, and in oter cases, suc as wind power generation prediction. Te remainder of tis paper is organized as follows. We present te local linear model for nonlinear time-series analysis in Section II. Te construction of SCB for time-varying coefficients wit simulated results is introduced in Section III. We use te simultaneous inference for analyzing a trace of solar intensity and weater data in Section IV. Section V concludes tis paper. II. LOCAL LINEAR MODEL FOR NONLINEAR TIME SERIES We consider te power generation from a renewable source as a continuous-time stocastic process Y (t) and a function of te meteorological variables X(t), wic is a continuous-time covariate process. It follows tat Y (t) = f( X(t)), t R. To identify te function f( ), it is straigtforward to try a linear model first, as Y (t) = X T (t) β(t) + ɛ(t), t R, (1) were X(t) = (1, X1 (t),..., X p 1 (t)) T and β(t) = (β 1 (t),..., β p (t)) T are bot p 1 vectors, and ɛ(t) is te error at time t. To use tis model, we need to predict te regression coefficients β(t) at eac time t, so tat we can calculate Y (t) given te forecast on weater variables X(t). It is not difficult to obtain β(t) using parametric smooting metods suc as multiple linear regression. Altoug tis model can indicate a certain level of interactions between te variables X(t) and te response Y (t), it cannot be used to represent te real underlying process, especially wen te metod treat continuous-time series simply as discrete data points. As we will sow in Section IV-D, te prediction wit te linear model is not satisfactory, and cannot be used in practice. From te linear model, we notice tat te linearity is fairly strong between X(t) and Y (t) witin a sort time period, i.e., several days or a week. If we take advantage of tis property and consider te process as continuous-time, te model would be closer to te real process. For t i close to t, we can ave β(t i ) β(t) + (t i t) β (t), were β (t) is te derivative of β(t), and tus for any time t i close to t, we ave te local linear model as [15] Y (t i ) = X T (t i )( β(t) + (t i t) β (t)) + ɛ(t i ), t i t ±, (2) were te bandwidt is te size of te local neigborood. Tis model divides te time series into periods and creates linear models using local data. Tis way, we treat te data as a continuous-time series, and exploit te strong correlations between close time periods in weater dependent systems. A. Local Linear Estimation To identify te time-varying coefficients β(t), te least squares metod for linear regression can be used. We also add some weigts on te terms considering tat contributions from different neigbors are different, wic means a closer neigbor would ave a stronger effect, wile a furter neigbor weaker effect. Usually a kernel function K( ) is assigned to eac point, wic is a symmetric density function defined on [-1,1] [15]. Here, we use a popular Epanecnikov kernel. { 3(1 a K(a) = 2 )/4, if a 1, if a > 1, wic decays fast for remote data point. We ten ave te following weigted least squares problem to solve. ( ) min : (Y (t i ) X T (t i )( β(t) (t i t) β ti t (t)))k. t i t± (3) At eac time t, we solve for te coefficients ˆ β (t) and ˆ β (t) under te bandwidt. Suppose te total number of observations is n, we can pick t i simply as t i = i/n, 1 i n, and denote Y (t i ) as y i and X(t i ) as x i. From [9], we can solve (3) by calculating te following matrices S k (t) and R k (t): n ( ) k ( ) S k (t) = x i x T ti t ti t i K /(n) (4) i=1 n ( ) k ( ) ti t ti t R k (t) = x i y i K /(n), (5) i=1 were k =, 1, 2,... We ten ave ( ) ˆ β (t) ˆ = β (t) B. Selection of Bandwidt ( S (t) S T 1 (t) S 1 (t) S 2 (t) ) 1 ( R (t) R 1 (t) ). (6) To solve problem (3) for te complete model using (4) to (6), we need to first fix bandwidt. As discussed, is te bandwidt