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A Ccombined TOPSIS and FA Based S a egic
Analysis o Technical Condi ion o High Powe
T ans o me s
Mikolaj BARTLOMIEJCZYK1,2, Mi osla GUTTEN3, S e an HAMACEK1
1Depa men o Elec ical Enginee ing, Facul y o Elec ical Enginee ing and Compu e Science,
VSB–Technical Uni e si y o Os a a, 17. lis opadu 15, 708 33 Os a a-Po uba, Czech Republic
2Depa men o Elec ical T anspo Enginee ing, Facul y o Elec ical and Con ol Enginee ing,
Gdansk Uni e si y o Technology, Sobieskiego 7, 802 16 Gdansk, Poland
3Depa men o Measu emen and Applied Elec ical Enginee ing, Facul y o Elec ical Enginee ing,
Uni e si y o Zilina, Uni e zi na 8215/1, 010 26 Zilina, Slo ak Republic
mba [email protected], [email p o ec ed], [email p o ec ed]
Abs ac . The pape p esen s ma hema ical model –
TOPSIS me hod, which was u ilized on insula ing s a e
o dis ibu ion ans o me o analyze and sensibili y o
indi idual measu emen s me hods mu ual compa ison.
We can uniquely de e mine he impo ance o hese
measu emen s me hods wi h his ma hema ical appa a-
us in hese measu emen s me hods in insula ing s a e
o ans o me s.
Keywo ds
Assigning indica o s, high powe ans o me s,
TOPSIS me hod.
1. In oduc ion
Wi h ega d o he de elopmen o he wo ld and na-
ional economies, also con ol, main enance and i s
analysis by ma hema ic calcula ions becomes an im-
po an subjec [1], [2], [4], [19], [20], [21], [22], [26].
This sphe e also includes powe ans o me s, whe e
hei p ope unc ion has a posi i e impac on he
ouble- ee supply o elec ici y and hea o indus-
ies and households. I is he e o e necessa y, in he
absence o scien i ic and esea ch po en ial in dis ibu-
ion u ili ies, o achie e he objec i es o he p oposed
ac i i ies, i.e. in-dep h analysis o undesi able impac s
on de ices condi ion, design o measu emen s and hei
e i ica ion, and design o new diagnos ic p ocedu es
o imp o ing eliabili y o powe ans o me s.
In case we wan o de e mine he eal insula ing
s a e o a ans o me and hen li e ime o insula ion, is
necessa y o analyze some measu emen s in indi idual
ypes o assays and hen de e mine hei exac ness and
eliabili y wi h ma hema ical models. We can exac ly
p o e he impo ance o hese assays by ma hema i-
cal and s a is ical models in he ield o analysis o he
insula ing s a e o ans o me s [23], [24].
Fo ma hema ical analyzing hese assays measu e-
men s we chose wi hin he ame o compa ison o he
deg ee o sensi i i y in single me hods o he insula ing
s a e o dis ibu ion ans o me s 110/22 kV:
•insula ion esis ance and pola izing index
R60/R15,
•dissipa ion ac o and capaci y: anδand C,
• ela i e change o sho -ci cui ol age dUk.
2. Desc ip ion o Chosen
Measu emen s
The oldes and easies me hod o inspec ing he s a e
o insula o s is by means o insula ion esis ance mea-
su ing. The main disad an age o his me hod is ha
insula ion esis ance does no only depend on he s a e
o insula ion bu also on i s ype and dimensions. In-
sula ion esis ance me hod can be used o e alua e he
s a e o insula ion o elec ic de ice only on he basis
o p e ious expe ience wi h he same insula ion on he
same de ices.
The me hod is based on he ollowing p inciple:
change in insula o s a e causes a change in ime de-
pendence o a cu en lowing h ough he insula o by
DC ol age [8]. A cu en lowing h ough an insula o
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consis s o a ime-dec easing abso p ion elemen and
s abilized elemen . The mo e wa e con en he e is
in he insula ion, mo e appa en inc ease o he s abi-
lized elemen o a cu en is he obse ed compa ing o
he abso p ion elemen . The abso p ion elemen o a
cu en has a low e ec on he cha ac e is ics o ime
dependency in ela ion o he cu en as well as he e-
sis ance, and la ens wi h inc easing humidi y (Fig. 1).
