Research Article 3D Modeling of Transformer Substation ...Research Article 3D Modeling of...

Research Article3D Modeling of Transformer Substation Based onMapping and 2D Images

Lei Sun Xuesong Suo Yifan Liu Meng Zhang and Lijuan Han

College of Mechanical amp Electrical Engineering Agricultural University of Hebei Baoding 071001 China

Correspondence should be addressed to Xuesong Suo 13903120861163com

Received 27 January 2016 Revised 8 June 2016 Accepted 30 August 2016

Academic Editor Mohammadreza Nasiriavanaki

Copyright copy 2016 Lei Sun et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new method for building 3D models of transformer substation based on mapping and 2D images is proposed in this paper Thismethod segments objects of equipment in 2D images by using 119896-means algorithm in determining the cluster centers dynamicallyto segment different shapes and then extracts feature parameters from the divided objects by using FFT and retrieves the similarobjects from 3D databases and then builds 3D models by computing the mapping data The method proposed in this paper canavoid the complex data collection and big workload by using 3D laser scanner The example analysis shows the method can buildcoarse 3D models efficiently which can meet the requirements for hazardous area classification and constructions representationsof transformer substation

1 Introduction

In recent years substations under centralized managementcontrol mode has also been exposed to many problems withthe increasing size of the grid such as the following thestation complex and diverse equipment are too difficult toobserve the staff cannot quickly master station equipmentenvironment [1ndash5] Creating a visual system is a great help forthe training of the operators However traditional workloadof 3D modeling for transformer substation is too great andthe work period is too long

In order to protect the security of staffs and facilitate theirtraining the 3D models of transformer substation by usinglaser scanning technology have increasingly aroused con-cerns of experts [6ndash9]The rise of 3D laser scanning technolo-gy can solve the problem of the data acquisition and precisionof the model while it may get a lot of points of informationthat leads to accurate modeling of the substation equipment[10ndash12] However the complex substation equipment struc-ture will bring high point density by using 3D laser scanningtechnology which will increase the difficulty and time datacollection and make the model work overload [13ndash15]

3D reconstruction is one of the main research contentsof the computer vision technology In engineering and many

other areas in order to get useful information for research weusually analyze the 3D structure of the objects The researchon 3D reconstruction has important practical significanceand has a broad range of applications such as archaeologyarchitecture materials processing industrial inspection andmedical imaging equipment industry [16ndash18]

In order to reduce the modeling workload researchershave raised lots of 3Dmodeling methods by using 2D images[19ndash21] Dr Macro proposed a 3D reconstruction method byusing a 2D detector under varying illumination conditions[22] The method proposed by Dr Macro is applied in headtracking which can extend the range of head motion andensure the reconstruction fast and reliable Literature [23]proposed a 3D modeling method based on 2D sketch Inliterature [24] a reconstructionmethod by using new Zernikecomparator is proposed which can provide a more accuratesimilarity measure together with the optimal rotation anglebetween the patterns while keeping the same complexity asthe classical approach The method can ensure the automatictranscription of sketched storyboards into reconstructed 3Dscenes These method are more focused on 3D modelingbased on texture map image however the 3D modelingof transformer substation is more focused on the external

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9320502 6 pageshttpdxdoiorg10115520169320502

2 Mathematical Problems in Engineering

Image captureand mapping

Object segmentation

Featureextraction

Objects retrieving in the 3D library

Size calculation and objects combination

Equipment location

calculation

3D scene generating

Figure 1 Flow chart of 3D modeling

structure and shape of power equipment in order that staffscan zone security area and understand the environment of thesubstation easily [25 26]

Themethod proposed in this paper is a 3D reconstructiontechnology based on 2D images which segments objects ofequipment in 2D images to consist of a correspondinglyshaped devicemodel that is retrieved from the objects libraryThen the size spacing and other parameters can be deter-mined by the device mapping dataThe 3Dmodel establishedby this method not only canmeet the requirements of zoningsecurity the substation shows and so on but also can reducemodeling costs and time greatly

