. 4
( 17)


transmission line of ¬‚at con¬guration and contains no line transpositions.


Power transformers are essential plant components of the power system. In general, they
provide the interface between sections of the network with different rating voltages, for
example a generating plant and the transmission network, a static VAR compensation (SVC)
and the transmission network. Transformers consist of two or three copper windings per
phase and one or more iron cores. They are normally contained in metallic enclosures (i.e.
tanks), and are immersed in high-grade oil for insulation purposes (Grainger and Stevenson,
From the modelling point of view, it is convenient to separate the electric circuit, formed
by the copper windings, from the magnetic circuit, formed by the iron core. The reactances
of the windings can be found from short-circuit tests, and the iron-core reactances can be
found from open-circuit tests. The three-phase windings of power transformers may be
connected in a number of ways, but in high-voltage transmission the most popular
connections are: (1) star“star, (2) delta“delta, and (3) star“delta. Furthermore, the star
point can be either solidly grounded, grounded through an earthing impedance, or it may be
In power transformers the magnetising current usually represents only a small percentage
of the load current. However, this current is rich in harmonics, and a detailed representation
of the magnetic circuit is mandatory in studies involving harmonic frequencies. In
fundamental frequency studies, such as power ¬‚ows, this requirement is not as severe and it
is waved in most cases, unless the study is aimed at conducting an accurate assessment of
power system losses (Acha and Madrigal, 2001; Arrillaga et al., 1997).

3.3.1 Single-phase Transformers

The starting point for developing comprehensive power ¬‚ow transformer models is the
schematic representation of the basic two-winding transformer shown in Figure 3.10. The
windings contain resistance but it is assumed that the core does not saturate and exhibits no
The two transformer windings, termed primary (p) and secondary (s) windings contain
Np and Ns turns, respectively. The voltages and currents existing in both windings are related
by a matrix of short-circuit (sc) impedance parameters, as given by the following

1 3
2 4

Two-winding transformer
Figure 3.10

! ! !
Vp Zsc p Zsc m Ip
¼ °3:68Þ
Vs Zsc m Zsc s Is
Zsc p ¼ Rp þ jXsc p ; >
Zsc s ¼ Rs þ jXsc s ; °3:69Þ
Zsc m ¼ jXsc m :
These parameters are obtained by measurements of the actual transformer, where Rp and Rs
are the resistances of the primary and secondary windings, respectively. The reactances
Xsc p ; Xsc s , and Xsc m are short-circuit reactances obtained by exciting two terminals of the
transformer shown in Figure 3.10, at reduced voltage, and short circuiting the other two. The
ratio of excitation voltage to short-circuit current gives the relevant short-circuit reactance:
E12 E34 E13
Xsc p ¼ ; Xsc s ¼ ; Xsc m ¼ : °3:70Þ
I34 I12 I24
From the point of view of system analysis there are advantages in expressing the short-
circuit impedance matrix of Equation (3.68) in admittance form:
! ! !
ÀYsc m Vp
Ip Ysc p
¼ ; °3:71Þ
ÀYsc m
Is Ysc s Vs
Zsc s
¼ >
Ysc p
Zsc p Zsc s À Zsc m > >
Zsc p
¼ ; °3:72Þ
Ysc s
Zsc p Zsc s À Zsc m >
Zsc m
¼ ;
Ysc m
Zsc p Zsc s À Zsc m

It is observed that up to three short-circuit tests may be required to characterise the matrices
of short-circuit parameters. However, the primary and secondary short-circuit admittances
are almost the same when expressed in per-unit values, say Ysc. Owing to the strong
magnetic coupling afforded by iron cores, the mutual admittance between primary and

secondary windings can also be taken to have a value of Ysc . Hence, the transformer short-
circuit admittance matrix, in per units, is:
! ! !
Ysc ÀYsc Vp
¼ : °3:73Þ
ÀYsc Ysc Vs

3.3.2 Simple Tap-changing Transformer

The effect of expressing the transformer parameters in the per-unit system is to transform
the original voltage ratio Np : Ns into a unity voltage ratio 1 : 1. This enables a simple
equivalent circuit consisting of the short-circuit admittance Ysc connected between the
primary bus (p) and the secondary bus (s) to describe adequately the system performance of
the two-winding transformer.
However, power transformers are often ¬tted with a tap-changing mechanism to enable a
degree of voltage magnitude regulation at one of the transformer terminals. This is achieved
by injecting a small variable voltage magnitude in phase (added or subtracted) with the
voltage magnitude at the output winding. Such transformers are termed load tap-changing
(ltc) transformers and play an important role in power ¬‚ow studies. The representation of an
ltc transformer may be achieved by the series connection of the short-circuit admittance
representing a per-unit transformer and an ideal transformer with taps ratio T : 1 (Laughton,
1968). This arrangement is shown in Figure 3.11.