Fig. 1: Time dependence o he insula ion esis ance.
U ilizing his knowledge o e alua ion o he insu-
la ion s a e does no equi e de e mining he ull ime
dependence o a cu en . I is enough o de e mine
he alue o a cu en ( esis ance) in wo di e en mo-
men s om he ime o connec ion o DC ol age. The
a io o hese wo alues de ines he s a e o insula-
ion and is called he pola izing index. Since i is a
non-dimensional pa ame e , i does no depend on he
dimension o insula ion. Pola izing index is measu ed
a e 1 and 10 min o a e 15 and 60 s.
So o he be e illus a e he change in alues o
he pola izing index, i needs o be exp essed by bo h
elemen s o a cu en - abso p ion elemen iaand s a-
bilized elemen i∞:
pi=R60
R15
=ia15 +i∞
ia60 +i∞
.(1)
The humid and con amina ed insula ion is de e -
mined by i∞, he e o e nume a o and denomina o
a e e y close alues and hei a ion ends owa ds 1.
On he o he hand, he d y and clean insula ion which
is in good condi ion has a e y low s abilized cu en
and he ime dependen elemen iais dominan . Thus,
he ac ion alue is no iceably highe han 1. The
pola izing index o new ans o me s be o e usage in
ope a ion should each a leas 1,3.
The measu emen s o he dissipa ion ac o ( an δ)
and he capaci ies o ans o me windings a e used o
addi ional de e mina ion o he insula ion quali y as
whole o only o some pa s o he ans o me . The
alue o an δindica es he p esence o pola and ion
compounds in oil and i also de e mina es he aging
o oil. The deg ee o oil humidi y can be measu ed by
empe a u e dependence o an δ[8].
Changes in he s a e o sho -ci cui ol age dUk
(impedance) exp ess geome ical winding mo emen s
and hei cons uc ion changes in ans o me s. This
echnical condi ion depends on he he mal and me-
chanical e ec s o sho -ci cui cu en s.
By means o measu emen o sho -ci cui ol age we
can iden i y he mechanical and insula ing de o ma ion
o he winding o a ans o me .
Absolu e alue o sho -ci cui ol ages usually a e
no su icien o quali y he condi ion o winding wi h-
ou knowledge o hei e olu ion in ime, so he analysis
is based on compa ison o alues o a speci ied ime o
ope a ion o a ans o me .
3. Composi e Indica o and
TOPSIS Me hod
A composi e indica o (CI) is a ma hema ical agg e-
ga ion o a se o indi idual indica o s ha measu e
a mul i-dimensional concep [25]. The e a e mcom-
pa ised al e na i es, each al e na i e consis s o nsub-
indica o s xij. Fo each al e na i e is e alua ed CI. CI
is used o he pe o mance measu emen s, benchma k-
ing, ia p o iding an agg ega ed pe o mance index in
a ious ields such as Human De elopmen Index, Road
Sa e y Index [2], [3], [16], [17], [18], [27].
The g aphical ep esen a ion o CI cons uc ion is
illus a ed on Eq. (2). The e a e mcompa ised al e -
na i es, each al e na i e consis nsub-indica o s xij .
Fo he each al e na i e is e alua ed CI. Sub-indica o s
usually ha e no common measu able uni s.
The TOPSIS me hod is used o analyze a mul i-
c i e ia decision making p oblem wi h mal e na i es
wi h nc i e ia. In he TOPSIS me hod, he bes al-
e na i e should ha e he sho es Euclidean dis ance
om he posi i e ideal solu ion (PIS) and he longes
dis ance om he nega i e ideal solu ion (NIS). The
PIS is a hypo he ical solu ion which maximum alues
om he da abase o all al e na i es, and he NIS is a
hypo he ical solu ion which minimum alues om he
da abase o all al e na i es. TOPSIS de ines an in-
dex called ela i e closeness o he PIS and emo eness
om he NIS [7]. This index can be used as a CI o
al e na i es.