2 Principle of the Modeling Method

The difficulty of building 3D models from 2D images liesin lack of depth data The method proposed in this papertakes shot from multiviews of equipment in the substationand then maps the position information of each device andgets equipment size equipment location coordinates andother information according to the scale After making pixeldivision of the equipment we retrieve the similar objectsfrom the objects library by using the extracted features The3D size dimension of the objects is calculated according to thepixel in the image Then determine the distance between thedevice and other aspects based on mapping data to establishthe 3D modeling of the transformer substation The user canalso adjust the color and the concrete models The process ofestablishing the 3D modeling is shown in Figure 1

3 Image Segmentation

This paper sets sides of objects as 119899 side and 119905 side Wesolve the energy function by using the improved 119896-means

algorithm for cluster of different connection componentsand then we can complete the image segmentation

119904 is the source point and 119905 is the terminal point the con-nection side between 119904 and 119905 point is the 119905 side 119875 and 119902 pointare neighbor points and the connection side between 119901 and119902 point is the 119899 side

We create a dynamic array 119866 = 1198740 1198741 1198742 to store

values of different segmentation shapes Every side has aweight value 119908 Segmentation cost is defined as the summa-tion weight value of all sides which is shown as

119862 (119874119899 119874119899+1

) = sum

119901isin119874119899119902isin119874119899+1

119908 (119901 119902) (1)

(119901 119902) means the connection side between 119901 and 119902 point119908(119901 119902) is the weight value of (119901 119902) 119862(119874

119899 119874119899+1

) is thesegmentation cost between the 119899th shape and (119899 + 1)thshape We define the segmentation cost as the sign of thesegmentation effect The less the segmentation cost is thebetter the segmentation effect is Equation (2) is Gibbsfunction that can minimize the segmentation cost

119864 (119883) = sum

119901isin120575

1198641(119909119901 119904 | 119905) + 120582 sum

(119901119902)isin120576

1198642(119909119901 119909119902) (2)

As shown in (2) energy value function 119864(119883) consisted ofthe 119905 side energy values and 119899 side energy values 119864

1(119909119901 119904 |

119905) is the energy values of 119905 side which is the connection sidebetween 119904 and 119905 point119864

2(119909119901 119909119902) is the energy values of 119899 side

which is the connection side between 119901 and 119902 point 120582 ge 0 isthe parameter used to balance the two values 119909

119901 119909119902are pixels

values of 119901 and 119902 point 120575 is the set of points and 120576 is the setof sides

The 119899th cluster center is defined as 119874119899 and the Euclidean

distance from one point to the 119899th cluster center is 119889119900119899

119901=

min119899119909119901

minus 119900119899 and to the previous cluster center is 119889

119900119899minus1

119901 =

min119899minus1

119909119901

minus 119900119899minus1

If the Euclidean distance is smaller thepoint will be more similar to the previous cluster center TheEuclidean distance from point 119901 notin 119874

119899 cup 119874

119899minus1 to the 119899th

cluster center is shown as

V =119889119900119899

119901

119889119900119899

119901 + 119889119900119899minus1

119901

(3)

We can get 1198641(119909119901 119904 | 119905) that is shown as

119901 isin 119874119899 119901 isin 119874

119899minus1 119901 notin 119874

119899 cup 119874

119899minus1

1198641(119909119901= 1) 0 infin V

1198641(119909119901= 0) infin 0 1 minus V

(4)

According to above-mentioned equations and analyticalresult we can get which energy values belong to119874

119899andwhich

Mathematical Problems in Engineering 3

853 (mm)

Figure 2 The 2D image of the sample

energy values belong to 119874119899minus1

If a point belongs to neither119874119899minus1

nor 119874119899 it will be defined as the (119899 + 1)th source point 119904

to repeat above process until all of the points are traversedFor the 119899 side we determine their respective section

according to the relationship with the surrounding pixelsThe more similar the 119901 and 119902 point are the higher the valueof 1198642is on the contrary the value of 119864

2will tend to 0

According to the color Euclidean distance between twopixelswe can define the energy value of 119899 side connection betweenthe 119901 and 119902 point as

1198642(119909119901 119909119902) =

1

radic2120587exp(minus

10038171003817100381710038171003817119909119901minus 119909119902

10038171003817100381710038171003817

2

2) (5)

At last by solving the energy function we can determinethe different outer contour shape and centroid of the objectsso that we can complete the image segmentation

