Energy and Power Engineering, 2010, 2, 238-247
doi:10.4236/epe.2010.24035 Published Online November 2010 (http://www.SciRP.org/journal/epe)
The Effects of Mutual Coupling and Transformer
Connection Type on Frequency Response of Unbalanced
Three Phase Electrical Distribution System
Omer Gül, Adnan Kaypmaz
Electrical Engineering Department, Istanbul Technical University, Maslak, Turkey
Received June 24, 2010; revised August 1, 2010; accepted September 3, 2010
Abstract
In this paper, a novel harmonic modeling technique by utilizing the concept of multi-terminal components is
presented and applied to frequency scan analysis in multiphase distribution system. The proposed modeling
technique is based on gathering the same phase busses and elements as a separate group (phase grouping
technique, PGT) and uses multi-terminal components to model three-phase distribution system. Using multi-
terminal component and PGT, distribution system elements, particularly, lines and transformers can effec-
tively be modeled even in harmonic domain. The proposed modeling technique is applied to a test system for
frequency scan analysis in order to show the frequency response of the test system in single and three-phase
conditions. Consequently, the effects of mutual coupling and transformer connection types on three-phase
frequency scan responses are analyzed for symmetrical and asymmetrical line configurations.
Keywords: Harmonic Resonance, Unbalanced Distribution System, Frequency Scan
1. Introduction
Harmonic studies have become an important aspect of
electrical distribution system analysis and design in re-
cent years largely due to the increasing presence of solid-
state electronic power converters. Moreover, shunt ca-
pacitors are extensively used in electrical distribution
systems (EDS) for power factor correction. Due to the
proliferation of nonlinear loads, awareness of harmonic
effects has been increasing [1,2]. It is therefore that the
possibility of resonance because of shunt capacitor
should then be analyzed by the utilities [3-5].
The first decision to make in any harmonic study of
distribution system is whether a three phase model is
required or a single phase model will be sufficient.
Three-phase distribution systems are generally unbal-
anced and asymmetrical. Hence, asymmetrical three-
phase distribution systems must be modeled by phase
co-ordinations and their analysis can be performed either
under sinusoidal or non-sinusoidal conditions [1,6-9].
Compared to the single-phase analysis, problem size
increases three-times in phase coordinated based model-
ing and analysis of EDS. In addition, when harmonics
are present in the system, the models must be realized for
each harmonic component, which requires new methods
in three-phase harmonic analysis of distribution systems
in order to decrease computation time and memory re-
quirement [7].
A number of different studies related to harmonic
modeling and analysis of EDS have been given in the
literature (e.g. [2,6]). Grainger [10] applied the matrix
factorization technique (MFT) to harmonic studies to
achieve a significant saving in computational effort. In
the paper, only the required columns of the bus imped-
ance matrix which represent those busses supplying non-
linear loads are obtained instead of performing a full
inverse.
As one of the most common and simple harmonic
analysis technique, frequency scan method is used to
identify the frequency response of EDS. However this is
not an easy task for some cases as shown in [5,6]. Firstly,
bus admittance matrix of EDS becomes both complicated
and large-scaled based on the number of busses and
three-phase system components. Secondly, the maximum
harmonic order to be considered is of importance in
terms of storage and computational effort for frequency
scan based harmonic analysis. If the maximum harmonic
order to be considered is as high as that of the number of

