Research ArticleNumerical Study of Heat Transfer in Trefoil Buried Cable withFluidized Thermal Backfill and Laying Parameter Optimization

Chen-Zhao Fu1 Wen-Rong Si 1 Lei Quan2 and Jian Yang 2

1State Grid Shanghai Electrical Power Research Institute Shanghai 200437 China2MOE Key Laboratory of Thermo-Fluid Science and Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Jian Yang yangjian81mailxjtueducn

Received 31 October 2018 Accepted 7 February 2019 Published 25 February 2019

Academic Editor Dragan Poljak

Copyright copy 2019 Chen-Zhao Fu 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

Trefoil buried cable is one of the important cable arrangements for the underground transmission line and its heat transferperformance is relatively poor By fillingwith fluidized thermal backfillmaterial (FTB) around trefoil buried cables the heat transferwould be efficiently enhanced while the filling cost should also be considered In the present study the heat transfer process in theFTB trefoil buried cables is numerically studied where the cable core loss and eddy current loss in the cable were coupled for thesimulation The heat transfer performances and ampacities for trefoil buried cables with different back fill materials were analysedand compared with each otherThen the laying parameters for the parabolic-type FTB trefoil buried cables were optimized with theradial basis function neural network (RBNN) and genetic algorithm (GA) Firstly it is found that with FTBmaterial themaximumtemperature in the cable core is obviously reduced and the cable ampacity is greatly improved as compared with the cables buriedaround natural soil (NS) Secondly when compared with flat-type FTBmodel the heat transfer rate in the cable with parabolic-typeFTB laying method would be slightly reduced while the FTB amount used for the buried cables is greatly reduced Finally as forparabolic-type FTB trefoil buried cables with proper design of geometric parameters (119904

1= 0290m 119904

2= 0302m and 119897 = 03mwith119868 = 1300A) for the FTB laying cross section the overall performance for the cable was optimized

1 Introduction

As global economics increases the demand for electricitysupply increases rapidly [1] The electrical energy is mainlytransported through power cables including overhead powercable and underground power cable The overhead lines areoften used for long distance electric power transmissionwhile the underground line usually plays an important role inthe area near the city or some other special areas For buriedcables the costs for laying andmaintenance are usually muchhigher than those of the overhead cables [2 3] Thereforefor buried cables the safety and cost issues should beconsidered simultaneously During actual operation processthe surrounding soil near the buried cable will be heatedand the soil will become dryer This will drastically reducethe soil conductivity and directly affect the heat transfer inthe cable system [4 5] In order to avoid soil desiccationand overheating of cables the fluidized thermal backfillmaterial (FTB) would be filled around the cables which

would efficiently improve heat transfer rate for the buriedcable system [6] FTBmaterial is an engineered slurry backfillmixture which comprises fine and coarse natural mineralaggregates constituting the bulk volume of the mixturecement providing the interparticle bond and strength andfluidizer to impart a homogeneous fluid consistency for easeof placement [4 6] However it should be noted that the costof FTB layingmethod is much higher than that of natural soil(NS) laying methodTherefore when the FTB laying methodis used not only heat transfer process in the cable systemshould be investigated but the laying parameters should alsobe optimized to reduce the FTB laying cost

Recently the heat transfer performance in the buriedcables has been widely studied by many researchers Ocłon etal [6] have numerically studied the heat transfer performancein the flat-type FTB trefoil buried cables It was found thatthe heat transfer rate was efficiently improved when theFTB material was used for the buried cables The effects ofthe backfill material amount and thermal conductivity on

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 4741871 13 pageshttpsdoiorg10115520194741871

2 Mathematical Problems in Engineering

the cable ampacity were numerically studied by Leon andAnders [7] with the finite element method (FEM) Theyfound that the cable ampacity would be improved as thebackfill material amount and thermal conductivity increasedThe soil desiccation phenomenon around buried cables hasbeen investigated by Gouda and Dein [8] The thermalproperties of eleven different backfill materials were analysedand the most suitable backfill material was obtained RerakandOcłon [9] have numerically investigated the heat transferperformance in the trefoil buried cables with FEM methodwhere the thermal conductivities of soil and backfill materialwere related to the temperature variations In their researchthe effects of the backfill material thermal conductivity andcable core current on the maximum temperature of cableswere carefully analysed Furthermore the moisture andtemperature distributions of the backfill material surroundthe buried cables were numerically studied by Anders andRadhakrishna [10] and their numerical results could agreewell with experimental data In addition researches on theoptimization of laying parameters for buried cables are alsopopular in recent years Ocłon et al [11] have optimizedthe laying parameters for flat-type buried cables by usingimproved Jaya algorithm combined with FEM method Itwas found that the improved Jaya algorithm would be moreaccurate and efficient for the optimizations Cichy et al[12 13] have optimized the laying parameters for both theflat-type and the trefoil-type buried cables by using geneticalgorithm (GA) With this method the optimum layingparameters were obtained for the minimum laying cost Saud[14] has optimized the laying parameters for the flat-typeburied cables by using particle swarm optimization method(PSO) With this method the optimum laying parameterswere obtained for different object functions including themaximum temperature laying cost and ampacity of buriedcables Furthermore Ocłon et al [4] have also optimizedthe laying parameters for the flat-type buried cables by usingparticle swarm optimization method (PSO) In their studythe thermal conductivities of soil and backfill material wererelated to the temperature variations and the optimum layingparameters were finally obtained combined with experimen-tal results [15]

Based on above literature survey it shows that theresearches on the heat transfer performance of buried cablesand the corresponding optimizations of laying parame-ters were popular in the recent years However all theseresearches weremainly focused on the flat-type buried cablesand the researches on the heat transfer and optimizationfor the trefoil-type buried cables were relatively few Trefoilburied cable is one of the important cable arrangementsfor the underground transmission line It is often used inthe situation with limited construction space and its heattransfer performance is relatively poor By filling with flu-idized thermal backfill material (FTB) around trefoil buriedcables the heat transfer would be efficiently enhanced [16]while the filling cost of FTB should also be consideredIn the present paper the heat transfer process in the flat-type FTB trefoil buried cables was numerically studied firstand the results were compared with those of flat-type NStrefoil buried cables Then the heat transfer process in the

parabolic-type FTB trefoil buried cables was numericallystudied and the results were compared with those of flat-typeFTB trefoil buried cables Finally the laying parameters forthe cross section of parabolic-type FTB trefoil buried cableswere optimized with the radial basis function neural network(RBNN) and genetic algorithm (GA) and the optimumlaying parameters and minimum total cost function wereobtained According to the authorsrsquo knowledge almost nostudy was performed on the heat transfer process in theparabolic-type FTB trefoil buried cables before and theresults would be meaningful for the optimal design for theFTB trefoil buried cables

2 Methodology

21 Model and Geometric Parameters The model of trefoilburied cables with fluidized thermal backfill material (FTB)is presented in Figure 1 It includes the flat-type FTB trefoilburied cables (Figure 1(a)) and parabolic-type FTB trefoilburied cables (Figure 1(b)) The length (119871) and height (H)of the computational domain are 20m and 10m respectivelyThree-phase cables are installed in a rectangular trench withwidth of 14m The distance between the trench top surfaceand the lower cable core is 1m As for the flat-type FTB trefoilburied cables (Figure 1(a)) the FTB thickness is ℎ and thedistance between the trench bottom surface and the lowercable core is 02m As for the parabolic-type FTB trefoilburied cables (Figure 1(b)) the distance between the FTB toppoint and the upper cable core is 119904

1 the distance between the

FTB bottom surface and the lower cable core is s2and the

FTB with is 2119897 Besides FTB region the other computationaldomain is filled with the natural soil (NS)

The arrangement of trefoil power cables and cable struc-ture are presented in Figure 2 It shows that the trefoil powercables are stacked in a triangular arrangement (Figure 2(a))where the cable diameter is 119889j In order to improve the com-putationalmesh quality near the contact points between cablesurfaces the cables were assumed to be stacked with verysmall gaps (119897

1= 2119889j) instead of contact points between each

other Furthermore as shown in Figure 2(b) the power cableis composed of copper conductor insulation layer sheathlayer and Jacket layer where the corresponding radiuses are119903c 119903i 119903s and 119903j respectively Typical geometric and physicalparameters of trefoil cables are listed in Table 1

22 Governing Equations and Computational Method In thepresent study the heat transfer in the buried cables can beregarded as two-dimensional steady heat conduction processThe power cable is composed of copper conductor insulationlayer sheath layer and Jacket layer and the heat loss isproduced in the copper conductor and sheath layer named asconductor loss and eddy loss respectively The heat transferequation for the computational domain is as follows

119896 sdot nabla2119879=

minus119902v Cable core and sheath layer

0 Insulation layer jacket layer NS and FTB regions

(1)

Mathematical Problems in Engineering 3

H=10m

L=20m

10m

NS

1m

02m

h

07m

FTB

o

y

x

(a)L=20m

H=10m

10m

NS

1m

s2

FTB

07m

l

s1

o

y

x

(b)

Figure 1 Model for FTB trefoil buried cables (a) Flat-type FTB trefoil buried cables (b) Parabolic-type FTB trefoil buried cables

l1

dj

(a)

Jacket

Sheath

XLPE Insulation

Copper Conductor

r＝

rs

rj

ri

(b)

Figure 2 Power cables in trefoil arrangement and cable structure (a) Power cables in trefoil arrangement (b) Power cable structure

where 119896 is the thermal conductivity 119902v is the heat loss ofpower cableThe thermal conductivities of NS and FTBmate-rial (119896NS and 119896FTB) are dependent on temperature variationswhich are defined as follows [6 17]

119896 (119879) = 119896dry + (119896wet minus 119896dry)sdot exp[minus119886

1[(119879 minus 119879ref)(1198862sdot 119879lim)]

2]1198861= 119879lim119879ref

1198862= 1 minus ( 1119886

1

)

(2)

where 119896dry and 119896wet are the thermal conductivities underdry and wet conditions for NS and FTB material (Table 2)119879lim is the limited temperature of power cable (363K) 119879refis the reference temperature of surrounding soil (293K) The

variations of the thermal conductivity of NS and FTB arepresented in Figure 3

The heat loss of power cable (119902v) is defined as follows

119902v =100381610038161003816100381610038161003816997888119869 1003816100381610038161003816100381610038162

120590 (3)

where 997888rarr119869 is the total current density in the cable and 120590 is theelectronic conductivity which are defined as follows

997888119869 = 997888119869119890+ 997888119869119904

997888119869 s gt 0 Cable core997888119869 s = 0 Sheath layer

(4)

997888119869 s = minus120590nabla120593997888119869 e = minus119895120596120590997888119860120590 = 1120588

0(1 + 120572 (119879 minus 119879ref))

(5)

4 Mathematical Problems in Engineering

Table 1 Typical geometric and physical parameters of trefoil cables (cable quadrature = 1600mm2)

Cable Radius [mm] Material Thermal conductivity [W(msdotK)] Electronic conductivity [Sm]Cable core 248 copper 400 5998 times 107

Insulation layer 553 XLPE 038 10 times 10minus15

Sheath layer 617 copper 400 5998 times 107

Jacket layer 668 HDPE 038 10 times 10minus15

300 320 340 360 380 400 420 4400

1

2

3

4

5

k(T)

[W(

mmiddotK

)]

kFTB

T [K]

kNS

Figure 3 Variations of 119896NS and 119896FTB with temperature

Table 2 119896dry and 119896wet for NS and FTB material [6 17]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100FTB 154 435

where 997888119869 e is the inductive current density997888119869 s is the source

current density 997888119860 j 120596 and 120593 are the magnetic vectorpotential unit of complex number angular frequency andelectric scalar potential respectively 120588

0and 120572 are the elec-

trical resistivity and temperature coefficient at a referencetemperature of 119879ref = 293K

The boundary conditions for the computational domainare set as follows

12059711987912059711990910038161003816100381610038161003816100381610038161003816x=0 or x=L = 0120597119879120597119910

10038161003816100381610038161003816100381610038161003816y=0 = 0119879|y=H = 293K

(6)

In the present study the above governing equations are solvedwith commercial code COMSOL MULTIPHYSICS and thePardiso solver is employed for the computationsThe currentfrequency is set to 50Hz The conservative interface fluxcondition for heat transfer is adopted at the cable-FTB andFTB-NS interfaces as well as the internal interfaces betweendifferent layers inside the cable which means that the heatflux on one-side of the interface was considered to be equalto the heat flux on the other-side of the interface between

different computational regions For convergence criteria allresiduals of the calculations are less than 10minus4

3 Grid Independence Test andModel Validations

Firstly the grid independence test was performed In thepresent study the parabolic-type FTB trefoil cables wereadopted for the test (119904

1= 35119903j 119897 = 8119903j) where the cable

core current is 1145A and the current frequency is 50HzAs presented in Figure 4 the self-adaptive tetrahedral meshwas used for the computations and the grids are intensifiedaround cable regions In order to improve the mesh qualitynear the contact points between cable surfaces according tothe report of Bu et al [18] the cables were stacked with verysmall gaps (119897