Gene ally, he s uc u e o CI can by exp essed by
he Eq. (3):
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al e na i e 1
al e na i e 2
.
.
.
al e na i e m





x11 x12 · · · x1n
x21 x22 · · · x2n
.
.
..
.
.....
.
.
xm1xm2· · · xmn





→





CI1
CI2
.
.
.
CIm





.(2)
CI =
n
X
i−1
wiIi,(3)
whe e wimeans weigh assigned o indica o i.
The main p ocedu e o he TOPSIS me hod is de-
sc ibed in he ollowing s eps:
S ep 1: De ine a decision ma ix:
The decision ma ix Do m×ndimension consis s o
alues o nsub-indica o s o mal e na i es.
S ep 2: No malize he decision ma ix:
The alues o sub-indica o s a e no malized o a scale
0–1. In case o ”bene i ype” indica o s, wha means
a highe alue is be e , as is used in he o mula:
x0
ij =xij −mini{xij}
maxi{xij} − mini{xij}.(4)
Wi h ”cos ype” sub-indica o s, wha means he
lowe alue is be e . They a e no malized in he ol-
lowing way:
x0
ij =maxi{xij} − xij
maxi{xij} − mini{xij}.(5)
As a esul is ob ained he no malized decision ma-
ix D’.
S ep 3: Compu e he weigh ed no malized
decision ma ix:
Elemen s o he no malized decision ma ix D’ a e mul-
iplied by weigh ec o W, which consis o n weigh
ac o s w. These ac o s exp ess he ela i ely impo -
ance o c i e ia. The elemen s o weigh ed no malized
decision ma ix Va e exp essed as:
ij =wjx0
ij.(6)
S ep 4: Iden i y he PIS and NIS:
The posi i e ideal solu ion A+and he nega i e ideal
solu ion A−can be exp essed as:
A+= (maxi{ i,1}, .., maxi{ i,n}) =  +
1, ..., +
n,(7)
A−= (mini{ i,1}, .., maxi{ i,n}) =  −
1, ..., −
n.(8)
S ep 5: Calcula e he dis ance o PIS and NIS:
Fo each al e na i e i he Euclidean dis ance d+
i o he
posi i e ideal solu ion and dis ance d−
i o he nega i e
ideal solu ion is de ined [7].
S ep 6: Compu e he ela i e closeness da a o
CI:
Values d+
iand d−
ia e combined o ela i e closeness
index Ci:
Ci=d−
i
d+
i−d−
i
.(9)
The Ciis a composed indica o CI o al e na i e i.
4. Composi e Indica o and
TOPSIS Me hod
To exp ess he subjec i eness and imp ecision o he
e alua ion p ocess, he sub-indica o s and weigh s a e
ep esen ed by a iangula uzzy numbe [7]. A i-
angula uzzy numbe ˜ncan be de ine by a iple (a,
b, c) shown in Fig. 2. The membe ship unc ion µ˜nis
de ined as:
µ˜n(x) =







x−a
b−a, a ≤x≤b
c−x
c−b, b ≤x≤c
0o he ise
,(10)
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whe e a<b<c. The bis he mos possible alue o a
uzzy numbe .
Le wo iangula posi i e iangula numbe s
˜n1= (a1, b1, c1), ˜n2= (a2, b2, c2) and a posi i e eal
numbe . Simila ly as in he case o eal numbe s, he
ope a ions o posi i e uzzy numbe s can be de ined as
ollows [7]:
˜n1(+)˜n2= (a1+a2, b1+b2, c1+c2),(11)
˜n1(×)˜n2= (a1a2, b1b2, c1c2),(12)
˜n1(/)˜n2= (a1/a2, b1/b2, c1/c2),(13)
1/˜n1= (1/c1,1/b1,1/a1),(14)
∗˜n1= ( a1, b1, c1).(15)
The dis ance be ween uzzy numbe s can be de ined:
D(˜n1,˜n2) = 1
3[(a2−a1)2+ (b2−b1)2+ (c2−c1)2].(16)
Used uzzy-TOPSIS model is simila o classic TOP-
SIS me hod. In s ep 1 decision ma ix is gene a ed, in
s ep 2 his ma ix is no malized. A e no maliza ion,
he eal alues in he decision ma ix and weigh alues
a e con e ed in o uzzy numbe s. The 7-le el scale o
uzzy numbe s exp essed in linguis ic e ms ha a e
used (Tab. 1).