determining te size of data used to estimate for a local linear model at time t. If is too small, many useful points are not included for estimation, wic may cause a larger error in te model; if it is too large, more remote points are included, wic increases te computational complexity and reduces te smootness of te model. Terefore, it is important to coose a proper. Popular bandwidt selection tecniques can be found in [15], [16] for different applications. Te tecniques are different for constant bandwidt and variable bandwidt. For constant bandwidt selection considered in our model, we adopt te generalized cross-validation (GCV) tecnique [16], wic is suitable for a wide range of applications. Similar to multiple linear regression, te coefficients β are estimated from te observed data Y and X. Tus, a square 61

3 Te First International Worksop on Smart Cities and Urban Informatics 215 at matrix H() exists for ˆ Y = H() Y [17], depending on te bandwidt. Ten we can coose te bandwidt by { ˆ } Y Y 2 ĥ = arg min n(1 tr{h()}/n) 2, (7) were, tr( ) is te trace of te matrix, and n is te number of total observations. III. SIMULTANEOUS CONFIDENCE BAND FOR TIME-VARYING COEFFICIENTS In tis section, we introduce te basic conditions and construction of te simultaneous confidence band (SCB) metod proposed in [13], and ten discuss its implications to furter understand te modeling and predicting for te power generation process from te renewable energy sources based on te weater data. A. Model Assumptions and Asymptotic Normality Different from most current models for time series, te approac of SCB analysis assumes locally stationary processes for bot X(t) and ɛ(t) [12]. Te locally stationary process guarantees te stationary property for local time series, and is useful for local linear estimation. It actually belongs to a special class of non-stationary time series as x i = G(t i, F i ), ɛ i = H(t i, F i ), i = 1, 2,..., n, (8) were G(t i, F i ) and H(t i, F i ) are measurable functions well defined on t i [, 1], F i = (..., ξ i 1, ξ i ) wit {ξ i } i Z are independent and identically distributed (i.i.d.) random variables, and E(ɛ i x i ) =. In our model for renewable power generation processes, we furter assume tat {ɛ i } i Z are i.i.d. and dependent of {ξ i } i Z. Based on te above assumptions, te central limit teorem for ˆ β(t) states tat: supposing n and n 7 [13], ten for any fixed t (, 1), (n) 1/2 { ˆ β(t) β(t) 2 β (t)µ/2} N{, Σ 2 (t)}, (9) were µ = R x 2 K(x) dx, (1) Σ(t) = (M 1 (t)λ(t)m 1 (t)) 1/2, (11) M(t) = E( G(t, F ) G(t, F ) T ). (12) Te covariance matrix Λ(t) can be furter approximated using tecniques proposed in Section III-B. B. Simultaneous Confidence Band Deriving from te central limit property and basic assumptions sown above, te 1(1 α)% asymptotic simultaneous confidence tube of β C (t) can be constructed using te following formula: β C, (t) + ˆq 1 α ˆΣC (t)b s, (13) were β C, (t) is te bias corrected estimator defined in (14), B s = { z R s : z 1} is te unit ball, and s is te rank Algoritm 1: Construction of SCB for Time-varying Coefficients 1 Find a proper bandwidt ĥ from te GCV selector (7); 2 Let = 2ĥ and calculate β C, (t) using (14) and (3); 3 Obtain te estimated (1 α)t quantile ˆq 1 α via te bootstrap metod; 4 Estimate ˆM(t) = S (t ) and ˆΛ(t) by (??), and calculate ˆΣ C (t) according to (15); 5 Construct te 1(1 α)% SCB of β C (t) using (13). of a matrix C p s, wic we use for coosing different linear combinations of β(t), and β C (t) = C T β(t). To obtain te SCB, we simply take s = 1 in (13), and te SCB is constructed similarly to te confidence interval of te coefficients of te multiple linear regression: ˆβ ± tα/2,n p se( ˆβ), were se( ˆβ) is te standard error of ˆβ, and t α/2,n p is te upper α/2 percentage point of