短期太阳能预测

Short-term solar power prediction using a support vector machine
Jianwu Zeng,Wei Qiao *
Department of Electrical Engineering,University of Nebraska-Lincoln,Lincoln,NE 68588-0511,USA
a r t i c l e i n f o
Article history:
Received 3March 2012Accepted 1October 2012
Available online 23November 2012Keywords:
Autoregressive (AR)model
Radial basis function neural network (RBFNN)Short term
Solar power prediction (SPP)Support vector machine (SVM)
a b s t r a c t
This paper proposes a least-square (LS)support vector machine (SVM)-based model for short-term solar power prediction (SPP).The input of the model includes historical data of atmospheric transmissivity in a novel two-dimensional (2D)form and other meteorological variables,including sky cover,relative humidity,and wind speed.The output of the model is the predicted atmospheric transmissivity,which then is converted to solar power according to the latitude of the site and the time of the puter simulations are carried out to validate the proposed model by using the data obtained from the National Solar Radiation Database (NSRDB).Results show that the proposed model not only signi ficantly outperforms a reference autoregressive (AR)model but also achieves better results than a radial basis function neural network (RBFNN)-based model in terms of prediction accuracy.The superiority of using transmissivity over sigmoid functions for data normalization is testi fied.Simulation studies also show that the use of additional meteorological variables,especially sky cover,improves the accuracy of SPP.
Ó2012Elsevier Ltd.All rights reserved.
1.Introduction
Due to the uncertainty and intermittency of solar energy,short-term solar power prediction (SPP)has become an important issue in increasing the penetration of solar energy sources in electric power grids.Accurate short-term SPP not only helps optimize the integration of solar energy into electric power grids,but also ensures a favorable trading performance of solar energy in elec-tricity markets [1].However,the SPP accuracy largely depends on the meteorological and climatic conditions of a location [2],which makes the SPP a challenging problem.
The goal of SPP is to predict solar radiation of a location at a speci fic time in the future.There are mainly two categories of models for SPP:physical models and statistical models.Physical models use mathe-matical equations to describe physics and dynamics of the atmo-sphere that in fluences solar radiation [1,3].They work well for medium-and long-term SPP.Statistical models are mainly based on time series analysis [4].They have lower complexity than physical models and can perform well for short-term SPP.Therefore,statistical models are of concern in this paper for short-term SPP.
Autoregressive (AR)and autoregressive moving average (ARMA)[5]are frequently used linear models in SPP [6,7].These linear models are superior to the traditional persistence model.Arti ficial intelligence-based nonlinear models,such as the Takagi-Sugeno
fuzzy model [8],wavelet-based models [9],and arti ficial neural network (ANN)-based models [10]have been shown their superi-ority to the linear models.Recently,the use of support vector machines (SVMs)for time series prediction has been studied [11],such as temperature prediction [12],short-term wind prediction [13],and solar flare forecasting [14].The success of using SVMs for time series prediction is largely due to its good generalization ability.However,the effectiveness of using SVMs for short-term SPP has not been studied yet.
In SPP,feature selection plays a key role in determining the performance of the prediction models.Improper features will lead to poor regression in an SPP model.It is known that solar radiation largely depends upon meteorological conditions.Some studies have shown that the SPP models using certain extra meteorological variables can achieve better performance than those using solar radiation data only [15].Sun duration,air temperature [16],cloudiness index (CI)[17],atmospheric transmissivity [18],day length,and time can be used as extra inputs for SPP mon approaches to predict solar power is first predicting CI or atmospheric transmissivity and then converting them to solar radiation according to known formula [19].Nevertheless,solar radiation was taken as a 1D time series in most of the existing work,which turned out to be inferior to a 2D representation [20].The 2D representation of solar radiation makes it possible to further combine image processing methods with nonlinear prediction methods to improve the accuracy of SPP [21].
This paper proposes a SVM-based model for short-term puter simulations are carried out to show the superiority of the
*Corresponding author.Tel.:þ14024729619;fax:þ14024724732.E-mail address:wqiao@ (W.