1= 2119889j) instead of contact points between each

other (Figure 2(a)) Four sets of grids were used for the testand the computational results are presented in Table 3 Itshows that the Grid-3 with total element number of 19346is good enough for the test based on the comparison ofthe maximum cable temperature (119879max) and heat flux onthe cable surface (120601) with different grids For Grid-3 theminimum length of the grid element in both the cable zone(Zone 1) and FTB zone (Zone 2) is 03mm and it is 04mmin the NS zone (Zone 3) Therefore similar grid settingsto the test grid of Grid-3 were employed for the followingsimulations

Subsequently the computational model and methodswere validated The heat transfer process in three-phaseburied cables [6] was restudied and the model is presentedin Figure 5 It shows that the power cables are parallelarranged and the cable core distance is 04m between eachother The distance between the ground and cable core ish1 In the near-cable region the NS is used as the backfill

material while the other regions are filled with multilayersoilThe symmetry boundary condition is adopted on the leftedge of the computational domain and the right edge andbottom surface of the computational domain are consideredto be adiabatic Furthermore the ground temperature isfixed at 293KThe thermal conductivities of different backfillmaterials are dependent on temperature variations whichcan be calculated with (2) and parameters listed in Table 4

When the cable core current is fixed at 1145A the varia-tions of temperature along the symmetry edge are presentedin Figure 6(a) It shows that the maximum temperature devi-ation between our present computations and those of Ocłonet al [6] is 14 K Meanwhile the variations of the maximumcable core temperature (119879max) with cable buried depth (ℎ

1)

are presented in Figure 6(b) It shows that the maximum

Mathematical Problems in Engineering 5

ZONE3

ZONE1 ZONE2

ZONE2

ZONE1

Figure 4 Typical computational mesh for parabolic-type FTB trefoil buried cables

Table 3 Computational results with different meshes

Grid Region Minimum length of gridelement [mm]

Maximum length of gridelement [mm] Total element number 119879max [K] 120601 [W]

Grid 1Zone 1 15 04

10228 32656 68899Zone 2 15 04Zone 3 15 05

Grid 2Zone 1 04 02

16337 329557 69892Zone 2 15 04Zone 3 15 04

Grid 3Zone 1 03 01

19346 329563 68903Zone 2 03 01Zone 3 04 02

Grid 4Zone 1 03 01

28394 329563 68903Zone 2 03 01Zone 3 035 015

6 Mathematical Problems in Engineering

NSNS

MS

Gr

SCLG

SCL

h1

02m

04m 07m

08m

09m

02m

07m

10m

Symmetry

Tref =293K

Adiabatic Adiabatic

Power cable

10m

NS Natural soil

MS Medium sand

Gr Gravel

SCLG Sand clay loamwith gravel SCL Sand clay loam

o

y

x

Figure 5 Model for validations [6]

Table 4 Thermal conductivity of different buried material inOcłonrsquos study [6]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100MS 021 228Gr 026 267SCLG 026 201SCL 017 145

deviation of 119879max between our present computations andthose of Ocłon et al [6] is 23 K

4 Results and Discussion

41 Performance Comparison between NS and FTB TrefoilBuried Cables When the cable core current is fixed at 1500Athe temperature distributions in the trefoil buried cables withNS and flat-type FTB are presented in Figure 7 It showsthat for the NS trefoil buried cables (Figure 7(a)) sincethe thermal conductivity of NS material is relatively low(Figure 3) the heat transfer rate is relatively low and themaximum temperature of the cable core is high (119879max = 448Kor 175∘C) while for the FTB trefoil buried cables (Figures7(b)ndash7(d)) since the thermal conductivity of FTB materialis relatively high (Figure 3) the heat transfer rate wouldbe significantly improved and the maximum temperatureof the cable core is obviously reduced When the thicknessof FTB is fixed at 04m 045m and 050m 119879max of thecable core are 362K (89∘C) 357K (84∘C) and 353K (80∘C)respectively The variations of the maximum cable coretemperature in the trefoil buried cables with NS and flat-type FTB are presented in Figure 8 It shows that with thesame cable core current 119879max of the cable core for the NStrefoil buried cables is obviously higher than that for the FTBtrefoil buried cables and 119879max of the cable core for the FTB

trefoil buried cables would be further reduced as the FTBthickness (ℎ) increases In the present study it is shown thatthe temperature distributions inside three-phase cables arequite similar for the same working condition Therefore theampacities in the three-phase cables would also be similarHere the cable ampacity is defined as the correspondingcurrent in the cable core when the maximum cable coretemperature reaches 90∘C (363K) It shows that the cableampacity for the NS trefoil buried cables is 12623 A while thecable ampacities for the FTB trefoil buried cables are 15135 A(ℎ = 040m) 15582 A (ℎ = 045m) 15872 A (ℎ = 050m) and16169 A (ℎ = 055m) respectively Therefore it is indicatedthat for trefoil buried cables the heat transfer rate wouldbe efficiently improved by using FTB material and the cableampacity would be significantly increased

The flat-type FTB trefoil buried cables as mentionedabove (Figure 1(a)) are quite easy to be constructed Howeverthis FTB laying method needs large amount of FTB layingmaterials which would lead to a certain waste Therefore inorder to reduce the FTB amount used for the trefoil buriedcables the parabolic-type FTB laying method (Figure 1(b))is adopted and the heat transfer performance and ampacityof the parabolic-type FTB trefoil buried cables are analysedand compared with those of the flat-type FTB trefoil buriedcables Typical geometric parameters for flat-type FTB andparabolic-type FTB trefoil buried cables are presented inTable 5 It shows that in the present study the cross sectionarea for the parabolic-type FTB model (119860FTB) is much lowerthan that for the flat-type FTB model

When the cable core current is fixed at 1500A thetemperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure 9It shows that the temperature distributions in the flat-typeFTB and parabolic-type FTB trefoil buried cables are similarwhere the corresponding 119879max of the cable core are 362K(89∘C) and 368K (95∘C) respectively This indicates withparabolic-type FTB laying method the heat transfer rate inthe cable would almost keep the same while the FTB amount(represented by 119860FTB) used for the trefoil buried cables canbe greatly reduced

The variations of the maximum cable core temperature(119879max) in the flat-type FTB and parabolic-type FTB trefoilburied cables are presented in Figure 10 It shows that withthe same cable core current 119879max of cable core for theparabolic-type FTB model is little higher than that for theflat-type FTB model and the corresponding ampacities are14775 A (119860FTB = 0351m2) and 15135 A (119860FTB = 0560m2)respectively This indicates that compared with flat-type FTBmodel the ampacity in the cable is slightly reduced withparabolic-type FTB laying method (reduced by 24 in thepresent study) while the FTB amount used for the buriedcables (represented by 119860FTB) is greatly reduced (reduced by373 in the present study Table 5) Therefore the parabolic-type FTB laying method is recommended for the trefoilburied cable applications

42 Laying Parameter Optimization for Parabolic-TypeFTB Trefoil Buried Cables The laying parameters for theparabolic-type FTB trefoil buried cables are presented in

Mathematical Problems in Engineering 7

0 2 4 6 8 10

295

300

305

310

315

320T

[K]

y [m]

Present simulation

Ref [6]

(a)

10 12 14 16 18 20 22 24 26

315

320

325

330

335

T max

[K]

h1 [m]

Present simulationRef [6]

(b)

Figure 6Model validations for temperatures withOcłon et al [6] (a) Temperatures along the symmetry edge (b)Themaximum temperatureof cable core

Table 5 Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables

FTB laying model 1199041[m] 119904

2[m] 119897 [m] ℎ [m] 119860FTB [m

2]FTB (Parabolic) 35119903j 02 8119903j 0351FTB (Flat) 04 0560

Figure 11 It shows that the FTB laying area (119860FTB) or FTBamount is closely related to the FTB laying parametersmainly including 119904

1 1199042 and 119897 In order to enhance the heat

transfer in trefoil buried cables and reduce the use of FTBmaterial the laying parameters should be optimized for theparabolic-type FTB trefoil buried cables In the present studythe total cost function (Fc) as reported by Ocłon et al [11] isadopted to access the overall performance for trefoil buriedcables which is defined as follows

Fc (119897 1199041 1199042)=

119860FTB + PF 119879max minus 119879opt gt 0 (119879opt = 338K)119860FTB 119879max minus 119879opt le 0 (119879opt = 338K)

(7)

where 119860FTB is the FTB laying area 119879max is the maximumtemperature of cable core 119879opt is the optimum workingtemperature for the cable (119879opt = 338K or 65∘C) PF is thepenalty function When the value of 119879max is higher than thatof 119879opt (338K) the penalty function (PF) should be set toguarantee the rationality of total cost function (Fc) where thePF is defined as follows

PF = 10 times (119879max minus 119879opt) (8)

Therefore it should be noted that when the value of 119879max islower than that of119879opt the total cost function (Fc) is depended

on 119860FTB Otherwise Fc should be both related to 119860FTB and119879maxIn the present study three geometric parameters of FTB

laying cross section including 1199041 1199042 and 119897 were selected

as the design variables and the total cost function (Fc) isadopted as the objective function for the optimization Theflowchart for the optimization procedure is presented inFigure 12 Firstly based on the design variables (119904

1 1199042 and 119897)

the geometric parameter combination samples were designedwith the Latin hypercube sampling method (LHS) [19 20]LHS is an effective method to design sample points withinthe design space by which the design points are equallyspaced of each variable in design space [19] When samplinga function of 119899 variables the range of each variable is dividedinto 119898 equal probable intervals For each variable designpoints are randomly selected from each interval Based onthe119898 points for each variable the sample points are obtainedby the random combination Hence the design points areequally spaced of each variable in design space [19] Withthis method 50 sets of random samples were selected as thedesign samples for the optimizations which are presented inTable 6 As shown in Table 6 both the ranges of 119904

1and 1199042are

from01m to 04mwhile the range of 119897 is from03m to 07mBased on these 50 sets of samples the heat transfer processesin the parabolic-type FTB trefoil buried cableswere simulatedwith commercial code COMSOL MULTIPHYSICS Thenthe maximum temperature (119879max) of the cable core and thetotal cost function (Fc) for the corresponding samples were

8 Mathematical Problems in Engineering

Table 6 Design samples with LHS method and the computational results of 119879max and Fc (119868 = 1300A)

Samples 1199041[m] 119904

2[m] 119897 [m] 119860FTB [m

2] 119879max [K] Fc1 03843 03769 03117 03233 3323360 032332 03181 03218 03565 03183 3346860 031833 01562 02716 06180 04077 3555410 17581774 03818 03014 05370 05316 3300740 053165 02238 02232 04746 03155 3381350 166556 01128 02896 03806 02221 3422830 4305217 01604 03373 05287 03921 3346920 039218 03733 01883 03684 02918 3381560 185189 03564 01961 04594 03687 3358680 0368710 03014 02172 04194 03140 3374600 0314011 02902 03613 06789 06546 3283190 0654612 01001 01387 06094 02479 3438940 59187913 01322 01752 04716 02255 3438250 58475514 01508 02070 05538 03093 3394900 15209315 03458 02273 04407 03641 3353500 0364116 02356 03250 04092 03281 3354600 0328117 01934 03718 04042 03261 3351620 0326118 01763 01190 05951 02859 3419510 39795919 01854 01328 05425 02735 3420180 40453520 02569 01070 03248 01667 3479650 99816721 01383 01705 03164 01380 3499190 119328022 02917 01161 03972 02364 3432150 52386423 03084 02500 05218 04286 3337690 0428624 03242 01794 04314 03155 3380940 1255525 02048 03864 06011 05264 3306140 0526426 03900 03173 06881 07151 3274050 0715127 01692 02671 06513 04393 3347680 0439328 03359 01279 05774 04059 3368790 0405929 02382 02923 06391 05106 3321900 0510630 03548 03105 06294 06152 3290470 0615231 02127 03023 05580 04290 3338130 0429032 03296 02413 06464 05518 3351400 0551833 02035 01481 05005 02713 3415770 36041334 03635 03457 05641 05801 3286610 0580135 03443 01867 03929 02980 3382780 3078036 02591 02779 04456 03472 3356430 0347237 02461 03439 03002 02413 3377180 0241338 02195 02851 04870 03622 3355940 0362239 02712 02026 03484 02329 3411970 32202940 02303 03541 05037 04297 3325610 0429741 01175 01257 03597 01312 3519180 139311242 02642 03307 06724 05971 3299080 0597143 03056 03823 06227 06271 3280290 0627144 01912 02420 03305 02009 3429110 49310945 02527 02565 06677 05163 3326100 0516346 01246 01592 03740 01583 3486010 106168347 02746 03970 03427 03188 3334970 0318848 01778 03630 05872 04737 3322980 0473749 01092 02612 06948 04105 3364770 0410550 03695 02309 06618 05919 3308430 05919

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 3

H=10m

L=20m

10m

NS

1m

02m

h

07m

FTB

o

y

x

(a)L=20m

H=10m

10m

NS

1m

s2

FTB

07m

l

s1

o

y

x

(b)

