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Spatial Load Forecasting by Data Fusion Technology
Hua Zhenga, Li Xieb and Lizi Zhangc
North China Electric Power University, China
Keywords: Data fusion, Grey theory, Least squares support vector machines, Spatial Load Forecasting
Abstract. Spatial load forecasting is one of key problems in the process of electric system planning.
But due to the special complexity of the power load, spatial load forecasting is still unsolved and need
to be studied further. Here spatial load forecasting is presented by data fusion technology. In detailed,
the samples are firstly classified by grey clustering, which results in the better-performanced samples
labeled with the grey leagues. After that, as limited samples as be concerned, various LS-SVMs are
trained by the corresponding classified samples. In this way, complicated nonlinear regression
modeling of spatial load and its influenced factors is accomplished by LS-SVM with more efficient
training. Finally, real data are employed to test the proposed approach.
Introduction
Under the new competitive electricity market, the market participants care much about the load
uncertainty, which has great influence on their planning and operation. In order to plan the efficient
operation and accomplish the economical benefits, the market participants must be able to evaluate
the system demand in the future. No doubted the accurate load forecasters can not only help market
participants identify the potential market changes during the future trade periods, but also provide
useful information to help their decision-making on various trades and operations. Especially for
middle-term and long-term planning, the error of annual load forecasting may increase the investment
cost and lead to excess or deficient supply. However different from short-term load forecasting, more
social and seasonal factors influence long-term loads, which are more volatile and unstable than the
influence factors to some extent. Moreover various demand forecasters have been widely applied in
market analysis fields, but due to the real-time balance of generation and consumption, the electric
power demand is more complicated than those of other commodities, even sometimes volatile, which
may change with accidental weather, consumption load and so on.
Till now, a wide variety of methods exist to forecast electric power load [1]-[4], such as auto
regressive integrated moving average (ARIMA), artificial intelligence [4] and so on. It should be
noted that most data fusion technologies especially commonly-used artificial neural networks, need a
large quantity of historical samples to ensure the model forecasting accuracy. However only limited
relative data can be applied on the spatial load forecasting for electric planning, which implies us a
typical limited-sampled problem.
So to solve above problem, spatial load forecaster is presented by a novel data fusion technology
that integrates Grey theory with least squares support vector machines (LS-SVM). The main
contribution of this paper is to combine two data fusion methods effectively, which are both suitable
for limited-sampled problem. In this study, firstly Grey theory is generalized for pattern identification
of the spatial load samples. Then LS-SVM offer new nonlinear regression modeling, trained with
classified samples having more latent regularity. Finally, real data are employed to test the proposed
approach.
Advanced Materials Research Vols. 219-220 (2011) pp 1625-1628Online available since 2011/Mar/28 at www.scientific.net© (2011) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.219-220.1625
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 129.93.16.3, University of Nebraska-Lincoln, Lincoln, United States of America-22/09/13,15:40:18)
Spatial Load Forecasting Based on Data Fusion
Framework of the Proposed Model. In this paper, two main problems faced when spatial loads are
forecasted have been considered and solved. The first is to select a small-sampled forecaster that will
be implemented by least squares support vector machines (abbr.LS-SVM). The other is to enhance the
training efficiency of standard LS-SVM and the forecasting performance, grey clustering is adopted
for the pattern identification with the small-sampled cases. Fig.1 shows the framework of the
proposed model.
Fig.1 the framework of spatial load forecasting by data fusion technology proposed in this paper
Spatial Load Pattern Identification by Grey Clustering. According to Grey theory, the
observation data including spatial load samples can be classified in terms of their grey league weights
that are computed by means of whitening function.
Supposing input vector ( )Tiniii xxx ,...,, 21=x denotes the load feature vector, where l is the sample
number, and the spatial load samples are classified into c grey leagues. As the commonly-used load
influence factors, popularity density, income per head, electric energy per head and electric power
price to fuel gas price increase ratios are selected and first standardized to have zero-mean and
unit-variance, whose data form the following sample matrix lM[5].
[ ]nlij
ll
n
n
l m
xxx
xxx
xxx
M×
=
=
ln21
22221
11211
�
����
�
�
(1)
The grey clustering weight ikσ for i-th sample belonging to the k-th grey league is computed by
( )∑=
=n
j
jijjkik mf
1
ησ (2)
where jη is the fixed weight of the j-th feature mainly determined by correlation analysis on these
features, and jkf is the whitening function. Then the grey statistic series of i-th load sample are
obtained to be ( )iniii σσσ ,...,, 21=σ . According to the classification rule of grey theory in Eq.3,
{ } *
1max
ikikCk
σσ =≤≤
(3)
we find the expected league k* to which i-th observation load is inferred to belong. In this way, the
spatial load samples are classified into C leagues with more obvious and similar characteristics.
Spatial Load Regression Modeling by LS-SVM. In this part, spatial load forecasting modeling is
accomplished to find the mapping relationship within each load league by applying LS-SVM to build
nonlinear regression model.
For random load league v (v=1,2,…,c), LS-SVM maps the influence factor data x into a higher
dimensional feature space through a nonlinear mapping and then a linear regression problem solved in
this feature space, according to statistical learning theory. No doubt, different load leagues respond to
different LS-SVM models. Given the observation samples Dv={(x1, y1), (x2, y2),…(xd, yd)}, with i-th
observation of system load and its influence factor vector being RyR i
n
i ∈∈ ,x , electricity load
model by LS-SVM is assumed to have the form[6].
