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SUMMARY
A procedure for calculation annual expanse for a distribution network per square
kilometre is described in this paper. The suppositions are: load per square kilometre is constant,
all transformers medium/low voltage have the same ratings, all transformers have nominal
loading, all low voltage lines have the same cross section, voltage drop at each low voltage line
does not exceed 3%.
Considering results obtained some conclusions were made. They can be useful in planning
and designing distribution networks on larger areas.
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1HND VR c MD R]QDaLPH SURVHaQDWD PHVHaQD FHQD QD L]JXEHQLRW kW SUL YUYQD
HOHNWULaQDPR`QRVWYRYRGRWDVR c MDR]QDaLPHSURVHaQDWDFHQDQDkWh L]JXEHQD
HOHNWULaQD HQHUJLMD YR YRGRW 6PHWDM`L GHND YUYQDWD PR`QRVW H HGQDNYD YR VLWH
PHVHFLRGJRGLQDWD]DYWRULRWYLGQDJRGL^QLWHWUR^RFLPRCHPHGDQDSL^HPH
P
H
D′′ = 12 ⋅ P ⋅ c + P ⋅ τ ⋅ c = P ⋅ (12 ⋅ c + τ ⋅ c )
YRG
Y
P
Y
H
Y
P
H
,PDM`LJLSUHGYLGUDYHQNLWHLSRYRYHGXYDZH
C = 12 ⋅ c + τ ⋅ c P
H
]DYNXSQLWHJRGL^QLWUR^RFL]DYRGRWVOHGXYD
D
YRG
= D′
YRG
+ D ′′ = p ⋅ ( d + h ⋅ F ) ⋅ l + ∆P ⋅ C.
YRG
Y
Y
3UHWSRVWDYXYDPH GHND VR GLVWULEXWLYQDWD PUHCD VH QDSRMXYD SRGUDaMH VR
NRQVWDQWQD SRYU^LQVND JXVWLQD QD SULYLGQRWR RSWRYDUXYDZH 3UL YUYQRWR
RSWRYDUXYDZH QD PUHCDWD WDD JXVWLQD `H MD R]QDaLPH VR σ 3ULWRD VHNRMD
WUDQVIRUPDWRUVND VWDQLFD VUHGHQQL]RN QDSRQ QDSRMXYD SRGUDaMH VR SRYU^LQD YR
YLGQDNYDGUDWLVHQDR_DYRQHJRYRWRVUHGL^WH2VYHQWRD`HSUHWSRVWDYLPHGHND
RG WUDQVIRUPDWRUVNDWD VWDQLFD L]OHJXYDDW n HGQDNYL QLVNRQDSRQVNL YRGRYL LOL
L]YRGL L GHND VLWH YRGRYL VH HGQDNYR RSWRYDUHQL 2SWRYDUXYDZHWR QD VHNRM RG
QLVNRQDSRQVNLWH YRGRYL H VR OLQHDUQR UDVWHaND UDVSUHGHOED QD RSWRYDUXYDZHWR
$NR YNXSQDWD DNWLYQD RWSRUQRVW QD HGHQ RG L]YRGLWH H R D VWUXMDWD QD QHJRYLRW
SRaHWRN H I1 WRJD^ VSRUHG [2] VWUDQLFD ]DJXEDWD QD DNWLYQDWD PR`QRVW YR
YRGRWH16
, ⋅ R ⋅ I1 6RWRDUDYHQNDWDPRCHGDVHQDSL^HYRYLG
2
D
YRG
= p ⋅ ( d + h ⋅ F ) ⋅ l + 1,6 ⋅ R ⋅ I12 ⋅ C.
Y
,PDM`L SUHGYLG GHND VR F H R]QDaHQD SOR^WLQDWD QD QDSUHaQLRW SUHVHN QD
HGHQRGID]QLWHVSURYRGQLFLQDYRGRWDNRVR ρ VHR]QDaLVSHFLILaQDWDRWSRUQRVW
QD PDWHULMDORW RG NRM VSURYRGQLFLWH VH QDSUDYHQL RG UDYHQNDWD ]D YNXSQLWH
JRGL^QLWUR^RFLQDHGHQYRGVOHGXYD
D
YRG
= p ⋅ ( d + h ⋅ F ) ⋅ l + 1,6 ⋅ ρ ⋅ C ⋅ I12 ⋅
Y
l
F
*RGL^QLWH WUR^RFL ]D VLWH n QLVNRQDSRQVNL YRGRYL L]YRGL QD HGQD
WUDQVIRUPDWRUVNDVWDQLFDVUHGHQQL]RNQDSRQ`HELGDW
D
YRGRYL
= n ⋅ p ⋅ (d + h ⋅ F ) ⋅ l + 1,6 ⋅ n ⋅ ρ ⋅ C ⋅ I12 ⋅
Y
l
F
$NRYNXSQLWHJRGL^QLWUR^RFL]DVLWH n YRGRYL VH GLIHUHQFLUDDW SR F L
GRELHQLRWL]UD]VHSULUDPQLQDQXOD`HVHGRELHUDYHQNDWD
p ⋅ h − 1,6 ⋅ ρ ⋅ C ⋅ I12 ⋅
Y
1
= 0.
F2
2GUDYHQNDWD]DRSWLPDOQLRWQDSUHaHQSUHVHNVOHGXYD
F
RSW
1,6 ⋅ ρ ⋅ C
p ⋅h
= I1 ⋅
Y
RGQRVQRRSWLPDOQDWDJXVWLQDQDVWUXMDL]QHVXYD
J
RSW
=
I1
p ⋅h
.
=
1,6 ⋅ ρ ⋅ C
F
Y
RSW
.DNR ^WR PRCH GD VH ]DEHOHCL RG UDYHQNDWD RSWLPDOQDWD JXVWLQD QD
VWUXMDWDHLVWD]DVLWHQLVNRQDSRQVNLL]YRGLNRLVHL]YHGHQLVRVSURYRGQLFLRGLVW
PDWHULMDO QH]DYLVQR RG QLYQLRW EURM L QDSUHaHQ SUHVHN 7DD JODYQR ]DYLVL RG
HNRQRPVNLWHSDUDPHWUL
$QDORJQRQDSUHWKRGQRWRLYNXSQLWHJRGL^QLWUR^RFL]DWUDQVIRUPDWRUVND
VWDQLFD VUHGHQQL]RN QDSRQ PRCDW GD VH SUHWVWDYDW NDNR VXPD RG GYD GHOD 'HORW
^WR QH ]DYLVL RG UHCLPRW QD NRULVWHZHWR QD WUDQVIRUPDWRUVNDWD VWDQLFD VH
GRELYD NRJD VR SURVHaQDWD JRGL^QD WUR^NRYQD NYRWD ]D RYRM YLG RSUHPD p VH
SRPQRCDWYNXSQLWHLQYHVWLFLRQLWUR^RFL]DWUDQVIRUPDWRUVNDWDVWDQLFDGDGHQL
VR UDYHQNDWD 9WRULRW GHO JR VRaLQXYDDW JRGL^QLWH WUR^RFL ]D ]DJXEHQDWD
HQHUJLMD YR WUDQVIRUPDWRURW 7DND ]D YNXSQLWH JRGL^QL WUR^RFL ]D HGQD
WUDQVIRUPDWRUVNDVWDQLFDPRCHGDVHQDSL^H
WV
D
WV
=p
WV
⋅ ( d '+ h'⋅S
QRP
) + ∆P
WU
⋅C NDGH ^WR VR ∆P H R]QDaHQD ]DJXEDWD QD DNWLYQDWD PR`QRVW YR WUDQVIRUPDWRURW
SULYUYQRWRRSWRYDUXYDZH
9U] RVQRYD QD UDYHQNLWH L YNXSQLWH JRGL^QL WUR^RFL ]D HGQD
WUDQVIRUPDWRUVND VWDQLFD VUHGHQQL]RN QDSRQ L ]D SULSDGQDWD QLVNRQDSRQVND
PUHCDPRCDWGDVHQDSL^DWYRYLG
WU
D1 = n ⋅ p ⋅ ( d + h ⋅ F ) ⋅ l + 1,6 ⋅ ρ ⋅ n ⋅ C ⋅ I12 ⋅
Y
l
+p
F
WV
⋅ ( d '+ h'⋅S
QRP
) + ∆P
WU
⋅C NDGH^WR
l=
1 S
⋅
σ
2
QRP
HSRORYLQDRGGROCLQDWDQDSRYU^LQDWDYRYLGQDNYDGUDW^WRVHQDSRMXYDRGHGHQ
WUDQVIRUPDWRU VUHGHQQL]RN QDSRQ RGQRVQR GROCLQD QD QLVNRQDSRQVNL L]YRG
GRGHNDVWUXMDWDQDSRaHWRNRWQDQLVNRQDSRQVNLRWL]YRGHRSUHGHOHQDVRUDYHQNDWD
I1 =
S
3 ⋅ n ⋅U
QRP
QRP
9.831,*2',>1,752>2&,=$75$16)250$7256.,
67$1,&,65('(11,=2.1$321,35,3$'1,7(
1,6.21$3216.,05(#,1$(',1,A1$3295>,1$
'DQDETXGXYDPHHGQRSRGUDaMHaLMDJXVWLQDQDSULYLGQRWRRSWRYDUXYDZHH σ 1D SRGUDaMHWR QHND LPD N WUDQVIRUPDWRUVNL VWDQLFL VUHGHQQL]RN QDSRQ
3OR^WLQDWD QD SRGUDaMHWR QHND H A 9NXSQLWH JRGL^QL WUR^RFL ]D VLWH
WUDQVIRUPDWRUVNLVWDQLFLLVRRGYHWQLWHQLVNRQDSRQVNLYRGRYLQDFHORWRSRGUDaMH
`HELGDW
l