Figure 1 Model for FTB trefoil buried cables (a) Flat-type FTB trefoil buried cables (b) Parabolic-type FTB trefoil buried cables

l1

dj

(a)

Jacket

Sheath

XLPE Insulation

Copper Conductor

r＝

rs

rj

ri

(b)

Figure 2 Power cables in trefoil arrangement and cable structure (a) Power cables in trefoil arrangement (b) Power cable structure

where 119896 is the thermal conductivity 119902v is the heat loss ofpower cableThe thermal conductivities of NS and FTBmate-rial (119896NS and 119896FTB) are dependent on temperature variationswhich are defined as follows [6 17]

119896 (119879) = 119896dry + (119896wet minus 119896dry)sdot exp[minus119886

1[(119879 minus 119879ref)(1198862sdot 119879lim)]

2]1198861= 119879lim119879ref

1198862= 1 minus ( 1119886

1

)

(2)

where 119896dry and 119896wet are the thermal conductivities underdry and wet conditions for NS and FTB material (Table 2)119879lim is the limited temperature of power cable (363K) 119879refis the reference temperature of surrounding soil (293K) The

variations of the thermal conductivity of NS and FTB arepresented in Figure 3

The heat loss of power cable (119902v) is defined as follows

119902v =100381610038161003816100381610038161003816997888119869 1003816100381610038161003816100381610038162

120590 (3)

where 997888rarr119869 is the total current density in the cable and 120590 is theelectronic conductivity which are defined as follows

997888119869 = 997888119869119890+ 997888119869119904

997888119869 s gt 0 Cable core997888119869 s = 0 Sheath layer

(4)

997888119869 s = minus120590nabla120593997888119869 e = minus119895120596120590997888119860120590 = 1120588

0(1 + 120572 (119879 minus 119879ref))

(5)

4 Mathematical Problems in Engineering

Table 1 Typical geometric and physical parameters of trefoil cables (cable quadrature = 1600mm2)

Cable Radius [mm] Material Thermal conductivity [W(msdotK)] Electronic conductivity [Sm]Cable core 248 copper 400 5998 times 107

Insulation layer 553 XLPE 038 10 times 10minus15

Sheath layer 617 copper 400 5998 times 107

Jacket layer 668 HDPE 038 10 times 10minus15

300 320 340 360 380 400 420 4400

1

2

3

4

5

k(T)

[W(

mmiddotK

)]

kFTB

T [K]

kNS

Figure 3 Variations of 119896NS and 119896FTB with temperature

Table 2 119896dry and 119896wet for NS and FTB material [6 17]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100FTB 154 435

where 997888119869 e is the inductive current density997888119869 s is the source

current density 997888119860 j 120596 and 120593 are the magnetic vectorpotential unit of complex number angular frequency andelectric scalar potential respectively 120588

0and 120572 are the elec-

trical resistivity and temperature coefficient at a referencetemperature of 119879ref = 293K

The boundary conditions for the computational domainare set as follows

12059711987912059711990910038161003816100381610038161003816100381610038161003816x=0 or x=L = 0120597119879120597119910

10038161003816100381610038161003816100381610038161003816y=0 = 0119879|y=H = 293K

(6)

In the present study the above governing equations are solvedwith commercial code COMSOL MULTIPHYSICS and thePardiso solver is employed for the computationsThe currentfrequency is set to 50Hz The conservative interface fluxcondition for heat transfer is adopted at the cable-FTB andFTB-NS interfaces as well as the internal interfaces betweendifferent layers inside the cable which means that the heatflux on one-side of the interface was considered to be equalto the heat flux on the other-side of the interface between

different computational regions For convergence criteria allresiduals of the calculations are less than 10minus4

3 Grid Independence Test andModel Validations

Firstly the grid independence test was performed In thepresent study the parabolic-type FTB trefoil cables wereadopted for the test (119904

1= 35119903j 119897 = 8119903j) where the cable

core current is 1145A and the current frequency is 50HzAs presented in Figure 4 the self-adaptive tetrahedral meshwas used for the computations and the grids are intensifiedaround cable regions In order to improve the mesh qualitynear the contact points between cable surfaces according tothe report of Bu et al [18] the cables were stacked with verysmall gaps (119897

1= 2119889j) instead of contact points between each

other (Figure 2(a)) Four sets of grids were used for the testand the computational results are presented in Table 3 Itshows that the Grid-3 with total element number of 19346is good enough for the test based on the comparison ofthe maximum cable temperature (119879max) and heat flux onthe cable surface (120601) with different grids For Grid-3 theminimum length of the grid element in both the cable zone(Zone 1) and FTB zone (Zone 2) is 03mm and it is 04mmin the NS zone (Zone 3) Therefore similar grid settingsto the test grid of Grid-3 were employed for the followingsimulations

Subsequently the computational model and methodswere validated The heat transfer process in three-phaseburied cables [6] was restudied and the model is presentedin Figure 5 It shows that the power cables are parallelarranged and the cable core distance is 04m between eachother The distance between the ground and cable core ish1 In the near-cable region the NS is used as the backfill

material while the other regions are filled with multilayersoilThe symmetry boundary condition is adopted on the leftedge of the computational domain and the right edge andbottom surface of the computational domain are consideredto be adiabatic Furthermore the ground temperature isfixed at 293KThe thermal conductivities of different backfillmaterials are dependent on temperature variations whichcan be calculated with (2) and parameters listed in Table 4

When the cable core current is fixed at 1145A the varia-tions of temperature along the symmetry edge are presentedin Figure 6(a) It shows that the maximum temperature devi-ation between our present computations and those of Ocłonet al [6] is 14 K Meanwhile the variations of the maximumcable core temperature (119879max) with cable buried depth (ℎ

1)

are presented in Figure 6(b) It shows that the maximum

Mathematical Problems in Engineering 5

ZONE3

ZONE1 ZONE2

ZONE2

ZONE1

Figure 4 Typical computational mesh for parabolic-type FTB trefoil buried cables

Table 3 Computational results with different meshes

Grid Region Minimum length of gridelement [mm]

Maximum length of gridelement [mm] Total element number 119879max [K] 120601 [W]

Grid 1Zone 1 15 04

10228 32656 68899Zone 2 15 04Zone 3 15 05

Grid 2Zone 1 04 02

16337 329557 69892Zone 2 15 04Zone 3 15 04

Grid 3Zone 1 03 01

19346 329563 68903Zone 2 03 01Zone 3 04 02

Grid 4Zone 1 03 01

28394 329563 68903Zone 2 03 01Zone 3 035 015

6 Mathematical Problems in Engineering

NSNS

MS

Gr

SCLG

SCL

h1

02m

04m 07m

08m

09m

02m

07m

10m

Symmetry

Tref =293K

Adiabatic Adiabatic

Power cable

10m

NS Natural soil

MS Medium sand

Gr Gravel

SCLG Sand clay loamwith gravel SCL Sand clay loam

o

y

x

Figure 5 Model for validations [6]

Table 4 Thermal conductivity of different buried material inOcłonrsquos study [6]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100MS 021 228Gr 026 267SCLG 026 201SCL 017 145

deviation of 119879max between our present computations andthose of Ocłon et al [6] is 23 K

4 Results and Discussion

41 Performance Comparison between NS and FTB TrefoilBuried Cables When the cable core current is fixed at 1500Athe temperature distributions in the trefoil buried cables withNS and flat-type FTB are presented in Figure 7 It showsthat for the NS trefoil buried cables (Figure 7(a)) sincethe thermal conductivity of NS material is relatively low(Figure 3) the heat transfer rate is relatively low and themaximum temperature of the cable core is high (119879max = 448Kor 175∘C) while for the FTB trefoil buried cables (Figures7(b)ndash7(d)) since the thermal conductivity of FTB materialis relatively high (Figure 3) the heat transfer rate wouldbe significantly improved and the maximum temperatureof the cable core is obviously reduced When the thicknessof FTB is fixed at 04m 045m and 050m 119879max of thecable core are 362K (89∘C) 357K (84∘C) and 353K (80∘C)respectively The variations of the maximum cable coretemperature in the trefoil buried cables with NS and flat-type FTB are presented in Figure 8 It shows that with thesame cable core current 119879max of the cable core for the NStrefoil buried cables is obviously higher than that for the FTBtrefoil buried cables and 119879max of the cable core for the FTB

trefoil buried cables would be further reduced as the FTBthickness (ℎ) increases In the present study it is shown thatthe temperature distributions inside three-phase cables arequite similar for the same working condition Therefore theampacities in the three-phase cables would also be similarHere the cable ampacity is defined as the correspondingcurrent in the cable core when the maximum cable coretemperature reaches 90∘C (363K) It shows that the cableampacity for the NS trefoil buried cables is 12623 A while thecable ampacities for the FTB trefoil buried cables are 15135 A(ℎ = 040m) 15582 A (ℎ = 045m) 15872 A (ℎ = 050m) and16169 A (ℎ = 055m) respectively Therefore it is indicatedthat for trefoil buried cables the heat transfer rate wouldbe efficiently improved by using FTB material and the cableampacity would be significantly increased

The flat-type FTB trefoil buried cables as mentionedabove (Figure 1(a)) are quite easy to be constructed Howeverthis FTB laying method needs large amount of FTB layingmaterials which would lead to a certain waste Therefore inorder to reduce the FTB amount used for the trefoil buriedcables the parabolic-type FTB laying method (Figure 1(b))is adopted and the heat transfer performance and ampacityof the parabolic-type FTB trefoil buried cables are analysedand compared with those of the flat-type FTB trefoil buriedcables Typical geometric parameters for flat-type FTB andparabolic-type FTB trefoil buried cables are presented inTable 5 It shows that in the present study the cross sectionarea for the parabolic-type FTB model (119860FTB) is much lowerthan that for the flat-type FTB model

When the cable core current is fixed at 1500A thetemperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure 9It shows that the temperature distributions in the flat-typeFTB and parabolic-type FTB trefoil buried cables are similarwhere the corresponding 119879max of the cable core are 362K(89∘C) and 368K (95∘C) respectively This indicates withparabolic-type FTB laying method the heat transfer rate inthe cable would almost keep the same while the FTB amount(represented by 119860FTB) used for the trefoil buried cables canbe greatly reduced

The variations of the maximum cable core temperature(119879max) in the flat-type FTB and parabolic-type FTB trefoilburied cables are presented in Figure 10 It shows that withthe same cable core current 119879max of cable core for theparabolic-type FTB model is little higher than that for theflat-type FTB model and the corresponding ampacities are14775 A (119860FTB = 0351m2) and 15135 A (119860FTB = 0560m2)respectively This indicates that compared with flat-type FTBmodel the ampacity in the cable is slightly reduced withparabolic-type FTB laying method (reduced by 24 in thepresent study) while the FTB amount used for the buriedcables (represented by 119860FTB) is greatly reduced (reduced by373 in the present study Table 5) Therefore the parabolic-type FTB laying method is recommended for the trefoilburied cable applications

42 Laying Parameter Optimization for Parabolic-TypeFTB Trefoil Buried Cables The laying parameters for theparabolic-type FTB trefoil buried cables are presented in

Mathematical Problems in Engineering 7

0 2 4 6 8 10

295

300

305

310

315

320T

[K]

y [m]

Present simulation

Ref [6]

(a)

10 12 14 16 18 20 22 24 26

315

320

325

330

335

T max

[K]

h1 [m]

Present simulationRef [6]

(b)

Figure 6Model validations for temperatures withOcłon et al [6] (a) Temperatures along the symmetry edge (b)Themaximum temperatureof cable core

Table 5 Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables

FTB laying model 1199041[m] 119904

2[m] 119897 [m] ℎ [m] 119860FTB [m

2]FTB (Parabolic) 35119903j 02 8119903j 0351FTB (Flat) 04 0560

Figure 11 It shows that the FTB laying area (119860FTB) or FTBamount is closely related to the FTB laying parametersmainly including 119904

1 1199042 and 119897 In order to enhance the heat

transfer in trefoil buried cables and reduce the use of FTBmaterial the laying parameters should be optimized for theparabolic-type FTB trefoil buried cables In the present studythe total cost function (Fc) as reported by Ocłon et al [11] isadopted to access the overall performance for trefoil buriedcables which is defined as follows

Fc (119897 1199041 1199042)=

119860FTB + PF 119879max minus 119879opt gt 0 (119879opt = 338K)119860FTB 119879max minus 119879opt le 0 (119879opt = 338K)

(7)

where 119860FTB is the FTB laying area 119879max is the maximumtemperature of cable core 119879opt is the optimum workingtemperature for the cable (119879opt = 338K or 65∘C) PF is thepenalty function When the value of 119879max is higher than thatof 119879opt (338K) the penalty function (PF) should be set toguarantee the rationality of total cost function (Fc) where thePF is defined as follows

PF = 10 times (119879max minus 119879opt) (8)