Fig. 2: T iangula uzzy numbe ˜
N.
Tab. 1: Table o con e sion eal alues in o he uzzy alues.
Real alue x Linguis ic alue Fuzzy alue ˜n
0≤x≤1/7 Ve y Low (0,0,1/6)
1/7≤x≤2/7 Low (0,1/6,2/6)
2/7≤x≤3/7 Medium Low (1/6,2/6,3/6)
3/7≤x≤4/7 Medium (2/6,3/6,4/6)
4/7≤x≤5/7 Medium High (3/6,4/6,5/6)
5/7≤x≤6/7 High (4/6,5/6,1)
6/7≤x≤7/7 Ve y High (5/6,1,1)
The calcula ions in s ep 3 a e p oceeding wi h he
uzzy alues. In s ep 4, he uzzy alues o PIS and
NIS a e de ined as:
A+
j=maxi˜n1
ij, maxi˜n2
ij, maxi˜n3
ij,(17)
A−
j=mini˜n1
ij, mini˜n2
ij, mini˜n3
ij,(18)
whe e ˜n1
ij, ˜n2
ij, ˜n3
ij a e uzzy alues o uzzy no malized
decision ma ix. In s ep 5 is calcula ed he dis ance o
PIS and NIS by o mula Eq. (18), in s ep 6 he ela i e
closeness in he es ima e by Eq. (11).
5. Assigning Indica o s
Weigh s by Fac o Analysis
The alues o he weigh s will be assigned by ac o
analysis. Fac o analysis me hod is based on a educing
he dimensions o he p oblem, whe e he ndimensions
a e ans o med in o a psmalle numbe unobse ed
a iables called ac o s. The idea o ac o analysis can
be desc ibed by he o mula:
X=FY +E, (19)
whe e X– ma ix o he inpu da a, Y– ma ix o un-
co ela ed common ac o s, F– ma ix o ac o load-
ings, E– ma ix o he speci ic ac o s.
The dimensionali y o ma ix Fdepends on he se-
lec ed numbe o ac o s. Each ac o explains a pa
o he a iance o he inpu da a.
The app oach o he calcula ions o weigh ac o s
sugges ed in [2] consis he ollowing s eps:
S ep 1: De ine a numbe o ac o s:
Chosen ac o s should explain 70–80 % o he a iance
o he inpu da a. Usually he e a e 2 o 3 ac o s.
S ep 2: De ine squa ed ac o loadings:
Squa ed ac o loadings can be desc ibed by he o -
mula:
uij =a2
ij
Pm
k=1 a2
kj
,(20)
whe e m– he numbe s o he ac o s.
S ep 3: Calcula e p elimina y weigh s:
The p elimina y alues o he weigh s can be exp essed
as:
w0
j=maxkukj
e0
k
,(21)
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whe e e’kis he ela i e a ia ion explained in he da a
shee :
e0
k=ek
e,(22)
whe e eis o al a ia ion explained by chosen m num-
be o he ac o s, ekis a ia ion explained by he k
ac o .
S ep 4: Rescaling o he weigh s:
The inal alues o he weigh s a e desc ibed by he
o mula:
wj=w0
j
Pm
i=1 wj
.(23)
6. Resul s o he Calcula ion
Assigning o he weigh s o he c i e ia by ac o anal-
ysis was he i s s ep o he calcula ion. Two ac o s
in ac o analysis we e chosen: ac o 1 ep esen s he
pa ame e s R60/R15 and C, and explains 38 % o he
o al a iance; ac o 2 ep esen s he pa ame e s dUk
and angen del a, and explains 34 % o he a iance.