te t n 2 distribution [17]. Similarly, te first term is te estimator of te time-varying coefficients corrected for bias by β C, (t) = C T β (t) ( = C T 2 ) β / 2 (t) ˆ β (t), (14) were βĥ(t) can also be acquired by solving (3) using a corresponding kernel function K (a) = 2 2K( 2a) K(a) and an updated bandwidt = 2ĥ of te GCV selector ĥ. Te second term in (13) ˆq 1 α is actually te upper α/2 percentage point of te normal distribution N{, Σ 2 (t)} defined in (9), wile te tird term ˆΣ C (t) is te estimated standard error. Te metod of wild bootstrap is applied to obtain ˆq 1 α. Firstly, generate a large number i.i.d. vectors v 1, v 2,..., N(, I s ), were v i R p and I s denotes te s s identity matrix, and ten calculate q = sup t 1 n i=1 v ik ((t i t)/ )/(n ) ; repeat te previous step for a large number of times (say, 5) to acquire te estimated 1(1 α)% quantile ˆq 1 α of q. Te estimate of te standard error in (13), ˆΣ C (t) is defined similarly as (11). ˆΣ C (t) = (C T ˆM 1 (t)ˆλ(t) ˆM 1 (t)c) 1/2. (15) We sall estimate ˆM(t) and ˆΛ(t) respectively. From te definition of M(t) in (12), it can be estimated by ˆM(t) = S (t ), were S ( ) is defined in (4), and t = max{, min(t, 1 )}. To obtain ˆΛ(t), we first define two p 1 vectors Z i = x iˆɛ i and W i = m j= m Z i+j, a matrix Ω i = W iw T i /(2m + 1), and a function g(t, i) = K((t i t)/τ)/ n k=1 K((t k t), were m and τ can be simply cosen as m = n 2/7 and τ = n 1/7. Ten ˆΛ(t) can be calculated by ˆΛ(t) = n i=1 g(t, i)ω i. Tis way, we are able to calculate te SCB using all te estimates. Te above steps for constructing te SCB are summarized in Algoritm 1. C. Furter Discussions Te SCB provides a dynamic and compreensive view on β(t). In simple linear regression, te confidence interval 62

4 Te First International Worksop on Smart Cities and Urban Informatics 215 provides a measure of te overall quality of te regression line [17]. Similarly, te SCB illustrates te overall pattern of β(t) and tus te accuracy of te model. Confidence bands wit smaller widt implies a better model wit smaller variability, wile too wide confidence bands are limited in use. Note tat te SCB is constructed under a complete analysis on te continuous-time assumption, wic is not merely te connections of te point-wise confidence intervals on isolated time instances. Furter, te SCB can also be used to test weter te coefficients β(t) are truly time-varying or not. If a orizontal line is covered by te SCB of a β k (t), we accept te ypotesis tat β k (t) is constant and not time-varying. Furtermore, in different cases, we can construct te SCB for different linear combinations of β k (t) s by setting a different matrix C p s. For example, if we set C p 1 = [1, 1,,...], we obtain te SCB of β C (t) = C T β(t) = β 1 (t) + β 2 (t); if we set C p 2 = [1,,,...;, 1,,...], te SCB of β 1 (t) and β 2 (t) becomes a tube at any time t. Tis is because wen s = 2, te unit ball B 2 turns to a unit circle from a unit interval. Tis provides us a convenient way to furter test te model. D. Algoritm Performance for Simulated Processes We now simulate a model wit X(t) and ɛ(t) locally stationary processes discussed in Sec. III-A, and construct te 9% and 95% SCB respectively for a given model, to test te correctness by comparing wit te true results. We use te following local linear model wit time-varying coefficient: y i = β 1 (i/n) + β 2 (i/n)x i + ɛ i, (16) were β 1 (t) = cos(2πt)/4, and β 2 (t) = exp{ (t 1/2) 2 }/2. Define H(t i, F i ) = (1/2) j= a(t i) j ξ i j, G(ti, F i ) = (1; j= b(t)j ε i j ), were ξ k and ε l are i.i.d. N(, 1). Ten x i and ɛ i can be generated using (8), for i = 1, 2,..., n. For te above setting, we generate 5 samples of size 5, and for eac sample SCB is constructed wit bandwidts setting from.1 to.3 of step.25. We use 