O. GÜL ET AL. 239
busses in the EDS to be analyzed a new approach to
solve such systems is needed in order to decrease com-
putation time and memory requirements.
In this paper, the following improvements are achi-
eved in modeling and computation techniques. Multi-
terminal component concept is used to find the mathe-
matical model of three-phase asymmetric EDS in har-
monic domain [11]. As for the mathematical models of
EDS, phase grouping technique, PGT is used [12]. The
technique is based on the separation of same phase buses
and components into different groups (PGT) so that more
understandable models can be constituted and savings in
memory use can be obtained. Moreover, MFT is pre-
ferred in this study to determine the frequency scan of
the EDS. Differing from Grainger, only the required
element of the bus impedance matrix on diagonal which
represent those busses supplying nonlinear loads are ob-
tained instead of performing a full inverse of the bus
admittance matrix [10]. In this paper, the aforementioned
ideas are combined to find a solution for multi-phase
frequency scan of asymmetric EDS.
2. Multiphase Distribution System Modeling
Obtaining the general model of electric circuits with the
aid of multi-terminal element is given in detail in modern
circuit theory. General form of algorithms given for
multi-terminal elements becomes more simple and un-
derstandable when it is used for mathematical modeling
of power systems. Graph and terminal equations associ-
ated with multi-terminal elements represent the mathe-
matical model of multi-terminal element and show the
whole features of it.
In this section, multi-terminal component models of a
distribution system, which is used in obtaining the har-
monic dependent modeling of EDS, is given together
with PGT. Harmonic dependent models are used in order
to find the frequency response of the network.
2.1. Basics of Multi-Terminal Approach for
Mathematical Modeling
To obtain the required models for power system analysis,
all buses in the system is generally desired to be shown
in the model and phase to ground voltages are needed for
power system modeling. As a result of this, the graph of
electric power systems that can be represented as a multi-
terminal element becomes “oriented graph” of which
common node denotes the ground and terminals of the
graph represent the buses of the system as shown in
Figure 1. Oriented graph of Figure 1(b) together with
Equation (1) gives the mathematical model of multi-
terminal component.
Multi-terminal component modeling technique can be
used in modeling of EDS for various aspects such as
single-phase, symmetrical components and phase coor-
dinated models of EDS without limitations [11,12].
Equation (1) gives the terminal equations.
(1,1)(1,2)(1, ) 11
(2,1)(2,2)(2, ) 22
(,1) (,2)(,)1
.
n
n
nn nn
nn
nx nxn

















 
YY Y
IV
YY Y
IV
YY Y IV





(1)
2.2. Mathematical Model of Electrical
Distribution System
Each element in an electric distribution system can be
represented as a multi-terminal component with its ma-
thematical model, explained in detail above. It is there-
fore an electric power network itself that can be modeled
as a multi terminal component as shown in Figure 7
through the combination of multi-terminal elements,
which is performed by using the parallel connection me-
thod of multi-terminal components.
The terminal equation of three-phase electric power
network in harmonic domain is given by Equation (2).
.
aaab aca
busbus bus busbus
bbabbcb
busbus bus busbus
ccacbc c
busbus bus busbus
hh











IYYY V
IYYY V
IYYY V
(2)
(n+1)-terminal
component
1
2
3
n
r (reference)
r (reference)
123
...
n
Figure 1. Multi-terminal modeling of n-bus EDS (a) and its
graph representation (b).
(3n+1)-terminal
component
a
1
a
2
a
3
b
1
b
2
b
3
a
n
bn
c
1
c
2
c
3
c
n
reference (ground)
...
.
.
.
.
.
.
a
1
a
2
a
3
an
b
1
b
2
b
3
bn
c
1
c
2
c
3
cn
reference
Figure 2. (a) Multi-terminal representation of a three-phase
electric power network; (b) Oriented graph of an electric
power network.

O. GÜL ET AL. 240
3. Proposed Approach
The proposed technique is based on both separation of
same phase buses and other power system components
into different groups, i.e., each phase group contains
same phase busses and elements, which can be repre-
sented as in the mathematical model of a three-phase line
model in Subsection 2.3. Since electric power networks
are composed of multi-terminal components connected
to each other, components models are firstly presented.
Hence, an electric power network itself can be modeled
as a multi-terminal structure which is a combination of
its constituents. Based on the topology of the system, the
combination procedure of multi-terminal elements rep-
resenting system constituents is carried out here through
parallel connection method of multi-terminal elements,
which is well-known in modern circuit theory [13].
Since the most common elements in electric power
networks are lines and transformers, the models of these
elements are given only in this paper. Yet, one can get
the others by following the procedure which is given in
the next section. After getting the element models in the
form of multi-terminal component, the mathematical
models are stemmed from the procedure as explained in
the following sections.
3.1. The Line Model
In general, the lines are represented as -equivalent cir-
cuit in most applications. The series impedance and
shunt admittance lumped- model representation of the
three-phase line is shown in Figure 3 [2,7]. To obtain a
symmetrical model of fundamental components, the lines
are generally transposed so as to eliminate the effect of
long lines. However, this aim can not be reached when
the system have harmonic components. Furthermore,
long line effect takes place in relatively short distances,
if the lines carry signals with high frequencies. Due to
these facts, it is a must to use phase coordinated models
in harmonic dependent line modeling.
In this case, three-phase representation of lines as mul-
ti-terminal component and its oriented graph are given
according to PGT in Figure 3.
According to proposed approach, the following pro-
cedure is given for obtaining the mathematical model of
lumped- model.
Step 1
Neglecting skin effects, harmonic dependent series
impedance matrix of a line or cable in Figure 3 is given
as in Equation (3)
22 2
11
abc abcabc
hrjhx
 