4 Image Reconstruction

This paper extracts feature parameters from the dividedobjects by using FFT For pixels are discrete in the 2D imagewe can descript the pixels in 119873 times 119872 image as

119891 (119909 119910)

=

[[[[[[

[

119891 (0 0) 119891 (0 1) sdot sdot sdot 119891 (0119873 minus 1)


sdot sdot sdot

119891 (119872 minus 1 0) 119891 (119872 minus 1 1) sdot sdot sdot 119891 (119872 minus 1119873 minus 1)

]]]]]]

]

(6)

We set 119873 = 119872 for convenience and then we can getdiscrete Fourier transform result as

119865 (119906 V) = I [119891 (119909 119910)]

=1

119873

119873minus1

sum

119909=0

119873minus1

sum

119910=0

119891 (119909 119910) 119890minus1198952120587((119906119909+V119910)119873)

(7)

Since the original image has been divided into simple 3Dgeometric objects that contain a small amount of informa-tion setting119873 = 256 can meet the requirement of retrievingThen we can get 119894th energy response as

119865119894= (

256

sum

119895=0

119860119895)

12

119894 = 0 1 2 256 (8)

Normalizing above equations we can get characteristiccomponent of the target object as

119878119865119894

=119865119894

sum256

119894=0119865119894

(9)

Then we can the similarity degree between the retrievedobject and the retrieval object by using Euclidean distance as

119878(119902119889)

= (

255

sum

119894=0

(119878119865119894119889minus 119878119865119894119902)2

)

12

(10)

Then use the most similar objects after setting their 3Dsize to reconstruct the equipment

5 Case Analysis

To verify the validity of the proposed method we give anexample of 3D model reconstruction analysis by combiningobjects The 2D image of power equipment models is shownin Figure 2

As mentioned above objects shown in Figure 2 arecomposed of basic shapes Segment Figure 2 as the methodproposed in this paper and we can get parts of objects assamples shown in Figure 3 In Figure 3 we can see that thebasic shapes retrieved from the objects library are similar tothe original basic shapes that can meet the requirements ofbuilding 3D transformer substation models

By calculating according to the pixel in the image we canget the 3D models of independent equipment as shown inFigure 4

Then determine the distance between the device andother aspects based on mapping data to establish the 3D


Figure 3 Parts of the divided objects

Figure 4 The 3D models image

DPI 600

1952

1 048 (mm)Pixel actual distance (mm)

Figure 5 The 3D combination chart of the three power equipment

model of the 2D image as shown in Figure 5 DPI of the imagein Figure 5 is 600 and the scale of it is a pixel to 048mmThedistance labeled in the image is 1952 pixels and the distanceconverted by the scale is 937mm while the actual distanceshown in Figure 2 is 853mm So the result of modeling canmeet basic needs

At last we built a 3Dmodeling of a transformer substationthat its panorama local view and local enlarged view areshown as Figures 6 7 and 8

To verify the superiority of the method proposed in thispaper we make the curves of precision ratio-times as shownin Figure 9 by using threemethods to retrieve a simplemodel


Figure 6 The 3D panorama of the transformer substation

Figure 7 The 3D local view of the transformer substation

Figure 8The 3D local enlarged view of the transformer substation

As shown in Figure 9 the speed of 3D modeling methodbased on 2D sketch is the fastest while its precision ratiois lowest 3D laser scanning technology can make the mostaccurate modeling however it will increase the difficulty andtime data collection and make the model work overload Themethod proposed in this paper can achieve a high accuracywith a little time which can verify the validity of the method

6 Conclusions

The diverse equipment in transformer station is too diffi-cult to observe and the staff cannot quickly master stationequipment environment In order to protect the security ofstaffs and facilitate their training this paper proposed a 3Dmodeling method based on mapping and 2D images Thismethod segments objects of equipment in 2D images by

0 50 100 150 200 250 300 350 400

00

02

04

06

08

10

Prec

ision

ratio

Time (s)

Method based on 2D sketchMethod proposed in this paper3D laser scanning technology

