Research Article Short-Term Power Load Point Prediction...

Research ArticleShort-Term Power Load Point Prediction Based on the SharpDegree and Chaotic RBF Neural Network

Dongxiao Niu Yan Lu Xiaomin Xu and Bingjie Li

School of Economics and Management North China Electric Power University Beijing 102206 China

Correspondence should be addressed to Yan Lu hdluyan163com

Received 21 October 2014 Revised 9 December 2014 Accepted 16 December 2014

Academic Editor Dan Simon

Copyright copy 2015 Dongxiao Niu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to realize the predicting and positioning of short-term load inflection point this paper made reference to related researchin the field of computer image recognition It got a load sharp degree sequence by the transformation of the original load sequencebased on the algorithm of sharp degree Then this paper designed a forecasting model based on the chaos theory and RBF neuralnetwork It predicted the load sharp degree sequence based on the forecasting model to realize the positioning of short-term loadinflection point Finally in the empirical example analysis this paper predicted the daily load point of a region using the actualload data of the certain region to verify the effectiveness and applicability of this method Prediction results showed that most ofthe test sample load points could be accurately predicted

1 Introduction

Short-term load forecasting (STLF) plays a key role inpower dispatching work It is the foundation of powergrid planning grid control system safety analysis and theeconomic operation The forecasting accuracy is closelyrelated to the grid running safety and economy [1 2] Atpresent many scholars and experts have already done a lotof theoretical researches and practical simulations on short-term load forecasting [3ndash20] More common short-term loadforecasting models include regression prediction model [3]time series prediction model [4] artificial neural networkprediction model [5] fuzzy logic and expert system [6 7]the wavelet analysis model [8] chaos theory [9] and com-bination prediction model [10ndash12] Literature [13] analyzedthe power factors and predicted the short-term load basedon rough set method Literature [14 15] introduced datamining techniques for short-term load forecasting Literature[16] proposed a novel combination prediction model forshort-term load forecasting which mainly studied the weightof combination prediction model with different scenariosLiterature [17] established a combination of modified fireflyalgorithm and support vector regression model which wasused for the STLF Literature [18] studied the short-termforecasting of categorical changes in wind power based on

Markov chain models Literature [19] studied the short-termload forecasting based on the wavelet transform and greymodel improved by PSO Literature [20] proposed a relevancevector machine short-term load forecasting model based onnonnegative matrix decomposition

On the whole research on short-term load forecastinghas been relatively matureThemodels are more complicatedand intelligent Combination forecasting model is a researchtrend [21] Inflection point is usually the most concern ofthe dispatch staff in short-term load forecasting How toidentify and predict inflection point of short-term load is achallenge problem At present there is no mature solution tothe short-term load forecasting of inflection pointThis papergets a load sharp degree sequence by the transformation ofthe original load sequence based on the algorithm of sharpdegree Then it forecasts the load sharp degree sequence bydesigning and training the forecasting model of chaotic RBFneural network to realize the short-term load forecasting ofinflection point

2 Inflection Point Identification AlgorithmBased on Sharp Degree

Inflection point is the most important features of loadcurve The inflection point in this paper is the point that

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 231765 8 pageshttpdxdoiorg1011552015231765

2 Mathematical Problems in Engineering

O

a

Pi+kPiminusk

Pi

Figure 1 Part outline figure of the load curve

changes the direction of load curve upward or downward ina certain time interval This paper identifies the inflectionpoint according to the sharp degree of each point on the loadcurve Set the function of load curve as 119875 = 119901(119905) in whichthe independent variable 119905 means time and the dependentvariable 119875 represents the load time series

Randomly select a point 119875119894on the curve as the center 119875

119894minus119896

is the point in front of 119875119894a distance of 119896 And 119875

119894+119896is the point

behind 119875119894a distance of 119896

Define the vector 119896

119886119894119896

= (119879 (119894) minus 119879 (119894 + 119896) 119875 (119894) minus 119875 (119894 + 119896))

119887119894119896

= (119879 (119894) minus 119879 (119894 minus 119896) 119875 (119894) minus 119875 (119894 minus 119896))

119888119894119896

= (119879 (119894 minus 119896) minus 119879 (119894 + 119896) 119875 (119894 minus 119896) minus 119875 (119894 + 119896))

(1)

Define 120572 the angle of 119886119894119896

and 119887119894119896 as the angle consisting

of 119875119894minus119896

119875119894 and 119875

119894+119896on the load curve Figure 1 shows the

part outline figure of the load curve The peripheral solidline is composed of load time series The dotted line is thecircular arc fitted by 119875

119894minus119896 119875119894 and 119875

119894+119896 And 119874 is the center of

circular arc Usually the value of 119896 is set in the range of 3ndash5 For the endpoint of load curve we will take 3 to 5 samplevalues forward and backward on the sample load data On theactual curve the points119875

119894minus119896119875119894 and119875

119894+119896can be approximately

regarded as three points on the circular arc as the interval ofthe three points is very small [22]

Assuming that |119875119894119875119894minus119896

| = |119875119894119875119894+119896

| then

sin(120572

2) =

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816 2

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816

=

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816 2

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

=

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

(2)

In this function 0 lt 120572 le 180When119875

119894minus119896119875119894 and119875

119894+119896are in a straight line 120572 = 180 then

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

= sin(120572

2) = sin (90) = 1 (3)

When the value of 120572 decreases tending to zero

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

= sin(120572

2) = sin (0) = 0 (4)

Therefore in a very small load curve the variable sharp canbe defined as sharp degree of 120572

sharp = 1 minus ang = 1 minus

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

(5)

The larger the value of sharp the smaller the value of 120572 andthe curve would be more sharp On the contrary the curvewould be more smooth Because of the continuity of loadthe identification of inflection point is closely related to theinterval of load curve sequence and the set threshold valueTherefore the threshold value can be set according to the loadcurve characteristics and actual situation If sharp (119875