Qiao).Contents lists available at SciVerse ScienceDirect
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Renewable Energy 52(2013)118e 127
SVM-based model over an AR model and a radial basis function neural network (RBFNN)model by using data obtained from the National Solar Radiation Database (NSRDB).The 2D transmissivity and other meteorological variables,including sky cover,relative humidity,and wind speed,are used as the inputs for different models to predict the atmospheric transmissivity,which then is converted to solar power according to the latitude of the site and the time of the day.The composition of this paper is as follows.Section 2describes different SPP models.The novel 2D solar radi-ation representation and data normalization are illustrated in Section puter simulations are provided in Section 4to demonstrate the superiority of the SVM-based model and investi-gate the factors that in fluence the performance of the model.Section 5provides concluding remarks of the paper.2.The SPP models
Consider the original dataset consisting of m variables,where one is the ground solar radiation time series y t (t ¼1,.,N ),and others are meteorological variables X t ,which is a N Â(m À1)
matrix.In this paper,X t (i )
is the i th (i ¼1,.,m À1)column of the matrix X t and represents a time series of the i th meteorological
variable.X (i )(t )represents the value of X t (i )
at the time t .
In this paper the following normalized time series of solar radiation is used as the input and output of the predict models.
y t ¼f ðy t Þ(1)
where f ($)is the normalization function to be de fined in Section 3.3;y t is the normalized values of the original time series y t .
Similarly,X ði Þ
t can be normalized to X ði Þt .2.1.The AR model
The AR model expresses a time series as a linear function of its past values.The order of the AR model indicates how many past values are used.An AR model,AR(m $D ),with an order of m $D can be written as:
where h is the prediction horizon;b
y ðt þh Þis the h -step-ahead predicted value of y ðt þh Þ,which is the h -step-ahead normalized
solar radiation;m is the number of input variables;a ij (i ¼1,.,m and j ¼1,.,D )is the AR coef ficient;x t ¼½X ð1Þ
ðt Þ;X
ð1Þ
ðt À1Þ;.;ð1Þ
ðt ÀD þ1Þ;.;X
ðm À1Þ
ðt Þ;X
ðm À1Þ
ðt À1Þ;.;X
ðm À1Þ
ðt ÀD þ1Þ;
y ðt Þ;y ðt À1Þ;.;y ðt ÀD þ1Þ T
are the current and past values of the time series ði Þ
t (i ¼1,.,m À1)and y t ;e (t )is a noise or error term,which is assumed to be a normally distributed random number.
2.2.The RBFNN-based model
The RBFNNs are a class of feed-forward ANNs constructed based on the function approximation theory.Fig.1shows the structure of
an RBFNN,where x t (i )
denotes the i th component of x t .It has three functionally distinct layers.The input layer is simply a set of sensory units.The second layer is a hidden layer of suf ficient dimension,which use RBFs as activation functions to perform a nonlinear transformation from the input space to a higher-dimensional hidden-unit space.The third layer performs a linear trans-formation from the hidden-unit space to the output space.The output of the RBFNN is given by:
b y ðt þh Þ¼
X n i ¼1
w i f ðx t ;c i ;s i Þþw 0
(3)
where x t ˛R m $D is the input regression vector,which is the same as that of the AR model;n is the number of neurons (i.e.,RBF units)in the hidden layer;w 0is a bias term;w i is the weight between the hidden and output layers;and f ($)is the RBF activation function in the hidden layer,which is de fined as:
f ðx t ;c i ;s i Þ¼exp
À
k x t Àc i k 2
2s 2
i
!
(4)
where c i and s i are the center and width of the RBF function,respectively.The values of c i and s i can be determined by different methods.The simplest method is to randomly choose a subset of the data points as the RBF centers.A more sophisticated approach is to cluster the data into an appropriate number of clusters,whose centers are then used as the centers of the RBF units.In this paper,a local Gaussian mixture model [22]with spherical covariance structure is created to determine the RBF centers by a K -means
clustering algorithm [23].The Gaussian mixture model is trained by using the Expectation Maximum (EM)algorithm [24]
.
Fig.1.The structure of an RBFNN.