Figure 9 The precision ratio of three methods

using 119896-means algorithm in determining the cluster centersdynamically to segment different shapes and then extractsfeature parameters from the divided objects by using FFTand retrieves the similar objects from 3D databases and thenbuilds 3D models by computing the mapping data The 3Dmodel established by this method not only can meet therequirements of zoning security the substation shows and soon but also can reduce modeling costs and time greatly

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the project National High-TechRampD Program (863 Program) of China (2015AA050603)


References

[1] H Zhang A Chen M Niyi and J Ding ldquoResearch on the keytechnology of smart substationmodel configuration and checkrdquoin Proceedings of the International Conference on AdvancedPower System Automation and Protection (APAP rsquo11) pp 291ndash294 October 2011

[2] G Romero J Maroto J Felez J M Cabanellas M L Martınezand A Carretero ldquoVirtual reality applied to a full simulator ofelectrical sub-stationsrdquo Electric Power Systems Research vol 78no 3 pp 409ndash417 2008

[3] S-Y Park and M Subbarao ldquoAn accurate and fast point-to-plane registration techniquerdquo Pattern Recognition Letters vol24 no 16 pp 2967ndash2976 2003

[4] G W Yan L Zhang and Y F Wang ldquoResearch and imple-mentation of the auto-generating system of three-dimensionalsubstation simulation scenerdquo in Proceedings of the IEEE Inter-national Conference on Intelligent Computing and IntelligentSystems (ICIS rsquo10) pp 768ndash771 IEEE Xiamen China October2010

[5] D Engel C Herdtweck B Browatzki and C Curio ldquoImageretrieval with semantic sketchesrdquo inHuman-Computer Interac-tion-INTERACT 2011 vol 6946 of Lecture Notes in ComputerScience pp 412ndash425 Springer Berlin Germany 2011

[6] K Xu H Zheng H Zhang D Cohen-Or L Liu and YXiong ldquoPhoto-inspired model-driven 3D object modelingrdquoACM Transactions on Graphics vol 30 no 4 article 80 2011

[7] Y Liu M Zhou and Y Fan ldquoUsing depth image in 3D modelretrieval systemrdquo Advanced Materials Research vol 268ndash270pp 981ndash987 2011

[8] C Goldberg T Chen F-L Zhang A Shamir and S-MHu ldquoData-driven object manipulation in imagesrdquo ComputerGraphics Forum vol 31 no 2 pp 265ndash274 2012

[9] T LanW Yang YWang et al ldquoImage retrieval with structuredobject queries using latent ranking SVMrdquo in Proceedings ofthe 12th European Conference on Computer Vision (ECCV rsquo12)Florence Italy October 2012 vol 7577 of Lecture Notes inComputer Science pp 129ndash142 Springer 2012

[10] J Gaspar Google Sketchup Pro 8 Step by Step VectorproPublisher 2011

[11] S Henrichs ldquo3ds max environment modeling1 procedur-al stonerdquo 2010 httpsaschahenrichsblogspotcomeg2010033dsmax-environment-modeling-1html

[12] J W H Tangelder and R C Veltkamp ldquoA survey of contentbased 3D shape retrieval methodsrdquo Multimedia Tools andApplications vol 39 no 3 pp 441ndash471 2008

[13] D Scharstein and R Szeliski ldquoA taxonomy and evaluation ofdense two-frame stereo correspondence algorithmsrdquo Interna-tional Journal of ComputerVision vol 47 no 1ndash3 pp 7ndash42 2002

[14] R Hartley and A ZissermanMultiple View Geometry in Com-puter Vision Cambridge University Press Cambridge UK 2ndedition 2003

[15] M Z Brown D Burschka and G D Hager ldquoAdvances incomputational stereordquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 25 no 8 pp 993ndash1008 2003

[16] O Faugeras and Q-T LuongThe Geometry of Multiple ImagesThe MIT Press Cambridge Mass USA 2001

[17] M Pantic and L J M Rothkrantz ldquoAutomatic analysis of facialexpressions the state of the artrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 22 no 12 pp 1424ndash14452000

[18] C Tomasi and T Kanade ldquoShape and motion from imagestreams under orthography a factorization methodrdquo Interna-tional Journal of Computer Vision vol 9 no 2 pp 137ndash154 1992