119894) gt 119879

the 119875119894can be regarded as inflection point

3 Prediction Model Based onChaotic RBF Neural Network

Artificial neural network is a common type of load fore-casting methods This method is highly intelligent whichhas strong ability to look for patterns [23] Radical basisfunction (RBF) neural network is a kind of highly efficientfeed-forward neural network It has the best approximationperformance and global optimal property And the structureis simple the training speed is fast [24] Chaos predictionalgorithm applied to power load is a very hot research fieldwhich is caused by the intrinsic characteristics of systemsIt can be predicted in the short term [25] It predictsdirectly according to the laws of the objective extracting datasequence itself and does not need to establish the mappingrelation model of influence factors In theory it not onlycan decrease the cost of prediction but can improve theprediction accuracy as well [26] The method combiningchaos and RBF neural network can make full use of therandomicity and initial value sensitivity of chaos In additionit can make full use of the function of RBF such as massivelyparallel processing self-organizing and self-adapting [27]

Although sharp degree sequence is turbulent it can beused to identify the inflection point Chaotic time series aresimilar to random noise which is chaotic and unpredictableBut its inherent nonlinear dynamics structure makes itmeet the short-term predictability [28] Use phase spacereconstruction to find the internal of regularity conversionsequences that can be described by a nonlinear mapping andthen approximate the nonlinear mapping by applying radialbasis to forecast the inflection pointThe process of themodelestablished in this paper is shown in Figure 2

First of all we can get the sharp degree sequence whichcan identify the inflection point on the basis of Section 2Then reconstruct phase space and identify chaos based onchaos theory This paper selects mutual information methodto calculate delay timeCaomethod to determine embeddingdimension and the computing of largest Lyapunov exponentto identify the chaos Thirdly the input layer of RBF neuralnetwork can be determined based on the delay time andembedding dimension Then train the RBF neural networkand predict the sharp degree sequence making use of thetrained network Finally determine the threshold and posi-tion the inflection point

Mathematical Problems in Engineering 3

Sharpdegree

sequence

Mutualinformation

method

Cao method

Computing thelargest Lyapunov

index

Delay timecalculated

Embeddingdimensiondetermined

Identify thechaos

Determinethe input

layer

RBF neuralnetwork trainand predict

Pointposition

Get the predictedsharp degreesequence

Figure 2 The steps of chaotic RBF neural network

31 Phase Space Reconstruction and Chaos IdentificationExtend the time series to three-dimension or higher dimen-sion space by using delay-coordinate method to reconstructstate space which can find information and rules containedin the original time series And by ldquoTakensrdquo theorem thedynamic characteristic of attractor of the system can berestored in the sense of topological equivalence [29 30]Assume that the time sequence is 119909

1 1199092 1199093 119909

119899 properly

select embedding dimension 119898 and time delay 120591 the recon-structed phase space is as follows

1198841= (1199091 1199091+120591

1199091+(119898minus1)120591

)119879

1198842= (1199092 1199092+120591

1199092+(119898minus1)120591

)119879

119884119873

= (119909119873 119909119873+120591

119909119873+(119898minus1)120591

)119879

(6)

of which 119873 = 119899 minus (119898 minus 1) lowast 120591 is the length of the vectorsequence

311 Delay Time Calculated by Mutual Information MethodMutual information method is a kind of commonly usedmethods to calculate delay time [31] It is suitable for largedata sets and nonlinear problem Set offline random variables119883 119884 The mutual information 119868(119883 119884) is as follows

119868 (119883 119884) = 119867 (119883) + 119867 (119884) minus 119867 (119883 119884)

119867 (119883) =

119902

sum

119894=1

119875 (119909119894) log119875 (119909

119894)

(7)

In the functions 119875(119909119894) is the probability of happening 119909

119894

119902 is the total number of possible things or states and thelogarithmic function often takes 2 for base number

For the time series 119909119894and delay time 120591 with an interval

of 119899 set 119875(119909119894) as the probability of the 119909

119894in 119909119894 and

119875(119909119894+120591

) the probability of 119909119894+120591

in 119909119894 can be calculated

through the frequency in the corresponding time sequenceand 119875(119909

119894 119909119894+120591

) the joint probability of common presenceof 119909119894and 119909

119894+120591in these two sequences can be found in the

corresponding grid on plane (119909119894 119909119894+120591

) Therefore the mutualinformation 119868(119909

119894 119909119894+120591

) can be regarded as the function of

delay time 120591 119868(120591) When the 119868(120591) meets the minimum forthe first time the lag time is the time delay of phase spacereconstruction

312 Embedding Dimension Determined by Cao MethodThere are somemethods to determine embedding dimensionsuch as the pseudo adjacent point method G-P algorithmand C-C method Cao algorithm is a mature method whichwas put forward by Cao Liangyue [32] When using Caomethod firstly set a delay time and the embedding dimen-sion and delay time are relatively independent [33] Define

119886 (119894 119898) =

1003817100381710038171003817119883119894 (119898 + 1) minus 119883119899(119894119898) (119898 + 1)

10038171003817100381710038171003817100381710038171003817119883119894 (119898) minus 119883

119899(119894119898) (119898)1003817100381710038171003817

(8)

In the function119883119894(119898+1) is a vector that 119894 is in a space of (119898+

1) dimensions 119899(119894 119898) is the integer that makes119883119899(119894119898)

(119898) bethe nearest neighbor domain of119883

119894(119898) in the119898-dimensional

reconstruction space 119899(119894 119898) depends on 119894 and 119898 and ∙

means the Euclidean distanceDefine the average of all 119886(119894 119898) as

119864 (119898) =1

119873 minus 119898120591

119873minus119898120591

sum

119894=1

119886 (119894 119898) (9)

Define the change of119898 dimensional to119898 + 1 dimensional as

1198641 (119898) =119864 (119898 + 1)

119864 (119898) (10)