Therefore it should be noted that when the value of 119879max islower than that of119879opt the total cost function (Fc) is depended

on 119860FTB Otherwise Fc should be both related to 119860FTB and119879maxIn the present study three geometric parameters of FTB

laying cross section including 1199041 1199042 and 119897 were selected

as the design variables and the total cost function (Fc) isadopted as the objective function for the optimization Theflowchart for the optimization procedure is presented inFigure 12 Firstly based on the design variables (119904

1 1199042 and 119897)

the geometric parameter combination samples were designedwith the Latin hypercube sampling method (LHS) [19 20]LHS is an effective method to design sample points withinthe design space by which the design points are equallyspaced of each variable in design space [19] When samplinga function of 119899 variables the range of each variable is dividedinto 119898 equal probable intervals For each variable designpoints are randomly selected from each interval Based onthe119898 points for each variable the sample points are obtainedby the random combination Hence the design points areequally spaced of each variable in design space [19] Withthis method 50 sets of random samples were selected as thedesign samples for the optimizations which are presented inTable 6 As shown in Table 6 both the ranges of 119904

1and 1199042are

from01m to 04mwhile the range of 119897 is from03m to 07mBased on these 50 sets of samples the heat transfer processesin the parabolic-type FTB trefoil buried cableswere simulatedwith commercial code COMSOL MULTIPHYSICS Thenthe maximum temperature (119879max) of the cable core and thetotal cost function (Fc) for the corresponding samples were

8 Mathematical Problems in Engineering

Table 6 Design samples with LHS method and the computational results of 119879max and Fc (119868 = 1300A)

Samples 1199041[m] 119904

2[m] 119897 [m] 119860FTB [m

2] 119879max [K] Fc1 03843 03769 03117 03233 3323360 032332 03181 03218 03565 03183 3346860 031833 01562 02716 06180 04077 3555410 17581774 03818 03014 05370 05316 3300740 053165 02238 02232 04746 03155 3381350 166556 01128 02896 03806 02221 3422830 4305217 01604 03373 05287 03921 3346920 039218 03733 01883 03684 02918 3381560 185189 03564 01961 04594 03687 3358680 0368710 03014 02172 04194 03140 3374600 0314011 02902 03613 06789 06546 3283190 0654612 01001 01387 06094 02479 3438940 59187913 01322 01752 04716 02255 3438250 58475514 01508 02070 05538 03093 3394900 15209315 03458 02273 04407 03641 3353500 0364116 02356 03250 04092 03281 3354600 0328117 01934 03718 04042 03261 3351620 0326118 01763 01190 05951 02859 3419510 39795919 01854 01328 05425 02735 3420180 40453520 02569 01070 03248 01667 3479650 99816721 01383 01705 03164 01380 3499190 119328022 02917 01161 03972 02364 3432150 52386423 03084 02500 05218 04286 3337690 0428624 03242 01794 04314 03155 3380940 1255525 02048 03864 06011 05264 3306140 0526426 03900 03173 06881 07151 3274050 0715127 01692 02671 06513 04393 3347680 0439328 03359 01279 05774 04059 3368790 0405929 02382 02923 06391 05106 3321900 0510630 03548 03105 06294 06152 3290470 0615231 02127 03023 05580 04290 3338130 0429032 03296 02413 06464 05518 3351400 0551833 02035 01481 05005 02713 3415770 36041334 03635 03457 05641 05801 3286610 0580135 03443 01867 03929 02980 3382780 3078036 02591 02779 04456 03472 3356430 0347237 02461 03439 03002 02413 3377180 0241338 02195 02851 04870 03622 3355940 0362239 02712 02026 03484 02329 3411970 32202940 02303 03541 05037 04297 3325610 0429741 01175 01257 03597 01312 3519180 139311242 02642 03307 06724 05971 3299080 0597143 03056 03823 06227 06271 3280290 0627144 01912 02420 03305 02009 3429110 49310945 02527 02565 06677 05163 3326100 0516346 01246 01592 03740 01583 3486010 106168347 02746 03970 03427 03188 3334970 0318848 01778 03630 05872 04737 3322980 0473749 01092 02612 06948 04105 3364770 0410550 03695 02309 06618 05919 3308430 05919

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

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Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

Table 1 Typical geometric and physical parameters of trefoil cables (cable quadrature = 1600mm2)

Cable Radius [mm] Material Thermal conductivity [W(msdotK)] Electronic conductivity [Sm]Cable core 248 copper 400 5998 times 107

Insulation layer 553 XLPE 038 10 times 10minus15

Sheath layer 617 copper 400 5998 times 107

Jacket layer 668 HDPE 038 10 times 10minus15

300 320 340 360 380 400 420 4400

1

2

3

4

5

k(T)

[W(

mmiddotK

)]

kFTB

T [K]

kNS

Figure 3 Variations of 119896NS and 119896FTB with temperature

Table 2 119896dry and 119896wet for NS and FTB material [6 17]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100FTB 154 435

where 997888119869 e is the inductive current density997888119869 s is the source

current density 997888119860 j 120596 and 120593 are the magnetic vectorpotential unit of complex number angular frequency andelectric scalar potential respectively 120588

0and 120572 are the elec-

trical resistivity and temperature coefficient at a referencetemperature of 119879ref = 293K

The boundary conditions for the computational domainare set as follows

12059711987912059711990910038161003816100381610038161003816100381610038161003816x=0 or x=L = 0120597119879120597119910

10038161003816100381610038161003816100381610038161003816y=0 = 0119879|y=H = 293K

(6)

In the present study the above governing equations are solvedwith commercial code COMSOL MULTIPHYSICS and thePardiso solver is employed for the computationsThe currentfrequency is set to 50Hz The conservative interface fluxcondition for heat transfer is adopted at the cable-FTB andFTB-NS interfaces as well as the internal interfaces betweendifferent layers inside the cable which means that the heatflux on one-side of the interface was considered to be equalto the heat flux on the other-side of the interface between

different computational regions For convergence criteria allresiduals of the calculations are less than 10minus4

3 Grid Independence Test andModel Validations

Firstly the grid independence test was performed In thepresent study the parabolic-type FTB trefoil cables wereadopted for the test (119904

1= 35119903j 119897 = 8119903j) where the cable

core current is 1145A and the current frequency is 50HzAs presented in Figure 4 the self-adaptive tetrahedral meshwas used for the computations and the grids are intensifiedaround cable regions In order to improve the mesh qualitynear the contact points between cable surfaces according tothe report of Bu et al [18] the cables were stacked with verysmall gaps (119897

1= 2119889j) instead of contact points between each

other (Figure 2(a)) Four sets of grids were used for the testand the computational results are presented in Table 3 Itshows that the Grid-3 with total element number of 19346is good enough for the test based on the comparison ofthe maximum cable temperature (119879max) and heat flux onthe cable surface (120601) with different grids For Grid-3 theminimum length of the grid element in both the cable zone(Zone 1) and FTB zone (Zone 2) is 03mm and it is 04mmin the NS zone (Zone 3) Therefore similar grid settingsto the test grid of Grid-3 were employed for the followingsimulations

Subsequently the computational model and methodswere validated The heat transfer process in three-phaseburied cables [6] was restudied and the model is presentedin Figure 5 It shows that the power cables are parallelarranged and the cable core distance is 04m between eachother The distance between the ground and cable core ish1 In the near-cable region the NS is used as the backfill

material while the other regions are filled with multilayersoilThe symmetry boundary condition is adopted on the leftedge of the computational domain and the right edge andbottom surface of the computational domain are consideredto be adiabatic Furthermore the ground temperature isfixed at 293KThe thermal conductivities of different backfillmaterials are dependent on temperature variations whichcan be calculated with (2) and parameters listed in Table 4

When the cable core current is fixed at 1145A the varia-tions of temperature along the symmetry edge are presentedin Figure 6(a) It shows that the maximum temperature devi-ation between our present computations and those of Ocłonet al [6] is 14 K Meanwhile the variations of the maximumcable core temperature (119879max) with cable buried depth (ℎ

1)

are presented in Figure 6(b) It shows that the maximum

Mathematical Problems in Engineering 5

ZONE3

ZONE1 ZONE2

ZONE2

ZONE1

Figure 4 Typical computational mesh for parabolic-type FTB trefoil buried cables

Table 3 Computational results with different meshes

Grid Region Minimum length of gridelement [mm]

Maximum length of gridelement [mm] Total element number 119879max [K] 120601 [W]

Grid 1Zone 1 15 04

10228 32656 68899Zone 2 15 04Zone 3 15 05

Grid 2Zone 1 04 02

16337 329557 69892Zone 2 15 04Zone 3 15 04

Grid 3Zone 1 03 01

19346 329563 68903Zone 2 03 01Zone 3 04 02

Grid 4Zone 1 03 01

28394 329563 68903Zone 2 03 01Zone 3 035 015

6 Mathematical Problems in Engineering

NSNS

MS

Gr

SCLG

SCL

h1

02m

04m 07m

08m

09m

02m

07m

10m

Symmetry

Tref =293K

Adiabatic Adiabatic

Power cable

10m

NS Natural soil

MS Medium sand

Gr Gravel

SCLG Sand clay loamwith gravel SCL Sand clay loam

o

y

x

Figure 5 Model for validations [6]

Table 4 Thermal conductivity of different buried material inOcłonrsquos study [6]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100MS 021 228Gr 026 267SCLG 026 201SCL 017 145

deviation of 119879max between our present computations andthose of Ocłon et al [6] is 23 K

4 Results and Discussion

41 Performance Comparison between NS and FTB TrefoilBuried Cables When the cable core current is fixed at 1500Athe temperature distributions in the trefoil buried cables withNS and flat-type FTB are presented in Figure 7 It showsthat for the NS trefoil buried cables (Figure 7(a)) sincethe thermal conductivity of NS material is relatively low(Figure 3) the heat transfer rate is relatively low and themaximum temperature of the cable core is high (119879max = 448Kor 175∘C) while for the FTB trefoil buried cables (Figures7(b)ndash7(d)) since the thermal conductivity of FTB materialis relatively high (Figure 3) the heat transfer rate wouldbe significantly improved and the maximum temperatureof the cable core is obviously reduced When the thicknessof FTB is fixed at 04m 045m and 050m 119879max of thecable core are 362K (89∘C) 357K (84∘C) and 353K (80∘C)respectively The variations of the maximum cable coretemperature in the trefoil buried cables with NS and flat-type FTB are presented in Figure 8 It shows that with thesame cable core current 119879max of the cable core for the NStrefoil buried cables is obviously higher than that for the FTBtrefoil buried cables and 119879max of the cable core for the FTB

trefoil buried cables would be further reduced as the FTBthickness (ℎ) increases In the present study it is shown thatthe temperature distributions inside three-phase cables arequite similar for the same working condition Therefore theampacities in the three-phase cables would also be similarHere the cable ampacity is defined as the correspondingcurrent in the cable core when the maximum cable coretemperature reaches 90∘C (363K) It shows that the cableampacity for the NS trefoil buried cables is 12623 A while thecable ampacities for the FTB trefoil buried cables are 15135 A(ℎ = 040m) 15582 A (ℎ = 045m) 15872 A (ℎ = 050m) and16169 A (ℎ = 055m) respectively Therefore it is indicatedthat for trefoil buried cables the heat transfer rate wouldbe efficiently improved by using FTB material and the cableampacity would be significantly increased

The flat-type FTB trefoil buried cables as mentionedabove (Figure 1(a)) are quite easy to be constructed Howeverthis FTB laying method needs large amount of FTB layingmaterials which would lead to a certain waste Therefore inorder to reduce the FTB amount used for the trefoil buriedcables the parabolic-type FTB laying method (Figure 1(b))is adopted and the heat transfer performance and ampacityof the parabolic-type FTB trefoil buried cables are analysedand compared with those of the flat-type FTB trefoil buriedcables Typical geometric parameters for flat-type FTB andparabolic-type FTB trefoil buried cables are presented inTable 5 It shows that in the present study the cross sectionarea for the parabolic-type FTB model (119860FTB) is much lowerthan that for the flat-type FTB model

When the cable core current is fixed at 1500A thetemperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure 9It shows that the temperature distributions in the flat-typeFTB and parabolic-type FTB trefoil buried cables are similarwhere the corresponding 119879max of the cable core are 362K(89∘C) and 368K (95∘C) respectively This indicates withparabolic-type FTB laying method the heat transfer rate inthe cable would almost keep the same while the FTB amount(represented by 119860FTB) used for the trefoil buried cables canbe greatly reduced

The variations of the maximum cable core temperature(119879max) in the flat-type FTB and parabolic-type FTB trefoilburied cables are presented in Figure 10 It shows that withthe same cable core current 119879max of cable core for theparabolic-type FTB model is little higher than that for theflat-type FTB model and the corresponding ampacities are14775 A (119860FTB = 0351m2) and 15135 A (119860FTB = 0560m2)respectively This indicates that compared with flat-type FTBmodel the ampacity in the cable is slightly reduced withparabolic-type FTB laying method (reduced by 24 in thepresent study) while the FTB amount used for the buriedcables (represented by 119860FTB) is greatly reduced (reduced by373 in the present study Table 5) Therefore the parabolic-type FTB laying method is recommended for the trefoilburied cable applications