J.Zeng,W.Qiao /Renewable Energy 52(2013)118e 127119
It has been shown [25]that setting the widths of the RBFs to be
equal to the variances of the corresponding mixture model tends to give poor results,because the widths are too small and there is insuf ficient overlap between the RBF functions.In this paper,all the widths are set at the same value,which is proportional to the maximum Euclidean distance,d max ,between RBF centers.
s i ¼k s $d max
(5)
where k s is a nonnegative scalar whose typical value is in the range of [0.1,0.2][26].Given a time series y t ,(3)can be written as:
b y t ¼F $w
(6)
where w ¼[w 0,w 1,.,w n ]T is the vector of the output weights and bias term;and F is the matrix of hidden-layer activations due to the input data y t .A sum-of-square error function is de fined by:
E ¼
12
b y t Ày t 2
(7)
Since this error function is a quadratic function of the vector w ,pseudo-inverse can be used to determine the optimal w to mini-mize the value of the error function.
w ¼F þ$y t
(8)
where F þ¼(F T F )À1F T .The Netlab toolbox [25]is used to construct the RBFNN in simulation studies of the paper.
2.3.The SVM-based model
The SVMs belong to the class of kernel methods.The use of an SVM for time series prediction can be expressed as follows.
b y ðt þh Þ¼v T 4ðx t Þþb
(9)
where b ˛R is a bias term;v ˛R M is the weight vector;and 4:R m $D /R M (M !m $D )is a nonlinear feature map,which transforms the input vector x t ˛R m $D to a higher-dimensional vector 4(x t )˛R M .Fig.2shows the structure of an SVM,where 4(i )(x t )denotes the i th component of 4(x t ).
In an SVM,the historical data of the time series is mapped into a higher-dimensional feature space via a nonlinear mapping 4;then linear regression is used in the high-dimensional feature space to predict the time series,which is equivalent to solve a nonlinear regression problem in the low-dimensional space of the original time series [27].The key issue to solve such a prediction problem is to find the optimal values of the SVM parameters w and b .This can be done by solving a constrained optimization problem [28].
min 12v T
v þg
X N t ¼1
e 2ðt Þs :t
y ðt þh Þ¼v T 4ðx t Þþb þe ðt Þ;
t ¼1;2;:::;N
(10)
where y ðt þh Þis the observed value of b y ðt þh Þ;e (t )is the predic-tion error at time t ;g is a regularization parameter,which balances
the fitting in the training stage and generalization in the imple-mentation stage.A too large or too small g might deteriorate the generalization ability of the SVM in the implementation stage.Problem (10)can be solved by using Lagrange multipliers and the solution is expressed in its dual form.Then the resulting SVM of (9)is called a least-square (LS)SVM and can be represented as follows:
b y ðt þh Þ¼
X N i ¼1
a i K ðx t ;x i Þþb
(11)
where a i (i ¼1,.,N )is the nonnegative Lagrange multiplier of Problem (10);and K (x t ,x i )¼4(x t )4(x i )is a positive-de finite kernel function.The input samples associated with the nonzero Lagrange multipliers in (11)are referred to as the support vectors (SVs).The value of a multiplier a i represents the contribution of the corre-sponding sample to the SVM,namely,a larger a i indicates that its corresponding SV is more important.
Commonly used kernel functions include linear,polynomial,and RBF kernels.An SVM with the following RBF kernel [29]is used in this paper.
K ðx t ;x i Þ¼exp
À
k x t Àx i k 2
s 2
!
(12)
where s is the width of the RBF kernel,which determines the in fluence area of the SVs over the data space.3.Data representation and preprocessing
The National Solar Radiation Database (NSRDB)[30]is used to validate the effectiveness of the proposed model.The data was recorded from 1991to 2005at 1454locations in the United States and contains 47variables,including hourly solar radiation and other meteorological data.In this paper,three sites located in different regions of the United States are selected,which are Seattle (Station ID:727930)in northwest,Denver (Station ID:724666)in Midwest,and Miami (Station ID:722020)in southeast.The Denver data is used in following illustration.3.1.2D representation
To visualize the bene fits of using the 2D representation,the average values of the 14-year NSRDB data (Jan.1,1991e Dec.31,2004)are firstly considered as a 1D time series and then as a 2D image formed in the raster scan form with the columns and rows corresponding to days and hours,respectively.Figs.3and 4show the 1D and 2D representations of the 14-year average solar radia-tion data,respectively,where each data point is the average value of the data points at the same time in the 14years.
In Fig.3,it is visually dif ficult to grasp the solar radiation char-acteristics within a day although the seasonal behavior is obvious.In Fig.4,both daily and seasonal behavior of solar radiation can be easily interpreted,where a larger value in the range of [0,900]indicates a stronger radiation.In winter,the sunrise to sunset period is shorter than that of summer.While in summer,radiation at noon achieves the strongest of the whole year.Such a 2D representation provides a signi ficant insight into not only
the
Fig.2.The structure of an SVM.