If the time series are generated by the chaotic attractor when1198641(119898) saturated with the increases of 119898 the value of 119898 + 1

is the minimum embedding dimension Define 1198642(119898) todistinguish deterministic chaotic signal and random signal

119864lowast(119898) =

1

119873 minus 119898120591

119873minus119898120591

sum

119894=1

1003816100381610038161003816119909119894+119898120591 minus 119909119899(119894119898)+119898120591

1003816100381610038161003816

1198642 (119898) =119864lowast(119898 + 1)

119864lowast (119898)

(11)

For the random sequence to all of the values of119898 the valuesof1198642(119898) always equal about 1 while for the chaotic sequencethe values of 1198642(119898) are related to values of119898 Namely valuesof1198642(119898) are not constant to all of119898 in chaotic sequence [34]


313 Identify the Chaos by Computing the Largest LyapunovExponent The fundamental characteristics of chaos are theextreme sensitivity to initial values The sensitivity canbe measured by the exponent of Lyapunov It reflects thedivergence speed close to the initial orbit The exponentof Lyapunov exponent reflects the contracting and tensileproperties of the system in all directionsThis paper calculatesLyapunov exponent based on the Wolf method to judgewhether the original sequence is with chaos characteristics[35]Wolf method is based on phase trajectory phase planeand phase volume The process is as follows

(1) Determine the embedding dimension and delay timeto reconstruct phase space

(2) Track the evolution of the distance from initial pointand its closest point Limit the short separation

(3) Continue to the front steps until the end of the timeseries

(4) Get the maximum Lyapunov exponent

32 Chaotic Time Series Predicted by RBF Neural Network

321 The Basic Principle of RBF Neural Network RBF neuralnetwork is a feed-forward neural network consisting of inputlayer hidden layer and output layer [36 37] as shown inFigure 3

Input layer to hidden layer is linear connection Andhidden layer to output layer is nonlinear connection

322 Chaotic RBF Neural Network Prediction Steps Thespecific steps of chaotic RBF neural network prediction areas follows

(1) Establish the Network Determine the input and outputaccording to the calculated embedding dimension 119898 anddelay time 120591The input is a vector [119909

119894 119909119894+120591

119909119894+(119898minus1)120591

]Theoutput is a data 119909

2+(119898minus1)120591 RBF neural network is a mapping

from119898-dimension to 1-dimension In this paper the numberof input layer nodes is 119898 The number of output layer nodesis 1 The input matrix119883 and output matrix 119884 are as follows

119883 =

[[[[

[

1199091

1199091+120591

sdot sdot sdot 1199091+(119898minus1)120591

1199092

1199092+120591


sdot sdot sdot sdot sdot sdot sdot sdot sdot

119909119899minus(119898minus1)120591

119909119899minus(119898minus2)120591

sdot sdot sdot 119909119899

]]]]

]

119884 =

[[[[

[

1199092+(119898minus1)120591

1199093+(119898minus1)120591

119909119899+1

]]]]

]

(12)

(2) Training Stage Select a certain length of time series setmatrix119883 as input and vector119884 as output to train the network

(3) Forecasting Stage Set 1198831015840 = [119909119899+1minus(119898minus1)120591

119909119899+1minus(119898minus2)120591

119909119899+1

] as the input of the neural network and the output

Input layer Hidden layer Output layer

X1

X2

XN

YN

Figure 3 RBF neural network structure

0 96 192 288 3841200

1400

1600

1800

2000

2200

2400Lo

ad (M

W)

May 2 May 9 May 16 May 23

Figure 4 The original load trend chart of a region

1198841015840= 119909119899+2

is the actual predictive value of the sequence Theniterate the networkwith the predictive value as a newnetworkinput

4 The Empirical Example Analysis

To do empirical analysis this paper selects the load on May2 2011 May 9 May 16 and May 23 and May 30 (Monday)of a region and collects 96 points per day a total of 480 loaddata to calculate Among them set the data of first four daysa total of 384 data as the training sample the data of last dayand a total of 96 as test samples to verify and analyze the loadprediction model of inflection points

Firstly compose the load data of May 2 9 16 and 23 to aload sequenceThen calculate the sharp degree of every pointof the load data according to the identification algorithm ofinflection point based on sharp degree that was mentionedabove The load sequence and sharp degree sequence arerespectively shown in Figures 4 and 5

As can be seen from Figure 5 the polarization phe-nomenon of sharp degree sequence is obvious The point


96 192 288 3840

010203040506070809

1

The s

harp

deg

ree

May 9 May 16 May 23May 2

Figure 5 The sharp degree of load sequence diagram chart

0 2 4 6 8 10 12 140

02

04

06

08

1

12

14

Dimension

E1

and E2

Cao method of minimum embedding dimension

E1

E2

Figure 6 Caomethod for minimum embedding dimension

of sharp degree sequence is corresponding to the point oforiginal sequence Most of the values in Figure 5 are less than01 which are corresponding to the flat load curve in Figure 4Only a few values in Figure 5 are between 01 and 1 which arecorresponding to the inflection points in Figure 4 Thereforeset the threshold as 01 The load point with a sharp degreevalue greater than 01 is the inflection point defined in thispaper Thus inflection point can be effectively identified bythe sharp degree sequence

According to the mutual information method the delaytime 120591 = 4 And calculate the minimum embeddingdimension with Cao method It can be seen in Figure 6 that1198641 tends to be saturate along with the increase of 119898 When119898 ge 5 1198641 remains the same So the minimum embeddingdimension is 6 (119898 = 5 + 1 = 6) At the same time it alsocan be seen in the diagram that with different values of 1198981198642 is not always equal to 1The value of Lyapunov exponent is03017 after being calculated byWolf method which is greaterthan zero So the conversion sequences of input can be judgedas chaotic sequence