 
z (3)
where, 21
abc
r


and 21
abc
x


are fundamental series re-
y3
a
y1
ay1 by1 cy3 by3 c
a1
b1
c1
a2
b2
c2
y3
ac
y1
ac
Z2
ac
Z2
ab
Z2
bc
y1
ab y1 bc y2 ab y2 bc
Z2
a
Z2
b
Z2
c
Figure 3. Lumped- model representation of the three-
phase line.
sistance and reactance matrices of a line or cable for the
phases a, b, c respectively and h is the harmonic order.
Harmonic dependent shunt admittance matrices of phas-
es a, b and c are given as below:


 
11 1
11
33 3
11
Re(1/) Im
Re(1/) Im
abc abcabc
h
abc abcabc
h
jh
jh

 


 

 


 
yy y
yy y
(4)
Real parts of Equation (4) are neglected for line and
cables.
Step 2
By inversion of primitive impedance matrix for Fig-
ure 2, series primitive admittance matrix in harmonic
domain are obtained as in Equation (5)
22 2
1
22 222
22 2
h
aabac
abcabcba b bc
h
ca cbc
h














yy y
yz yyy
yy y
(5)
By gathering the primitive admittance values of same
phase elements in one group, the primitive admittance
matrix of Figure 2 and its short form are given as Equa-
tion (6) and (7), respectively.
11 1
22 2
33 3
111
222
333
11 1
22 2
33 3
00 00 00
000000
00 0000
00 00 00
000000
00 0000
00 00 00
000000
00 0000
aabac
aabac
aabac
ba bbc
abc bc b bc
h
bcb bc
ca cbc
ca cbc
ca cbc
h































yy y
yy y
yyy
yyy
yyyy
yyy
yy y
yyy
yyy
(6)

aabac
babbc
h
ca cbc
h











yy y
yyyy
yy y
(7)

O. GÜL ET AL. 241
where ,
ab


yy
and c


y are the sub admittance
matrices of phase elements which are grouped based on
the phases a, b, c. k


y is a mutual admittance matrix
between k and  (k = a, b, c; k  ;  = a, b, c).
Step 3
Each system element of Figure 3(b) is first assumed
to be excited by a voltage source in order to obtain the
terminal equation of three-phase line, which is repre-
sented in the form of multi terminal component in Fig-
ure 3(a). In this case, the closed loop equations for ori-
ented graph of three-phase line whose components have
been excited by different voltage sources can be arranged
by gathering the same phase terminals together as in
Equation (8). In the same way, basic cut-set equations
can be written as Equation (9).
()
()
00 00
()
0000 0
00 00
a
b
a
c
b
a
c
b
c
BU
BU
BU









 








V
V
V
V
V
V
(8)
()
()
0000 ()
000 00
000 0
a
b
a
c
b
a
c
b
c
UQ
UQ
UQ



















I
I
I
I
I
I
(9)
Where






,,
abc
BBB are basic loop matrices re-
lated to phases, a, b and c, and ,,
abc
QQQ


are
basic cut-set matrices related to phases, a, b and c, re-
spectively. Moreover, as known, the expression
 