J.Zeng,W.Qiao /Renewable Energy 52(2013)118e 127
120
radiation pattern as a function of time,but also the horizontal and vertical correlations within the 2D data.As shown in Fig.4,the solar radiation values are close to zero from 8pm to 6am of the next day because there is no sunlight at night.It is unnecessary to use all recorded data for either training or testing.Therefore,only the data recorded during 6am e 8pm are used in this paper.3.2.Input dimension determination
The embedding dimension D of the input of the prediction model is determined by the autocorrelation coef ficients of y t in historical 14years (Jan.1,1991e Dec.31,2004)de fined as follows.
r k ¼
1
ðN Àk Þ$s 2y X N t ¼k þ1
y ðt ÞÀm y $ y ðt Àk ÞÀm y (13)
where r k (k ¼1,2,.,60)is called the autocorrelation coef ficient at lag k ;m y and s y are the mean and standard deviation of the time series y t ,respectively;N is the total number of samples of the time series y t and the value of N is 14Â365Â15in this paper.Due to its diurnal characteristic,i.e.,the radiation from 9pm to 5am can be taken as zero,15-h data instead of 24-h data is used in a day.The autocorrelation coef ficients between two consecutive days can be
easily calculated by replacing k with 24in (13).Table 1shows the autocorrelation coef ficients between current solar radiation and historical solar radiation obtained from the 14-year average data.
In Table 1,i denotes an hour in a day (e.g.,1pm)and j denotes a day (e.g.,September 1)in a year.The value in cell (i À2,j À3)represents the autocorrelation coef ficient between the latest solar radiation data and the data obtained 3days and 2h (i.e.,74h)ago,which is 0.6470.An important observation from Table 1is that there are strong correlations between the radiations not only in consecutive hours,but also in some hours of consecutive days.The correlation between two consecutive days in the same hour is stronger than that between the current hour and 2h ahead of the same day.Therefore,when constructing a prediction model,the data from the same hour of the previous day should be used with a higher priority than the data of the previous 2h in the same day.In this study,the radiation data at the same hour of the former two days and the latest data are used as the input of the prediction model.
3.3.Normalization
Two functions f 1($)and f 2($)are used for normalization of y t in this paper.
p t ¼f 1ðy t Þ¼
1
1þexp
À
y t Àm y
s y
(14)
where p t is the normalized value of y t using the sigmoid function f 1($),which is also used for normalizing other meteorological
variables X t (i )
(i ¼1,.,m À1).
Another method for data normalization is based on the concept of transmissivity [31],which is de fined as the ratio between the solar radiation received on the ground surface and the incoming solar radiation (extraterrestrial radiation)on the top of atmosphere.
s t ¼f 2ðy t Þ¼
y t t
(15)
where s t is the time series of the transmissivity;y t and R t are the time series of the ground and extraterrestrial radiations,respec-tively.The extraterrestrial solar radiation can be accurately calcu-lated by using geometry factors (latitude and longitude),day of the year (DOY),and time of the day (TOD).Therefore,once the trans-missivity is known,the ground radiation can be easily obtained.
The transmissivity takes time and weather variations into account.It not only re flects the ground radiation,but also contains certain weather information.Fig.5compares two normalization methods for radiation data.Fig.5(a)shows the original radiation observed at the Denver station on July 16,2000and Feb.26,2001,where the R t curve indicates seasonal variations of the solar radi-ation.The extraterrestrial radiation on July 16is much stronger than that on Feb.26.The ground radiation y t curve re flects the effect of the weather condition on the solar radiation.For example,the sky on Feb.26was clearer than that on July 16;otherwise,the ground radiation on July 16should be much larger than that on
Feb.
Hour of the day
D a y o f t h e y e a r
5
101520
50
100150
200
250
300
350
100
200300400500
600
700800900
Fig.4.A 2D image view of the solar radiation data.
Table 1
Autocorrelation coef ficients of solar radiation.Hours
Days j À3
j À2j À1j i À30.27500.27660.28150.2916i À20.64700.64870.65440.6679i À10.89410.89560.90240.9184i
0.9712
0.9724
0.9792
1
Fig.3.Hourly solar radiation data in a 1D time plot.