Reconstruct the phase space according to the minimumembedding dimension and delay time Then we can predictthe sharp degree of load sequence based on RBF neuralnetwork The most critical part of determining RBF network

0 50 100 150 200 250

Best training performance is 000099767 at epoch 285

285 epochs

TrainBestGoal

Mea

n sq

uare

d er

ror (

mse

)

101

100

10minus1

10minus2

10minus3

10minus4

Figure 7 The convergence curve in training process

is the choice of nonlinear excitation function In some typicalexcitation functions this paper uses the Gaussian function asthe excitation function for its radial symmetry smoothnessand simple expression formula Set the error of trainingsystem to 0001 Select the self-organizing selection centermethod to determine the network center width and weightThen train the training sample with RBF neural networkTheconvergence curve in training process is shown in Figure 7We can see that the blue trace is the convergence curve oftraining The red horizontal line which parallels the 119909-axisand equals 10minus3 is the goal of training The green vertical linewhich parallels the 119910-axis and equals 285 epochs at the farright side of the plot means that best training performanceis 000099767 at epoch 285 The forecasting result of testsamples of the load sharp degree sequence is shown inFigure 8

This paper uses the average absolute percentage errorMAPE and the root mean square error RMSE to evaluate theforecasting effect and the formula is

MAPE =1

119873

119873119899

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119910119899minus 119910119899

119910119899

10038161003816100381610038161003816100381610038161003816

RMSE = (1

119873

119873

sum

119899=1

(119910119899minus 119910119899)2)

12

(13)

The calculation shows that the MAPE is 5293 and theRMSE is 6391 The predicting is with a high accuracy

The forecasting error curve of load sharp degree sequencetest samples based on chaotic RBF is shown in Figure 9 It canbe seen that there is one point whose error value is bigger thanothers The point is the ninth point of prediction sequenceThe actual value of this point is less than 0001 However thepredicted value is 0576 The reason of this situation is thatthe ninth point of 96 points in the historical everyday data isinflection pointThe ninth point of forecasting sequence is anabnormal point


0 10 20 30 40 50 60 70 80 900

02

04

06

08

1

Point

Real valuePredictive value

Shar

p de

gree

of v

alue

Figure 8 Contrast figure of chaotic RBF inflection point sequencepredicted values and the real values

0 10 20 30 40 50 60 70 80 90

0

01

02

03

04

05

06

Point

Error curve

The e

rror

val

ue

minus01

Figure 9 Error graph of chaotic RBF inflection

This paper selects BP neural network model in orderto verify the effectiveness and superiority of the proposedalgorithm BP neural network is one of the most commonlyused kinds of artificial neural networkThe forecasting resultof test samples based on chaotic RBF and BP is shown inFigure 10 The MAPE of the results based on BP neuralnetwork is 9149 which is much higher than the MAPE ofchaotic RBF Thus the superiority of chaotic RBF algorithmis evident particularly

Set the inflection point threshold 119879 = 01 extract thepoints with a predictive value greater than 01 in sharp degreesequence and then locate and highlight the points in theknown load curve on May 30 as shown in Figure 11

It can be seen from Figure 11 that most of the inflectionpoints of test samples can be accurately predicted by eitherusing inflection point identification algorithm based oncontour sharp degree or forecasting model based on chaoticRBF neural network and that verified that inflection pointprediction model proposed in this paper is of high accuracy

In order to further validate the applicability and effective-ness of model this paper selected representative load curvesof different seasons to carry out the empirical researches

0 10 20 30 40 50 60 70 80 900

010203040506070809

1

Point

Predictive value based on C-RBFPredictive valu based on BP

Shar

p de

gree

of v

alue

Real value

Figure 10 Contrast figure of the real values predicted values basedon chaotic RBF and predicted values based on BP

0 10 20 30 40 50 60 70 80 901200

1400

1600

1800

2000

2200

2400

Point

Load

(MW

)

The original load curve

Figure 11 The inflection point location map according to the pre-dicted results

The prediction training sample contains 96 points per day ofthe same week type 4 weeks before the prediction and thetraining data totally contains 384 pointsThe prediction errorresults are shown in Table 1

The forecasting of 16 groups data shows that the averageMAPE of the model proposed in this paper is 425 Theaverage RMSE is 3897 It suggests that the forecast effectis fine and the model is with strong practicability andapplicability

5 Conclusion

This paper puts forward a feasible load inflection point pre-diction method by proposing an inflection point forecastingmodel that can realize effective prediction of inflection pointbased on the algorithm of sharp degree and chaotic RBFneural network on the basis of research on short-term powerload

In the calculation example the paper proposed an empir-ical analysis method of 96 points per day and the algorithmcan be also applied to the inflection point prediction of a


Table 1 Results of empirical study in 2011

Number Time MAPE RMSE Number Time MAPE RMSE1 January 3 6322 2647 9 August 9 3095 21912 January 11 3769 597 10 August 16 3616 20943 January 19 6937 3969 11 August 25 231 59674 January 28 4723 6162 12 August 31 3202 19875 May 4 311 2217 13 October 8 2459 59676 May 18 8468 4392 14 October 19 4193 22287 May 24 4271 2344 15 October 22 2719 59698 May 31 5187 6153 16 October 28 3616 2092

The mean of MAPE 425 The mean of RMSE 3897

shorter or a longer time interval However the time intervaland different threshold selection will directly affect theinflection point identification based on the forecasted sharpdegree sequence According to actual needs the original loadtime interval and the value of threshold can be adjusted

Although forecasting of the load curve contains theinformation of inflection point during the ordinary loadforecasting the prediction error of inflection points isslightly larger than other points This paper transformedthe original load sequence based on the sharp degree Andafter transformation the polarization phenomenon of thesequence is obvious that highlights the regularity of thechange of the inflection point In the later research workthe inflection point forecast correction the improvement oforiginal load curve forecasting results and mining of theinfluence factors of inflection point to increase the inputdimension of inflection point prediction can be consideredto further improve the prediction accuracy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is supported by the National Science Foundationof China (Grant nos 71071052 and 71471059)