T
BQ is valid for all phases.
Step 4
By using the Equation (7), Equation (8) and Equation
(9), terminal equations in harmonic domain can be given
as Equation (10), which is well known in circuit theory
[16]. Oriented graph of Figure 3(b) together with Equa-
tion (10) gives the harmonic dependent mathematical
model of multi-terminal line component.
11
22
11
22
11
22
.
aa
aa
aabac
bus bus bus
bb
ba bbc
bus bus bus
bb
ca cb c
bus bus bus
cc
h
cc
hh
 
 
 

 


 

 


 
 
 
 
IV
IV
YYY
IV
YYY
IV
YYY
IV
IV
(10)
Phase sub-admittance and sub-mutual admittance ma-
trices in Equation (10) are given as Equation (11) and
(12) for hth-harmonic order, respectively.
 
T
kkkk
bus BB, ka, b, c
   


Yy (11)
 
T
kkk
bus BB,
ka, b, c; (k); a, b, c
 




Yy


(12)
Equation (11) and (12) represent 9 different sub-ma-
trices for each harmonic order, or rather these matrices
simply form the terminal equation of harmonic depend-
ent three-phase line in the newly proposed method.
These 9 matrices for each harmonic order can be calcu-
lated independently and for this reason the matrix calcu-
lations can be done at the same time by more than one
computer in parallel processing.
Simplified assumptions.
1) a) As shown in Equation (13), the primitive har-
monic admittance matrices associated with each phase
can be assumed to be equal on the condition that the
phase characteristics such as conductor size, line length,
phase material and the number of component belong to
each phase are same.
a b cphase
hhh h





yyyy (13)
b) As shown in Equation (14), the mutual admittance
matrices of hth-order can be assumed to be equal on the
conditions that long line effect is neglected and the lines
are transposed, which leads to symmetrical mutual cou-
pling between lines.
ab ac ba bc
h
hhh
ca cb m
hhh





 

 
yyyy
yyy
(14)
As a consequence, when assumptions are made related
to Equations (13) and (14) it is enough to form the primi-
tive admittance matrix representing the whole three-
phase line (as in Equation (10)) by just using both one-
phase primitive admittance matrix and a mutual admit-
tance matrix.
2) As shown in Equation (15), basic loop matrices are
assumed to be equal when the topology of phases is iden-
tical.







ab
BBBB
c
 (15)
Instead of using the Equations (11)-(12), which con-
tain 9 different equations for each harmonic, only the
Equations (16) and (17) can be used in order to form the
harmonic dependent terminal equation of the line on the
conditions that the performance equations of a primitive
network are as Equations (13)-(14) and the topology of
phases are identical.

O. GÜL ET AL. 242
 
T
ab cphase
bus bus bush
hhh
Y=Y=Y=By B
 

 (16)
 
ab acba bc
bus busbus bus
hhhh
T
ca cbm
bus bush
hh
Y=Y=Y=Y
=Y=Y=ByB





(17)
The algorithm given above for lines is also valid for
power networks composed of more than one line. In that
case, if a mutual coupling between different three-phase
lines exists, all these lines must be modeled as a single
multi-terminal component. For this reason, the formation
of mathematical model is realized as explained above.
Consequently, as one might expect that the proposed
algorithm can be applied to a system whose bus numbers
are different at different phases, which show another
merit of the proposed modeling technique.
3.2. Three-Phase Transformer Model
Magnetizing current in transformers leads to harmonic
currents due to its saturated core. Due to the fact that the
transformers should be modelled in harmonic domain so
that harmonic currents are required to take place in the
model [2,7]. However, harmonic currents is not included
in the transformer model since our interest in this study
is to determine the frequency response of EDS.
Mathematical model associated with any of trans-
former can be obtained by utilizing the concept of mul-
ti-terminal component. However, the most common
transformers in use, i.e., Y- connected three-phase
transformers are preferred here to show the potential
application of the proposed method.
In respect of the proposed method, a transformer
model is given here in the case that the mutual coupling
between phases of primary and secondary windings is
not neglected. Y- connected three-phase transformer is
illustrated in Figure 4.
Step 1
Y- connected three-phase transformer is represented
as a multi-terminal component in Figure 5. When it is
desired to form the mathematical model of the trans-
formers with isolated neutral point, one should take the
neutral point into consideration. As a result, the terminal
number in multi-terminal representation of transformer is
increased from 7 to 8, which leads to increased dimen-
sion in oriented graph and terminal equations.
Step 2
In multi-phase system representation, the power trans-
former are represented by reactance and resistance ma-
trices for each pair of windings. According to proposed
PGT, the primitive admittance matrix of the transformer
hth harmonic order is given by Equation (18).
7-terminal
component
c1
c2 a2
a1
b1b2
r
c2
a1
a2
b1b2
c1
r
Figure 4. (a) Three-phase representation of a line in the
form of multi-terminal component; (b) Oriented graph of a
three-phase line.
Ia1
Ib1
Ic1
Ia2
Ib2
Ic2
Va2
Vb2
Vc2
Vc1
Va1
Vb1
Figure 5. Y- connected three-phase transformer with sin-
gle core.
1
2
1
2
1
2
y
mmmmm
m mmmm
mm mmm
abc
hmm mmm
mmmm m
mmm mh

 






  






 







  








 




y yyyyy
yyyyyy
yy yyyy
yy yyyy
yyyy y y
yy yyyy
(18)
where,
1
y is the admittance at primary side
2
y is the admittance at secondary side
m
yis the mutual admittance between primary and sec-
ondary side at same phases
'
m
y is the mutual admittance between primary coils.
''
m
yis the mutual admittance between primary and sec-
ondary coil on different cores.
'''
m
y is the mutual admittance between secondary coil.
As a result of PGT, the primitive admittance matrix in
Equation (18) has a simple and symmetrical structure,
which can easily be shown in its short form as in Equa-
tion (19).
phase mm
m phase m
mm phase
abc
h
h













yyy
yyyy
yyy
(19)

O. GÜL ET AL. 243
m


y is generally taken as zero for each harmonic. It
is therefore that the primitive matrix abc
h


y becomes
equal to the primitive matrix of three single-phase trans-
formers connected to each other, based on their connec-
tion group.
Step 3
Transformers’ performance equations in real values
are converted to expressions in p.u. by using the turns
ratio of transformers. Moreover, phase shifting origi-
nated from connection group of three-phase transformer
(Delta-Wye) should be included in mathematical model
of the transformers. Furthermore, instead of modelling
the voltage regulators individually, phase shifting of
voltage regulators can also be included in transformers’
mathematical model on the condition that the model of
voltage regulator itself is not needed.
In this study, transformers’ performance equations are
assumed to be given in p.u. and the equivalent circuit of
Figure 6 is used in order to include both magnitude and
phase shifting in the mathematical model.
Apart from these, the phase shifting operations in
phase shifting three-phase transformers are included in
the mathematical model in the same way.
When the aforementioned conditions, which represent
the very general form of transformer model, are taken
into consideration, voltage ratio for each phase in three
-phase transformer are given as Equation (20) with the
assumption that phases have different complex turns ra-
tio. As for the current turns ratio of the transformer, it is
given in Equation (21) as complex conjugate of voltage
turns ratio.
k
kv
vk
v
10
, ka, b, c
01





α
δβ (20)
*
kk
iv
, ka, b, c



δδ (21)
With these assumptions, sub-matrices of primitive
admittance matrix of hth harmonics are given in general
form as Equations (22) and (23).
kkkphase k
iv
.., ka, b, c
 


 
yδyδ (22)
8-terminal
component
c1
c2 a2
a1
b1b2
r
d1
r
c2
a1
a2
b1b2
c1
d1
Figure 6. (a) Multi-terminal representation of Y- connect-
ed three-phase transformer; (b) oriented graph of Y- con-
nected transformer.
 
Y
I2 I1
V1V2
Figure 7. Basic equivalent circuit in p.u. for coupling be-
tween primary and secondary coils with both primary and
secondary off-nominal tap ratios of  and .
kkm
iv
.., ka, b, c; (k); a, b, c


   
yδyδ
 
(23)
Since phase shifts in voltage and current equations due
to transformer connection type are same for all phases,
the primitive admittance matrices can be given as Equa-
tions (24) and (25). As a result of -connection of the
transformer, changes in phase angles and magnitudes in
these equations are same. For this reason, the structure of
primitive admittance matrix has become simple and
powerful in computation.
 
*
ab cphase
hv v
hhh
..
 

 
yyyδyδ (24)
 
*
vv
..
ab ac ba bc ca
hhhhh
cb m
h
h





 



yyyyy
yδyδ
(25)
In this study, phase shifting due to transformers con-
nection group is included in the mathematical model
through above equations.
Step 4
To form the terminal equations (or mathematical mo-
del) of three phase transformer, closed loop equations for
voltage should be obtained as Equation (26).

*
abcd
B U0





V
V (26)
As shown in Equation (27), let


abcd
B be divided
into two sub-matrices as


abc
B and


d
B, which are
sub-loop matrix related to phases and basic loop matrix
associated with the nodes of star connection, respec-
tively.




abcdabc d
BB B (27)
The matrix


d
B exist when the star connection node
is not grounded. As for the matrix


abc
B, it can be
written as Equation (28) for the six most common con-
nections of three-phase transformers (Yg-Yg, Yg-Y,
Yg-, Y-Y, Y-, -).

abc
UB O
BOUB
BOU














(28)

O. GÜL ET AL. 244
where, the sub-matrix


B given as below is a matrix
associated with -connection of three-phase transform-
ers.

(1) 0
B0(1)







(29)
In Equation (29), the coefficients  and  take the val-
ues “−1” or “0” based on the transformers connection
type. Whilst  is “−1” when the primary windings of
transformer is wye,  is “0” when the primary side of the
transformer is delta. As for the coefficients , it takes the
value “−1” when secondary side is wye, and takes the
value “0” when secondary side is delta. Furthermore, the
sub-matrices


U and


O are given as below
 
10 00
U and O
01 00
 

 
 
(30)
Basic sub-loop matrix


d
B in Equation (27) can be
given as Equation (31), if the star connection nodes are
not grounded.

T
d ddd
B bbb

 (31)
If star connection node in one of the primary and sec-
ondary windings is not grounded, the sub-matrix


d
b
in Equation (31) can be given as Equation (32).




d
bλ
γ
 (32)
If the star connection nodes in both sides are not
grounded, then, the sub-matrix


d
b
is given by Equa-
tion (33)

dλ0
b0
γ



(33)
Step 5
As a consequence, bus admittance matrix of a trans-
former is given as a function of primitive admittance
matrix and loop equations as Equation (34).
 
T
abcdabcd abcabcd
bus h
hB..B
 


Yy (34)
3.3. Other Components
Loading should be included in the system representation
because of its damping effect near resonant frequencies.
However, an accurate model for the system load is diffi-
cult to determine because the frequency-dependent char-
acteristics are usually unknown and the load itself is
changing continuously. In general, if a load is linear, the
load is represented as an admittance using the CIGRE
load model at the interested frequencies [14]. If load is
nonlinear, the load is represented as an open circuit in
frequency scan study.
Capacitors are often placed in distribution networks to
regulate voltage levels and reduce real power loss. Ca-
pacitor bank size and locations are the most important
factors in determining the response of distribution system
to a harmonic source. For accurate representation of ca-
pacitors, a shunt capacitor can be modelled as wye con-
nected or delta connected constant admittance [2,3,7].
For harmonic studies of EDS, it is usually sufficient to
represent the entire transmission system by its 50 Hz
short-circuit equivalent resistance and inductance at the
high side of the substation transformer [7].
4. Three Phase Frequency Scan Analysis
An electric energy system simply consists of the resis-
tances (R), inductances (L) and capacitances (C). All
circuits containing both capacitance and inductance have
one or more natural resonant frequencies [5,14-17].
Normally electric energy systems are designed to operate
at frequencies of 50 Hz so as not to be under resonance
for fundamental frequency. However, certain types of
loads produce currents and voltages with frequencies that
are integer multiples of the 50 Hz fundamental frequency.
These higher frequencies are a form of electrical distor-
tion known as power system harmonics. When one of the
natural frequencies corresponds to an exciting frequency
produced by non-linear loads, harmonic resonance can
occur. Voltage and current will be dominated by the
resonant frequency and can be highly distorted. It is
therefore that for all effective harmonic frequencies, the
system should be analyzed for whether a resonance is
going to occur or not. Frequency scan analysis is the
most common and effective method to detect the har-
monic condition in a network. The simplest way to de-
termine the frequency response of a network is to im-
plement frequency or impedance scan study. The process
of frequency scan study can be performed by solving the
network equation for the frequencies of h.fo. Here, f0 is
the fundamental frequency and h is the harmonic order.






hhh
VZI (36)
where


h
V is the nodal vector,


h
I is the current
vector.
The aim of the frequency scan study is to determine
impedance as a function of frequency. The frequency
scan technique basically involves following steps:
A current injection, which is a scan of sine waves of
magnitude one, is firstly applied to the bus of interest.
Secondly a resultant voltage of that bus is measured.
hhhhh h
1 0

VIZVZ Z (37)
For large-scaled systems, h Z is simply derived from
bus impedance matrix, which is defined by taking the

O. GÜL ET AL. 245
inverse of bus admittance matrix as below:
 
-1
hh
ZY (38)
and the expanded form of nodal impedance matrix at any
frequency is

h
h






  



11 121n
21 222n
n1 n2nn
ZZ Z
ZZ Z
Z
ZZZ
(39)
where
ij
Z = transfer impedance between nodes i and j
ii
Z= driving point impedance at node i, r.
The effects of, the transformer connection types and
the mutual coupling between lines on frequency response
are examined on a three-bus industrial test system shown
in Figure 8 by using the frequency scan technique.
The test system consists of three busses utility, IND1
and IND2. IND1 and IND2 busses are connected through
a short three-phase and four-wire line. The system is
supplied by the utility through 69/13.8 kV transformer.
While a motor and linear load are connected on bus
IND1, a harmonic producing nonlinear load and a linear
load are connected on bus IND2. Harmonic currents of
the nonlinear load are given in the Table 1.
Since zero sequence harmonics are not found and
since only one harmonic source is present in the test sys-
tem, the system can be assumed to be balanced and
symmetric. That is why single phase analysis can be used
to solve this system.
The values on Table 1 are calculated in pu system.
The selected base quantities are 10.000 kVA and 13.8 kV.
The data and calculations are available on the web site
http://www.ee.ualberta.ca/pwrsys/IEEE/download.html.
The following assumptions are made in the analysis. 1)
The load points are supplied from an infinite bus system.
2) The linear loads are modeled with its series resistance
and reactance. 3) For the motor loads, a locked rotor im-
pedance are used [11,12,13].
In this study, three-phase models are used and follow-
ing four-cases are considered in the frequency scan si-
mulation of the test system. The frequency responses
with three-phase models are given in comparison to sin-
gle-phase models in the Figures 9-12 for each of the
following cases. Firstly, the transformer connection type
is selected as wye-grounded/wye-grounded.
In the second case, the mutual coupling between lines
is taken into consideration for the transformer connection
type as wye-grounded/wye-grounded and the value of
mutual impedance is taken as one-third of phase imped-
ance. The system is still symmetric in this case. So, only
one-phase frequency response analysis is enough for the
system.
69kV
13.8kV
IND1 IND2
1
2
3
Figure 8. The considered three-bus industrial test system.
Figure 9. Frequency responses of single and three-phase
EDS (transformer connection type is wye-grounded/wye-
grounded).
Figure 10. Frequency responses of single and three-phase
EDS (transformer connection type is wye-grounded/wye-
grounded and there is a mutual coupling between lines).
Figure 11. Frequency responses of single and three-phase
EDS.

O. GÜL ET AL. 246
Table 1. Harmonic current spectrum of nonlinear load at
bus IND2.
h 5 7 11 13 17 19 23 25
%Ic1 0.2 0.143 0.091 0.077 0.059 0.053 0.0430.04
Ich 0.119 0.085 0.054 0.046 0.035 0.031 0.0260.024
h − 0 − 0 − 0 −0
h 29 31 35 37 41 43 47 49
%Ic1 0.034 0.032 0.029 0.027 0.024 0.023 0.0210.02
Ich 0.020 0.019 0.017 0.016 0.014 0.014 0.0120.012
h − 0 − 0 − 0 −0
In the third case, the transformer connection type is
selected as wye-grounded/delta and it is shown in Figure
11 that delta connection of transformer has an effect on
frequency response similar to that of case-2.
Finally we had the system modified so as to have an
asymmetric three-phase network. The asymmetry in the
fourth case is obtained by changing the compensation
capacitors values for the phases a, b and c as
a
C
X0.455j p.u, j068.0X b
C p.u. and c C
X0.193j
p.u respectively.
As in the second case, mutual impedance is taken as
one-third of phase impedance and the transformer con-
nection type is wye-grounded/wye-grounded too. It is
shown in Figure 12 that frequency responses of all
phases are different from each other. Hence, frequency
responses of each phase in asymmetric networks should
be determined individually. For large-scale distribution
systems modeled and analyzed by phase coordinates, the
modified MFT will reduce the number of computation
for large-scaled networks since the MFT is capable of
computing only the necessary elements in impedance
matrix instead of taking a full inverse of the bus admit-
tance matrix
Because of the asymmetric compensation capacitors
for the phases, three-phase frequency response becomes
different from single phase response on the condition
that coupling between lines and/or transformer connec-
tion types are taken into consideration in three-phase
modeling. The frequency responses of all phases in asy-
mmetric networks are different from each other.
5. Conclusions
In this paper, to provide savings in storage and computa-
tion time, frequency scan analysis in multiphase asym-
metric distribution system is realized either by combin-
ing PGT and MFT or individually. The solution algo-
rithms are based on PGT and MFT. Whilst the PGT uses
05 10 15 2025 30 35 40 45 50
0.5
1
1.5
2
2.5
3
3.5
h
Z (p.u.)
3 phase, a
(a)
05 10 15 20 25 30 35 40 4550
1
2
3
4
5
6
7
h
Z (p.u.)
3 phase, b
(b)
05 10 15 2025 30 35 4045 50
1
2
3
4
5
6
7
h
Z (p.u.)
3 phase, c
(c)
Figure 12. Frequency responses of asymmetric three-phase
network (a) frequency response of phase a; (b) frequency
response of phase b; (c) frequency response of phase c.
the concept of multi-terminal component modeling tech-
nique to obtain the harmonic dependent model of EDS in
terms of frequency scan, the MFT uses only the required
element of the bus impedance matrix on diagonal to de-
termine the frequency scan of the EDS. Therefore an
MFT based algorithm is given to find the latter.
In order to show the accuracy of the proposed tech-
nique, a symmetric three-phase industrial system with
3-bus is preferred for simplicity. As a result, single-phase
and three-phase frequency responses of EDS are ob-

O. GÜL ET AL. 247
tained. The results show that three-phase frequency re-
sponse becomes different from single phase response on
the condition that coupling between lines and/or trans-
former connection types are taken into consideration in
three-phase modeling. In addition, frequency responses
of all phases in asymmetric networks are different from
each other. Consequently, beside in asymmetrical mod-
eling, one can easily extract that a phase-coordinated
based model must be used to detect the frequency re-
sponse of EDS, even in the symmetrical modeling.
6. References
[1] IEEE Std. 519-1992, IEEE Recommended Practice and
Requirements for Harmonic Control in Electric Power
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[2] J. Arrillaga, B. C. Smith and N. R. Watson, “Power Sys-
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[3] IEEE Guide for Application of Shunt Power Capacitors,
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List of the Symbols and Abbreviations
EDS Electrical Distribution Systemb
PGT Phase Grouping Technique
MFT Matrix Factorization Technique
α Primary off-nominal tap ratings
β Secondary off-nominal tap ratings
V
Voltage phasor
I Current phasor
Y
Admittance phasor
Z Impedance phasor
Y Primitive admittance
Z Primitive impedance
B Basic loop matrices
N Bus numbers
d Star point index

Delta connection index
a, b, c Line indexes
M Mutual coupling
h harmonic index
* Complex conjugate

Branch index
j Imaginary unit
V Voltage index
I Current index


Matrix

T Matrix transpose