D = N ⋅  n ⋅ p ⋅ (d + h ⋅ F ) ⋅ l + 1,6 ⋅ ρ ⋅ n ⋅ C ⋅ I12 ⋅ + p
F

Y
WV
⋅ (d '+ h'⋅S
QRP
) + ∆P
WU

⋅ C  
3UHWSRVWDYXYDPH GHND VLWH WUDQVIRUPDWRUL VH RSWRYDUHQL VR QRPLQDOQD
PR`QRVW7RJD^YNXSQRWRRSWRYDUXYDZHQDFHODWDSRYU^LQDHHGQDNYRQDVXPDWDQD
QRPLQDOQLWHPR`QRVWLQDVLWHWUDQVIRUPDWRULLPRCHGDVHQDSL^H
N ⋅S
QRP
= A ⋅σ RGNDGH^WRVOHGXYD
N = A⋅
S
QRP
σ
$NR L]UD]RW ]D N RG UDYHQNDWD L L]UD]RW ]D I1 RG UDYHQNDWD VH
]DPHQDW YRUDYHQNDWD SR GHOHZHWR VR A ]D VSHFLILaQLWH JRGL^QL WUR^RFL
QD GLVWULEXWLYQD PUHCD WH JRGL^QL WUR^RFL ]D GLVWULEXWLYQD PUHCD QD
SRYU^LQDVRSOR^WLQDHGHQNYDGUDWHQNLORPHWDUGRELYDPH
D′ =
σ
S