References

[1] D X Niu S H Cao and J C H Lu Power Load ForecastingTechnique and Its Application China Electric Power Press 2009

[2] C H Q Kang Q Xia andM Liu Power System Load Forecast-ing China Electric Power Press 2007

[3] T Hong and P Wang ldquoFuzzy interaction regression for shortterm load forecastingrdquo Fuzzy Optimization and Decision Mak-ing vol 13 no 1 pp 91ndash103 2014

[4] N Amjady ldquoShort-term hourly load forecasting using time-series modeling with peak load estimation capabilityrdquo IEEETransactions on Power Systems vol 16 no 3 pp 498ndash505 2001

[5] L Hernandez C Baladron J M Aguiar et al ldquoArtificialneural networks for short-term load forecasting in microgridsenvironmentrdquo Energy vol 75 pp 252ndash264 2014

[6] M Y Chow andH Tram ldquoApplication of fuzzy logic technologyfor spatial load forecastingrdquo IEEE Transactions on Power Sys-tems vol 12 no 3 pp 1360ndash1366 1997

[7] D Srinivasan and S S Tan ldquoParallel neural network-fuzzyexpert system strategy for short-term load forecasting systemimplementation and performance evaluationrdquo IEEE Transac-tions on Power Systems vol 14 no 3 pp 1100ndash1106 1999

[8] W J Staszewski and K Worden ldquoWavelet analysis of time-series coherent structures chaos and noiserdquo InternationalJournal of Bifurcation and Chaos vol 9 no 3 pp 455ndash471 1999

[9] H Mori and S Vrano ldquoShort-term load forecasting withchaos time series analysisrdquo in Proceedings of the InternationalConference on Intelligent Systems Applications to Power Systemspp 283ndash287 2007

[10] J X Che and J Z Wang ldquoShort-term load forecasting usinga kernel-based support vector regression combination modelrdquoApplied Energy vol 132 pp 602ndash609 2014

[11] A Deihimi O Orang and H Showkati ldquoShort-term electricload and temperature forecasting using wavelet echo statenetworks with neural reconstructionrdquo Energy vol 57 pp 382ndash401 2013

[12] H J Sadaei R Enayatifar A H Abdullah and A Gani ldquoShort-term load forecasting using a hybrid model with a refinedexponentially weighted fuzzy time series and an improvedharmony searchrdquo International Journal of Electrical Power andEnergy Systems vol 62 pp 118ndash129 2014

[13] Z H Xiao and S J Ye ldquoRough set method for short-term loadforecastingrdquo Journal of Systems Engineering vol 24 no 2 pp143ndash149 2009

[14] C H Kim B G Koo and J H Park ldquoShort-term electric loadforecasting using data mining techniquerdquo Journal of ElectricalEngineering and Technology vol 7 no 6 pp 807ndash813 2012

[15] L Z H Zhu ldquoShort-term electric load forecasting with com-bined data mining algorithmrdquo Automation of Electric PowerSystems vol 30 no 14 pp 82ndash86 2006

[16] D X Niu and Y NWei ldquoShort-term power load combinatorialforecast adaptively weighted by FHNN similar-day clusteringrdquoAutomation of Electric Power Systems vol 37 no 3 pp 54ndash572013

[17] A Kavousi-Fard H Samet and F Marzbani ldquoA new hybridModified Firefly Algorithm and Support Vector Regressionmodel for accurate Short Term Load Forecastingrdquo ExpertSystems with Applications vol 41 no 13 pp 6047ndash6056 2014

[18] M Yoder A S Hering W C Navidi and K Larson ldquoShort-term forecasting of categorical changes in wind power withMarkov chain modelsrdquo Wind Energy vol 17 no 9 pp 1425ndash1439 2014


[19] S Bahrami R-A Hooshmand andM Parastegari ldquoShort termelectric load forecasting by wavelet transform and grey modelimproved by PSO (particle swarm optimization) algorithmrdquoEnergy vol 72 pp 434ndash442 2014

[20] S H D Pan Z H N Wei Z H Gao et al ldquoA short-termload forecasting model based on relevance vector machine withnonnegative matrix factorizationrdquoAutomation of Electric PowerSystems vol 36 no 11 pp 62ndash66 2012

[21] Q-H Zhang ldquoAmodel for short-term load forecasting in powersystem based on multi-AI methodsrdquo System EngineeringTheoryand Practice vol 33 no 2 pp 354ndash362 2013

[22] Z G Qian and X Z Lin ldquoDetection algorithm of image cornerbased on contour sharp degreerdquo Computer Engineering vol 34no 6 pp 202ndash204 2008

[23] B Amrouche and X Le Pivert ldquoArtificial neural networkbased daily local forecasting for global solar radiationrdquo AppliedEnergy vol 130 pp 333ndash341 2014

[24] RMohammadi SM T F Ghomi and F Zeinali ldquoA new hybridevolutionary based RBF networks method for forecasting timeseries a case study of forecasting emergency supply demandtime seriesrdquo Engineering Applications of Artificial Intelligencevol 36 pp 204ndash214 2014

[25] H Y Luo T Q Liu and X Y Li ldquoChaotic forecastingmethod ofshort-termwind speed in wind farmrdquo Power SystemTechnologyvol 33 no 9 pp 67ndash71 2009

[26] M D Alfaro J M Sepulveda and J A Ulloa ldquoForecastingchaotic series in manufacturing systems by vector supportmachine regression and neural networksrdquo International Journalof Computers Communications amp Control vol 8 no 1 pp 8ndash172013

[27] Z Y Zhang T Wang and X G Liu ldquoMelt index prediction byaggregated RBF neural networks trained with chaotic theoryrdquoNeurocomputing vol 131 pp 368ndash376 2014