The alues o weigh s a e p esen ed in Tab. 2.
Tab. 2: Calcula ed alues o he weigh s.
Pa ame e Weigh
R60/R15 0,24
dUk0,36
angen del a 0,17
C 0,22
Assigned weigh s o he c i e ia we e applied o uzzy
TOPSIS model. Tab. 3 p esen s he esul s o he cal-
cula ions as well as he da a o ou sub-indica o s.
To in es iga e he impac o c i e ia weigh s was e-
alized he sensi i i y analysis – we e calcula ed alues
o CI o di e en se s o weigh s. The 11 expe imen s
we e conduc ed, he se s o weigh s a e p esen ed in
Tab. 4, he esul s o he sensi i i y analysis is showed
in Fig. 3. The e is shown ange o s anda d de ia ion
o CI calcula ed o 11 se s o weigh s and CI calcula ed
in he p e ious pa o he pape .
I is no iceable, ha he assigned alues o CI a e
placed in he ange o he s anda d de ia ion, wha
con i m he eliabili y o he uzzy TOPSIS me hod.
In Fig. 4 is p esen ed sensi i i y analysis ega ding he
inal ou come anking – he echnical condi ion o each
ans o me e e ed o he o he ans o me s.
Be e posi ion in anking means be e echnical
condi ion – highe CI. The a e age posi ion in sensi-
i e analysis in e e y case is loca ed closely o posi ion
based on he p e ious assigned alue o CI. In some
cases is isible signi ican be ween he maximal and he
minimal posi ion – columns T8, T10, T12. I can be
explained by he disp opo ion be ween se e al echni-
cal pa ame e s o ans o me . Tha ac indica es a
p e equisi e o ill na u ed echnical condi ion o ans-
o me and can be used o iden i y a ailu e.
To in es iga e he accu acy o p esen ed uzzy TOP-
SIS me hod was ealized he clus e analysis. Clus-
e analysis is a mul i a ia e echnique which in o ms
abou he simila i y in he da a se . Clus e ing is a
ask o assigning objec s in o g oups – clus e . The
objec s in he same clus e a e mo e simila o each
o he han o hose in o he clus e s.
Fig. 3: The esul s o he sensi i i y analysis - alues o CI and
s anda d de ia ion.
Fig. 4: The esul s o he sensi i i y analysis - anking o ans-
o me s, max, min, a - he highes , lowes , a e age
anking posi ion ob ained in sensi i i y es , cal - posi-
ion co esponding wi h CI alue, be e posi ion means
be e CI alu.
The classi ica ion aims o educe he dimensionali y
o a da a se by inding he simila i ies be ween classes
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Tab. 3: Fou sub-indica o sand CI alues o he 13 ans o me s.
TR Sub-indica o s Resul s o he calcula ions
R60/R15 dUk[h] an δC [pF] d+d−CI
T1 1,36 0,47 0,0217 2881,4 0,53 0,41 0,43
T2 1,37 8 0,0186 4746,7 0,51 0,44 0,46
T3 1,58 4,5 0,0123 2996,5 0,12 0,81 0,87
T4 1,44 7,9 0,0075 2731,5 0,41 0,49 0,54
T5 1,31 42,7 0,0046 1957,5 0,74 0,21 0,22
T6 1,55 32,4 0,0135 3815,5 0,27 0,76 0,74
T7 1,25 0 0,0424 3940,0 0,75 0,35 0,32
T8 1,31 21,4 0,0177 4882,0 0,66 0,31 0,32
T9 1,30 8,95 0,0160 4235,3 0,64 0,39 0,38
T10 1,39 2,41 0,0122 2236,0 0,54 0,43 0,44
T11 1,38 8,93 0,0153 3825,2 0,51 0,43 0,46
T12 1,32 20,4 0,0126 4030,7 0,66 0,31 0,32
T13 1,31 3,64 0,0187 3775,0 0,64 0,39 0,38
Tab. 4: Se o weigh s o sesi i i y analysis.