C⋅ρ ⋅l ⋅S2
⋅ n ⋅ p ⋅ (d + h ⋅ F ) ⋅ l + 1,6 ⋅
+p
3⋅U 2 ⋅ n ⋅ F

QRP
Y
QRP
QRP
WV
⋅ (d '+ h'⋅S
QRP
) + ∆P
WU

⋅ C  . 
5(=8/7$7,2'35(60(7.,7(
.RULVWHM`LJRUD]YLHQLRWPDWHPDWLaNLPRGHOQDSUDYHQLVHSUHVPHWNLVRFHO
GDVHLVWUDCLYOLMDQLHWRQDRGGHOQLJROHPLQLYU]VSHFLILaQLWHJRGL^QLWUR^RFL
]DGLVWULEXWLYQDPUHCD
2VYHQ WLH ^WR VH SRUDQR QDYHGHQLWH NRULVWHQL VH L VOHGQLYH SDUDPHWUL
U
= 400 V ρ = 0,030 Ω / m / mm 2 p = 0,12 p = 0,15 9VX^QRVW YR QDSUDYHQLWH SUHVPHWNL QDPHVWR GD VH NRULVWDW SRGDWRFLWH ]D
SOR^WLQDWD QD QDSUHaQLRW SUHVHN QD NDEHORW L VSHFLILaQDWD RWSRUQRVW QD
PDWHULMDORWRGNRM^WRWRMHQDSUDYHQNRULVWHQHNDWDOR^NLRWSRGDWRN]DDNWLYQD
RWSRUQRVWQDNDEHORWSRHGLQLFDGROCLQD
3UHVPHWNLWHQDVSHFLILaQLJRGL^QLWUR^RFLVHQDSUDYHQL]HPDM`L
Á UD]OLaQLSRYU^LQVNLJXVWLQLQDSULYLGQRWRYUYQRRSWRYDUXYDZH
Á UD]OLaQLQRPLQDOQLPR`QRVWLQDWUDQVIRUPDWRULWHVUHGHQQL]RNQDSRQ
QRP
Y
WV
Á UD]OLaQL FHQL ]D kW YUYQD PR`QRVW UD]OLaQL FHQL ]D kWh ]DJXEHQD
HOHNWULaQDHQHUJLMDYRPUHCDWDLUD]OLaQLìvremiwaQD]DJXELí
− ]DJXEDWDQDQDSRQRWYRQLVNRQDSRQVNLWHL]YRGLQHVPHWDM`LJLRWFHSLWH
GDQHHSRJROHPDRG
2SWLPDOQLWH JXVWLQL QD VWUXMD SUHVPHWDQL VR UDYHQNDWD ]D UD]QL
YUHGQRVWLQDC, VHGDGHQLYRWDEHODWDIII.
7DEHODIII 2SWLPDOQLJXVWLQLQDVWUXMD]DUD]QLYUHGQRVWLQDC
C GHQDUL(kW⋅JRGLQD)
2
2
2
2SWLPDOQDJXVWLQDQDVWUXMD 1,533 A/mm 1,084 A/mm 0,766 A/mm 0,626 A/mm2
3RUDGL IDNWRW ^WR SUL SURPHQD QD C VH PHQXYD L RSWLPDOQDWD JXVWLQD QD
VWUXMDSULSUHVPHWNLWHQDVSHFLILaQLJRGL^QLWUR^RFL]DGLVWULEXWLYQDPUHCD
EURMRWQDQLVNRQDSRQVNLWHL]YRGLSRWUDQVIRUPDWRULEURMRWQDNDEOLSRL]YRGH
RSUHGHOXYDQVRFHOGDVHSRVWLJQHSULEOLCQRHGQDNYDJXVWLQDQDVWUXMDYRL]YRGLWH
EOLVND GR RSWLPDOQDWD QH]DYLVQR RG QRPLQDOQDWD PR`QRVW QD WUDQVIRUPDWRURW
VUHGHQQL]RNQDSRQ
3UHVPHWDQLWH YUHGQRVWL QD VSHFLILaQLWH JRGL^QL WUR^RFL ]D GLVWULEX
WLYQD PUHCD VH SULNDCDQL YR WDEHOLWH IV L V 9R WDEHOLWH IV L V ]D VHNRMD
QRPLQDOQD PR`QRVW QD WUDQVIRUPDWRU VUHGHQQL]RN QDSRQ H GDGHQ EURMRW QD
QLVNRQDSRQVNLWHL]YRGLLEURMRWQDNDEOLWHSRL]YRG3ULWRD×R]QDaXYDGHND
LPD QLVNRQDSRQVNL L]YRGL SR WUDQVIRUPDWRU L GHND VHNRM L]YRG H L]YHGHQ VR SR
HGHQ NDEHO aLM QDSUHaHQ SUHVHN QD ID]QLWH CLOL H mm2 GRGHND VR ×× H
R]QDaHQRGHNDLPDQLVNRQDSRQVNLL]YRGLSRWUDQVIRUPDWRULGHNDVHNRMRGQLYH
L]YHGHQ VR SR GYD NDEOD aLM QDSUHaHQ SUHVHN QD ID]QLWH CLOL H mm2 $NR YR
SROHWR SUHGYLGHQR ]D VSHFLILaQLWH JRGL^QL WUR^RFL VWRL FUWLaND WRD R]QDaXYD
GHND VR L]EUDQLWH SDUDPHWUL ]DJXEDWD QD QDSRQRW YR QLVNRQDSRQVNLWH L]YRGL EH]
RWFHSLHSRJROHPDRG
5H]XOWDWLWHYRWDEHODWDIVVHRGQHVXYDDWQDPUHCLWHVRUD]OLaQLQRPLQDOQL
PR`QRVWL QD WUDQVIRUPDWRUL 1R EURMRW QD QLVNRQDSRQVNLWH L]YRGL L QLYQLRW
QDSUHaHQSUHVHNNDMWUDQVIRUPDWRUVNLWHVWDQLFLVRUD]OLaQLQRPLQDOQLPR`QRVWL
QD WUDQVIRUPDWRULWH VH L]ELUDQL WDND ^WR JXVWLQDWD QD VWUXMDWD GD ELGH
SULEOLCQR HGQDNYD L PQRJX GD QH VH UD]OLNXYD RG RSWLPDOQDWD 9R WDEHODWD IV VH
SULNDCDQL GYH JUXSL UH]XOWDWL (GQLWH VH GRELHQL ]HPDM`L GHND H C=1
GHQDULkW⋅JRGLQD GRGHND GUXJLWH VH GRELHQL ]HPDM`L GHND H C=2
GHQDULkW⋅JRGLQD 6SRUHG RFHQNDWD QD DYWRULWH YR QD^LWH XVORYL YUHGQRVWD QD
RYRMSDUDPHWDUHRNROXGHQDULkW⋅JRGLQD1RUHDOQRHGDVHRaHNXYDSRUDVW
NDNRQDFHQLWH]DHOHNWULaQDWDPR`QRVWLHQHUJLMDWDNDLQDYUHPHWRQD]DJXELWH
3RUDGL WRD VH SULNDCDQL L UH]XOWDWLWH ]D C=2GHQDULkW⋅JRGLQD 7RD H
QDSUDYHQR VR FHO GD VH VRJOHGD GDOL SUL WLH XVORYL GRD_D GR SURPHQL YR
SULRULWHWRWQDWUDQVIRUPDWRULWH
5H]XOWDWLWHYRWDEHODWDVVHRGQHVXYDDWQDPUHCLWHVRUD]OLaQLQRPLQDOQL
PR`QRVWLQDWUDQVIRUPDWRULSUL^WREURMRWQDQLVNRQDSRQVNLWHL]YRGLLQLYQLRW
QDSUHaHQ SUHVHN VH L]ELUDQL WDND ^WR JXVWLQDWD QD VWUXMDWD GD ELGH SULEOLCQR
HGQDNYD ]D VHNRMD JUXSD LVSLWXYDQL VOXaDL 3ULNDCDQL VH GYD VOXaDMD 9R SUYLRW
VOXaDMJXVWLQLWHQDVWUXLWHVHGRVWDSRJROHPLRGRSWLPDOQDWDDYRYWRULRWWLHVH
PDONXSRPDOLRGRSWLPDOQDWD
7DEHODIV 6SHFLILaQLJRGL^QLWUR^RFL]DGLVWULEXWLYQLPUHCL
VRPH_XVHEQRHGQDNYLJXVWLQLQDVWUXMDEOLVNLGRRSWLPDOQLWH
YRLOMDGLGHQDULkm2
σ
kVA/
km2
C=GHQDULkW⋅JRGLQD
C=2GHQDULkW⋅JRGLQD
1RPLQDOQDPR`QRVWQDWUDQVIRUPDW 1RPLQDOQDPR`QRVWQDWUDQVIRUPDW
kVA kVA kVA kVA kVA kVA kVA kVA
%URMLSUHVHNQDL]YRGLWH
%URMLSUHVHNQDL]YRGLWH
×
×
×
×
×
×
× ××
2
3RVWLJQDWDJXVWLQDQDVWUXMDA/mm ) 3RVWLJQDWDJXVWLQDQDVWUXMDA/mm2)
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
7DEHODV 6SHFLILaQLJRGL^QLWUR^RFL]DGLVWULEXWLYQLPUHCL
VRUD]OLaQLJXVWLQLQDVWUXMDYRLOMDGLGHQDULkm2
σ
kVA/
km2
C=GHQDULkW⋅JRGLQD
C=1GHQDULkW⋅JRGLQD
1RPLQDOQDPR`QRVWQDWUDQVIRUPDW 1RPLQDOQDPR`QRVWQDWUDQVIRUPDW
kVA kVA kVA kVA kVA kVA kVA kVA
%URMLSUHVHNQDL]YRGLWH
%URMLSUHVHNQDL]YRGLWH
×
×
×
×
×
×
× ××
2
3RVWLJQDWDJXVWLQDQDVWUXMDA/mm ) 3RVWLJQDWDJXVWLQDQDVWUXMDA/mm2)
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
=$./8A2&,
,]ORCHQLRW PHWRG H XSRWUHEHQ ]D SUHVPHWND QD VSHFLILaQL JRGL^QL
WUR^RFL ]D GLVWULEXWLYQL PUHCL SUL UD]OLaQL SRYU^LQVNL JXVWLQL QD
RSWRYDUXYDZHWR L VR UD]OLaQL WHKQLaNL L HNRQRPVNL SDUDPHWUL 'RELHQLWH
UH]XOWDWLRYR]PRCXYDDWGDVHL]YOHaDWQHNRONXRS^WL]DNOXaRFL
2GWDEHOLWHIV L V VHJOHGDGHNDRGGYHGLVWULEXWLYQLPUHCLVRHGQDNYL
QRPLQDOQL PR`QRVWL QD WUDQVIRUPDWRULWH VUHGHQQL]RN QDSRQ L VR
HGQDNYL VLWH RVWDQDWL SDUDPHWUL RVYHQ QDSUHaQLWH SUHVHFL QD QLVNRQD
SRQVNLWHL]YRGLSRQLVNLVSHFLILaQLJRGL^QLWUR^RFLLPDPUHCDWDNDM
NRMD JXVWLQDWD QD VWUXMDWD YR QLVNRQDSRQVNLWH L]YRGL SRPDONX VH
UD]OLNXYD RG RSWLPDOQDWD 1R L SUL JROHPD UHODWLYQD SURPHQD QD
JXVWLQLWHQDVWUXLWHNDNYD^WRHSULNDCDQDYRWDEHODWDVQHPRUDGDVH
MDYDWJROHPLUHODWLYQLSURPHQLQDVSHFLILaQLWHJRGL^QLWUR^RFLLDNR
HPRCQDSURPHQDYRSULRULWHWRWQDWUDQVIRUPDWRULWH
2G VLWH GLVWULEXWLYQL PUHCL NDM NRL JXVWLQLWH QD VWUXLWH YR QLVNRQD
SRQVNLWH L]YRGL VH PH_XVHEQR HGQDNYL L EOLVNL GR RSWLPDOQDWD SUL
SRYU^LQVNL JXVWLQL QD RSWRYDUXYDZHWR SRJROHPL RG kVA/km2 QDM
PDOL VSHFLILaQL JRGL^QL WUR^RFL LPDDW PUHCLWH YR NRL VH NRULVWDW
WUDQVIRUPDWRULVUHGHQQL]RNQDSRQVRQRPLQDOQDPR`QRVWRGkVA
=D VHNRMD SRYU^LQVND JXVWLQD QD RSWRYDUXYDZHWR L ]D JXVWLQL QD VWUXMD
EOLVNL GR RSWLPDOQLWH SRVWRL QRPLQDOQD PR`QRVW QD WUDQVIRUPDWRU
VUHGHQQL]RN QDSRQ VR NRM VH SRVWLJQXYDDW QDMPDOL VSHFLILaQL JRGL^QL
WUR^RFL ]D GLVWULEXWLYQDWD PUHCD 1R ]DVOXCXYD GD VH QDJODVL GHND YR
PQRJXVOXaDLUHODWLYQRHPDODUD]OLNDWD SRPH_X VSHFLILaQLWH JRGL^QL
WUR^RFL]DPUHCLVRUD]OLaQL WUDQVIRUPDWRUL 9R QHNRL VOXaDL NDNRY
^WRHHGQLRWRGVOXaDLWHSULNDCDQYRWDEHODWDV]DQLHGQDSRYU^LQVND
JXVWLQD QD RSWRYDUXYDZHWR PUHCLWH VR WUDQVIRUPDWRUL VR QRPLQDOQD
PR`QRVWRGkVA QHPDDWSRPDOLVSHFLILaQLWUR^RFLRGPUHCLWH YR
NRLVHNRULVWDWWUDQVIRUPDWRULVRGUXJLQRPLQDOQLPR`QRVWL
2GWDEHODWDIV VHJOHGDGHNDSULUHODWLYQRJROHPDSURPHQDQDYUHGQRVWD
QD C RG 1GHQDULkW⋅JRGLQD QD GHQDULkW⋅JRGLQD QDVWDQX
YDDW ]QDaLWHOQL SURPHQL QD VSHFLILaQLWH JRGL^QL WUR^RFL ]D RGGHOQL
GLVWULEXWLYQL PUHCL 0H_XWRD PDOL VH SURPHQLWH YR SULRULWHWRW QD
WUDQVIRUPDWRULWH
%/$*2'$51267
$YWRULWH PX ]DEODJRGDUXYDDW QD SURI GU 5LVWR $aNRYVNL ]D SRPR^WD SUL
UHDOL]DFLMDQDGHORGLVWUDCXYDZHWR
/,7(5$785$
[1]
[2]
5$aNRYVNL3UHQRVQL L GLVWULEXWLYQL VLVWHPL 8QLYHU]LWHW ì6Y .LULO L
0HWRGLMí, Elektrotehni~ki fakultet, Skopje 1995 godina.
'5DMLaL`9RYHG YR GLVWULEXWLYQL HOHNWURHQHUJHWVNL VLVWHPL 8QLYHU]L
WHWì6Y.LULOL0HWRGLMí, Elektrotehni~ki fakultet, Skopje 1995 godina.
0$.('216.,.20,7(7=$*2/(0,(/(.75,A1,6,67(0,
6,*5(6.23-(
97252629(789$:(1$0$.26,*5(
GU'UDJRVODY5DMLaL`6NRSMH
PU0LUNR7RGRURYVNLGLSOHOLQC,&(,SUL0$18−6NRSMH
35(767$989$:(1$75$16)250$725,
35,35(60(7.$1$.86,7(956.,
92',675,%87,91,7(05(#,
.5$7.$62'5#,1$
(GQD RG QDMHILNDVQLWH SRVWDSNL ]D SUHVPHWND QD VWUXL L QDSRQL SUL NXVL
YUVNLLOLVSRHYLVR]HPMDYRPUHCLVRUDGLMDOQDWRSROR^NDVWUXNWXUDHSRVWDSNDWD
YRNRMD VHNRULVWLPHWRGRWQDVXPLUDZHDGPLWDQFLL0H_XWRDYRSRVWRMQDWD OLWH
UDWXUD QH H RSL^DQR NDNR GD VH SUHWVWDYXYDDW HOHNWURQHUJHWVNL WUDQVIRUPDWRUL
]DSUHWVWDYDWDGDELGHVRRGYHWQD]DQDYHGHQLRWPHWRG9RRYRMWUXGHL]ORCHQHGHQ
SULVWDS]DGRELYDZHWDNYDSUHWVWDYD2GOLNDWDQDRYRMSULVWDSH^WRJRXYDCXYDQH
VDPR YLVWLQVNLRW SUHQRVHQ RGQRV QD WUDQVIRUPDWRURW WXNX L QDaLQRW QD
VRHGLQXYDZH QD QHJRYLWH QDPRWNL .RULVWHM`L JR WRM PRGHO QDSUDYHQL VH SRYH`H
SUHVPHWNL QD WHVW PUHCL 3ULNDCDQLRW GHO RG GRELHQLWH UH]XOWDWL JR LOXVWULUD
YOLMDQLHWRQDSUHQRVQLRWRGQRVYU]JROHPLQDWDQDVWUXMDWDQDNXVDWDYUVND
,VWLRW QDaLQ QD SUHWVWDYXYDZH WUDQVIRUPDWRUL PRCH GD VH NRULVWL L SUL
SUHVPHWNDQDQDSRQLLVWUXLYRUHCLPLWHEH]NXVLYUVNL
SUMMARY
The admittance summation method is one of the most efficient for calculation short
circuits currents and voltages in radial networks. However, in the existing literature one can not
find any explanation how to represent power transformers in that approach. A suitable
transformer model is developed and described in this paper. The model takes into account actual
transformer ratio and type of connections of its windings. A large number of calculations were
made using that approach. As an illustration, some of the results for single fault currents are
presented in this paper. According to the results it is not correct to ignore the influence of the
transformer ratio on short circuit currents.
The same transformer model can be used in calculation voltages and currents by
admittance summation method in radial networks without short circuits.
.OXaQL]ERURYL.XVLYUVNL6SRHYLVR]HPMD5DGLMDOQDPUHCD3RGUHGXYDZHQD
HOHPHQWLQDUDGLMDOQDPUHCD
929('
=DSUHVPHWNDQDVWUXLWHLQDSRQLWHSULNXVLYUVNLLVSRHYLVR]HPMDQDNXVR
JUH^NL SRVWRMDW QHNRONX PHWRGL 3RYH`HWR RG QLY NRULVWDW VLPHWULaQL NRPSR
QHQWL 3RMDYD QD HYWLQL L EU]L GLJLWDOQL NRPSMXWHUL RYR]PRCLOD VÚ SRaHVWR
SUHVPHWNLWH QD UHCLPLWH QD GLVWULEXWLYQLWH PUHCL GD VH L]YHGXYDDW YR ID]HQ
GRPHQ7RDYDCLNDNR]DVWDFLRQDUQLWHUHCLPLEH]JUH^NDWDNDL]DUHCLPLWHVR
JUH^ND7DNYLRWSULVWDSRYR]PRCXYDXYDCXYDZHQDQHVLPHWULMDRGNDNRYLGDHYLG
2VREHQRHSULNODGHQ]DSUHVPHWNDQDVWUXLLQDSRQLSULSRYH`HNUDWQLJUH^NL
9U]RVQRYDQDPHWRGLWHUD]YLHQLYRWUXGRYLWH[1] L[2]YRWUXGRW[3] HL]OR
CHQ HGHQ RG SRQRYLWH SULVWDSL ]D SUHVPHWND QD NXVLWH YUVNL YR GLVWULEXWLYQLWH
PUHCL 9R SULVWDSRW VH NRULVWL PHWRGRW SR]QDW SRG LPHWR PHWRG QD VXPLUDZH
VWUXL ^WR VSD_D PH_X QDMHILNDVQLWH PHWRGL ]D DQDOL]D QD GLVWULEXWLYQL PUHCL
7RDRVREHQRYDCL]DDQDOL]DQDVWDFLRQDUQLUHCLPLEH]JUH^ND0H_XWRDYR[3,4]
VH SUHGODJD SUL SUHVPHWNL QD NXVLWH YUVNL YR GLVWULEXWLYQLWH PUHCL
SRWUR^XYDaLWHGDVHSUHWVWDYXYDDWVRNRQVWDQWQLLPSHGDQFLLWHLPSHGDQFLL^WR
QH ]DYLVDW RG QDSRQRW 7RNPX YR WDNYLWH VOXaDL PHWRGRW QD VXPLUDZH VWUXL LPD
]QDaLWHOQR SROR^L NDUDNWHULVWLNL RG PHWRGRW QD VXPLUDZH DGPLWDQFLL. 7RD ELO
GRYROHQPRWLYGDVHSULVWDSLNRQUD]YLYDZHQRYDSRVWDSND]DSUHVPHWNDQDQDSRQL
LVWUXLSULNXVLYUVNLYRGLVWULEXWLYQLPUHCLYRNRMDELVHNRULVWHOPHWRGRWQD
VXPLUDZHDGPLWDQFLL[6,7](GQDWDNYDSRVWDSNDHRSL^DQDYRWUXGRW[8]1HM]LQLWH
SUHGQRVWL YR RGQRV QD SRVWDSNDWD L]ORCHQD YR WUXGRYLWH [3,4] VH GDYD SRWRaQL
UH]XOWDWL SUHVPHWNLWH VH L]YHGXYDDW ]D SRNXVR YUHPH QHPD SRWUHED VHNRMD NXVD
YUVND LOL ]HPHQ VSRM GD VH WUHWLUD NDNR GRSROQLWHOQD NRQWXUD QD SRHGQRVWDYHQ
QDaLQ VH ]HPDDW SUHGYLG QDSUHaQLWH HOHPHQWL QD PUHCDWD NDNR ^WR VH
SRWUR^XYDaLLNRQGHQ]DWRUVNLEDWHULL
'D SRGYOHaHPH GHND L YR [3] L YR [8] SUHVPHWNLWH VH L]YHGXYDDW YR ID]HQ
GRPHQ'YDWDVHSULPHQOLYLQDSURL]YROHQEURMHGQRYUHPHQLNXVLYUVNLRGNDNRYL
GD H YLG QD NRL L GD ELOR ORNDFLL 3RNUDM WRD QD VHNRH PHVWR QD NXVD YUVND
LPSHGDQFLMDWDSUHNXNRMDQDVWDSLODNXVDWDYUVNDPRCHGDLPDSURL]YROQDYUHGQRVW
9R SUHWKRGQLWH WUXGRYL QH H REMDVQHWR NDNR GD VH SUHWVWDYXYDDW HOHNWUR
HQHUJHWVNL WUDQVIRUPDWRUL NRJD ]D SUHVPHWNLWH VH NRULVWL PHWRGRW QD VXPLUDZH
DGPLWDQFLL 7RD H SUHGPHW QD RYRM WUXG 1R SUHG GD ELGH REMDVQHWR NDNR GD VH
SUHWVWDYXYDDWWUDQVIRUPDWRULWHQDMQDSUHG`HELGHGDGHQNXVRVYUWYU]SRVWDSNDWD
]DSUHVPHWNDQDNXVLWHYUVNLL]ORCHQDYRWUXGRW[8]
2695795=3267$3.$7$,=/2#(1$92758'27[8]
3RVWDSNDWD H UD]YLHQD SUHWSRVWDYXYDM`L GHND HOHPHQWLWH L MD]OLWH QD UDGL
MDOQDWDPUHCDVHSRGUHGHQLVSRUHGSUDYLODWDRG[5]
6RFHOGDVHRYR]PRCLDQDOL]DQDQHXUDPQRWHCHQLLQHVLPHWULaQLPUHCLVH
NRULVWLWULID]QDSUHWVWDYDNDNRQDHOHPHQWLWHQDPUHCDWDWDNDLQDSRWUR^XYD
aLWH0DWULFDWDQDNRPSOHNVQLWHLPSHGDQFLLQDWULID]QLRWHOHPHQWk H
ZS ak ,b,c
ZS aa
k
 ba
= ZS k
 ZS ca
 k
ZS ab
k
bb
ZS k
ZS cb
k