[28] H B Bi and Y B Zhang ldquoApplication of the chaotic RBF neuralnetwork on electrical loads predictionrdquo Science Technology andEngineering vol 9 no 24 pp 7480ndash7492 2009

[29] W Y Zhang W C Hong Y Dong G Tsai J T Sung and GF Fan ldquoApplication of SVR with chaotic GASA algorithm incyclic electric load forecastingrdquo Energy vol 45 no 1 pp 850ndash858 2012

[30] S Kouhi F Keynia and S N Ravadanegh ldquoA new short-termload forecast method based on neuro-evolutionary algorithmand chaotic feature selectionrdquo International Journal of ElectricalPower amp Energy Systems vol 62 pp 862ndash867 2014

[31] X-Q Lu B Cao M Zeng S-S Huang and X-G LiuldquoAlgorithm of selecting delay time in the mutual informationmethodrdquo Chinese Journal of Computational Physics vol 23 no2 pp 184ndash188 2006

[32] K Aihara T Takabe and M Toyoda ldquoChaotic neural net-worksrdquo Physics Letters A vol 144 no 6-7 pp 333ndash340 1990

[33] Y R Cheng and S B Guo ldquoStock price prediction based onanalysis of chaotic time seriesrdquo Journal of UEST of China vol32 no 4 pp 469ndash472 2003

[34] S-Q Zhang J Jia M Gao and X Han ldquoStudy on theparameters determination for reconstructing phase-space inchaos time seriesrdquo Acta Physica Sinica vol 59 no 3 pp 1576ndash1582 2010

[35] D Kressner M Pleinger and C Tobler ldquoA preconditioned low-rank CG method for parameter-dependent Lyapunov matrixequationsrdquo Numerical Linear Algebra with Applications vol 21no 5 pp 666ndash684 2014

[36] J Z Zhou Y C ZhangQQ Li and J Guo ldquoProbabilistic short-term load forecasting based on dynamic self-adaptive radialbasis function networkrdquo Power System Technology vol 34 no3 pp 37ndash41 2010

[37] H L Deng and X Q Li ldquoStock price inflection point predictionmethod Based on chaotic time series analysisrdquo Statistics andDecision no 5 pp 19ndash20 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


O

a

Pi+kPiminusk

Pi

Figure 1 Part outline figure of the load curve

changes the direction of load curve upward or downward ina certain time interval This paper identifies the inflectionpoint according to the sharp degree of each point on the loadcurve Set the function of load curve as 119875 = 119901(119905) in whichthe independent variable 119905 means time and the dependentvariable 119875 represents the load time series

Randomly select a point 119875119894on the curve as the center 119875

119894minus119896

is the point in front of 119875119894a distance of 119896 And 119875

119894+119896is the point

behind 119875119894a distance of 119896

Define the vector 119896

119886119894119896

= (119879 (119894) minus 119879 (119894 + 119896) 119875 (119894) minus 119875 (119894 + 119896))

119887119894119896

= (119879 (119894) minus 119879 (119894 minus 119896) 119875 (119894) minus 119875 (119894 minus 119896))

119888119894119896

= (119879 (119894 minus 119896) minus 119879 (119894 + 119896) 119875 (119894 minus 119896) minus 119875 (119894 + 119896))

(1)

Define 120572 the angle of 119886119894119896

and 119887119894119896 as the angle consisting

of 119875119894minus119896

119875119894 and 119875

119894+119896on the load curve Figure 1 shows the

part outline figure of the load curve The peripheral solidline is composed of load time series The dotted line is thecircular arc fitted by 119875

119894minus119896 119875119894 and 119875

119894+119896 And 119874 is the center of

circular arc Usually the value of 119896 is set in the range of 3ndash5 For the endpoint of load curve we will take 3 to 5 samplevalues forward and backward on the sample load data On theactual curve the points119875

119894minus119896119875119894 and119875

119894+119896can be approximately

regarded as three points on the circular arc as the interval ofthe three points is very small [22]

Assuming that |119875119894119875119894minus119896

| = |119875119894119875119894+119896

| then

sin(120572

2) =

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816 2

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816

=

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816 2

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

=

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

(2)

In this function 0 lt 120572 le 180When119875

119894minus119896119875119894 and119875

119894+119896are in a straight line 120572 = 180 then

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

= sin(120572

2) = sin (90) = 1 (3)

When the value of 120572 decreases tending to zero

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

= sin(120572

2) = sin (0) = 0 (4)

Therefore in a very small load curve the variable sharp canbe defined as sharp degree of 120572

sharp = 1 minus ang = 1 minus

1003816100381610038161003816119875119894minus119896119875119894+1198961003816100381610038161003816

1003816100381610038161003816119875119894119875119894minus1198961003816100381610038161003816 +

1003816100381610038161003816119875119894119875119894+1198961003816100381610038161003816

(5)

The larger the value of sharp the smaller the value of 120572 andthe curve would be more sharp On the contrary the curvewould be more smooth Because of the continuity of loadthe identification of inflection point is closely related to theinterval of load curve sequence and the set threshold valueTherefore the threshold value can be set according to the loadcurve characteristics and actual situation If sharp (119875

119894) gt 119879

the 119875119894can be regarded as inflection point

3 Prediction Model Based onChaotic RBF Neural Network

Artificial neural network is a common type of load fore-casting methods This method is highly intelligent whichhas strong ability to look for patterns [23] Radical basisfunction (RBF) neural network is a kind of highly efficientfeed-forward neural network It has the best approximationperformance and global optimal property And the structureis simple the training speed is fast [24] Chaos predictionalgorithm applied to power load is a very hot research fieldwhich is caused by the intrinsic characteristics of systemsIt can be predicted in the short term [25] It predictsdirectly according to the laws of the objective extracting datasequence itself and does not need to establish the mappingrelation model of influence factors In theory it not onlycan decrease the cost of prediction but can improve theprediction accuracy as well [26] The method combiningchaos and RBF neural network can make full use of therandomicity and initial value sensitivity of chaos In additionit can make full use of the function of RBF such as massivelyparallel processing self-organizing and self-adapting [27]