42 Laying Parameter Optimization for Parabolic-TypeFTB Trefoil Buried Cables The laying parameters for theparabolic-type FTB trefoil buried cables are presented in

Mathematical Problems in Engineering 7

0 2 4 6 8 10

295

300

305

310

315

320T

[K]

y [m]

Present simulation

Ref [6]

(a)

10 12 14 16 18 20 22 24 26

315

320

325

330

335

T max

[K]

h1 [m]

Present simulationRef [6]

(b)

Figure 6Model validations for temperatures withOcłon et al [6] (a) Temperatures along the symmetry edge (b)Themaximum temperatureof cable core

Table 5 Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables

FTB laying model 1199041[m] 119904

2[m] 119897 [m] ℎ [m] 119860FTB [m

2]FTB (Parabolic) 35119903j 02 8119903j 0351FTB (Flat) 04 0560

Figure 11 It shows that the FTB laying area (119860FTB) or FTBamount is closely related to the FTB laying parametersmainly including 119904

1 1199042 and 119897 In order to enhance the heat

transfer in trefoil buried cables and reduce the use of FTBmaterial the laying parameters should be optimized for theparabolic-type FTB trefoil buried cables In the present studythe total cost function (Fc) as reported by Ocłon et al [11] isadopted to access the overall performance for trefoil buriedcables which is defined as follows

Fc (119897 1199041 1199042)=

119860FTB + PF 119879max minus 119879opt gt 0 (119879opt = 338K)119860FTB 119879max minus 119879opt le 0 (119879opt = 338K)

(7)

where 119860FTB is the FTB laying area 119879max is the maximumtemperature of cable core 119879opt is the optimum workingtemperature for the cable (119879opt = 338K or 65∘C) PF is thepenalty function When the value of 119879max is higher than thatof 119879opt (338K) the penalty function (PF) should be set toguarantee the rationality of total cost function (Fc) where thePF is defined as follows

PF = 10 times (119879max minus 119879opt) (8)

Therefore it should be noted that when the value of 119879max islower than that of119879opt the total cost function (Fc) is depended

on 119860FTB Otherwise Fc should be both related to 119860FTB and119879maxIn the present study three geometric parameters of FTB

laying cross section including 1199041 1199042 and 119897 were selected

as the design variables and the total cost function (Fc) isadopted as the objective function for the optimization Theflowchart for the optimization procedure is presented inFigure 12 Firstly based on the design variables (119904

1 1199042 and 119897)

the geometric parameter combination samples were designedwith the Latin hypercube sampling method (LHS) [19 20]LHS is an effective method to design sample points withinthe design space by which the design points are equallyspaced of each variable in design space [19] When samplinga function of 119899 variables the range of each variable is dividedinto 119898 equal probable intervals For each variable designpoints are randomly selected from each interval Based onthe119898 points for each variable the sample points are obtainedby the random combination Hence the design points areequally spaced of each variable in design space [19] Withthis method 50 sets of random samples were selected as thedesign samples for the optimizations which are presented inTable 6 As shown in Table 6 both the ranges of 119904

1and 1199042are

from01m to 04mwhile the range of 119897 is from03m to 07mBased on these 50 sets of samples the heat transfer processesin the parabolic-type FTB trefoil buried cableswere simulatedwith commercial code COMSOL MULTIPHYSICS Thenthe maximum temperature (119879max) of the cable core and thetotal cost function (Fc) for the corresponding samples were

8 Mathematical Problems in Engineering

Table 6 Design samples with LHS method and the computational results of 119879max and Fc (119868 = 1300A)

Samples 1199041[m] 119904

2[m] 119897 [m] 119860FTB [m

2] 119879max [K] Fc1 03843 03769 03117 03233 3323360 032332 03181 03218 03565 03183 3346860 031833 01562 02716 06180 04077 3555410 17581774 03818 03014 05370 05316 3300740 053165 02238 02232 04746 03155 3381350 166556 01128 02896 03806 02221 3422830 4305217 01604 03373 05287 03921 3346920 039218 03733 01883 03684 02918 3381560 185189 03564 01961 04594 03687 3358680 0368710 03014 02172 04194 03140 3374600 0314011 02902 03613 06789 06546 3283190 0654612 01001 01387 06094 02479 3438940 59187913 01322 01752 04716 02255 3438250 58475514 01508 02070 05538 03093 3394900 15209315 03458 02273 04407 03641 3353500 0364116 02356 03250 04092 03281 3354600 0328117 01934 03718 04042 03261 3351620 0326118 01763 01190 05951 02859 3419510 39795919 01854 01328 05425 02735 3420180 40453520 02569 01070 03248 01667 3479650 99816721 01383 01705 03164 01380 3499190 119328022 02917 01161 03972 02364 3432150 52386423 03084 02500 05218 04286 3337690 0428624 03242 01794 04314 03155 3380940 1255525 02048 03864 06011 05264 3306140 0526426 03900 03173 06881 07151 3274050 0715127 01692 02671 06513 04393 3347680 0439328 03359 01279 05774 04059 3368790 0405929 02382 02923 06391 05106 3321900 0510630 03548 03105 06294 06152 3290470 0615231 02127 03023 05580 04290 3338130 0429032 03296 02413 06464 05518 3351400 0551833 02035 01481 05005 02713 3415770 36041334 03635 03457 05641 05801 3286610 0580135 03443 01867 03929 02980 3382780 3078036 02591 02779 04456 03472 3356430 0347237 02461 03439 03002 02413 3377180 0241338 02195 02851 04870 03622 3355940 0362239 02712 02026 03484 02329 3411970 32202940 02303 03541 05037 04297 3325610 0429741 01175 01257 03597 01312 3519180 139311242 02642 03307 06724 05971 3299080 0597143 03056 03823 06227 06271 3280290 0627144 01912 02420 03305 02009 3429110 49310945 02527 02565 06677 05163 3326100 0516346 01246 01592 03740 01583 3486010 106168347 02746 03970 03427 03188 3334970 0318848 01778 03630 05872 04737 3322980 0473749 01092 02612 06948 04105 3364770 0410550 03695 02309 06618 05919 3308430 05919

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

ZONE3

ZONE1 ZONE2

ZONE2

ZONE1

Figure 4 Typical computational mesh for parabolic-type FTB trefoil buried cables

Table 3 Computational results with different meshes

Grid Region Minimum length of gridelement [mm]

Maximum length of gridelement [mm] Total element number 119879max [K] 120601 [W]

Grid 1Zone 1 15 04

10228 32656 68899Zone 2 15 04Zone 3 15 05

Grid 2Zone 1 04 02

16337 329557 69892Zone 2 15 04Zone 3 15 04

Grid 3Zone 1 03 01

19346 329563 68903Zone 2 03 01Zone 3 04 02

Grid 4Zone 1 03 01

28394 329563 68903Zone 2 03 01Zone 3 035 015

6 Mathematical Problems in Engineering

NSNS

MS

Gr

SCLG

SCL

h1

02m

04m 07m

08m

09m

02m

07m

10m

Symmetry

Tref =293K

Adiabatic Adiabatic

Power cable

10m

NS Natural soil

MS Medium sand

Gr Gravel

SCLG Sand clay loamwith gravel SCL Sand clay loam

o

y

x

Figure 5 Model for validations [6]

Table 4 Thermal conductivity of different buried material inOcłonrsquos study [6]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100MS 021 228Gr 026 267SCLG 026 201SCL 017 145

deviation of 119879max between our present computations andthose of Ocłon et al [6] is 23 K

4 Results and Discussion

41 Performance Comparison between NS and FTB TrefoilBuried Cables When the cable core current is fixed at 1500Athe temperature distributions in the trefoil buried cables withNS and flat-type FTB are presented in Figure 7 It showsthat for the NS trefoil buried cables (Figure 7(a)) sincethe thermal conductivity of NS material is relatively low(Figure 3) the heat transfer rate is relatively low and themaximum temperature of the cable core is high (119879max = 448Kor 175∘C) while for the FTB trefoil buried cables (Figures7(b)ndash7(d)) since the thermal conductivity of FTB materialis relatively high (Figure 3) the heat transfer rate wouldbe significantly improved and the maximum temperatureof the cable core is obviously reduced When the thicknessof FTB is fixed at 04m 045m and 050m 119879max of thecable core are 362K (89∘C) 357K (84∘C) and 353K (80∘C)respectively The variations of the maximum cable coretemperature in the trefoil buried cables with NS and flat-type FTB are presented in Figure 8 It shows that with thesame cable core current 119879max of the cable core for the NStrefoil buried cables is obviously higher than that for the FTBtrefoil buried cables and 119879max of the cable core for the FTB

trefoil buried cables would be further reduced as the FTBthickness (ℎ) increases In the present study it is shown thatthe temperature distributions inside three-phase cables arequite similar for the same working condition Therefore theampacities in the three-phase cables would also be similarHere the cable ampacity is defined as the correspondingcurrent in the cable core when the maximum cable coretemperature reaches 90∘C (363K) It shows that the cableampacity for the NS trefoil buried cables is 12623 A while thecable ampacities for the FTB trefoil buried cables are 15135 A(ℎ = 040m) 15582 A (ℎ = 045m) 15872 A (ℎ = 050m) and16169 A (ℎ = 055m) respectively Therefore it is indicatedthat for trefoil buried cables the heat transfer rate wouldbe efficiently improved by using FTB material and the cableampacity would be significantly increased

The flat-type FTB trefoil buried cables as mentionedabove (Figure 1(a)) are quite easy to be constructed Howeverthis FTB laying method needs large amount of FTB layingmaterials which would lead to a certain waste Therefore inorder to reduce the FTB amount used for the trefoil buriedcables the parabolic-type FTB laying method (Figure 1(b))is adopted and the heat transfer performance and ampacityof the parabolic-type FTB trefoil buried cables are analysedand compared with those of the flat-type FTB trefoil buriedcables Typical geometric parameters for flat-type FTB andparabolic-type FTB trefoil buried cables are presented inTable 5 It shows that in the present study the cross sectionarea for the parabolic-type FTB model (119860FTB) is much lowerthan that for the flat-type FTB model

When the cable core current is fixed at 1500A thetemperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure 9It shows that the temperature distributions in the flat-typeFTB and parabolic-type FTB trefoil buried cables are similarwhere the corresponding 119879max of the cable core are 362K(89∘C) and 368K (95∘C) respectively This indicates withparabolic-type FTB laying method the heat transfer rate inthe cable would almost keep the same while the FTB amount(represented by 119860FTB) used for the trefoil buried cables canbe greatly reduced

The variations of the maximum cable core temperature(119879max) in the flat-type FTB and parabolic-type FTB trefoilburied cables are presented in Figure 10 It shows that withthe same cable core current 119879max of cable core for theparabolic-type FTB model is little higher than that for theflat-type FTB model and the corresponding ampacities are14775 A (119860FTB = 0351m2) and 15135 A (119860FTB = 0560m2)respectively This indicates that compared with flat-type FTBmodel the ampacity in the cable is slightly reduced withparabolic-type FTB laying method (reduced by 24 in thepresent study) while the FTB amount used for the buriedcables (represented by 119860FTB) is greatly reduced (reduced by373 in the present study Table 5) Therefore the parabolic-type FTB laying method is recommended for the trefoilburied cable applications

42 Laying Parameter Optimization for Parabolic-TypeFTB Trefoil Buried Cables The laying parameters for theparabolic-type FTB trefoil buried cables are presented in

Mathematical Problems in Engineering 7

0 2 4 6 8 10

295

300

305

310

315

320T

[K]

y [m]

Present simulation

Ref [6]

(a)

10 12 14 16 18 20 22 24 26

315

320

325

330

335

T max

[K]

h1 [m]

Present simulationRef [6]

(b)

Figure 6Model validations for temperatures withOcłon et al [6] (a) Temperatures along the symmetry edge (b)Themaximum temperatureof cable core

Table 5 Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables

FTB laying model 1199041[m] 119904

2[m] 119897 [m] ℎ [m] 119860FTB [m

2]FTB (Parabolic) 35119903j 02 8119903j 0351FTB (Flat) 04 0560

Figure 11 It shows that the FTB laying area (119860FTB) or FTBamount is closely related to the FTB laying parametersmainly including 119904

1 1199042 and 119897 In order to enhance the heat

transfer in trefoil buried cables and reduce the use of FTBmaterial the laying parameters should be optimized for theparabolic-type FTB trefoil buried cables In the present studythe total cost function (Fc) as reported by Ocłon et al [11] isadopted to access the overall performance for trefoil buriedcables which is defined as follows

Fc (119897 1199041 1199042)=

119860FTB + PF 119879max minus 119879opt gt 0 (119879opt = 338K)119860FTB 119879max minus 119879opt le 0 (119879opt = 338K)

(7)

where 119860FTB is the FTB laying area 119879max is the maximumtemperature of cable core 119879opt is the optimum workingtemperature for the cable (119879opt = 338K or 65∘C) PF is thepenalty function When the value of 119879max is higher than thatof 119879opt (338K) the penalty function (PF) should be set toguarantee the rationality of total cost function (Fc) where thePF is defined as follows