Pa ame e R60/R15 dUk[h] an C [pF]
Se 1 0,25 0,25 0,25 0,25
Se 2 0,5 0,167 0,167 0,0167
Se 3 0,167 0,5 0,167 0,167
Se 4 0,167 0,167 0,5 0,167
Se 5 0,167 0,167 0,167 0,5
Se 6 0,4 0,4 0,1 0,1
Se 7 0,1 0,4 0,4 0,1
Se 8 0,1 0,1 0,4 0,4
Se 9 0,4 0,1 0,1 0,4
Se 10 0,1 0,4 0,4 0,1
Se 11 0,1 0,4 0,4 0,1
Se 12 0,1 0,4 0,4 0,1
Se 13 0,24 30,36 0,017 0,22
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[2], [5], [6]. A dendog am is he esul o me hod and
illus a es he ela ionships be ween objec s. On Fig. 5
is shown dendog am ob ained by he clus e ing.
I is seen, ha he loca ion o he ans o me s on
dendog am co esponds wi h he alues o CI. T ans-
o me s wi h he simila alue o CI a e loca ed nea
on he dendog am, e.g. T8 and T12, T9 and T13. The
mos “sepa a ely” loca ed a e cases T3, T6, T5 and
T7. I may be explained by CI alues: T3 and T6 go
he bes a ing, T5 and T7 one o he wo s . Thus,
he posi ion o poin s can be used as an indica o o
ans o me insula ion condi ion.
Fig. 5: Dendog am based on hie a chical clus e ing.
7. Conclusion
On he basis o summa y esul s o he ma hema ical
CI model, he e can be se op imized mode n ech-
niques o he diagnosis o insula ion s a e chosen oil
ans o me s, he eby a highe quali y o ouble- ee
dis ibu ion o hea and elec ici y will be achie ed.
A composed indica o has been accep ed as a use ul
ool in many non- echnical a eas, such economy, soci-
e y, and en i onmen [9], [10], [11], [12], [13], [14],[15].
In his pape is p esen ed he applica ion o CI in he
ield o echnical sciences. Beside his, o he p esen ed
me hods o MCDA (e.g. hie a chical clus e ing) can be
used o e alua ion he echnical condi ion o elec ical
equipmen .
Acknowledgmen
This pape has been elabo a ed in he amewo k o
he p ojec Oppo uni y o young esea che s, eg.
no. CZ.1.07/2.3.00/30.0016, suppo ed by Ope a ional
P o-g amme Educa ion o Compe i i eness and co-
inanced by he Eu opean Social Fund and he s a e
budge o he Czech Republic, p ojec SP2013/47 and
p ojec MSMT KONTAKT II: LH11125 - In es iga ion
o he g ound cu en ields a ound he elec i ied lines.
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Abou Au ho s
Mikolaj BARTLOMIEJCZYK bo ned in 1983
in Gdansk, in 2007 g adua ed Facul y o Elec ical
and Con ol Enginee ing o Technical Uni e si y o
Gdansk, in 2011 eached Ph.D. deg ee in he same
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acul y. His a ea o esea ch is elec ical ac ion, es-
pecially supply sys ems o amways and olleybuses.
Beside esea ch ac i i es wo ks in The T olleybus
T anspo Company o Gdynia om 2003 yea .
Mi osla GUTTEN was bo n in Zilina. He
g adua ed a Uni e si y o Zilina. He ac s on De-
pa men o Measu emen and Applied Elec ical
Enginee ing a Uni e si y o Zilina.
S e an HAMACEK was bo n in Cadca.
He g adua ed SOUS Cadca in 2004.
In 2008 he g adua ed VSB–Technical Uni e si y o
Os a a, Facul y o Elec ical Enginee ing and Com-
pu e Science. Today he is scien i ic esea che in he
Depa men o Elec ical Enginee ing, VSB–Technical
Uni e si y o Os a a and he applies himsel o he
issue o medium- ol age lines wi h co e ed conduc o s
and p oblems associa ed wi h aul s de ec ion o
co e ed conduc o s. I also deals wi h he p oblems o
ac ion ca hena y and esea ch dissemina ion o s ay
cu en s in he a ea o ac ion.
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