Although sharp degree sequence is turbulent it can beused to identify the inflection point Chaotic time series aresimilar to random noise which is chaotic and unpredictableBut its inherent nonlinear dynamics structure makes itmeet the short-term predictability [28] Use phase spacereconstruction to find the internal of regularity conversionsequences that can be described by a nonlinear mapping andthen approximate the nonlinear mapping by applying radialbasis to forecast the inflection pointThe process of themodelestablished in this paper is shown in Figure 2

First of all we can get the sharp degree sequence whichcan identify the inflection point on the basis of Section 2Then reconstruct phase space and identify chaos based onchaos theory This paper selects mutual information methodto calculate delay timeCaomethod to determine embeddingdimension and the computing of largest Lyapunov exponentto identify the chaos Thirdly the input layer of RBF neuralnetwork can be determined based on the delay time andembedding dimension Then train the RBF neural networkand predict the sharp degree sequence making use of thetrained network Finally determine the threshold and posi-tion the inflection point


Sharpdegree

sequence

Mutualinformation

method

Cao method


index



Identify thechaos

Determinethe input

layer


Pointposition




1 1199092 1199093 119909

119899 properly


1198841= (1199091 1199091+120591

1199091+(119898minus1)120591

)119879

1198842= (1199092 1199092+120591

1199092+(119898minus1)120591

)119879

119884119873

= (119909119873 119909119873+120591

119909119873+(119898minus1)120591

)119879

(6)



119868 (119883 119884) = 119867 (119883) + 119867 (119884) minus 119867 (119883 119884)

119867 (119883) =

119902

sum

119894=1

119875 (119909119894) log119875 (119909

119894)

(7)


119894




119894in 119909119894 and

119875(119909119894+120591




119894 119909119894+120591





119894 119909119894+120591




119886 (119894 119898) =

1003817100381710038171003817119883119894 (119898 + 1) minus 119883119899(119894119898) (119898 + 1)

10038171003817100381710038171003817100381710038171003817119883119894 (119898) minus 119883

119899(119894119898) (119898)1003817100381710038171003817

(8)







119864 (119898) =1

119873 minus 119898120591

119873minus119898120591

sum

119894=1

119886 (119894 119898) (9)


1198641 (119898) =119864 (119898 + 1)

119864 (119898) (10)



119864lowast(119898) =

1

119873 minus 119898120591

119873minus119898120591

sum

119894=1

1003816100381610038161003816119909119894+119898120591 minus 119909119899(119894119898)+119898120591

1003816100381610038161003816

1198642 (119898) =119864lowast(119898 + 1)

119864lowast (119898)

(11)













119894 119909119894+120591

119909119894+(119898minus1)120591




119883 =

[[[[

[

1199091

1199091+120591


1199092

1199092+120591



119909119899minus(119898minus1)120591

119909119899minus(119898minus2)120591


]]]]

]

119884 =

[[[[

[

1199092+(119898minus1)120591

1199093+(119898minus1)120591

119909119899+1

]]]]

]

(12)



119909119899+1minus(119898minus2)120591

119909119899+1



X1

X2

XN

YN


0 96 192 288 3841200

1400

1600

1800

2000

2200

2400Lo

ad (M

W)



1198841015840= 119909119899+2







96 192 288 3840

010203040506070809

1

The s

harp

deg

ree



0 2 4 6 8 10 12 140

02

04

06

08

1

12

14

Dimension

E1

and E2


E1

E2





0 50 100 150 200 250


285 epochs

TrainBestGoal

Mea

n sq

uare

d er

ror (

mse

)

101

100

10minus1

10minus2

10minus3

10minus4




MAPE =1

119873

119873119899

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119910119899minus 119910119899

119910119899

10038161003816100381610038161003816100381610038161003816

RMSE = (1

119873

119873

sum

119899=1

(119910119899minus 119910119899)2)

12

(13)




0 10 20 30 40 50 60 70 80 900

02

04

06

08

1

Point


Shar

p de

gree

of v

alue


0 10 20 30 40 50 60 70 80 90

0

01

02

03

04

05

06

Point

Error curve

The e

rror

val

ue

minus01






0 10 20 30 40 50 60 70 80 900

010203040506070809

1

Point


Shar

p de

gree

of v

alue

Real value


0 10 20 30 40 50 60 70 80 901200

1400

1600

1800

2000

2200

2400

Point

Load

(MW

)





5 Conclusion











Acknowledgment


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sharpdegree

sequence

Mutualinformation

method

Cao method


index



Identify thechaos

Determinethe input

layer


Pointposition




1 1199092 1199093 119909

119899 properly


1198841= (1199091 1199091+120591

1199091+(119898minus1)120591

)119879

1198842= (1199092 1199092+120591

1199092+(119898minus1)120591

)119879

119884119873

= (119909119873 119909119873+120591

119909119873+(119898minus1)120591

)119879

(6)



119868 (119883 119884) = 119867 (119883) + 119867 (119884) minus 119867 (119883 119884)

119867 (119883) =

119902

sum

119894=1

119875 (119909119894) log119875 (119909

119894)

(7)


119894




119894in 119909119894 and

119875(119909119894+120591




119894 119909119894+120591





119894 119909119894+120591




119886 (119894 119898) =

1003817100381710038171003817119883119894 (119898 + 1) minus 119883119899(119894119898) (119898 + 1)

10038171003817100381710038171003817100381710038171003817119883119894 (119898) minus 119883

119899(119894119898) (119898)1003817100381710038171003817

(8)







119864 (119898) =1

119873 minus 119898120591

119873minus119898120591

sum

119894=1

119886 (119894 119898) (9)


1198641 (119898) =119864 (119898 + 1)