Input samples Pattern 2
…
Pattern k
Spatial
Load Pattern
Identificatio
n by Grey
Clustering
Spatial load regression modeling by LS-SVM 1
Spatial load regression modeling by LS-SVM 2
…
Spatial load regression modeling by LS-SVM k
Pattern 1
1626 Advanced Research on Information Science, Automation and MaterialSystem
( ) vv
Tby += xwv ϕ (4)
where ( )xvϕ is the mapping to a higher-dimensional feature space by which nonlinear regression is
converted to a linear one. Eq.4 can be inferred by minimizing the cost function
( )
( ) vivv
T
i
d
i
iv
T
v
diebyts
eG
,,2,1..
2
1
2
1,min
1
2
,
…=++=
+= ∑=
iv
vvv
xw
www
ϕ
γξξω (5)
By introducing the Lagrange multipliers viα for the equality constraints in Eq.5, the following
Lagrange function is constructed
( ) ( ) ( )[ ]iviviv
Td
i
vivvvvv yebeGebL −++−= ∑=
xww vv ϕααω1
,,,, (6)
whose solution can be obtained by partially differentiating with w,αvi, ζvi, and bv
0=∂∂
vw
L, 0=∂∂
vb
L, 0=∂∂
vi
L
ξ, 0=∂∂
vi
L
α (7)
Eq.7 is rewritten as the following linear equations in matrix form:
=
Λ+ yαK vv
0
1
10 v
v
T b�
(8)
where
[ ] [ ] [ ]TT
d
T
vdvv
dd
v yyydiag 1,,1,11,,,,,,1,,
1,1
2121 …�
……� ===
=Λ
×
yαv αααγγγ
(9)
with
( ) ( ) ( )jvi
T
vjivij Kk xxxx ϕϕ== , (10)
Mercer’s condition is applied with kernel function matrix
=
dddd
d
d
kkk
kkk
kkk
�
����
�
�
21
22221
11211
vK (11)
The final simulation model of electricity load is derived to be
( ) ( ) v
d
i
ivivv bKfy +== ∑=1
, xxx α (12)
To ensure the validity of the trained model, the parameters of LS-SVMs are tuned including the
regularization parameter and the kernel parameter. The proposed model is more flexible and practical
than traditional regression methods in terms of its non-linear mapping abilities, generalization ability
and applicability to small-sample cases. Till now, the spatial load forecaster labeled with patterns by
above data fusion method will give a good performance in mapping with the appropriate parameters.
Case Studies.
In the case studies, real spatial load data of some areas in China are applied to test the proposed
model. As for the spatial load forecasting model, the influence factors are chosen as popularity density,
income per head, electric energy per head and electric power price to fuel gas price increase ratios.
The expected is the annual load density increase ratio of this area.
According to the proposed model, by using grey clustering algorithm, the samples of the training
dataset are clustered into three classifications by their grey clustering weight vector ( )321 ,, iiii σσσσ = ,
where i=1, 2, …, 15. The clustering data are listed in Table 1.
Advanced Materials Research Vols. 219-220 1627
Table 1 Grey clustering data of spatial load samples
1iσ 2iσ 3iσ Grey league
Sample 1 0.15 0.63 0.27 2
Sample 2 0.22 0.49 0.23 2
… … … … …
As followed, different LS-SVMs are trained using the corresponding samples labeled with their
classifications. The forecasting data of the obtained model and original data are shown in Fig.2.
Besides, the mean prediction abstract error as below is computed to evaluate the forecasting
performance.
%1001
1
∑=
×′−
=d
i i
ii
y
yy
dMAPE (13)
where yi and y’i are real and forecasted load density increase ratios respectively. MAPE of this
forecaster is 4.93%. Then we find that the proposed work is suitable for spatial load forecasting in
such type of areas.
Fig.2 Annual load density increase ratios of forecasting data and the original data
Summary
Spatial load forecaster used for electric planning is proposed by combining grey clustering with
LS-SVM in the paper, where the sample patterns are identified from “grey system” point of view. In
addition, our results provide empirical evidence that the data fusion technology proposed has good
generalization ability. From this study, we also find that it is still early in the development of data
fusion technology; however the results of the case studies reported are encouraging and indicative of
many further successful applications.
Acknowledgment
The authors gratefully acknowledge the financial support of He Bei Province Natural Science Fund
program (F2010001712).
References
[1] H.Wang and N.N.Schulz: submitted to Electric Power Syst.Res.(2006)
[2] A.Azadeh, M.Saberi, S.F.Ghaderi, A. Gitiforouz and V.Ebrahimipour: submitted to Energy
Convers. Manage(2008)
[3] LUO Zhi-qiang, ZHANG Yan, and ZHU Jie: submitted to Power System Technology(2004)
[4] Hung-Chih Wu and Chan-Nan Lu: submitted to IEEE Trans. on Power Syst. (2002).
[5] DENG Julong, in: Grey theory, HuaZhong University Press of Science and Technology(2003)
[6]J.A.K.Suykens, J.De Brabanter, L.Lukas and J.Vandewalle: submitted to Neurocomputing(2002)
1628 Advanced Research on Information Science, Automation and MaterialSystem
Advanced Research on Information Science, Automation and Material System 10.4028/www.scientific.net/AMR.219-220 Spatial Load Forecasting by Data Fusion Technology 10.4028/www.scientific.net/AMR.219-220.1625
DOI References
[4] Hung-Chih Wu and Chan-Nan Lu: submitted to IEEE Trans. on Power Syst. (2002).
doi:10.1109/TPWRS.2002.1007927