ZS ac
k
bc 
ZS k  
ZS cc
k 
7ULID]HQ SRWUR^XYDa NDM MD]RORW k VH SUHWVWDYXYD VR PDWULFDWD QD
NRPSOHNVQLWHDGPLWDQFLL
L ak ,b,c
 L aa
k
 ba
= L k
 L ca
 k
L ab
k
bb
Lk
L cb
k

L ac
k
bc 
Lk  
L cc
k 
.DNRYLGDHYLGJUH^NDNDMMD]RORWk PRCHGDVHSUHWVWDYLVRHGQDPDWULFDQD
NRPSOHNVQL DGPLWDQFLL RG WUHW UHG YF ak , b, c (OHPHQWLWH QD WDD PDWULFD PRCDW GD
LPDDWSURL]YROQDYUHGQRVW=DUD]OLaQLYLGRYLNXVLYUVNLVRRGYHWQLWHPDWULFLQD
DGPLWDQFLLQDJUDQNDWDQDJUH^NDVHGDGHQLYR[8]7DEHODIHSUH]HPHQDRGWRMWUXG
7DEHODI0DWULFLQDDGPLWDQFLL]DUD]OLaQLYLGRYLJUH^NL[8]
9LGQDJUH^ND
a
b
YF
YF
YF
YF ak ,b,c
Yg
7ULID]QDVR]HPMD
a
b
YF
0DWULFDQDDGPLWDQFLL
c
2Y + Y
g
 F
YF
=
−
Y

F
3Y F + Y g 
 −YF
YF
b
YF
YF ak ,b,c
YF
YF ak ,b,c
Yg
'YRID]QDVR]HPMD
b


YF
=

2Y F + Y g 

0
0
0
0
YF +Yg
−YF
0 

−YF 
Y F + Y g 
c
YF
YF
'YRID]QD
a
 2 − 1 − 1
YF 

=
− 1 2 − 1

3
− 1 − 1 2 
c
YF
a
−YF 

−YF 
2Y F + Y g 
c
7ULID]QD
a
−YF
2Y F + Y g
−YF
b
YF ak ,b,c
0
0
0
YF 

=
0
1 − 1

2
0 − 1
1
c
YF
(GQRID]QD
YF ak ,b,c
Y F

= 0
 0
0 0

0 0
0 0
=DGDVHRYR]PRCLHILNDVQRL]YHGXYDZHQDSUHVPHWNLWHNRQVHNRMMD]ROk VH
SULGUXCXYDDW SR GYH PDWULFL YL ak , b, c L YN ak , b, c 0DWULFDWD YL ak , b, c VH GRELYD NDNR
VXPD QD VLWH PDWULFL QD QDSUHaQLWH WULID]QL HOHPHQWL ^WR VH SULGUXCHQL NRQ
MD]RORW k 9R WDD VXPD VH YNOXaXYDDW NDNR PDWULFDWD QD LPSHGDQFLLWH VR NRMD H
SUHWVWDYHQ SRWUR^XYDaRW NDM MD]RORW k WDND L PDWULFDWD QD JUDQNDWD QD JUH^ND
GRNRONXNDMWRMMD]ROVHMDYLODJUH^ND
0DWULFDWD YN ak , b, c HYVX^QRVWPDWULFDQDNRPSOHNVQLHNYLYDOHQWQLDGPLWDQ
FLL QD GHORW QD UDGLMDOQDWD WULID]QD PUHCD ^WR VH QDSRMXYD SUHNX MD]RORW k
]HPDM`LJLWXNDSUHGYLGVLWHQDSUHaQLDGPLWDQFLLNDMWRMMD]RO
=D SROHVQR UD]ELUDZH QD QDWDPR^QDWD SRVWDSND H NRULVQR GD JL QDYHGHPH
RVQRYQLWH RGOLNL QD VSRPQDWRWR SRGUHGXYDZH 9R SURFHVRW QD SRGUHGXYDZHWR QD
VHNRM QDSUHaHQ HOHPHQW L QD VHNRM MD]RO PX VH GRGHOXYD LQGHNV 3ULWRD ]D VHNRM
HOHPHQW VH GHILQLUD SRaHWHQ MD]RO L NUDHQ MD]RO 3RaHWHQ H RQRM RG MD]OLWH QD
HOHPHQWRWaLMSDWGRQDSRMQLRWMD]RORGQRVQRMD]RORWVRSR]QDWQDSRQSRPRGXO L
ID]HQ DJRO QH JR VRGUCL VDPLRW HOHPHQW 'UXJLRW MD]RO H NUDHQ ,QGHNVRW QD
SRaHWQLRW MD]RO H VHNRJD^ SRPDO RG LQGHNVRW QD NUDMQLRW MD]RO ,QGHNVRW QD
NUDMQLRW MD]RO H HGQDNRY VR LQGHNVRW QD HOHPHQWRW $NR UDGLMDOQDWD PUHCD LPD n
QDSUHaQL HOHPHQWL QLP LP VH GRGHOXYDDW LQGHNVL RG HGHQ GR n 1D MD]OLWH LP VH
GRGHOXYDDWLQGHNVLRGQXODGRn
3UHVPHWNDWDQDVWUXLWHLQDSRQLWHVHVRVWRLRGWULaHNRUL9RSUYLRWaHNRU
VHSUHVPHWXYDDWPDWULFLWH YL ak , b, c k = 1,, n 3RWRDVH]HPD
,c
a ,b ,c
;
YN ak ,(bSRaHWQD
YUHGQRVW ) = YL k
k = 1, , n.
9R YWRULRW aHNRU VH SUHVPHWXYDDW PDWULFLWH YN ak , b, c ( k = n,,1) 7RD VH
L]YHGXYD VR REUDERWND HOHPHQW SR HOHPHQW WUJQXYDM`L RG HOHPHQWRW VR QDMJROHP
LQGHNV3ULREUDERWNDQDHOHPHQWRWkMDRSUHGHOXYDPHSRPR^QDWDPDWULFD
(
D ak ,b,c = E + ZS ak ,b,c ⋅ YN ak ,b,c
)−1 NDGH^WRVREHR]QDaHQDHGLQLaQDPDWULFDRGWUHWUHG3RWRDSUHVPHWXYDPH
b ,c
a ,b ,c
a ,b,c
⋅ D ak ,b,c ,
YN ai(,QRYD
YUHGQRVW ) = YN i ( VWDUD YUHGQRVW ) + YN k
NDGH ^WR i H LQGHNV QD SRaHWQLRW MD]RO QD HOHPHQWRW k 'ROQDWD R]QDND ìVWDUD
YUHGQRVWí VH NRULVWL NDNR VNXVHQD LQIRUPDFLMD QDPHVWR ]ERURYLWH ìpred
REUDERWNDWD QD HOHPHQWRW kí $QDORJQR R]QDNDWD ìQRYD YUHGQRVWí VH NRULVWL
QDPHVWR]ERURYLWHìpo REUDERWNDWDQDHOHPHQWRWkí
9RWUHWLRWaHNRUVHSUHVPHWXYDDWQDSRQLWHQDMD]OLWH7RDVHL]YHGXYDSUHNX
SURFHV YR NRM SDN VH REUDERWXYD HOHPHQW SR HOHPHQW QR RYRMSDW VH WUJQXYD RG
HOHPHQWRW VR LQGHNV HGQDNRY QD HGLQLFD 3UL REUDERWND QD HOHPHQWRW k VH
RSUHGHOXYDQDSRQRWQDQHJRYLRWNUDHQMD]ROaLM^WRLQGHNVHSDNk
U ak ,b,c = D ak ,b,c ⋅ U ai ,b,c NDGH ^WR i H SRaHWHQ MD]RO QD HOHPHQWRW k D D ak , b, c H VRRGYHWQD SRPR^QD PDWULFD
RSUHGHOHQDYRYWRULRWaHNRU
6R SR]QDWL QDSRQL QD MD]OLWH QH H WH^NR GD VH RSUHGHODW VWUXLWH QD
HOHPHQWLWH6WUXMDWDYRHOHPHQWRWk H
I ak ,b,c = YN ak ,b,c ⋅ U ak ,b,c $NRJUH^NDWDQDVWDQDODNDMMD]RORWmVWUXMDWDQDJUH^NDWDVHSUHVPHWXYDVR
UDYHQNDWD
I am,b,c = YF am,b,c ⋅ U am,b,c .
35(767$989$:(1$75$16)250$725,
.DNR ^WR YH`H EH^H QDYHGHQR YR SUHWKRGQRWR SRJODYMH YR RSL^DQDWD
SRVWDSND ]D SUHVPHWND QD VWUXL L QDSRQL SUL NXVLWH YUVNL VH UDERWL YR ID]HQ
GRPHQ3RUDGLWRDHQHRSKRGQRGDVHUDVSRODJDVRPRGHOQDWUDQVIRUPDWRUYRID]HQ
GRPHQ , QH VDPR WRD 0RGHORW WUHED GD H SULNODGHQ ]D SULPHQD QD PHWRGRW QD
VXPLUDZH DGPLWDQFLL 3UL UD]YLYDZH QD WDNRY PRGHO `H WUJQHPH RG PRGHORW
L]ORCHQYR[9]NRMVRRGYHWQR`HJRPRGLILFLUDPH'REUDWDVWUDQDQDWRMSULVWDSH
^WRNDNRRVQRYDJRLPDPRGHORWQDWUDQVIRUPDWRUYRVLPHWULaQLNRPSRQHQWLNRM
HHGQRVWDYHQLGREURSR]QDW
,]YHGXYDZHWR QD PRGHORW QD WUDQVIRUPDWRU YR ID]HQ GRPHQ `H JR
GHPRQVWULUDPH ]HPDM`L NDNR SULPHU WUDQVIRUPDWRU VR VSUHJD Yd5 6R DQDORJQD
SRVWDSNDPRCHGDVHGRMGHGRPRGHOQDWUDQVIRUPDWRUVRNDNYDLGDHVSUHJD
(NYLYDOHQWQLWH^HPLQDHOHNWURHQHUJHWVNLWUDQVIRUPDWRUVRVSUHJDYd5 ]D
GLUHNWHQLQYHU]HQLQXOWLVLVWHPVHSULNDCDQLQDVOLNLWHDELYVRRGYHWQR
I (p1)
U (p1)
Yk
E (p1)
I (p2)
I (s1)
m(1):1
E (s1)
U (s1)
D=DGLUHNWHQUHGRVOHG
U (p2)
Yk
E (p2)
m(2 ):1
I (s2)
E (s2)
U (s2)
E=DLQYHU]HQUHGRVOHG
Y=DQXOWLUHGRVOHG
6OLND(NYLYDOHQWQL^HPLQDHOHNWURHQHUJHWVNLWUDQVIRUPDWRU
1DVOLNDWDVR Y k HR]QDaHQDDGPLWDQFLMDWDQDNXVDWDYUVNDQDWUDQVIRUPD
WRURW D VR E( ) LQGXFLUDQ QDSRQ *ROHPLQLWH SULGUXCHQL ]D SULPDUQDWD QDPRWND
LPDDW GROHQ LQGHNV p D JROHPLQLWH SULGUXCHQL ]D VHNXQGDUQDWD QDPRWND LPDDW
GROHQ LQGHNV s =D RYRM VOXaDM NRPSOHNVQLWH SUHQRVQL RGQRVL VH GHILQLUDQL VR
VOHGQLYHUDYHQVWYD
m(1) = 3 ⋅
Np
Ns
( )
m(2) = m(1) *
⋅e
j5
π
6
NDGH^WR VR Np L Ns VH R]QDaHQL EURHYLWH QD QDYLYNLWH QD SULPDUQDWD L
VHNXQGDUQDWD QDPRWND VRRGYHWQR 'D QDJODVLPH GHND ]D WUDQVIRUPDWRUL VR
UD]OLaQL VSUHJL NRPSOHNVQLRW SUHQRVHQ RGQRV m(1) QHPD HGQDNYL YUHGQRVWL $NR
VSUHJDWDHYd5 DUJXPHQWRWQD m(1) H 5 ⋅
π
NDNR^WRHYR9RVOXaDMQDVSUHJDYd11
6
π
3UDYLORWRVSRUHGNRHVHRSUHGHOXYDDUJXPHQWRWQD m(1) H
6
VRVHPDHGQRVWDYQRVSUHCQLRWEURMVHSRPQRCXYDVRπ,VWRWRSUDYLORYDCLL]D
GUXJLWH YLGRYL VSUHJL =D PRGXORW QD m(1) YDCL VOHGQRYR ]D VSUHJD Yy L Dd H
DUJXPHQWRW`HELGH11 ⋅
HGQDNRYQD N p N s ]DVSUHJDYd HHGQDNRY QD 3 ⋅ N p N s D]DVSUHJDDy HHGQDNRYQD
3 ⋅ Ns =D JROHPLQLWH RG HNYLYDOHQWQLWH ^HPL SULNDCDQL QD VOLNDWD YDCDW
VOHGQLYHUDYHQNL
Á ]DGLUHNWHQUHGRVOHG
Np
(1)
(1)
U p = Ep +
I (p1)
Yk
U (s1) = E (s1) ;
;
E (p1) = m(1) ⋅ E (s1) I (s1) = m(2) ⋅ I (p1) Á ]DLQYHU]HQUHGRVOHG
(2 )
(2 )
U p = Ep +
I (p2)
Yk
;
U (s2) = E (s2) ;
I (s2 ) = m(1) ⋅ I (p2) E (p2) = m(2) E (s2) ;
Á ]DQXOWLUHGRVOHG
I (p0 ) = Y k ⋅ U (p0) I (s0 ) = 0 2GUDYHQNLWHVHGRELYD
I (p1) = Y k ⋅ U (p1) − m(1) ⋅ Y k ⋅ U (s1) DRGLVOHGXYD
I (s1) = m(2) ⋅ Y k ⋅ U (p1) − m2 ⋅ Y k ⋅ U (s1) 1DWDPXRGUDYHQNLWHVHGRELYD
I (p2 ) = Y k ⋅ U (p2) − m(2) ⋅ Y k ⋅ U (s2) DRGLVOHGXYD
I (s2 ) = m(1) ⋅ Y k ⋅ U (p2) − m2 ⋅ Y k ⋅ U (s1) 5DYHQNLWH RG GR IRUPLUDDW HGHQ VLVWHP UDYHQNL ^WR PRCH GD VH
QDSL^HYRPDWULaQDIRUPDQDVOHGQLRYQDaLQ
 I (p0)  Y k
 (1)  
I p   0
 I (2)   0
 p =
 I (s0)   0
 (1)   0
Is  
 I (2)   0
 s 
0
0
0
Yk
0
0
Yk
0
0
m(2) ⋅ Y k
0
m(1) ⋅ Y k
0
0
(1)
0 − m ⋅Y k
0
0
0
0
2
0 − m ⋅Y k
0
0
(0)
 U p 


(1) 
U

p 

(2)

(
− m ⋅ Y k  U p2) 
 ⋅  (0)  0
 U s 
 U (1) 
0
  s 
− m2 ⋅ Y k  U (2) 
 s 
0
0
$NR ]D QD]QDaHQLWH EORNRYL QD PDWULFLWH RG PDWULaQDWD UDYHQND VH
YRYHGDWVRRGYHWQLR]QDNLWDDPRCHGDVHQDSL^HYRYLG
 I 0p,1, 2 
 0,1, 2  =
 I s 
Y 2 0,1, 2  U 0p,1, 2 

⋅
Y 4 0,1, 2  U 0s ,1, 2 
I 0p,1, 2 = Y10,1,2 ⋅ U 0p,1,2 + Y 2 0,1,2 ⋅ U 0s ,1,2
I 0s ,1, 2 = Y 30,1,2 ⋅ U 0p,1, 2 + Y 4 0,1,2 ⋅ U 0s ,1,2  Y10,1, 2
 0,1, 2
Y 3
LOL
1D^LRWLQWHUHVHGDQDMGHPHQDaLQNDNRSULSR]QDWDDGPLWDQFLMDSULNOXaHQD
QDVHNXQGDURWQDWUDQVIRUPDWRURWGDMDRSUHGHOLPHHNYLYDOHQWQDWDDGPLWDQFLMDQD
SULPDUQDWD VWUDQD 1HND HNYLYDOHQWQDWD DGPLWDQFLMD SULNOXaHQD QD VHNXQGDURW H
YN 0s ,1,2 7RJD^PRCHGDVHQDSL^H
I 0s ,1, 2 = YN 0s ,1,2 ⋅ U 0s ,1,2 LRGLVOHGXYD
(
U 0s ,1, 2 = YN 0s ,1, 2 − Y 40,1,2
)−1 ⋅ Y 30,1,2 ⋅ U 0p,1,2 3RWRDRGLVHGRELYD
(
(
I 0p,1, 2 = Y10,1,2 + Y 20,1,2 ⋅ YN 0s ,1, 2 − Y 40,1,2
)−1 ⋅ Y 30,1,2 ) ⋅ U 0p,1,2 RGNDGH^WR]DEDUDQDWDHNYLYDOHQWQDDGPLWDQFLMDQDSULPDUQDWDVWUDQDVOHGXYD
(
YN 0p,1,2 = Y10,1, 2 + Y 20,1,2 ⋅ YN 0s ,1,2 − Y 40,1,2
)−1 ⋅ Y 30,1,2 .DNR^WRHSR]QDWR[10]HGQDPDWULFDRGGRPHQRWQDVLPHWULaQLNRPSRQHQWL
PRCH GD VH WUDQVIRUPLUD YR ID]HQ GRPHQ DNR PDWULaQR VH SRPQRCL RGOHYR VR
PDWULFDWD T sim. LRGGHVQRVR (T sim. ) NDGH^WRH
−1
1 1 1 
1  2

T sim. = ⋅ 1 a a  a = e j 2π / 3 3
1 a a2 
7DNDVHGRELYD
(
)−1 (
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[1] D. Shirmohammadi, H. W. Hong, A. Semlyen, G. X. Luo: "A Compensation-Based Power
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FUZZY RADIAL DISTRIBUTION LOAD FLOW
ABSTRACT
This paper presents a new approach for load flow analysis in radial distribution networks
based on the fuzzy set theory. The uncertainty of the input parameters is modeled by using fuzzy
numbers, which in turn are special form of fuzzy sets. The mathematical operations in this case
are defined by the Extension Principle and the results obtained are also in a form of fuzzy
numbers. For practical purposes, trapezoidal or triangular fuzzy numbers are used and the
computation is much simpler. This approach gives us a complete solution of the possible system
operating states that result from including the uncertainty in the analysis.
Keywords: radial distribution networks, fuzzy load flow, fuzzy sets, fuzzy numbers,
uncertainty.
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YUHGQRVWL [a1, a4] L LQWHUYDO QD QDMRaHNXYDQL YUHGQRVWL [a2, a3]
9RVSHFLMDOQLRWVOXaDMNRJDa2 = a3RGWUDSH]RLGQLRW)%VHGRELYDWULDJROHQ
)%6OLaQRVHNRMLQWHUYDOHQEURMHVDPRVSHFLMDOHQVOXaDMQDWUDSH]RLGHQ)%SUL
a1 = a2 L a3 = a4.RQHaQRNRJDa1 = a2 = a3 = a4VHGRELYDRELaHQUHDOHQEURM1DWRM
QDaLQUHDOQLWHLLQWHUYDOQLWHEURHYLVHVDPRVSHFLMDOQL)%
)XQNFLLRGID]LEURHYL
7UDGLFLRQDOQLRW PDWHPDWLaNL NRQFHSW ]D IXQNFLMD H SUR^LUHQ L YR ID]L
GRPHQRW VR SRPR^ QD WQDU SULQFLS QD HNVWHQ]LMD extension principle [7, 8] 6R
SULPHQDWDQDRYRMSULQFLSPRCHGDVHSRNDCHGHNDDULWPHWLaNLWHRSHUDFLLVR)%
VHHGQDNYLQDVRRGYHWQLWHLQWHUYDOQLDULWPHWLaNLRSHUDFLL]DVHNRMαSUHVHN
~
~
α
α α
=ELURW QD GYD )% A L B VR αSUHVHFL A α = [a1α , a α
2 ] L B = [b1 , b 2 ] ~
UHVSHNWLYQRGDYD)% C aLM^WRαSUHVHNCαHGDGHQVR
α α
α α
α
α α
α
Cα = A α + Bα = [c1α , c α
2 ] = [a1 , a 2 ] + [b1 , b 2 ] = [a1 + b1 , a 2 + b 2 ] .
(1)
~
~
6OLaQR ]D RVWDQDWLWH DULWPHWLaNL RSHUDFLL SRPH_X GYD )% A L B VH
~
GRELYDDWVOHGQLWHL]UD]L]DVHNRMαSUHVHNQDUH]XOWDQWQLRW)% C :
α α
α α
α
α α
α
Cα = A α − Bα = [c1α , c α
2 ] = [a1 , a 2 ] − [b1 , b 2 ] = [a1 − b 2 , a 2 − b1 ] ,
(2)
α α
α α
α α α α
Cα = A α ⋅ Bα = [c1α , c α
2 ] = [a1 , a 2 ] ⋅ [b1 , b 2 ] = [a1 ⋅ b1 , a 2 ⋅ b 2 ] ,
(3)
α α
α α
α α α α
Cα = A α / Bα = [c1α , c α
2 ] = [a1 , a 2 ] /[b1 , b 2 ] = [a1 / b 2 , a 2 / b1 ] .
(4)
2SHUDFLMDWD YR QH H GHILQLUDQD DNR LQWHUYDORW Bα MD VRGUCL QXODWD SD
WDNDWRMPRCHGDVRGUCLVDPRSR]LWLYQLLOLVDPRQHJDWLYQLEURHYL
~
~ ~ ~
~ ~
~
7UHEDGDVHQDJODVLGHNDDNR]DWUL)% A B L C YDCL C A B )% A QH
~ ~ ~
PRCH GD VH GRELH QD]DG NDNR A C B =D WDD FHO WUHED GD VH SULPHQL WQDU
~ ~
~
RSHUDFLMDGHNRQYROXFLMDREWSFMVE[IXSRGQRVQR A C ( B [3]2YDHUH]XOWDWQD
~
~
IDNWRW^WR)% B L C QHVHPH_XVHEQRQH]DYLVQLWXNXVHYRHGHQYLGNRUHODFLMD
~ ~ ~
~
6OLaQRYDCLLYRVOXaDMRWQDSURL]YRGRW C A ⋅ B )% A PRCH GD VH GRELH
~ ~
~
QD]DGSUHNXGHNRQYROXFLMDQDPQRCHZHWRRGQRVQR A C B =ELURW L UD]OLNDWD NDNR OLQHDUQL RSHUDFLL JR ]DGUCXYDDW REOLNRW QD
IXQNFLLWH QD SULSDGQRVW L UH]XOWLUDDW VR WUDSH]RLGHQ )% DNR VH L]YHGHQL YU]
WUDSH]RLGQL)%*HQHUDOQRRYDQHHVOXaDMVRSURL]YRGRWLNROLaQLNRWNDNRLVR
VLWH RVWDQDWL QHOLQHDUQL WUDQVIRUPDFLL 6HSDN RG SUDNWLaQL SULaLQL
UH]XOWDWLWHYRRELaDHQRVHDSURNVLPLUDDWVRWUDSH]RLGQLLOLWULDJROQL)%1D
WRMQDaLQSRWUHEQRHGDVHSUHVPHWDDWVDPRUH]XOWDQWQLWHαSUHVHFL]Dα Lα 6SRPHQDWLRW SULQFLS QD HNVWHQ]LMD H JHQHUDOHQ L PRCH GD VH SULPHQL YU]
VHNDNYD IXQNFLMD RG HGHQ LOL SRYH`H DUJXPHQWL ]D GD VH SUHVPHWD IXQNFLMDWD QD
SULSDGQRVW QD UH]XOWDQWQDWD ]DYLVQD SURPHQOLYD 2YD PRCH GD VH QDSUDYL VR
SUHVPHWNDQDSRHGLQLWHαSUHVHFLQDVOLaHQQDaLQNDNRLSRJRUH1DSULPHUVR
~
~
HGDGHQαSUHVHNRWCαQDUH]XOWDQWQLRW)% C GRELHQVRVWHSHQXYDZHQD)% A QD
VWHSHQn, nSULURGHQEURM
[ {
} {
α n
α n
n
C α = min (a1α ) n , (a α
2 ) , max (a 1 ) , (a 2 )
}]
(5)
9R SUHWKRGQLRW L]UD] H SUHWSRVWDYHQR GHND LQWHUYDORW Aα QH MD VRGUCL
QXODWD9RVSURWLYQRGROQDWDJUDQLFDQDLQWHUYDORWCα`HELGHQXODNRJDnHSDUHQ
EURM
~
6OLaQRVRHGDGHQαSUHVHNRWCαQDUH]XOWDQWQLRW)% C GRELHQNDNRnWL
~
NRUHQQD)% A , nSULURGHQEURM

C α = n a 1α , n a α
, n a 3α , n a α
2
4


(6)
*HQHUDOQR ]D VHNRMD IXQNFLMD f NRMD ^WR H NRQWLQXLUDQD L QHRSD_DaND YR
LQWHUYDORW[a1, a4]YDCL
~
f (A) = [ f (a1 ), f (a 2 ), f (a 3 ), f (a 4 )] .
(7)
6OLaQR ]D VHNRMD IXQNFLMD g NRMD ^WR H NRQWLQXLUDQD L QHUDVWHaND YR
LQWHUYDORW[a1, a4]YDCL
~
g (A) = [g (a 4 ), g (a 3 ), g (a 2 ), g (a 1 )].
(8)
9R VSHFLMDOQLRW VOXaDM NRJD )% GHJHQHULUDDW YR RELaQL UHDOQL EURHYL
QDUHaHQL X^WH L ID]L VLQJOWRQL fuzzy singletons VLWH RSHUDFLL L IXQNFLL JR
LPDDW YRRELaDHQRWR ]QDaHZH 9R WDD VPLVOD ID]L DQDOL]DWD L ID]L PDWHPDWLNDWD
JL RSID`DDW UHDOQDWD L LQWHUYDOQDWD DQDOL]D L PDWHPDWLND NDNR VSHFLMDOQL
VOXaDL
)D]LSUHVPHWNDQDR-XHOHPHQW
3ULPHQDWD QD )% ]D SUHVPHWND QD VRVWRMEDWD YR 5'0 `H MD LOXVWULUDPH
QDMSUYR QD SULPHURW QD HOHNWULaQDWD SUHVPHWND QD HGHQ SN YRG SULNDCDQ QD
VOLNDWDNRMPRCHGDVHWUHWLUDLNDNRQDMSURVWREOLNQD5'0
1D VOLNDWD VH SULNDCDQL L QRPLQDOQLWH SRGDWRFL SRWUHEQL ]D SUHVPHWNDWD
QD]DJXEDWDQDQDSRQL]DJXEDWDQDPR`QRVWYR10 kVYRG7DND]DGDGHQLWHSRGDWRFL
VHVDPRSULEOLCQLLQLHELPRCHOHGDMDPRGHOLUDPHWDDQHVLJXUQRVWVRSRPR^QD
)%QDVOHGQLRWQDaLQ
1
2
10 kV 6 km
0.83+j0.33 Ω/km
600 kVA
cosϕ=0.9
6OLND (GQRVWDYQD 5'0 NDNR SULPHU ]D ID]L SUHVPHWND
$NR QD SULPHU UHJXODFLRQLRW WUDQVIRUPDWRU LPD UHJXODFLRQD SUHNORSND
VRVWHSHQLQDUHJXODFLMD ±×NRMDJRNRQWUROLUDQDSRQRWQDYUHGQRVW 10.5 kV
WRJD^ QHJRYDWD PDNVLPDOQD YDULMDFLMD `H L]QHVXYD 10.5±0.15 kV 2aHNXYDQDWD
YDULMDFLMD EL PRCHOD GD ELGH SRORYLQD RG WDD YUHGQRVW 1D WRM QDaLQ QDSRQRW YR
MD]HORW`HJRSUHWVWDYLPHVR)%
~
U1 = [10.35, 10.425, 10.575, 10.65] kV.
3RQDWDPXDNRGR]YROLPHGHNDYRGROCLQDWDQDYRGRWPRCHGDLPDJUH^NDRG
WDDELPRCHODGDVHSUHWVWDYLNDNR
~
l = [5.88, 5.94, 6.06, 6.12] km.
1D WRM QDaLQ SRG SUHWSRVWDYND GHND LPSHGDQFLMDWD QD YRGRW SR HGLQLFD
GROCLQDz=0.83+j0.33Ω/kmHWRaQDYNXSQDWDLPSHGDQFLMDQDYRGRW`HL]QHVXYD
~ ~ ~
~
Z = z ⋅ l = R + jX [4.880, 4.930, 5.030, 5.080] + j[1.940, 1.960, 2.000, 2.020]) Ω.
.RQHaQRLPRCQDWDYDULMDFLMDQDPR`QRVWDQDSRWUR^XYDaRWYRMD]HORW`H
MDSUHWVWDYLPHVRVOHGQLWHGYD)%]DSULYLGQDWDPR`QRVWLQHM]LQLRWIDNWRU
~
~ = [0.85, 0.88, 0.92, 0.95].
S = [500, 540, 560, 600] kVAL cos ϕ
2
2
~
3R DQDORJLMD VR GHWHUPLQLVWLaNDWD SUHVPHWND, QDSRQRW YR MD]HORW U 2 VH
GRELYDNDNRUH^HQLHQDVOHGQDWDELNYDGUDWQDUDYHQND[4]
~ ~
~
~ ~
~
~
~ +X
~ ) +~
U 42 − U 22 U12 − 2S2 (R ⋅ cos ϕ
⋅ sin ϕ
S22 Z2 = 0 ,
(9)
2
2
[
]
RGQRVQR
~
U 2 = [10.02, 10.13, 10.30, 10.40] kV.
~ YR VH RSUHGHOXYD RG )% cos ϕ
~ VR SRVOHGRYDWHOQD SULPHQD QD
)% sin ϕ
2
2
IXQNFLLWHarccosLsinNRULVWHM`LJLSULWRDL]UD]LWHL
~
=DJXEDWDQDQDSRQYRYRGRW ∆U `HELGHGDGHQDVRVOHGQDWDGHNRQYROXFLMD
~ ~
~
∆U = U1 (−) U 2 =[0.25, 0.28, 0.30, 0.33] kV.
.RPSOHNVQDWD ]DJXED QD PR`QRVW YR YRGRW NDNR L YR GHWHUPLQLVWLaNLRW
VOXaDMHGDGHQDVR
~
~ S22 ~ 2 ~ 2
(10)
∆ S = ~ ⋅ R + jX ,
U 22
(
)
RGQRVQR
~
∆S = [11.28, 13.55, 15.38, 18.20] + j[4.48, 5.39, 6.12, 7.24] kVA.
)$=,0(72'=$35(60(7.$1$5'0
,]ORCHQDWD SRVWDSND YR SUHWKRGQDWD JODYD ]D ID]L SUHVPHWND QD HGHQ R-X
HOHPHQW OHCL YR RVQRYDWD QD LWHUDWLYQLRW PHWRG ]D SUHVPHWND QD VRVWRMEDWD QD
SURL]YROQD 5'0 0HWRGRW H YVX^QRVW ID]L SUHVOLNXYDZH QD HGQD YDULMDQWD QD
WQDU PHWRG VXPLUDZH QD PR`QRVWL [4] 3RVWDSNDWD SUHWSRVWDYXYD GHND YR
PUHCDWDSUHWKRGQRHL]YU^HQRSRGUHGXYDZHQDHOHPHQWLWHLVHVRVWRLRGVOHGQLWH
aHNRUL
•1DMQDSUHG SUHG GD ]DSRaQH LWHUDWLYQLRW SURFHV QDSRQLWH YR VLWH MD]OL
GRELYDDWSRaHWQLYUHGQRVWLHGQDNYLQDQDSRQRWYRQDSRMQLRWMD]RO9U]RVQRYDQD
WDND SUHWSRVWDYHQLWH QDSRQL LOL SUHVPHWDQLWH YUHGQRVWL RG SUHWKRGQDWD
LWHUDFLMD VH SUHVPHWXYDDW YNXSQLWH ID]L LQMHNWLUDQL PR`QRVWL YR MD]OLWH
PR`QRVWLWH QD SRWUR^XYDaLWH L SRSUHaQLWH DGPLWDQFLL 7LH LVWRYUHPHQR JL
RGUHGXYDDWLSRaHWQLWHYUHGQRVWLQDPR`QRVWLWH^WRWHaDWQDNUDMRWR]QDNDŽRG
UHGQDWD JUDQND YR π]DPHQVNDWD ^HPD 7DD PR`QRVW ]D HOHPHQWRW i-j YR
LWHUDFLMDWDk`HL]QHVXYD
~ ( k ) ~ (k ) ~ ( k )
~ (k ) ~ * ~
S"i − j = Pi′′− j + jQ′i′− j = − S j + Y jΣ ⋅ ( U i( k −1) ) 2 .
(11)
~
NDGH VR Y jΣ H R]QDaHQ ]ELURW QD VLWH SRSUHaQL ID]L DGPLWDQFLL SULNOXaHQL YR
MD]RORW j RQLH RG π]DPHQVNLWH ^HPL QD HOHPHQWLWH NDNR L DGPLWDQFLLWH QD
~ (k )
HYHQWXDOQLWH NRQGHQ]DWRUVNL EDWHULL UHDNWRUL L VO D VR S j LQMHNWLUDQDWD
ID]L NRPSOHNVQD PR`QRVW YR MD]RORW j YR LWHUDFLMDWD k <YH]GLaNDWD YR
VXSHUVNULSWR]QDaXYDNRQMXJDFLMDQDNRPSOHNVQDWDYHOLaLQD
•3RWRD WUJQXYDM`L RG HOHPHQWRW VR QDMJROHP UHGHQ EURM YR PUHCDWD
VXNFHVLYQR VH SUHVPHWXYDDW PR`QRVWLWH QD NUDMRW YR VRRGYHWQLWH QDSRMQL JUDQNL
backward sweepVRGRGDYDZHQDPR`QRVWDL]DJXELWHQDPR`QRVWYRREUDERWXYDQDWD
JUDQND7DND]DJUDQNDWDi-jaLMD^WRQDSRMQDJUDQNDHQDSULPHUJUDQNDWDl-iL
NRMD^WRJRLPDUHGQLRWEURMQDMD]RORWiVHSUHVPHWXYD
2
(k ) 
~
~ ( k ) ~ (k ) ~ (k ) ~
 Si′′− j 
S"l − i := S"l − i + S"i − j + Zi − j ⋅  ~
 .
 U (jk ) 


(12)
•2WNRJD `H VH SUHVPHWDDW PR`QRVWLWH YR JUDQNLWH WUJQXYDM`L RG SUYDWD
JUDQNDYROLVWDWDQDSRGUHGHQLJUDQNLVHSULVWDSXYDNRQSUHVPHWNDQDQDSRQLWHYR
MD]OLWH forward sweep 1DSRQRW YR NUDMQLRW MD]RO QD JUDQNDWD i-j SUL RYDD
SUHVPHWNDHGDGHQVR
~ (k ) ~ ( k )
Pi′′− j − jQ′i′− j
~ ( k ) ~ (k ) ~
,
(13)
U j = U i − Zi − j ⋅
~*
Uj
RGQRVQR SR VUHGXYDZHWR VR UD]GYRMXYDZH QD UHDOQLRW L LPDJLQDUQLRW GHO L
L]HGQDaXYDZHWRQDPRGXOLWHVHGRELYD
(k )  2
~
′
′
S
 i− j 
~
~
~
~
~
~
~
(14)
( U (jk ) ) 2 = ( U i( k ) ) 2 − 2 ⋅ (Pi′′−(kj ) ⋅ R i − j + Q′i′−( kj) ⋅ X i − j ) − Z i2− j ⋅  ~
  U (jk ) 


~
LNRQHaQRELNYDGUDWQDWDUDYHQNDSRQHSR]QDWDWD U j [
]
~ (k ) ~
~ (k )
~ (k )
~ (k )
~ (k ) ~
~ (k )
~
( U j ) 4 − ( U i ) 2 − 2 ⋅ (Pi′′− j ⋅ R i− j + Q ′i′− j ⋅ X i − j ) ⋅ ( U j ) 2 + Z i2− j ⋅ (Si′′− j ) 2 = 0.
(15)
7UHED GD VH QDSRPHQH GHND GRNRONX VH NRULVWL L]UD]RW ]D SUHVPHWND QD
~
QDSRQLWHYRVHNRMDLWHUDFLMDGHOHZHWRVR U j YVX^QRVWSUHWVWDYXYDGHNRQYROXFLMD
QDPQRCHZH
35,0(5
3UHWKRGQR L]QHVHQDWD SRVWDSND `H MD LOXVWULUDPH QD SULPHU SUHY]HPHQ RG
[5] 0UHCDWD H SULNDCDQD QD VOLNDWD 3 D SRGDWRFLWH ]D HOHPHQWLWH YR HGLQLaQL
YUHGQRVWL HY VH GDGHQL YR 7DEHODWD I 0UHCDWD LPD MD]OL YNOXaXYDM`L JR L
QDSRMQLRWRGNRLQDVHSULNOXaHQLSRWUR^XYDaLNDNR^WRHWRDSULNDCDQRQD
VOLNDWD 2SWRYDUXYDZDWD QD VLWH SRWUR^XYDaL LPDDW LVWD QRPLQDOQD YUHGQRVW
0,16+j0,08HY1DSRQRWYRQDSRMQLRWMD]HO"0"L]QHVXYD1,05HY
=DSRWUHELWHQDID]LSUHVPHWNDWDRSWRYDUXYDZDWDQDSRWUR^XYDaLWH `HJL
ID]LIL]LUDPH QD WRM QDaLQ ^WR `H JL SRPQRCLPH VR )% [0.9, 0.95, 1.05, 1.10] D
QDSRQRW YR QDSRMQLRW MD]HO `H JR SUHWVWDYLPH VR WULDJROQLRW )% [0.9, 1, 1, 1.05]
3DUDPHWULWHQDHOHPHQWLWHQDPUHCDWDRVWDQXYDDWRELaQLYUHGQRVWLNDNR^WRVH
GDGHQLYRWDEHODWDYRSULORJRW
5H]XOWDWLWHRGID]LSUHVPHWNDWD]DQDSRQLWHYRMD]OLWHNDNRL]DDNWLYQLWH
L UHDNWLYQLWH PR`QRVWL YR HOHPHQWLWH QD NUDHYLWH QD VRRGYHWQLWH MD]OL VH
SULNDCDQLYR7DEHODWDII9RWDEHODWDVHSULNDCDQLVDPRNDUDNWHULVWLaQLWHWRaNL
QDWUDSH]RLGQLWHIXQNFLLQDSULSDGQRVWRGQRVQRJUDQLFLWHQDαSUHVHFLWH]Dα Lα 3URFHVRW QD SUHVPHWXYDZH YR NRQNUHWQLRY VOXaDM LVNRQYHUJLUDO SR LWHUDFLL3ULWRD NDNR NULWHULXP QD WRaQRVW H XVYRHQR GD ]ELURW QD DSVROXWQLWH
YUHGQRVWLQDSULUDVWLWHQDSUHVHFLWHQDYNXSQLWHDNWLYQLLUHDNWLYQLPR`QRVWL
YRVLVWHPRWELGHSRPDORG 7UHEDGDVHQDSRPHQHGHNDYRVOXaDMRWNRJDVLWHYOH]QL)%GHJHQHULUDDWYR
RELaQL UHDOQL EURHYL a1 = a2 = a3 = a4 VH GRELYDDW LVWL UH]XOWDWL NDNR L VR
RELaQDWD GHWHUPLQLVWLaND SUHVPHWND QD 5'0 1D WRM QDaLQ RYDD YWRUDWD
SUHWVWDYXYDVDPRVSHFLMDOHQVOXaDMQDSUYDWD
$%(/$ I. 2'$72&, =$ 6,67(027
0
7
1
3
4
2
5
6
18
7
8
19
21
9
11
10
20
12
22
23
24
25
26
27
28
29
13
14
15
16
17
6OLND 5DGLMDOQD GLVWULEXWLYQD PUHCD RG
[5]
3
JUDQND
0-1
1-2
2-3
2-4
2-5
5-6
5-7
5-8
5-9
9 - 10
9 - 11
9 - 12
9 - 13
13 - 14
13 - 15
13 - 16
13 - 17
2 - 18
18 - 19
18 - 20
18 - 21
21 - 22
21 - 23
21 - 24
24 - 25
24 - 26
24 - 27
27 - 28
27 - 29
R [HY]
0.0236
0.00027
0.0051
0.0062
0.00316
0.00297
0.00297
0.00786
0.00128
0.0033
0.00502
0.00273
0.00078
0.00249
0.00262
0.00646
0.0041
0.00117
0.00114
0.00609
0.00119
0.0008
0.00342
0.0009
0.00296
0.00318
0.00092
0.00604
0.00159
X [HY]
0.0233
0.0002
0.0005
0.0006
0.00114
0.00031
0.00031
0.00081
0.0008
0.00034
0.00052
0.00028
0.00049
0.00026
0.00027
0.00066
0.00042
0.00073
0.00012
0.00062
0.00075
0.00026
0.00033
0.00055
0.00030
0.00033
0.00058
0.00068
0.00016
=$./8A2.
)D]L EURHYLWH L ID]L PDWHPDWLNDWD SUHWVWDYXYDDW QDaLQ QD PRGHOLUDZH QD
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YNOXaXYDZH YR SUHVPHWNLWH QD VRVWRMEDWD QD VLVWHPRW 2YRM SULVWDS PRCH GD VH
UD]JOHGXYD L NDNR SUR^LUXYDZH YR GODERaLQD QD LQWHUYDOQLWH EURHYL L
LQWHUYDOQDWDPDWHPDWLND7RMQHSUHWVWDYXYD]DPHQD]DVWRKDVWLaNLRWSULVWDSWXNX
QHJRYNYDOLWDWLYHQNRPSOHPHQW
5H]XOWDWLWHGRELHQLVRID]LSUHVPHWNDQDVRVWRMEDWDLQWHJUDOQRJLVRGUCDW
VLWH UH^HQLMD]DYNOXaHQLWH QHVLJXUQRVWL 1DSRURW VR NRM WRD H QDSUDYHQR H VDPR
PDONX SRJROHP YR VSRUHGED VR RQRM SRWUHEHQ ]D RELaQDWD GHWHUPLQLVWLaND
SUHVPHWND 2G GUXJD VWUDQD ID]L SULVWDSRW GDYD NRPSOHWQD VOLND ]D VLWH PRCQL
VRVWRMELQDUDERWDQDVLVWHPRW6HaLQLGHNDRYRMSULVWDSRWYRUDQRYLKRUL]RQWL]D
NYDOLWDWLYQRSUR^LUXYDZHLSRGREUXYDZHQDDQDOL]DWDQD((6
$%(/$ II (=8/7$7, 2' )$=, 35(60(7.$7$ 1$ 05( $7$ 35,.$ $1$ 1$ 6/,.$
7
C
5
~
U (HY)
α
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
0
0.9000
0.7243
0.7225
0.7212
0.7209
0.7132
0.7125
0.7125
0.7112
0.7103
0.7094
0.7090
0.7096
0.7093
0.7086
0.7086
0.7076
0.7082
0.7195
0.7192
0.7179
0.7171
0.7169
0.7163
0.7160
0.7152
0.7152
0.7154
0.7138
0.7150
1
1.0000
0.8598
0.8584
0.8573
0.8571
0.8510
0.8504
0.8504
0.8493
0.8486
0.8480
0.8476
0.8481
0.8478
0.8473
0.8473
0.8465
0.8470
0.8559
0.8557
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[4] ' 5DMLaL` 9RYHG YR GLVWULEXWLYQL HOHNWURHQHUJHWVNL VLVWHPL 8QLYHU]LWHW
6Y.LULOL0HWRGLM(OHNWURWHKQLaNLIDNXOWHW6NRSMH
[5] S. Rajagopalan, "A New Computational Algorithm for Load Flow Study of Radial
Distribution System", Comput. and Elect. Engng., Vol. 5, 1978, pp. 225-230
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[8] H. -J. Zimmermann, Fuzzy Set Theory - and Its Applications, 2nd revised edition, Kluwer
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Theory in Power Systems”, IEEE Trans. on PWRS, Vol. 10, No. 3, Aug. 1995, pp.
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SUMMARY
In this paper the results from the analyses of the conditions of the high voltage network in the
Polog region are briefly presented. As the situation of the above mentioned regional network is
critical, the solutions for its expansion have been suggested according to the results of the
previous analysis.
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