PF = 10 times (119879max minus 119879opt) (8)

Therefore it should be noted that when the value of 119879max islower than that of119879opt the total cost function (Fc) is depended

on 119860FTB Otherwise Fc should be both related to 119860FTB and119879maxIn the present study three geometric parameters of FTB

laying cross section including 1199041 1199042 and 119897 were selected

as the design variables and the total cost function (Fc) isadopted as the objective function for the optimization Theflowchart for the optimization procedure is presented inFigure 12 Firstly based on the design variables (119904

1 1199042 and 119897)

the geometric parameter combination samples were designedwith the Latin hypercube sampling method (LHS) [19 20]LHS is an effective method to design sample points withinthe design space by which the design points are equallyspaced of each variable in design space [19] When samplinga function of 119899 variables the range of each variable is dividedinto 119898 equal probable intervals For each variable designpoints are randomly selected from each interval Based onthe119898 points for each variable the sample points are obtainedby the random combination Hence the design points areequally spaced of each variable in design space [19] Withthis method 50 sets of random samples were selected as thedesign samples for the optimizations which are presented inTable 6 As shown in Table 6 both the ranges of 119904

1and 1199042are

from01m to 04mwhile the range of 119897 is from03m to 07mBased on these 50 sets of samples the heat transfer processesin the parabolic-type FTB trefoil buried cableswere simulatedwith commercial code COMSOL MULTIPHYSICS Thenthe maximum temperature (119879max) of the cable core and thetotal cost function (Fc) for the corresponding samples were

8 Mathematical Problems in Engineering

Table 6 Design samples with LHS method and the computational results of 119879max and Fc (119868 = 1300A)

Samples 1199041[m] 119904

2[m] 119897 [m] 119860FTB [m

2] 119879max [K] Fc1 03843 03769 03117 03233 3323360 032332 03181 03218 03565 03183 3346860 031833 01562 02716 06180 04077 3555410 17581774 03818 03014 05370 05316 3300740 053165 02238 02232 04746 03155 3381350 166556 01128 02896 03806 02221 3422830 4305217 01604 03373 05287 03921 3346920 039218 03733 01883 03684 02918 3381560 185189 03564 01961 04594 03687 3358680 0368710 03014 02172 04194 03140 3374600 0314011 02902 03613 06789 06546 3283190 0654612 01001 01387 06094 02479 3438940 59187913 01322 01752 04716 02255 3438250 58475514 01508 02070 05538 03093 3394900 15209315 03458 02273 04407 03641 3353500 0364116 02356 03250 04092 03281 3354600 0328117 01934 03718 04042 03261 3351620 0326118 01763 01190 05951 02859 3419510 39795919 01854 01328 05425 02735 3420180 40453520 02569 01070 03248 01667 3479650 99816721 01383 01705 03164 01380 3499190 119328022 02917 01161 03972 02364 3432150 52386423 03084 02500 05218 04286 3337690 0428624 03242 01794 04314 03155 3380940 1255525 02048 03864 06011 05264 3306140 0526426 03900 03173 06881 07151 3274050 0715127 01692 02671 06513 04393 3347680 0439328 03359 01279 05774 04059 3368790 0405929 02382 02923 06391 05106 3321900 0510630 03548 03105 06294 06152 3290470 0615231 02127 03023 05580 04290 3338130 0429032 03296 02413 06464 05518 3351400 0551833 02035 01481 05005 02713 3415770 36041334 03635 03457 05641 05801 3286610 0580135 03443 01867 03929 02980 3382780 3078036 02591 02779 04456 03472 3356430 0347237 02461 03439 03002 02413 3377180 0241338 02195 02851 04870 03622 3355940 0362239 02712 02026 03484 02329 3411970 32202940 02303 03541 05037 04297 3325610 0429741 01175 01257 03597 01312 3519180 139311242 02642 03307 06724 05971 3299080 0597143 03056 03823 06227 06271 3280290 0627144 01912 02420 03305 02009 3429110 49310945 02527 02565 06677 05163 3326100 0516346 01246 01592 03740 01583 3486010 106168347 02746 03970 03427 03188 3334970 0318848 01778 03630 05872 04737 3322980 0473749 01092 02612 06948 04105 3364770 0410550 03695 02309 06618 05919 3308430 05919

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

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Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

NSNS

MS

Gr

SCLG

SCL

h1

02m

04m 07m

08m

09m

02m

07m

10m

Symmetry

Tref =293K

Adiabatic Adiabatic

Power cable

10m

NS Natural soil

MS Medium sand

Gr Gravel

SCLG Sand clay loamwith gravel SCL Sand clay loam

o

y

x

Figure 5 Model for validations [6]

Table 4 Thermal conductivity of different buried material inOcłonrsquos study [6]

Material 119896dry [W(msdotK)] 119896wet [W(msdotK)]NS 030 100MS 021 228Gr 026 267SCLG 026 201SCL 017 145

deviation of 119879max between our present computations andthose of Ocłon et al [6] is 23 K

4 Results and Discussion

41 Performance Comparison between NS and FTB TrefoilBuried Cables When the cable core current is fixed at 1500Athe temperature distributions in the trefoil buried cables withNS and flat-type FTB are presented in Figure 7 It showsthat for the NS trefoil buried cables (Figure 7(a)) sincethe thermal conductivity of NS material is relatively low(Figure 3) the heat transfer rate is relatively low and themaximum temperature of the cable core is high (119879max = 448Kor 175∘C) while for the FTB trefoil buried cables (Figures7(b)ndash7(d)) since the thermal conductivity of FTB materialis relatively high (Figure 3) the heat transfer rate wouldbe significantly improved and the maximum temperatureof the cable core is obviously reduced When the thicknessof FTB is fixed at 04m 045m and 050m 119879max of thecable core are 362K (89∘C) 357K (84∘C) and 353K (80∘C)respectively The variations of the maximum cable coretemperature in the trefoil buried cables with NS and flat-type FTB are presented in Figure 8 It shows that with thesame cable core current 119879max of the cable core for the NStrefoil buried cables is obviously higher than that for the FTBtrefoil buried cables and 119879max of the cable core for the FTB

trefoil buried cables would be further reduced as the FTBthickness (ℎ) increases In the present study it is shown thatthe temperature distributions inside three-phase cables arequite similar for the same working condition Therefore theampacities in the three-phase cables would also be similarHere the cable ampacity is defined as the correspondingcurrent in the cable core when the maximum cable coretemperature reaches 90∘C (363K) It shows that the cableampacity for the NS trefoil buried cables is 12623 A while thecable ampacities for the FTB trefoil buried cables are 15135 A(ℎ = 040m) 15582 A (ℎ = 045m) 15872 A (ℎ = 050m) and16169 A (ℎ = 055m) respectively Therefore it is indicatedthat for trefoil buried cables the heat transfer rate wouldbe efficiently improved by using FTB material and the cableampacity would be significantly increased

The flat-type FTB trefoil buried cables as mentionedabove (Figure 1(a)) are quite easy to be constructed Howeverthis FTB laying method needs large amount of FTB layingmaterials which would lead to a certain waste Therefore inorder to reduce the FTB amount used for the trefoil buriedcables the parabolic-type FTB laying method (Figure 1(b))is adopted and the heat transfer performance and ampacityof the parabolic-type FTB trefoil buried cables are analysedand compared with those of the flat-type FTB trefoil buriedcables Typical geometric parameters for flat-type FTB andparabolic-type FTB trefoil buried cables are presented inTable 5 It shows that in the present study the cross sectionarea for the parabolic-type FTB model (119860FTB) is much lowerthan that for the flat-type FTB model

When the cable core current is fixed at 1500A thetemperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure 9It shows that the temperature distributions in the flat-typeFTB and parabolic-type FTB trefoil buried cables are similarwhere the corresponding 119879max of the cable core are 362K(89∘C) and 368K (95∘C) respectively This indicates withparabolic-type FTB laying method the heat transfer rate inthe cable would almost keep the same while the FTB amount(represented by 119860FTB) used for the trefoil buried cables canbe greatly reduced

The variations of the maximum cable core temperature(119879max) in the flat-type FTB and parabolic-type FTB trefoilburied cables are presented in Figure 10 It shows that withthe same cable core current 119879max of cable core for theparabolic-type FTB model is little higher than that for theflat-type FTB model and the corresponding ampacities are14775 A (119860FTB = 0351m2) and 15135 A (119860FTB = 0560m2)respectively This indicates that compared with flat-type FTBmodel the ampacity in the cable is slightly reduced withparabolic-type FTB laying method (reduced by 24 in thepresent study) while the FTB amount used for the buriedcables (represented by 119860FTB) is greatly reduced (reduced by373 in the present study Table 5) Therefore the parabolic-type FTB laying method is recommended for the trefoilburied cable applications

42 Laying Parameter Optimization for Parabolic-TypeFTB Trefoil Buried Cables The laying parameters for theparabolic-type FTB trefoil buried cables are presented in

Mathematical Problems in Engineering 7

0 2 4 6 8 10

295

300

305

310

315

320T

[K]

y [m]

Present simulation

Ref [6]

(a)

10 12 14 16 18 20 22 24 26

315

320

325

330

335

T max

[K]

h1 [m]

Present simulationRef [6]

(b)

Figure 6Model validations for temperatures withOcłon et al [6] (a) Temperatures along the symmetry edge (b)Themaximum temperatureof cable core

Table 5 Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables

FTB laying model 1199041[m] 119904

2[m] 119897 [m] ℎ [m] 119860FTB [m

2]FTB (Parabolic) 35119903j 02 8119903j 0351FTB (Flat) 04 0560

Figure 11 It shows that the FTB laying area (119860FTB) or FTBamount is closely related to the FTB laying parametersmainly including 119904

1 1199042 and 119897 In order to enhance the heat

transfer in trefoil buried cables and reduce the use of FTBmaterial the laying parameters should be optimized for theparabolic-type FTB trefoil buried cables In the present studythe total cost function (Fc) as reported by Ocłon et al [11] isadopted to access the overall performance for trefoil buriedcables which is defined as follows

Fc (119897 1199041 1199042)=

119860FTB + PF 119879max minus 119879opt gt 0 (119879opt = 338K)119860FTB 119879max minus 119879opt le 0 (119879opt = 338K)

(7)

where 119860FTB is the FTB laying area 119879max is the maximumtemperature of cable core 119879opt is the optimum workingtemperature for the cable (119879opt = 338K or 65∘C) PF is thepenalty function When the value of 119879max is higher than thatof 119879opt (338K) the penalty function (PF) should be set toguarantee the rationality of total cost function (Fc) where thePF is defined as follows

PF = 10 times (119879max minus 119879opt) (8)

Therefore it should be noted that when the value of 119879max islower than that of119879opt the total cost function (Fc) is depended

on 119860FTB Otherwise Fc should be both related to 119860FTB and119879maxIn the present study three geometric parameters of FTB

laying cross section including 1199041 1199042 and 119897 were selected

as the design variables and the total cost function (Fc) isadopted as the objective function for the optimization Theflowchart for the optimization procedure is presented inFigure 12 Firstly based on the design variables (119904

1 1199042 and 119897)

the geometric parameter combination samples were designedwith the Latin hypercube sampling method (LHS) [19 20]LHS is an effective method to design sample points withinthe design space by which the design points are equallyspaced of each variable in design space [19] When samplinga function of 119899 variables the range of each variable is dividedinto 119898 equal probable intervals For each variable designpoints are randomly selected from each interval Based onthe119898 points for each variable the sample points are obtainedby the random combination Hence the design points areequally spaced of each variable in design space [19] Withthis method 50 sets of random samples were selected as thedesign samples for the optimizations which are presented inTable 6 As shown in Table 6 both the ranges of 119904

1and 1199042are

from01m to 04mwhile the range of 119897 is from03m to 07mBased on these 50 sets of samples the heat transfer processesin the parabolic-type FTB trefoil buried cableswere simulatedwith commercial code COMSOL MULTIPHYSICS Thenthe maximum temperature (119879max) of the cable core and thetotal cost function (Fc) for the corresponding samples were

8 Mathematical Problems in Engineering

Table 6 Design samples with LHS method and the computational results of 119879max and Fc (119868 = 1300A)