119864 (119898) (10)



119864lowast(119898) =

1

119873 minus 119898120591

119873minus119898120591

sum

119894=1

1003816100381610038161003816119909119894+119898120591 minus 119909119899(119894119898)+119898120591

1003816100381610038161003816

1198642 (119898) =119864lowast(119898 + 1)

119864lowast (119898)

(11)













119894 119909119894+120591

119909119894+(119898minus1)120591




119883 =

[[[[

[

1199091

1199091+120591


1199092

1199092+120591



119909119899minus(119898minus1)120591

119909119899minus(119898minus2)120591


]]]]

]

119884 =

[[[[

[

1199092+(119898minus1)120591

1199093+(119898minus1)120591

119909119899+1

]]]]

]

(12)



119909119899+1minus(119898minus2)120591

119909119899+1



X1

X2

XN

YN


0 96 192 288 3841200

1400

1600

1800

2000

2200

2400Lo

ad (M

W)



1198841015840= 119909119899+2







96 192 288 3840

010203040506070809

1

The s

harp

deg

ree



0 2 4 6 8 10 12 140

02

04

06

08

1

12

14

Dimension

E1

and E2


E1

E2





0 50 100 150 200 250


285 epochs

TrainBestGoal

Mea

n sq

uare

d er

ror (

mse

)

101

100

10minus1

10minus2

10minus3

10minus4




MAPE =1

119873

119873119899

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119910119899minus 119910119899

119910119899

10038161003816100381610038161003816100381610038161003816

RMSE = (1

119873

119873

sum

119899=1

(119910119899minus 119910119899)2)

12

(13)




0 10 20 30 40 50 60 70 80 900

02

04

06

08

1

Point


Shar

p de

gree

of v

alue


0 10 20 30 40 50 60 70 80 90

0

01

02

03

04

05

06

Point

Error curve

The e

rror

val

ue

minus01






0 10 20 30 40 50 60 70 80 900

010203040506070809

1

Point


Shar

p de

gree

of v

alue

Real value


0 10 20 30 40 50 60 70 80 901200

1400

1600

1800

2000

2200

2400

Point

Load

(MW

)





5 Conclusion











Acknowledgment


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















119894 119909119894+120591

119909119894+(119898minus1)120591




119883 =

[[[[

[

1199091

1199091+120591


1199092

1199092+120591



119909119899minus(119898minus1)120591

119909119899minus(119898minus2)120591


]]]]

]

119884 =

[[[[

[

1199092+(119898minus1)120591

1199093+(119898minus1)120591

119909119899+1

]]]]

]

(12)



119909119899+1minus(119898minus2)120591

119909119899+1



X1

X2

XN

YN


0 96 192 288 3841200

1400

1600

1800

2000

2200

2400Lo

ad (M

W)



1198841015840= 119909119899+2







96 192 288 3840

010203040506070809

1

The s

harp

deg

ree



0 2 4 6 8 10 12 140

02

04

06

08

1

12

14

Dimension

E1

and E2


E1

E2





0 50 100 150 200 250


285 epochs

TrainBestGoal

Mea

n sq

uare

d er

ror (

mse

)

101

100

10minus1

10minus2

10minus3

10minus4




MAPE =1

119873

119873119899

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119910119899minus 119910119899

119910119899

10038161003816100381610038161003816100381610038161003816

RMSE = (1

119873

119873

sum

119899=1

(119910119899minus 119910119899)2)

12

(13)




0 10 20 30 40 50 60 70 80 900

02

04

06

08

1

Point


Shar

p de

gree

of v

alue


0 10 20 30 40 50 60 70 80 90

0

01

02

03

04

05

06

Point

Error curve

The e

rror

val

ue

minus01






0 10 20 30 40 50 60 70 80 900

010203040506070809

1

Point


Shar

p de

gree

of v

alue

Real value


0 10 20 30 40 50 60 70 80 901200

1400

1600

1800

2000

2200

2400

Point

Load

(MW

)





5 Conclusion











Acknowledgment


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










96 192 288 3840

010203040506070809

1

The s

harp

deg

ree



0 2 4 6 8 10 12 140

02

04

06

08

1

12

14

Dimension

E1

and E2


E1

E2





0 50 100 150 200 250


285 epochs

TrainBestGoal

Mea

n sq

uare

d er

ror (

mse

)

101

100

10minus1

10minus2

10minus3

10minus4




MAPE =1

119873

119873119899

sum

119899=1

10038161003816100381610038161003816100381610038161003816

119910119899minus 119910119899

119910119899

10038161003816100381610038161003816100381610038161003816

RMSE = (1

119873

119873

sum

119899=1

(119910119899minus 119910119899)2)

12

(13)




0 10 20 30 40 50 60 70 80 900

02

04

06

08

1

Point


Shar

p de

gree

of v

alue


0 10 20 30 40 50 60 70 80 90

0

01

02

03

04

05

06

Point

Error curve

The e

rror

val

ue

minus01






0 10 20 30 40 50 60 70 80 900

010203040506070809

1

Point


Shar

p de

gree

of v

alue

Real value


0 10 20 30 40 50 60 70 80 901200

1400

1600

1800

2000

2200

2400

Point

Load

(MW

)





5 Conclusion











Acknowledgment


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 900

02

04

06

08

1

Point


Shar

p de

gree

of v

alue


0 10 20 30 40 50 60 70 80 90

0

01

02

03

04

05

06

Point

Error curve

The e

rror

val

ue

minus01






0 10 20 30 40 50 60 70 80 900

010203040506070809

1

Point


Shar

p de

gree

of v

alue

Real value


0 10 20 30 40 50 60 70 80 901200

1400

1600

1800

2000

2200

2400

Point

Load

(MW

)





5 Conclusion











Acknowledgment


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Acknowledgment


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Short-Term Power Load Point Prediction...

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Transcript of Research Article Short-Term Power Load Point Prediction...