[19] N Iyer S Jayanti K Lou Y Kalyanaraman and K RamanildquoShape-based searching for product lifecycle applicationsrdquoComputer Aided Design vol 37 no 13 pp 1435ndash1446 2005

[20] C Migniot and F Ababsa ldquoHybrid 3D-2D human tracking ina top viewrdquo Journal of Real-Time Image Processing vol 11 no 4pp 769ndash784 2016

[21] R Ohbuchi and T Furuya ldquoScale-weighted dense bag of visualfeatures for 3D model retrieval from a partial view 3D modelrdquoin Proceedings of the 12th IEEE International Conference onComputer Vision Workshops (ICCV rsquo09) pp 63ndash70 IEEEPiscataway NJ USA May 2009

[22] M L Cascia S Sclaroff and V Athitsos ldquoFast reliable headtracking under varying illumination an approach based onregistration of texture-mapped 3D modelsrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 22 no 4 pp322ndash336 2000

[23] X Xiaohua ldquoThree dimensional shape retrieval based on twodimensional skechesrdquo Journal of Integration Technology vol 4no 2 pp 22ndash33 2015

[24] J Revaud G Lavoue and A Baskurt ldquoImproving Zernikemoments comparison for optimal similarity and rotation angleretrievalrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 31 no 4 pp 627ndash636 2009

[25] W Rende D Yong and S Xiaojun ldquoExpectation and review onsubstation 3D modeling methodsrdquo North China Electric Powervol 2 pp 19ndash23 2015

[26] W Xianbing Z Xuedong H Tao et al ldquoDigital visualizationmanagement andmonitoring system for 3D virtual transformersubstationsrdquo Engineering Journal of Wuhan University vol 44no 6 pp 786ndash791 2011

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

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International Journal of


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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


Image captureand mapping

Object segmentation

Featureextraction

Objects retrieving in the 3D library

Size calculation and objects combination

Equipment location

calculation

3D scene generating

Figure 1 Flow chart of 3D modeling

structure and shape of power equipment in order that staffscan zone security area and understand the environment of thesubstation easily [25 26]

Themethod proposed in this paper is a 3D reconstructiontechnology based on 2D images which segments objects ofequipment in 2D images to consist of a correspondinglyshaped devicemodel that is retrieved from the objects libraryThen the size spacing and other parameters can be deter-mined by the device mapping dataThe 3Dmodel establishedby this method not only canmeet the requirements of zoningsecurity the substation shows and so on but also can reducemodeling costs and time greatly

2 Principle of the Modeling Method

The difficulty of building 3D models from 2D images liesin lack of depth data The method proposed in this papertakes shot from multiviews of equipment in the substationand then maps the position information of each device andgets equipment size equipment location coordinates andother information according to the scale After making pixeldivision of the equipment we retrieve the similar objectsfrom the objects library by using the extracted features The3D size dimension of the objects is calculated according to thepixel in the image Then determine the distance between thedevice and other aspects based on mapping data to establishthe 3D modeling of the transformer substation The user canalso adjust the color and the concrete models The process ofestablishing the 3D modeling is shown in Figure 1

3 Image Segmentation

This paper sets sides of objects as 119899 side and 119905 side Wesolve the energy function by using the improved 119896-means

algorithm for cluster of different connection componentsand then we can complete the image segmentation

119904 is the source point and 119905 is the terminal point the con-nection side between 119904 and 119905 point is the 119905 side 119875 and 119902 pointare neighbor points and the connection side between 119901 and119902 point is the 119899 side

We create a dynamic array 119866 = 1198740 1198741 1198742 to store

values of different segmentation shapes Every side has aweight value 119908 Segmentation cost is defined as the summa-tion weight value of all sides which is shown as

119862 (119874119899 119874119899+1

) = sum

119901isin119874119899119902isin119874119899+1

119908 (119901 119902) (1)

(119901 119902) means the connection side between 119901 and 119902 point119908(119901 119902) is the weight value of (119901 119902) 119862(119874

119899 119874119899+1

) is thesegmentation cost between the 119899th shape and (119899 + 1)thshape We define the segmentation cost as the sign of thesegmentation effect The less the segmentation cost is thebetter the segmentation effect is Equation (2) is Gibbsfunction that can minimize the segmentation cost