3 and 5 bootstrap samples to estimate ˆq 1 α for α =.1 and α =.5 to sow te effect of te sample size on te results. Te simulation results are sown in Table I, were te coverage rate and widt of SCB for β 2 (t) wit different bandwidts at 9% and 95% levels are listed. It sows tat te coverage rate is close to te nominal level wit most bandwidts. And te bandwidt selected by GCV is.22, wic yield fairly good results. We also notice tat ˆq 1 α are affected by te bootstrap sample size, and its value affects te widt of SCB directly. Terefore, for practical application, a large size of bootstrap samples is very important. According to our numerical studies, at least 5 samples are suggested. IV. APPLICATION TO SOLAR ENERGY GENERATION In tis section, we apply te SCB analysis to modeling te solar power generation process and predicting te generation amount based on weater data. We also compare results wit tat of oter metods. TABLE I THE COVERAGE PROBABILITIES OF SCB FOR β 2 (t) AND QUANTILES OF ˆq AT NOMINAL LEVEL OF 9% AND 95% Solar Intensity β 2 (t) 3 Samples 5 Samples bandwidt 9% 95% ˆq.9 ˆq.95 ˆq.9 ˆq A. Data Description Fig. 1. Daily solar intensity for 21 and 211. As an application, we consider te data from te UMASS Trace Repository [18], wic records te solar power generation by solar intensity in watts/m 2, and te data of several weater metrics from January, 21 to February, 213. It recorded te weater data every five minutes. Many weater parameters were observed in details. We consider five main variables, including temperature, umidity, dew point, wind speed, and precipitation. Te data as been used in [6] for a study of te statistical relationsip between te weater variables and solar power generation. Te paper also predicted solar power generation using multiple linear regression and Support Vector Macines regression. Our purpose is also to investigate te dynamic association between te weater variables and solar power generation, and to elp better predict te solar power generation. We plot te daily solar intensity for 21 and 211 in Fig. 1. An apparent seasonal pattern is sown wit peak in summer and valley in winter. It would be elpful to consider te seasonal patterns for forecast. It is also interesting to see a similar pattern for daily observations, and similar patterns can also be seen for te oter weater variables, suc as temperature, umidity, and dew point (See [6]). Fig. 1 also 63

5 Te First International Worksop on Smart Cities and Urban Informatics 215 sows a strong correlation between two consecutive days, wic means te solar generation process is not i.i.d. As discussed in Section III-A, we do not require te observations to be i.i.d. to construct te SCB. B. Prediction Model As in Section (II), we use te following local linear model. y i = β 1 (i/n) + 5 β p (i/n) x p,i + ɛ i, for i = 1,..., n, (17) p=2 were y i is te solar intensity, x p,i, p = 2, 3, 4, 5, represent te series of temperature in Fareneit, umidity in percentage, dew point in Fareneit, wind speed in miles per our, and precipitation in inces, respectively. We use n = 73 observations of 211 and 212 for local linear regression and te model is represented in a daily pattern. Note our model and analysis can be built on any time scale. We take daily pattern for an application example ere, were β 1 ( ) is te intercept and β p ( ) are te associated coefficients for x p,i. C. Simultaneous Inference for Time-varying Coefficients We now perform te SCB analysis. We center all te weater variables on teir averages so tat te intercept β 1 ( ) can be interpreted as te expected solar intensity. From GCV, we select te bandwidt =.25. Te 95% SCB of te coefficients β p ( ) are sown from Fig. 2 to 7. In eac figure, te middle tick solid curve is te estimated series for te variable; te