J.Zeng,W.Qiao /Renewable Energy 52(2013)118e 127121
26.Fig.5(b)shows the normalized values of ground radiation by using the two methods.It can be seen that the sigmoid normalized ground radiation values (p t )in both days are similar;it fails to ‘discover ’the weather difference.On the other hand,the trans-missivity ‘recognizes ’the weather condition,a larger s is equivalent to a clearer sky,which plays an important role in SPP.Due to this physical meaning,the normalization by the transmissivity is superior to that by the sigmoid function.
Fig.6illustrates the flowchart of the overall SPP system,which includes four blocks,i.e.,normalization,feature representation,prediction,and denormalization.The input of the SPP system includes the ground solar radiation y t and other meteorological variables X t .They are normalized by using transmissivity or
sigmoid function,and the results are y t and X ði Þ
t ,respectively,where y t ¼p t if sigmoid function is used for normalization and y t ¼s t if transmissivity is used for normalization.After data normalization,the input vector x t of the prediction model is generated by the feature representation block.The output of the prediction model is b y ðt þh Þ,which is the h -step-ahead predicted value of the normalized solar radiation.It is converted to the ground solar radiation by the denormalization block.3.4.Performance evaluation
The mean absolute error (MAE),mean absolute percentage error (MAPE),and correlation coef ficient (r )of the observed and
07/16/2000 02/26/2001
R a d i a t i o n (W /m 2)
07/16/2000 02/26/2001
N o r m a l i z e d v a l u e
a
b
parison of the two methods for normalization
of the radiation data:(a)the original ground and extraterrestrial radiation curves;(b)the normalized ground radiation curve using sigmoid function and transmissivity.
Fig.6.Flowchart of the overall SPP system.
Fig.7.Training and testing set generation.
J.Zeng,W.Qiao /Renewable Energy 52(2013)118e 127
122
predicted values are used to evaluate the performance of the SPP models.Their de finitions are expressed as follows.
MAE ¼1X N t ¼1 b y ðt ÞÀy ðt Þ (16)
MAPE ¼1N X N t ¼1
b y ðt ÞÀy ðt Þy ðt Þ
(17)
r ¼P N
t ¼1
b y ðt ÞÀm b y
$ y ðt ÞÀm y
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP N t ¼1 b y ðt ÞÀm b y
2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP N t ¼1
y ðt ÞÀm y 2r (18)
where y and b y are the observed and predicted values of ground
solar radiation,respectively;m b y
represents the mean values of b y .Smaller values of MAE and MAPE imply a superior prediction performance of the model.r (0 j r j 1)is a measure of linear correlation between two variables.A larger positive r indicates that the samples are more correlated.
In order to evaluate the improvement of one model to another,a parameter called skill is de fined as follows:
12
34
3638
40
Training length (years)
M A E (W /m 2)
0246810
5
5
M A P E (%)
Fig.8.The MAE and MAPE as functions of the training length for the Denver dataset.
Fig.9.One-hour-ahead prediction in Denver using SVM.
Predicted radiation (W/m 2)
A c t u a l r a d i a t i o n (W /m 2)
Fig.10.Correlation between the real and predicted solar radiations in Denver.
J.Zeng,W.Qiao /Renewable Energy 52(2013)118e 127
123
Fig.11.One-hour-ahead prediction in Seattle using SVM.
20406080100
M A E (W /m 2
)
Denver
10
152025303540M A P E (%)
2040
60
80
100
Seattle
M A E (W /m 2)
20
253035404550M A P E (%)
Fig.12.SVM-based J.Zeng,W.Qiao /Renewable Energy 52(2013)118e 127
124
skill ¼
j e 0Àe 1j
e 0
Â100%(19)
where e 1and e 0are the MAE of the SPP using a new model and the reference model,respectively.A larger skill value indicates more improvement of the new model over the reference model.4.Simulation results
In this section,simulations are carried out for short-term SPP using the NSRDB.The original data is divided into two sets;one is the training set and the other is the testing set.Fig.7shows the division of the data of (k þ1)years that are used for training and testing,where L is the length of the testing set;s 0represents the first testing data sample;m 0¼s 0þL /2is the middle point of the testing set;then p 00and p 01,which have the bilateral symmetric structure with respect to m 0,represent the first and the last data samples of the season,respectively;L j 1is the length of the data that are selected for training in the j th year,where j ¼0,1,.,k ,and k is the number of historical years selected for generating training data.In this paper,L 11¼L 21¼/¼L k 1.The data samples in [p j 0,p j 1]of the j th year (j ¼1,.,k )as well as in [p 00,s 0)of the testing year are selected to form the training set.In this study,the testing set is selected to be the data from Sept.1,2005to Sept.10,2005,which has moderate numbers of sunny and cloudy days.Given the number of historical years k ,the training data is automatically generated by the method shown in Fig.7.Then the training set contains the data from July 17to Oct.20in previous years and from July 17to Aug.31in 2005.
The inputs of all three models include the latest observed solar radiation,radiations at the hour of prediction in the previous two consecutive days,and the latest observations of some meteorological features,including sky cover,wind speed,and relative humidity.In
addition,since there is no radiation at night,only the observations from 5am to 9pm are used.Therefore,for the 1-h prediction,the radiations from 6am to 9pm in a day are predicted.During testing,all of the predicted values are true out-of-sample forecasts,inwhich only the historical data samples are used.The predicted data is then compared to the actual measured value.The procedure is repeated for the next hour until it runs over the entire testing dataset.4.1.Short-term prediction
Fig.8shows the MAE and MAPE as functions of the length of the training data (called the training length)for a prediction horizon of 1h in Denver.As shown in Fig.8,it is not true that the longer the better for the training length.The MAE and MAPE decrease dras-tically with the increase of the training length up to six years.However,after six years the MAE and MAPE increase with the training length.Therefore,six year is selected as the best training length for the Denver dataset.
Figs.9and 10show the 1-h-ahead prediction results in Denver using the SVM-based model,where the normalized error is de fined as:ð1Àb y t =y t ÞÂ100%.As shown in Fig.9,the SVM-based model works well especially during clear days (the 45th e 105th hours),where the predicted values closely follow the rge prediction errors mainly occur in those days when the ground radiation drastically changes.For instance,the weather conditions during the 30th e 45th and 105th e 120th hours make the prediction less accurate than in the 45th e 105th hours.The error distribution shows that the majority of the prediction errors concentrate in a small range.More than 70%of the normalized errors are less than 10%in the case of 1-h-ahead SPP.Fig.10shows the correlation between the real and predicted solar radiations in Denver.As aforementioned,r (j r j 1)is a measure of linear correlation between two variables.r ¼0.974indicates that the predicted and actual radiations are highly correlated.Moreover,such a conclusion can also be drawn from the slope of the fitting line,which is close to one or 45 in this case.Therefore,the predicted values closely match the actual data,which indicates an accurate prediction.Fig.11shows the 1-h-ahead prediction in Seattle,which has less clear days than Denver,by using the SVM-based model.Similarly,more than 50%of the normalized errors are less than 10%and approximately 45%of the normalized errors concentrate within 5%parison of three SPP models
The AR model,which has been shown to be superior to the persistence model,is used as the reference model in this paper.Fig.12compares the AR,RBFNN and SVM-based models for SPP using the data in Denver and Seattle.The RBFNN and SVM achieved much better results than the AR model in terms of accuracy.In Denver ’s 1-h-ahead prediction,the MAE of the AR model is 62W/m 2,while the MAE of the RBFNN and SVM are 43W/m 2and 33.7W/m 2,respectively.The inferiority of the AR model shows that the linear model is worse than the nonlinear models to capture the nonlinear characteristic of the solar radiation.Moreover,the prediction accuracy improvement brought by the proposed SVM-
09/05/2005 Time (h) 09/06/2005
R a d i a t i o n (W /m 2)
parison of the 1-h-ahead predicted values of the three models with the observations in two consecutive days in Seattle.
Table 2
Comparison of MAES using different normalization methods.Prediction horizon (h)
Sigmoid function (transmissivity)Seattle (W/m 2)
Denver (W/m 2)Miami (W/m 2)134.372(34.230)46.839(33.773)64.504(62.864)249.645(48.530)59.344(48.707)79.570(78.285)3
64.169(62.275)
67.934(58.174)
92.565(91.476)
Table 3
Comparison of MAES without (with)meterological variables.Prediction horizon (h)
Without (with)meteorological variables Seattle (W/m 2)
Denver (W/m 2)Miami (W/m 2)134.342(34.230)42.352(33.773)58.882(62.864)249.969(48.530)58.648(48.707)75.181(78.285)3
67.405(64.169)
72.266(58.174)
90.485(91.476)
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125。