Samples 1199041[m] 119904

2[m] 119897 [m] 119860FTB [m

2] 119879max [K] Fc1 03843 03769 03117 03233 3323360 032332 03181 03218 03565 03183 3346860 031833 01562 02716 06180 04077 3555410 17581774 03818 03014 05370 05316 3300740 053165 02238 02232 04746 03155 3381350 166556 01128 02896 03806 02221 3422830 4305217 01604 03373 05287 03921 3346920 039218 03733 01883 03684 02918 3381560 185189 03564 01961 04594 03687 3358680 0368710 03014 02172 04194 03140 3374600 0314011 02902 03613 06789 06546 3283190 0654612 01001 01387 06094 02479 3438940 59187913 01322 01752 04716 02255 3438250 58475514 01508 02070 05538 03093 3394900 15209315 03458 02273 04407 03641 3353500 0364116 02356 03250 04092 03281 3354600 0328117 01934 03718 04042 03261 3351620 0326118 01763 01190 05951 02859 3419510 39795919 01854 01328 05425 02735 3420180 40453520 02569 01070 03248 01667 3479650 99816721 01383 01705 03164 01380 3499190 119328022 02917 01161 03972 02364 3432150 52386423 03084 02500 05218 04286 3337690 0428624 03242 01794 04314 03155 3380940 1255525 02048 03864 06011 05264 3306140 0526426 03900 03173 06881 07151 3274050 0715127 01692 02671 06513 04393 3347680 0439328 03359 01279 05774 04059 3368790 0405929 02382 02923 06391 05106 3321900 0510630 03548 03105 06294 06152 3290470 0615231 02127 03023 05580 04290 3338130 0429032 03296 02413 06464 05518 3351400 0551833 02035 01481 05005 02713 3415770 36041334 03635 03457 05641 05801 3286610 0580135 03443 01867 03929 02980 3382780 3078036 02591 02779 04456 03472 3356430 0347237 02461 03439 03002 02413 3377180 0241338 02195 02851 04870 03622 3355940 0362239 02712 02026 03484 02329 3411970 32202940 02303 03541 05037 04297 3325610 0429741 01175 01257 03597 01312 3519180 139311242 02642 03307 06724 05971 3299080 0597143 03056 03823 06227 06271 3280290 0627144 01912 02420 03305 02009 3429110 49310945 02527 02565 06677 05163 3326100 0516346 01246 01592 03740 01583 3486010 106168347 02746 03970 03427 03188 3334970 0318848 01778 03630 05872 04737 3322980 0473749 01092 02612 06948 04105 3364770 0410550 03695 02309 06618 05919 3308430 05919

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

0 2 4 6 8 10

295

300

305

310

315

320T

[K]

y [m]

Present simulation

Ref [6]

(a)

10 12 14 16 18 20 22 24 26

315

320

325

330

335

T max

[K]

h1 [m]

Present simulationRef [6]

(b)

Figure 6Model validations for temperatures withOcłon et al [6] (a) Temperatures along the symmetry edge (b)Themaximum temperatureof cable core

Table 5 Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables

FTB laying model 1199041[m] 119904

2[m] 119897 [m] ℎ [m] 119860FTB [m

2]FTB (Parabolic) 35119903j 02 8119903j 0351FTB (Flat) 04 0560

Figure 11 It shows that the FTB laying area (119860FTB) or FTBamount is closely related to the FTB laying parametersmainly including 119904

1 1199042 and 119897 In order to enhance the heat

transfer in trefoil buried cables and reduce the use of FTBmaterial the laying parameters should be optimized for theparabolic-type FTB trefoil buried cables In the present studythe total cost function (Fc) as reported by Ocłon et al [11] isadopted to access the overall performance for trefoil buriedcables which is defined as follows

Fc (119897 1199041 1199042)=

119860FTB + PF 119879max minus 119879opt gt 0 (119879opt = 338K)119860FTB 119879max minus 119879opt le 0 (119879opt = 338K)

(7)

where 119860FTB is the FTB laying area 119879max is the maximumtemperature of cable core 119879opt is the optimum workingtemperature for the cable (119879opt = 338K or 65∘C) PF is thepenalty function When the value of 119879max is higher than thatof 119879opt (338K) the penalty function (PF) should be set toguarantee the rationality of total cost function (Fc) where thePF is defined as follows

PF = 10 times (119879max minus 119879opt) (8)

Therefore it should be noted that when the value of 119879max islower than that of119879opt the total cost function (Fc) is depended

on 119860FTB Otherwise Fc should be both related to 119860FTB and119879maxIn the present study three geometric parameters of FTB

laying cross section including 1199041 1199042 and 119897 were selected

as the design variables and the total cost function (Fc) isadopted as the objective function for the optimization Theflowchart for the optimization procedure is presented inFigure 12 Firstly based on the design variables (119904

1 1199042 and 119897)

the geometric parameter combination samples were designedwith the Latin hypercube sampling method (LHS) [19 20]LHS is an effective method to design sample points withinthe design space by which the design points are equallyspaced of each variable in design space [19] When samplinga function of 119899 variables the range of each variable is dividedinto 119898 equal probable intervals For each variable designpoints are randomly selected from each interval Based onthe119898 points for each variable the sample points are obtainedby the random combination Hence the design points areequally spaced of each variable in design space [19] Withthis method 50 sets of random samples were selected as thedesign samples for the optimizations which are presented inTable 6 As shown in Table 6 both the ranges of 119904

1and 1199042are

from01m to 04mwhile the range of 119897 is from03m to 07mBased on these 50 sets of samples the heat transfer processesin the parabolic-type FTB trefoil buried cableswere simulatedwith commercial code COMSOL MULTIPHYSICS Thenthe maximum temperature (119879max) of the cable core and thetotal cost function (Fc) for the corresponding samples were

8 Mathematical Problems in Engineering

Table 6 Design samples with LHS method and the computational results of 119879max and Fc (119868 = 1300A)

Samples 1199041[m] 119904

2[m] 119897 [m] 119860FTB [m

2] 119879max [K] Fc1 03843 03769 03117 03233 3323360 032332 03181 03218 03565 03183 3346860 031833 01562 02716 06180 04077 3555410 17581774 03818 03014 05370 05316 3300740 053165 02238 02232 04746 03155 3381350 166556 01128 02896 03806 02221 3422830 4305217 01604 03373 05287 03921 3346920 039218 03733 01883 03684 02918 3381560 185189 03564 01961 04594 03687 3358680 0368710 03014 02172 04194 03140 3374600 0314011 02902 03613 06789 06546 3283190 0654612 01001 01387 06094 02479 3438940 59187913 01322 01752 04716 02255 3438250 58475514 01508 02070 05538 03093 3394900 15209315 03458 02273 04407 03641 3353500 0364116 02356 03250 04092 03281 3354600 0328117 01934 03718 04042 03261 3351620 0326118 01763 01190 05951 02859 3419510 39795919 01854 01328 05425 02735 3420180 40453520 02569 01070 03248 01667 3479650 99816721 01383 01705 03164 01380 3499190 119328022 02917 01161 03972 02364 3432150 52386423 03084 02500 05218 04286 3337690 0428624 03242 01794 04314 03155 3380940 1255525 02048 03864 06011 05264 3306140 0526426 03900 03173 06881 07151 3274050 0715127 01692 02671 06513 04393 3347680 0439328 03359 01279 05774 04059 3368790 0405929 02382 02923 06391 05106 3321900 0510630 03548 03105 06294 06152 3290470 0615231 02127 03023 05580 04290 3338130 0429032 03296 02413 06464 05518 3351400 0551833 02035 01481 05005 02713 3415770 36041334 03635 03457 05641 05801 3286610 0580135 03443 01867 03929 02980 3382780 3078036 02591 02779 04456 03472 3356430 0347237 02461 03439 03002 02413 3377180 0241338 02195 02851 04870 03622 3355940 0362239 02712 02026 03484 02329 3411970 32202940 02303 03541 05037 04297 3325610 0429741 01175 01257 03597 01312 3519180 139311242 02642 03307 06724 05971 3299080 0597143 03056 03823 06227 06271 3280290 0627144 01912 02420 03305 02009 3429110 49310945 02527 02565 06677 05163 3326100 0516346 01246 01592 03740 01583 3486010 106168347 02746 03970 03427 03188 3334970 0318848 01778 03630 05872 04737 3322980 0473749 01092 02612 06948 04105 3364770 0410550 03695 02309 06618 05919 3308430 05919

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

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Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Table 6 Design samples with LHS method and the computational results of 119879max and Fc (119868 = 1300A)

Samples 1199041[m] 119904

2[m] 119897 [m] 119860FTB [m

2] 119879max [K] Fc1 03843 03769 03117 03233 3323360 032332 03181 03218 03565 03183 3346860 031833 01562 02716 06180 04077 3555410 17581774 03818 03014 05370 05316 3300740 053165 02238 02232 04746 03155 3381350 166556 01128 02896 03806 02221 3422830 4305217 01604 03373 05287 03921 3346920 039218 03733 01883 03684 02918 3381560 185189 03564 01961 04594 03687 3358680 0368710 03014 02172 04194 03140 3374600 0314011 02902 03613 06789 06546 3283190 0654612 01001 01387 06094 02479 3438940 59187913 01322 01752 04716 02255 3438250 58475514 01508 02070 05538 03093 3394900 15209315 03458 02273 04407 03641 3353500 0364116 02356 03250 04092 03281 3354600 0328117 01934 03718 04042 03261 3351620 0326118 01763 01190 05951 02859 3419510 39795919 01854 01328 05425 02735 3420180 40453520 02569 01070 03248 01667 3479650 99816721 01383 01705 03164 01380 3499190 119328022 02917 01161 03972 02364 3432150 52386423 03084 02500 05218 04286 3337690 0428624 03242 01794 04314 03155 3380940 1255525 02048 03864 06011 05264 3306140 0526426 03900 03173 06881 07151 3274050 0715127 01692 02671 06513 04393 3347680 0439328 03359 01279 05774 04059 3368790 0405929 02382 02923 06391 05106 3321900 0510630 03548 03105 06294 06152 3290470 0615231 02127 03023 05580 04290 3338130 0429032 03296 02413 06464 05518 3351400 0551833 02035 01481 05005 02713 3415770 36041334 03635 03457 05641 05801 3286610 0580135 03443 01867 03929 02980 3382780 3078036 02591 02779 04456 03472 3356430 0347237 02461 03439 03002 02413 3377180 0241338 02195 02851 04870 03622 3355940 0362239 02712 02026 03484 02329 3411970 32202940 02303 03541 05037 04297 3325610 0429741 01175 01257 03597 01312 3519180 139311242 02642 03307 06724 05971 3299080 0597143 03056 03823 06227 06271 3280290 0627144 01912 02420 03305 02009 3429110 49310945 02527 02565 06677 05163 3326100 0516346 01246 01592 03740 01583 3486010 106168347 02746 03970 03427 03188 3334970 0318848 01778 03630 05872 04737 3322980 0473749 01092 02612 06948 04105 3364770 0410550 03695 02309 06618 05919 3308430 05919

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

[K]320 360 400 440

(a)

[K]300 320 340 360

(b)

[K]300 320 340 360

(c)

[K]300 320 340 360

(d)

Figure 7 Temperature distributions in the trefoil buried cables with NS and flat-type FTB (119868 = 1500A) (a) NS buried cables (b) Flat-typeFTB buried cables (ℎ = 040m) (c) Flat-type FTB buried cables (ℎ = 045m) (d) Flat-type FTB buried cables (ℎ = 050m)

calculated based on the simulation results which are alsopresented in Table 6

Secondly based on the sample data and calculation resultsas presented in Table 6 the geometric parameters for theparabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897) were

trainedwith the radial basis functionneural network (RBNN)[19ndash21] RBNN is a two-layer network with strong ability toapproximate objective function and rapid convergence rateincluding a hidden layer of radial basis function (RBF) and alinear output layer [19] Figure 13 shows the model of radialbasis function neuronThe input 119899 to the radial basis functionis the vector distance between the input vector x and theweight vector w multiplied with a threshold value 119887

119899 = 119889119894119904119905 sdot 119887 = w minus x sdot 119887 (9)Gaussian function is employed as the radial basis function

119910 = 119903119886119889119887119886119904 (119899) = 119890minusn2 (10)

When 119899 approaches zero which means the input vector xand weight vector w are approximative the output of the

radial basis function will be a relatively large value close to1 thus the network has a strong ability of local perceptionFigure 14 shows the whole structure of the radial basis neuralnetwork The output of the network is a weighted sum of theoutputs of hidden layers The RBNN is established by usingmathematical software MATLAB The function of RBNN inMATLAB is newrb Besides the sample data of inputs andoutputs a spread constant and the maximum number ofneurons need to be determined Spread constant is relatedto the threshold value 119887 that controls the sensitivity of radialbasis function neuron Its value should be large enough toensure the response within all the design space However itshould not be so large that all the neurons have significantresponse

The maximum temperature (119879max) of the cable core waspredicted with RBNN and validated with commercial codeCOMSOL MULTIPHYSICS The predicted values of 119879maxwith RBNN and COMSOL MULTIPHYSICS for 10 sets ofrandom samples are presented in Table 7 It is found thatthe maximum deviation of 119879max with RBNN and COMSOL

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

Table 7 The comparison of 119879max with RBNN and COMSOL (119868 = 1300A)

Samples Design variables 119879max [K]1199041[m] 119904

2[m] 119897 [m] RBNN COMSOL

1 03152 02035 03393 3407534 3400952 01936 01294 05348 3421368 3420753 03079 03779 04763 3283667 3303144 03557 02250 05911 3326455 3323485 02578 02849 05785 3334514 3328566 02280 02790 04530 3367406 3363327 03755 03126 04107 3320261 3322298 01740 03672 06314 3336398 3316129 01432 01645 06692 3407285 33948610 01196 01443 03564 3493775 350652