upper and lower solid curves are te envelops for te simultaneous confidence band for eac variable. From te SCB, we are able to test weter a coefficients is significantly associated wit te solar intensity, wic equals to test: H : β p (t) =, t [, 1]; v.s. H 1 : β p (t), t [, 1]. If te zero line is included in te SCB, we accept te ypotesis tat te coefficient is not significant and could be omitted from te model; oterwise, we keep it in te model. We can also test weter te coefficients are constant, by attempting to include a constant orizontal line into te SCB. Tis is equal to testing: H : β p (t) = c p, t [, 1]; v.s. H 1 : β p (t) c p, t [, 1], were c p is a constant of eac p. If te line is covered, we accept tat te coefficient is constant; oterwise, it is not. In Fig. 2, te curve indicates te expected solar intensity for two years, and illustrates an obvious seasonal pattern. Te widt of te 95% SCB of β 1 (t) is so narrow tat no orizontal line can be covered, and even a iger level of 98% SCB cannot cover a orizontal line. We are confident tat te solar power generation is time-varying, te same as te natural process. As we center all te weater variables on teir averages, te SCB of te β p (t) actually indicates te effect on te solar intensity. In eac figure of Fig. 3 to 5, te zero line is not covered, wile in Fig. 6 and 7, te zero line is covered by te 95% SCB. Terefore, we can conclude tat for a level of 95%, temperature, umidity and dew point ave a TABLE II RMS-ERRORS IN watts/m 2 FOR TLLE, SVM AND MLR TLLE SVM MLR RMS-Error strong effect on solar generation, but te effect from wind and precipitation are weak. Also, we accept β 1 (t) to β 4 (t) as timevarying coefficients, because a constant orizontal line cannot be covered entirely in tose SCBs. Note te SCB associated wit wind in Fig. 6 sows some variations. Altoug te zero line may not be covered by a narrower SCB, say 9% SCB, at te 95% significant level, we do not accept β 5 (t) as a non-zero function.te SCB associated wit precipitation in Fig. 7 is also too wide for β 6 (t) to be accepted as a non-zero function. D. Comparisons wit Oter Models on Prediction Results From te above discussions, we could exclude te variable of precipitation and simplify te model for better prediction. We use te model to predict te daily solar intensity for January and February in 213. Te weater information of te previous year is used as te weater forecast, and te time for prediction is set from day 366 to 423, using data around January and February in 212. In oter words, te predictions are made using te data around te same time in te previous year. Te results are sown in Fig. 8. Te upper grap is te prediction curve made by te time-varying local linear model (TLLM) and te actual observations; te lower one sows te results from SVM regression used in [6]. We also perform te multiple linear regression (MLR) as in (1). But te prediction is too poor to be sown as a comparison ere. From Fig. 8, we can see tat te TLLE predicted curve tracks te actual observations better tan te SVM regression. And it is also sown in Table II tat te root mean squares error (RMS-Error) between te predicted series and te observations for TLLE is watts/m 2, smaller tan tat of SVM and MLR, wic are watts/m 2 and watts/m 2 respectively. Note tat te SVM regression depending igly on te selection of te parameters and te kernels, and tus, is not practical in many cases lacking a compreensive understanding of te real model. For example, te kernel function cosen for daily prediction is not guarantied to perform well in weekly prediction. However, te TLLE analyzes te model using simultaneous inference, wic reflects te overall pattern of te dynamic pattern of te regression functions. Terefore, it can be used in many oter applications, suc as te ourly or weekly solar power generation forecast were te time scale is set in our or week. V. CONCLUSIONS In tis paper, we proposed te simultaneous inference for weater dependent power generation from renewable energy sources, suc as solar and wind. We first introduced te local linear model for time series, and presented te construction of te SCB for time-varying coefficients. We ten performed te SCB analysis wit a trace of solar intensity and weater 64