119864 (119883) = sum

119901isin120575

1198641(119909119901 119904 | 119905) + 120582 sum

(119901119902)isin120576

1198642(119909119901 119909119902) (2)

As shown in (2) energy value function 119864(119883) consisted ofthe 119905 side energy values and 119899 side energy values 119864

1(119909119901 119904 |

119905) is the energy values of 119905 side which is the connection sidebetween 119904 and 119905 point119864

2(119909119901 119909119902) is the energy values of 119899 side

which is the connection side between 119901 and 119902 point 120582 ge 0 isthe parameter used to balance the two values 119909

119901 119909119902are pixels

values of 119901 and 119902 point 120575 is the set of points and 120576 is the setof sides

The 119899th cluster center is defined as 119874119899 and the Euclidean

distance from one point to the 119899th cluster center is 119889119900119899

119901=

min119899119909119901

minus 119900119899 and to the previous cluster center is 119889

119900119899minus1

119901 =

min119899minus1

119909119901

minus 119900119899minus1

If the Euclidean distance is smaller thepoint will be more similar to the previous cluster center TheEuclidean distance from point 119901 notin 119874

119899 cup 119874

119899minus1 to the 119899th

cluster center is shown as

V =119889119900119899

119901

119889119900119899

119901 + 119889119900119899minus1

119901

(3)

We can get 1198641(119909119901 119904 | 119905) that is shown as

119901 isin 119874119899 119901 isin 119874

119899minus1 119901 notin 119874

119899 cup 119874

119899minus1

1198641(119909119901= 1) 0 infin V

1198641(119909119901= 0) infin 0 1 minus V

(4)

According to above-mentioned equations and analyticalresult we can get which energy values belong to119874

119899andwhich


853 (mm)







2will tend to 0


1198642(119909119901 119909119902) =

1


10038171003817100381710038171003817119909119901minus 119909119902

10038171003817100381710038171003817

2

2) (5)




119891 (119909 119910)

=

[[[[[[

[



sdot sdot sdot


]]]]]]

]

(6)


119865 (119906 V) = I [119891 (119909 119910)]

=1

119873

119873minus1

sum

119909=0

119873minus1

sum

119910=0

119891 (119909 119910) 119890minus1198952120587((119906119909+V119910)119873)

(7)


119865119894= (

256

sum

119895=0

119860119895)

12

119894 = 0 1 2 256 (8)


119878119865119894

=119865119894

sum256

119894=0119865119894

(9)


119878(119902119889)

= (

255

sum

119894=0

(119878119865119894119889minus 119878119865119894119902)2

)

12

(10)


5 Case Analysis








DPI 600

1952











6 Conclusions


0 50 100 150 200 250 300 350 400

00

02

04

06

08

10

Prec

ision

ratio

Time (s)




Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










853 (mm)







2will tend to 0


1198642(119909119901 119909119902) =

1


10038171003817100381710038171003817119909119901minus 119909119902

10038171003817100381710038171003817

2

2) (5)




119891 (119909 119910)

=

[[[[[[

[



sdot sdot sdot


]]]]]]

]

(6)


119865 (119906 V) = I [119891 (119909 119910)]

=1

119873

119873minus1

sum

119909=0

119873minus1

sum

119910=0

119891 (119909 119910) 119890minus1198952120587((119906119909+V119910)119873)

(7)


119865119894= (

256

sum

119895=0

119860119895)

12

119894 = 0 1 2 256 (8)


119878119865119894

=119865119894

sum256

119894=0119865119894

(9)


119878(119902119889)

= (

255

sum

119894=0

(119878119865119894119889minus 119878119865119894119902)2

)

12

(10)


5 Case Analysis








DPI 600

1952











6 Conclusions


0 50 100 150 200 250 300 350 400

00

02

04

06

08

10

Prec

ision

ratio

Time (s)




Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












DPI 600

1952











6 Conclusions


0 50 100 150 200 250 300 350 400

00

02

04

06

08

10

Prec

ision

ratio

Time (s)




Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














6 Conclusions


0 50 100 150 200 250 300 350 400

00

02

04

06

08

10

Prec

ision

ratio

Time (s)




Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article 3D Modeling of Transformer Substation ...Research Article 3D Modeling of...

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