Table 8 The optimum results with GA for parabolic-type FTB trefoil buried cables (119868 = 1300A)

Population Optimal design variables (GA) 119879max [K] Fc1199041[m] 119904

2[m] 119897 [m] GA COMSOL GA

100 0298 0303 03 337799 337675 02456200 0290 0302 03 337998 338076 02420

1000 1200 1400 1600 1800 2000

300

400

500

600

700

800

900

T max

[K]

Cable core current [A]

with NS with flat-type FTB h=040 m with flat-type FTB h=045 m with flat-type FTB h=050 m with flat-type FTB h=055 m

363 K

Figure 8 Variations of the maximum cable core temperature in thetrefoil buried cables with NS and flat-type FTB

MULTIPHYSICS is 2 KTherefore the prediction results withRBNN for the present study are reliable

Finally combined with RBNN the genetic algorithm(GA) [22 23] is used to optimize the geometric parametersfor the parabolic-type FTB trefoil buried cables (119904

1 1199042 and 119897)

where the total cost function (Fc) is adopted as the objectivefunction for the optimizations Genetic algorithm (GA) canbe easily used to get the maximum value or minimum valueof a specific function In the optimization when GA is

performed the RBNN acts as the objective function to beminimized GA uses the theory of biological evolution andit obtains the optimized point by a number of generationsevolving from the initial population Individuals of eachgeneration are decided by the value of fitness function ofeach individual of the last generation and the randomnessof selection crossover and mutation The standard geneticalgorithm (SGA) is conducted using the genetic algorithmtoolbox of MATLAB The predicted results of 119879max withGA were also validated with commercial code COMSOLMULTIPHYSICS When the cable core current is fixed at1300A the optimum results with GA for the parabolic-typeFTB trefoil buried cables are presented in Table 8 It showsthat the values of the optimum geometric parameters (119904

1 1199042

and 119897) of FTB cross section with GA are almost the same atpopulations of 100 and 200 When the population numberis 200 the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively and the maximum temper-ature (119879max) of the cable core is 337998K The deviationof 119879max predicted with GA and COMSOL MULTIPHYSICSis 0078K and 119879max is also quite close to the optimumcable working temperature (119879opt = 338K or 65∘C) With theoptimum geometric parameters (119904

1= 0290m 119904

2= 0302m

and 119897 = 03m) theminimum total cost function (Fc) is 02420for parabolic-type FTB trefoil buried cables

5 Conclusions

In the present paper the heat transfer process in theparabolic-type FTB trefoil buried cables is numerically stud-ied where the cable core loss and eddy current loss inthe cable were coupled for the simulation and the thermalconductivity of the backfill material was considered to berelated to temperature variations The heat transfer perfor-mances and ampacities for the flat-type NS flat-type FTB

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

[K]300 320 340 360

(a) Flat-type FTB buried cables (119860FTB = 0560m2)

[K]300 320 340 360

(b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

Figure 9 Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables (119868 = 1500A) (a) Flat-type FTB buriedcables (119860FTB = 0560m2) (b) Parabolic-type FTB buried cables (119860FTB = 0351m2)

1000 1200 1400 1600 1800 2000

300

400

500

600

T max

[K]

Cable current core [A]

FTB (Parabolic) FTB (Flat)

363K

Figure 10 Variations of the maximum cable core temperature inflat-type FTB and parabolic-type FTB trefoil buried cables

and parabolic-type FTB trefoil buried cables were carefullyanalysed and compared with each other Then the layingparameters for the cross section of parabolic-type FTB trefoilburied cables were optimized with the radial basis functionneural network (RBNN) and genetic algorithm (GA) Themain findings are as follows

(1) Since the thermal conductivity of FTB material isrelatively high the heat transfer rate in the FTB trefoil buriedcables would be significantly improved and the maximumtemperature in the cable core is obviously reducedThereforethe ampacity for the FTB trefoil buried cables is much higherthan that for the NS trefoil buried cables

s1

s2

l

l

s

FTB

Figure 11 The geometric parameters of parabolic-type FTB trefoilburied cables

1 Problem setup(Objective function amp Design variables)

2 Sample design and simulation(LHS Comsol Multiphysics)

3 Construction of surrogate and test (RBNN Comsol Multiphysics)

4 Search for optimal point and test (GA Comsol Multiphysics)

Figure 12 Flow chart for the optimization procedure

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

x1

x2

xm

hellip

w1h

w2h

wmh

n

b

ydist

Figure 13 The model of radial basis function neuron

x1

x2

xm

y1

y2

ym

Figure 14 The structure of radial basis neural network

(2) When compared with flat-type FTB model the heattransfer rate in the cable with parabolic-type FTB layingmethod would be slightly reduced while the FTB amountused for the buried cables would be greatly reduced Whenthe cable core current is fixed at 1500A the ampacityin the cable is only reduced by 24 with parabolic-typeFTB laying method while the FTB amount is reduced by373 Therefore the parabolic-type FTB laying method isrecommended for the trefoil buried cable applications

(3) For parabolic-type FTB trefoil buried cables withproper design of geometric parameters (119904

1 1199042 and 119897) for the

FTB laying cross section the overall performance of the cablewould be optimized When the cable core current is fixedat 1300A the optimum values of 119904

1 1199042 and 119897 are 0290m

0302m and 03m respectively the maximum temperature(119879max) of the cable core is 337998K and the minimum totalcost function (Fc) is 02420

Nomenclature

1198861 1198862 Model coefficients in (2)119887 Threshold value in (9)119860FTB FTB area m2997888119860 Magnetic vector potential Wbm119889j The diameter of cable m119865c Total cost functionℎ The thickness of the flat-type FTB layingmodel mℎ

1 The distance between the cables and

ground in [6] m119867 The height of the computational model m119868 Cable core current A119895 Unit of complex number997888119869 Current density Am2997888rarr119869 e Inductive current density Am2997888rarr119869 s Source current density Am2119896dry Dry thermal conductivity of backfillmaterial W(msdotK)

119896FTB Thermal conductivity of FTB W(msdotK)119896NS Thermal conductivity of NS W(msdotK)119896wet Wet thermal conductivity of backfillmaterial W(msdotK)119897 The distance between the edge of FTB andthe axis of symmetry in the parabolic-typeFTB laying model m119897

1 Distance of the gap between cables m119897s Distance between two neighbor cable cores

m119871 The width of the computational model m119899 Input to the radial basis function in (9)119902v Heat loss Wm3119903c The radius of conductor m119903i The radius of insulation layer m119903j The radius of jacket layer m119903s The radius of sheath layer m1199041 The distance between upper cable core and

the top of FTB in the parabolic-type FTBlaying model m119904

2 The distance between the lower cable core

and bottom of FTB in the parabolic-typeFTB laying model m119879 Temperature K119879lim The limited temperature K119879max The maximum temperature K119879opt The optimum temperature K119879ref The reference temperature K

w Weight vector in (9)x Input vector in (9)

Greek Letters

120572 Reference temperature coefficient in (5)1205880 Reference electrical conductivity Sm2120590 Electrical conductivity Sm2120601 Heat flux W120593 Electric scalar potential V120596 Angular frequency rads

Subscripts

c Conductori Insulation layerj Jacket layermax The maximum valueopt The optimum valueref The reference values Sheath layer

Abbreviations

FEM Finite element methodFTB Fluidized thermal backfillGA Genetic algorithmLHS Latin hypercube samplingNS Natural soilPF Penalty functionPSO Particle swarm optimization

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 13

RBF Radial basis functionRBNN Radial basis neural networkSGA Standard genetic algorithmXLPE Cross-linked polyethylene

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was financially supported by the project ldquoResearchand demonstration application of temperature rise algorithmfor buried (direct buried and pipe laying) cable grouprdquofrom State Grid Corporation of China (SGCC) under Grant52094018001K

References

[1] J M Wang J Zhang and J Nie ldquoAn improved artificialcolony algorithm model for forecasting Chinese electricityconsumption and analyzing effect mechanismrdquo MathematicalProblems in Engineering vol 2016 Article ID 8496971 14 pages2016

[2] F Aras C Oysu and G Yilmaz ldquoAn assessment of the methodsfor calculating ampacity of underground power cablesrdquo ElectricMachines and Power Systems vol 33 no 12 pp 1385ndash1402 2005

[3] I A Metwally A H Al-Badi and A S Al Farsi ldquoFactorsinfluencing ampacity and temperature of underground powercablesrdquo Electrical Engineering vol 95 no 4 pp 383ndash392 2013

[4] P Ocłon P Cisek D Taler M Pilarczyk and T SzwarcldquoOptimizing of the underground power cable bedding usingmomentum-type particle swarmoptimizationmethodrdquoEnergyvol 92 pp 230ndash239 2015

[5] O E Gouda A Z El Dein G M Amer et al ldquoEffect of theformation of the dry zone around underground power cableson their ratingsrdquo IEEE Transactions on Power Delivery vol 26no 2 pp 972ndash978 2011

[6] P Ocłon P Cisek M Pilarczyk and D Taler ldquoNumericalsimulation of heat dissipation processes in underground powercable system situated in thermal backfill and buried in amultilayered soilrdquo Energy Conversion and Management vol 95pp 352ndash370 2015

[7] F de Leon and G J Anders ldquoEffects of backfilling on cableampacity analyzed with the finite element methodrdquo IEEETransactions on Power Delivery vol 23 no 2 pp 537ndash543 2008

[8] O E Gouda and A Z El Dein ldquoImproving undergroundpower distribution capacity using artificial backfill materialsrdquoIET Generation Transmission amp Distribution vol 9 no 15 pp2180ndash2187 2015

[9] M Rerak and P Ocłon ldquoThermal analysis of undergroundpower cable systemrdquo Journal of Thermal Science vol 26 no 5pp 465ndash471 2017

[10] G J Anders and H S Radhakrishna ldquoComputation of tem-perature field and moisture content in the vicinity of currentcarrying underground power cablesrdquo IEEE Proceedings vol 135no 1 pp 51ndash62 1988

[11] P Ocłon P Cisek M Rerak et al ldquoThermal performanceoptimization of the underground power cable system by usinga modified Jaya algorithmrdquo International Journal of ThermalSciences vol 123 pp 162ndash180 2018

[12] A Cichy B Sakowicz and M Kaminski ldquoEconomic opti-mization of an underground power cable installationrdquo IEEETransactions on Power Delivery vol 33 no 4 pp 1124ndash11332018

[13] A Cichy B Sakowicz and M Kaminski ldquoDetailed model forcalculation of life-cycle cost of cable ownership and comparisonwith the IEC formulardquo Electric Power Systems Research vol 154pp 463ndash473 2018

[14] M S A Saud ldquoParticle swarm optimization of power cableperformance in complex surroundingsrdquo IET Generation Trans-mission and Distribution vol 12 no 10 pp 2452ndash2461 2018

[15] P Ocłon M Bittelli P Cisek et al ldquoThe performance analysisof a new thermal backfill material for underground power cablesystemrdquo Applied Thermal Engineering vol 108 pp 233ndash2502016

[16] L Quan C Z Fu W R Si et al ldquoNumerical study of heattransfer in underground power cable systemrdquo in Proceedings ofthe 10th International Conference on Applied Energy (ICAE rsquo18)Hong Kong August 2018

[17] P Ocuon D Taler P Cisek and M Pilarczyk ldquoFem-basedthermal analysis of underground power cables located inbackfills made of different materialsrdquo Strength of Materials vol47 no 5 pp 770ndash780 2015

[18] S S Bu J YangM Zhou et al ldquoOn contact point modificationsfor forced convective heat transfer analysis in a structuredpacked bed of spheresrdquo Nuclear Engineering and Design vol270 no 5 pp 21ndash33 2014

[19] H-N Shi T Ma W-X Chu and Q-W Wang ldquoOptimizationof inlet part of a microchannel ceramic heat exchanger usingsurrogate model coupled with genetic algorithmrdquo Energy Con-version and Management vol 149 pp 988ndash996 2017

[20] G-W Koo S-M Lee and K-Y Kim ldquoShape optimization ofinlet part of a printed circuit heat exchanger using surrogatemodelingrdquo Applied Thermal Engineering vol 72 no 1 pp 90ndash96 2014

[21] K-D Lee D-W Choi andK-Y Kim ldquoOptimization of ejectionangles of double-jet film-cooling holes using RBNN modelrdquoInternational Journal of Thermal Sciences vol 73 no 4 pp 69ndash78 2013

[22] A Baghernejad and M Yaghoubi ldquoExergoeconomic analysisand optimization of an Integrated Solar Combined CycleSystem (ISCCS) using genetic algorithmrdquo Energy Conversionand Management vol 52 no 5 pp 2193ndash2203 2011

[23] L Gosselin M Tye-Gingras and F Mathieu-Potvin ldquoReviewof utilization of genetic algorithms in heat transfer problemsrdquoInternational Journal of Heat and Mass Transfer vol 52 no 9-10 pp 2169ndash2188 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom