POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 |NUMBER: 5 |2015 |DECEMBER
O e ol age P opaga ion om T ansmission Line
in o T ans o me Winding
Vacla KOTLAN, Zdenka BENESOVA
Depa men o Theo y o Elec ical Enginee ing, Facul y o Elec ical Enginee ing,
Uni e si y o Wes Bohemia in Pilsen, Uni e zi ni 26, 306 14 Pilsen, Czech Republic
k[email p o ec ed], b[email p o ec ed]
DOI: 10.15598/aeee. 13i5.1421
Abs ac . The pape deals wi h e y as ansien phe-
nomena in a sys em consis ing o wo pa s: a cable
line and a ans o me winding. In his case an ade-
qua e model should be conside ed as a ci cui wi h dis-
ibu ed pa ame e s. I s desc ip ion is gi en by a sys-
em o pa ial di e en ial equa ions o hype bolic ype.
Ou app oach is based on a nume ical solu ion in he
ime domain and he me hod FDTD has been used. I
allows ob aining esul s in a o m o ime-space ol age
and cu en wa e dis ibu ion along he cable line and
he ans o me winding. This dis ibu ion is depend-
ing on many ac o s some o hem we e s udied in his
pape .
Keywo ds
O e ol age on ansmission line, ansien s in
ans o me winding.
1. In oduc ion
I is a common p ac ice o connec a ans o me in
a subs a ion o he incoming line h ough a cable. A
su ge o e ol age wa e induced by swi ching-on and
swi ching-o p ocesses a els no only along he ca-
ble bu i p opaga es in o he ans o me winding.
These p ocesses can cause ans o me ailu es and can
be e y dange ous o i s isola ion sys em. Mo eo e ,
he shape o he a elling wa e is s ongly depending
on pa ame e s o bo h pa s. Due o di e en su ge
impedances o bo h, he cable and he ans o me
winding, e lec ions appea a he ans o me wind-
ing inpu . Because o hese e lec ions he ol age o
cu en peak alue can each highe alues han i was
conside ed in he design o he isola ion and p o ec ion
sys ems. The o he ac o s which ha e a simila impac
on he su ge wa e shape a e he manne o ans o me
winding ou pu and he cable leng h.
Many au ho s ha e been in e es ed in he analysis o
e y as ansien s in a ans o me winding and hey
used a ious me hods. Some o hem u ilized a simula-
ion in EMTP o ATP so wa e [1], [2]. O he au ho s
used ansmission line model sol ed in he equency
domain [3], [4]. In [5] he nume ical analysis based
on FDTD me hod in he ime domain was p esen ed.
Ve y impo an ole in ansien s analysis play he pa-
ame e s o winding, hose p oblems we e discussed in
[4], [6], [7]. Simula ion o a ious swi ching condi ions
has shown ha se e e in e nal o e ol age can a ise
be ween adjacen coils o he high ol age windings,
while he e minal ol age o he ans o me emains
below he gua an eed basic insula ion le el [7]. In such
cases su ge a es e canno p o ide a su icien p o ec-
ion since high equency oscilla ions may cause pa ial
winding esonance. All au ho s o hese pape s anal-
ysed only ansien s in a ans o me winding.
Acco ding o p ac ical expe iences he dis ibu ion
o ol age and cu en wa e in a ans o me winding
depends no only on ans o me pa ame e s bu also
on he shape o a a elling su ge wa e which is p op-
aga ing om he ansmission line connec ed o he
ans o me . In his case is also necessa y o s udy an
in luence o line pa ame e s. Fo such a complex analy-
sis i is impo an o conside he whole sys em consis -
ing o a ansmission line (cable-line) and a ans o me
winding.
In his pape an algo i hm o a complex analysis
is p esen ed. Ou app oach published in [8] is based
on modelling o he sys em consis ing o a ansmis-
sion line and a ans o me winding as a ci cui wi h
dis ibu ed pa ame e s. A gene al basic line elemen
has been in oduced i allows o o mula e one com-
mon ma hema ical model o bo h pa s. The ob ained
equa ions can be e y simply modi ied o he line and
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he ans o me winding. This app oach allows a o -
mula ion o a common algo i hm o nume ical compu-
a ion.
1.1. Ma hema ical Model
A gene al basic elemen espec ing a ansmission line
elemen and a ans o me winding elemen is depic ed
in he Fig. 1, all pa ame e s a e pe uni leng h.
iL(x, )
R(x) xd
R2(x) xdK(x)/ xd
L(x) xd
C(x) xdG(x) xd
u(x, ) U(x, )+( ) xd∂u(x, )/ ∂
dx
x
Fig. 1: Basic elemen a dis ance x om he line beginning.
In he p oposal ci cui elemen induc ance L(x)in-
ol es sel -induc ance and in he case o a ans o me
winding also mu ual induc ances u n o u n.
Capaci ance K(x) espec s u n o u n capaci ances
and i is in ol ed only in he ans o me winding pa .
Capaci ance C(x) espec s a capaci ance o he ea h.
Resis ance R(x)co esponds o Joule’s losses and esis-
ance R2(x)in e up s a edundan capaci ance loop.
The elemen on he Fig. 1 can be desc ibed wi h
ollowing equa ions:
−∂u
∂x =L(x)∂iL
∂ +R(x)iL,(1)
−∂iL
∂x =C(x)∂u
∂ +K(x)∂2uK
∂x∂ +G(x)u, (2)
−uK=∂u
∂x +R2(x)K(x)∂uK
∂ .(3)
The Eq. (1), Eq. (2) and Eq. (3) o e come o he
known ansmission line equa ions in he case ha
R2→ ∞ and K=0. To ind he solu ion o hese
pa ial di e en ial equa ions he knowledge o ini ial
and bounda y condi ions is needed. The compu a ion
s a s om he ze o ini ial condi ion, bounda y condi-
ions desc ibe he ela ionship a he inpu o he line
and he way o ans o me winding end.
1.2. Algo i hm o Nume ical
Solu ion
To de i e a sys em o di e ence equa ions o he pa -
ial di e en ial equa ions Eq. (1), Eq. (2) and Eq. (3)
he s able implici Wend o ’s o mula [9] was u ilized.
The i s o de de i a i es ha e been eplaced by he
ollowing di e ences:
∂ (x, )
∂ k,l
=1
2 l
k− l−1
k
∆ + l
k+1 − l−1
k+1
∆ !,(4)
∂ (x, )
∂ k,l
=1
2 l
k− l
k+1
∆ + l−1
k− l−1
k+1
∆ !,
k= 1,2, ..., N.
(5)
A di e ence scheme o he second o de de i a i e
in Eq. (2) was e alua ed in he o m:
∂ 2(x, )
∂x∂ 
l
k
∼
=
∂ (x, )
∂ 
l
k+1 −∂ (x, )
∂ 
l
k
∆x=
=1
∆x l
k+1 − l−1
k+1
∆ − l
k− l−1
k
∆ !.
(6)
Supposing ha he leng h o he cable line and he
ans o me winding is di ided in o Nleng h ele-
men s and applying hese di e ence app oxima ions o
Eq. (1), Eq. (2) and Eq. (3) we ecei e a sys em o
algeb aic equa ions o nodal alues o cu en s iLk in
induc ance L, ol ages ukac oss capaci ance Cand
ol ages uKk ac oss capaci ance Ka he ime l- h
le el. These disc e e nodal alues a e a anged in he
ec o (l)= [{iLk},{uk},{uKk}]. This sys em o al-
geb aic equa ions is han supplemen ed by bounda y
condi ions a he inpu and ou pu o he sys em ac-
co ding o Fig. 2. I esul s in o 3N+ 3 equa ions in
a ma ix o m:
A· (l)=B· (l−1) +D.(7)
Elemen s o ma ix Aand Bdepend on pa ame e s
o he cable and he ans o me winding, elemen s o
ma ix D espec a ol age sou ce. Ma ix Eq. (7)
enables o e alua e unknown disc e e alues o ol ages
and cu en s a l- h ime le el om he known alues a
p e ious (l-1)- h le el, he compu a ions we e ca ied
ou in MATLAB.
2. Illus a i e Examples
The p oposal algo i hm has been used o he anal-
ysis o e y as ansien phenomena in one phase
1 MVA ans o me connec ed o a 22 kV cable line.
The unloaded HV ans o me winding was conside ed
because i is he wo s case wi h ega d o an o e -
ol age peak alue. The in luence o some impo an
ac o s was s udied, he esul s ob ained o a ious ca-
ble leng h and a ious ol age wa e shape a e discussed
in he nex pa s.
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2.1. O e ol age Induced by S ep
Vol age Su ge Wa e
A i s he ol age sou ce p oducing a s ep ol age o
peak alue Un= 22 kV wi h a a ious slope was con-
nec ed o he sys em on he Fig. 2. The e alua ion was
ca ied ou o cable pa ame e s: R= 0.35 mΩ·m−1,L
= 0.529 µH·m−1, C = 218 pF·m−1, he a ious cable
leng h lC= 0, 100, 200, 300 and 400 m was supposed.
ansmission
line
ans o me
winding
RLCG,,, RLCG,,,, K2
u
0()
+
lw
lc
Fig. 2: Sys em consis ing o line and ans o me winding sup-
plied om ol age sou ce.
The ime space dis ibu ion o ol age and cu en
su ge wa e is depic ed on Fig. 3 and Fig. 4. The ca-
ble line and he ans o me winding ha e di e en
su ge impedances o his eason he e lec ions a he
ans o me inpu cause he highe magni ude o ol -
age wa e p opaga ing in o winding and opposi e he
cu en wa e in he ans o me winding eaches lowe
magni ude. The e y high alues o cu en wa e a
he line inpu a e caused by a small impedance o he
ol age sou ce (nea o sho -ci cui ed end).
Fig. 3: Time-space ol age dis ibu ion, s ep ol age sou ce.
On he Fig. 5 is shown he dependence o he ou -
pu ol age in % uni on he slope o sou ce ol age
ise and on he cable leng h. Vol age a winding inpu
inc eases due o su ge wa e p opaga ion h ough cable
o a g ea e a e o ise. I esul s in e y high peak
alue a winding ou pu Uou > 3.2 Un.
The ime ol age dis ibu ion a he inpu
and ou pu ans o me winding (lC= 400 m,
slope = 10 kV·µs−1) is depic ed on he Fig. 6. I is seen
ha no only he high ol age peak alue bu also high
equency oscilla ions depending on he slope alue ap-
pea . Fo compa ison he ime ol age dis ibu ion a
he ans o me inpu o a smalle pa ame e s (lC=
Fig. 4: Time-space cu en dis ibu ion, s ep ol age sou ce.
1 2 3 4 5 6 7 8 9 10
200
220
240
260
280
300
320
340
360
380
Slope [kV/µs]
Umax/Un [%]
Maximal ol age on he winding ou pu s ep ol age
lc=0m
lc=100m
lc=200m
lc=300m
lc=400m
Fig. 5: Dependence o maximal ol age peak alue Uou on ca-
ble leng h and s ep ol age slope.
Fig. 6: Time ol age dis ibu ion o ans o me winding inpu
and ou pu , s ep ol age sou ce: slope = 10 kV·µs−1,
lC= 400 m.
100 m, slope = 1 kV·µs−1) is shown on he Fig. 7. The
oscilla ions p opaga ing om he cable line a e much
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Fig. 7: Time ol age dis ibu ion a ans o me winding inpu
o s ep ol age sou ce: slope = 1 kV·µs−1, cable leng h
lC= 100 m.
lowe and sho e , he peak alue is nea o Un= 22 kV.
In his case he impac o wa e a elling along he ca-
ble is insigni ican . F om ob ained esul s ollows ha
he o e ol age in he ans o me winding can each
dange ous peak alue o a high a e o ol age ise and
a longe cable leng h. This in luence should be aken
in o accoun in a p o ec ion sys em design.
2.2. O e ol age Induced by Vol age
Su ge Wa e
The p opaga ion o he su ge wa e 1.2/50 µs wi h peak
alue Un= 22 kV along he cable in o he ans o me
winding has been s udied in his pa . A i s he
su ge ol age wa e was applied di ec ly a he ans-
o me inpu . Secondly, he same su ge wa e p opaga -
ing h ough he cable o he leng h 100 m and 400 m
was in es iga ed. The pa ame e s a e he same as in
he p e ious example.
The ol age ime dis ibu ion a he ans o me
winding inpu is depic ed on he Fig. 8. I is seen ha
due o p opaga ion o ol age su ge wa e along he
cable high equency oscilla ions occu and he peak
alue is nea ly wice highe han Un. I is caused by
e lec ions a he connec ion poin o bo h pa s. The
ol age ime dis ibu ion a he ans o me winding
ou pu is depic ed on he Fig. 9.
Tab. 1: Vol ages on he inpu and ou pu o a ious cable
leng hs.
cable line Uin [kV] Uou [kV]
0 21.993 53.050
100 37.101 78.419
400 37.483 76.519
The e is seen how long needs he wa e o each wind-
ing ou pu and oscilla ions caused by e lec ions a he
winding inpu and ou pu .
Compa ing he ime ol age dis ibu ion on he
Fig. 8 and Fig. 9 we ind ha i is s ongly depend-
ing on he cable leng h. No only he ol age peak
0 1 2 3 4 5 6 7 8 9
x 10−4
0
0.5
1
1.5
2
2.5x 104
[s]
uin [V]
lc = 0 m
0 1 2 3 4 5 6 7 8 9
x 10−4
−2
−1
0
1
2
3
4x 104
[s]
uin [V]
lc = 100 m
0 1 2 3 4 5 6 7 8 9
x 10−4
−3
−2
−1
0
1
2
3
4x 104
[s]
uin [V]
lc = 400 m
Fig. 8: Time ol age dis ibu ion a ans o me winding inpu
o su ge ol age wa e and cable leng h lC∈(0, 100,
400) m.
alue bu also he equency o oscilla ions di e s o
a ious cable leng hs. The e y high peak alues a e
seen in he ollowing Tab. 1.
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0 1 2 3 4 5 6 7 8 9
x 10−4
−6
−4
−2
0
2
4
6x 104
[s]
uou [V]
lc = 0 m
0 1 2 3 4 5 6 7 8 9
x 10−4
−8
−6
−4
−2
0
2
4
6
8
10x 104
[s]
uou [V]
lc = 100 m
0 1 2 3 4 5 6 7 8 9
x 10−4
−6
−4
−2
0
2
4
6
8x 104
[s]
uou [V]
lc = 400 m
Fig. 9: Time ol age dis ibu ion a ans o me winding ou -
pu o su ge ol age wa e 1.2/50 µs and leng h lC∈
(0, 100, 400) m.
3. Conclusion
An e icien algo i hm o a nume ical analysis o e y
as ansien s in he ans o me winding connec ed
o he ansmission line was in oduced. I esul s in
ime-space ol age and cu en su ge wa e dis ibu ion
and allows assessing o dange ous swi ching condi ions
in powe sys ems. The in luence o he shape su ge
wa e, i s slope and he line leng h was s udied. I
was ound ha besides he high o e ol age peak alue
also oscilla ions wi h e y high equency can appea .
These ac s a e e y dange ous o he isola ion sys em
and can cause damage o he ans o me winding. The
p oposed algo i hm p o ides a deepe iew in o e y
as ansien phenomena and can be e y help ul o
a co ec design o he p o ec ing and isola ion sys em
o ans o me .
Re e ences
[1] POPOV, M. and E. ACHA. O e ol age due o
swi ching o an unloaded ans o me wi h a
akuum b eake . IEEE T ansac ions on Powe
Deli e y. 1999, ol. 14, iss. 4, pp. 1317–1326. ISSN
0885-8977. DOI: 10.1109/61.796224.
[2] MARTI, J. R. and L. R. LINARES. Real-
ime EMTP-based ansien s simula ion.
IEEE T ansac ions on Powe Sys ems. 1994,
ol. 9, iss. 3, pp. 1309–1317. ISSN 0885-8950.
DOI: 10.1109/59.336135.
[3] POPOV, M., L. VAN DER SLUIS, R. P. P.
SMEETS and J. L. ROLDAN. Analysis o Ve y
Fas T ansien s in Laye -Type T ans o me Wind-
ings. IEEE T ansac ions on Powe Deli e y. 2007,
ol. 22, iss. 1, pp. 238–247. ISSN 0885-8977.
DOI: 10.1109/TPWRD.2006.881605.
[4] SHIBUYA, Y. and S. FUJITA. High equency
model and ansien esponse o ans o me
windings. In: T ansmission and Dis ibu ion
Con e ence and Exhibi ion 2002. Yokohama:
IEEE, 2002, pp. 1839–1844. ISBN 0-7803-7525-4.
DOI: 10.1109/TDC.2002.1177736.
[5] PREDOTA, A. and Z. BENESOVA. Modelling
o T ansien s in T ans o me Winding. P zeglad
Elek o echniczny. 2010, ol. 86, no. 1, pp. 14–16.
ISSN 0033-2097.
[6] HOSSEINI, S. M. H., M. VAKILIAN and G. B.
GHAREHPETIAN. Compa ison o T ans o me
De ailed Models o Fas and Ve y Fas T ansien
S udies. IEEE T ansac ions on Powe Deli e y.
2008, ol. 23, iss. 2, pp. 733–741. ISSN 0885-8977.
DOI: 10.1109/TPWRD.2008.915795.
[7] SOYSAL, O. A. Vol age s esses in a dis ibu ion
ans o me unde nonideal swi ching condi ions.
In: Powe Enginee ing Socie y 1999 Win e Mee -
ing. New Yo k: IEEE, 1999, pp. 1031–1035. ISBN
0-7803-4893-1. DOI: 10.1109/PESW.1999.747339.
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POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 13 |NUMBER: 5 |2015 |DECEMBER
[8] BENESOVA, Z. and V. KOTLAN. New app oach
o su ge phenomena analysis in ans o me wind-
ing. In: IEEE In e na ional Powe Modula o and
High Vol age Con e ence (IPMHVC). San a Fe:
IEEE, 2014, pp. 263–266. ISBN 978-1-4673-7323-
4. DOI: 10.1109/IPMHVC.2014.7287259.
[9] BENESOVA, Z. and V. KOTLAN. P opaga ion
o Su ge Wa es on In e connec ed T ansmission
Lines Induced by Ligh ning S oke. Ac a Technica
CSAV. 2006, ol. 51, iss. 3, pp. 301–316. ISSN
0001-7043.
Abou Au ho s
Vacla KOTLAN (was bo n in Pilsen in he Czech
Republic (1947). She g adua ed om Uni e si y
o Wes Bohemia in Pilsen, Facul y o Elec ical
Enginee ing in 1970; in 1985 she ecei ed he CSc.
deg ee Ph.D. She has wo ked as an Associa e P o esso
a Uni e si y o Wes Bohemia in Pilsen since 1994
and as a P o esso since 2002. He esea ch in e es
conce ns on nume ical me hods o elec omagne ic
ield analysis and on heo e ical p oblems o powe
elec ical ene gy sys ems ( ansmission line pa am-
e e s, nume ical me hods o analysis o ansien
phenomena on ansmission lines, ans o me winding
and EMC). She published mo e han 90 pape s and
con ibu ions in a ious scien i ic and con e ence
p oceedings.
Zdenka BENESOVA (1979) g adua ed om
he Facul y o Elec ical Enginee ing (Uni e si y o
Wes Bohemia Pilsen, Czech Republic) in 2003. In
2008 he ecei ed his Ph.D. deg ee in he ield o
Elec ical Powe Enginee ing. His esea ch in e es s
a e aimed a heo e ical p oblems o elec ical powe
sys ems ( ansmission line pa ame e s, nume ical
me hods o analysis o ansien phenomena on ans-
mission lines) and in he las yea s he also deal wi h
nume ical solu ion o elec omagne ic and coupled
ields. He published abou 80 pape s in scien i ic
jou nals and con e ence p oceedings.
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