6 Te First International Worksop on Smart Cities and Urban Informatics Intercept Temperature 2 1 Humidity Fig % SCB for Intercept Fig % SCB for Temperature Fig % SCB for Humidity x Dew Point Wind Percipitation Fig % SCB for Dew Point Fig % SCB for Wind Fig % SCB for Precipitation. Solar Intensity (watt\m2) Solar Intensity (watt\m2) Observed TLLE Observed SVM Fig. 8. Comparisons of predictions on solar intensity. data to validate te efficacy of te proposed approac. Te presented model was sown to outperform an existing metod for solar intensity prediction. ACKNOWLEDGMENT Tis work is supported in part by te US National Science Foundation under Grant CNS REFERENCES [1] X. Fang, S. Misra, G. Xue, and D. Yang, Smart grid te new and improved power grid: A survey, IEEE Commun. Surveys Tuts., vol. 14, no. 4, pp , Apr [2] Y. Wang, S. Mao, and R. Nelms, Distributed online algoritm for optimal real-time energy distribution in te smart grid, IEEE Internet Tings J., vol. 1, no. 1, pp. 7 8, Feb [3] M. Caouc, Clustering-based improvement of nonparametric functional time series forecasting: Application to intra-day ouseold-level load curves, IEEE Trans. Smart Grid, vol. 5, no. 1, pp , Jan [4] V. Dordonnat, S.J. Koopman, and M. Ooms, Dynamic factors in periodic time-varying regressions wit an application to ourly electricity load modelling, Comput. Stat. Data Anal., vol. 56, no. 11, pp , 212. [5] H.S. Hippert, C.E. Pedreira, and R.C. Souza, Neural networks for sortterm load forecasting: A review and evaluation, IEEE Trans. Power Syst., vol. 16, no. 1, pp , Feb. 21. [6] N. Sarma, P. Sarma, D. Irwin, and P. Senoy, Predicting solar generation from weater forecasts using macine learning, in Proc. IEEE SmartGridComm 11, Oct. 211, pp [7] S. Zu, M. Yang, M. Liu, and W.J. Lee, One parametric approac for sort-term jpdf forecast of wind generation, in Proc. 213 IEEE IAS Annual Meeting, Orlando, FL, Oct. 213, pp [8] F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis: Teory and Practice. New York: Springer-Verlag, 26. [9] W. Hardle, Applied nonparametric regression, Econometric Teory, vol. 8, no. 3, pp , Sept [1] G. Cao, L. Yang, and D. Todem, Simultaneous inference for te mean function based on dense functional data, J. Nonparametric Stat., vol. 24, no. 2, pp , June 212. [11] Z. Zou and W.B. Wu, Local linear quantile estimation of nonstationary time series, Ann. Statist., vol. 37, no. 5B, pp , July 29. [12] D. Dragicescu, S. Guillas, and W.B. Wu, Quantile curve estimation and visualization for nonstationary time series, J. Comput. Grap. Statist., vol. 18, no. 1, pp. 1 2, 29. [13] Z. Zou and W.B. Wu, Simultaneous inference of linear models wit time varying coefficients, Journal of te Royal Statistical Society: Series B (Statistical Metodology), vol. 72, no. 4, pp , Sept. 21. [14] G. Cao, Simultaneous confidence bands for derivatives of dependent functional data, Elec. J. Stat., vol. 8, no. 2, pp , Dec [15] J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications. New York: Capman and Hall, [16] P. Craven and G. Waba, Smooting noisy data wit spline functions, Numer. Mat, vol. 31, no. 4, pp , Dec [17] E. Montgomery, C.and Peck and G. Vining, Introduction to Linear Regression Analysis, 4t ed. A Jon Wiley & Sons, Inc., July [18] [online] Available: ttp://traces.cs.umass.edu/. 65