Vp Vs
Ip Is

T :1
Simple tap-changing transformer
Figure 3.11

The following relationships exist in the ideal transformer,
T I0
¼; ¼: °3:74Þ
Vs 1 1 I
The current across the admittance Ysc is:
I ¼ Ysc Vp À V ¼ Ysc Vp À TVs ¼ Ip : °3:75Þ
I 0 ¼ TI ¼ Ysc TVp À T 2 Vs ¼ ÀIs : °3:76Þ

Combining Equations (3.75) and (3.76) in matrix form gives:
! ! !
ÀTYsc Vp
Ip Ysc
¼ : °3:77Þ
ÀTYsc T 2 Ysc
Is Vs

3.3.3 Advanced Tap-changing Transformer

Following the same line of reasoning, a comprehensive power system transformer model is
derived for a single-phase three-winding transformer (Acha, Ambriz-Perez, and Fuerte-
Esquivel, 2000). Each winding is represented as the series combination of a short-circuit
admittance and an ideal transformer. Furthermore, each winding is provided with a complex
tap-changing mechanism to allow for tap-changing and phase-shifting facilities. Moreover,
the magnetising branch of the transformer is included to account for the core losses.
Figure 3.12 shows the equivalent circuit of the three-winding transformer.

I2′ Ysc s
V2 I2 Is
Ysc p I ′
Ip I1 V0

V3 I3 Ysc t
I0 It
Ti:1 1:Wv

Figure 3.12 Comprehensive tap-changing transformer. Reproduced, with permission, from E. Acha,
H. Ambriz-Perez, and C.R. Fuerte-Esquivel, ˜Advanced Transformer Control Modelling in an Optimal
Power Flow using Newton™s Method™, IEEE Trans. Power Systems 15(1) 290“298, # 2000 IEEE

The primary winding is represented as an ideal transformer having complex tap ratios
Tv : 1 and Ti : 1 in series with the admittance Ysc p , where Tv ¼ Tià ¼ t þ j ¼ T¬t . The
symbol à denotes the conjugate operation. The secondary winding is represented as an ideal
transformer having complex tap ratios Uv : 1 and Ui : 1 in series with the admittance Ysc s ,
where Uv ¼ Uià ¼ u þ j ¼ U¬u . Similarly, the ideal transformer in the tertiary winding
has complex tap ratios Wv : 1 and Wi : 1 in series with an admittance Ysc t , where
Wv ¼ Wià ¼ w þ j
¼ W¬w . It is assumed here that Ysc p , Ysc s and Ysc t are the short-circuit
admittances of the primary, secondary, and tertiary windings, respectively. The magnetising
branch of the transformer is represented by the admittance Y0 ¼ G0 þ jB0 .
The resistive path of the magnetising branch is directly related to the iron losses, and its
conductance G0 draws a current that varies linearly with the voltage across the magnetising
branch. However, in the inductive path the relationship between the current and the voltage
is dictated by the rms V“I characteristic, which under saturating conditions becomes non-

The following relationships exist in the ideal primary, secondary, and tertiary
V1 Tv Ti I1
¼ ; and ¼; °3:78Þ
V0 1 1 I1
V2 Uv Ui I2
¼ ; ¼; °3:79Þ
V0 1 1 I2
V3 Wv Wi I3
¼ ; ¼: °3:80Þ
V0 1 1 I3

The currents across the admittances Ysc p, Ysc s, and Ysc t are, respectively,
I1 ¼ Ysc p Vp À V1 ¼ Ysc p Vp À Tv V0 ¼ Ip ; °3:81Þ
I2 ¼ Ysc s °Vs À V2 Þ ¼ Ysc s °Vs À Uv V0 Þ ¼ Is ; °3:82Þ
I3 ¼ Ysc t °Vt À V3 Þ ¼ Ysc t °Vt À Wv V0 Þ ¼ It ; °3:83Þ

and at the centre of the transformer the following relationship holds:
0 0 0
0 ¼ I1 þ I2 þ I3 À I0 ¼ Ti I1 þ Ui I2 þ Wi I3 À I0 : °3:84Þ
Substituting Equations (3.81)“(3.83) into Equation (3.84) gives:
À2 Á
à à Ã
0 ¼ ÀTv Ysc p Vp À Uv Ysc s Vs À Wv Ysc t Vt þ Tv Ysc p þ Uv Ysc s þ Wv Ysc t þ Y0 V0 : °3:85Þ
2 2

Putting Equations (3.81)“(3.83) and (3.85) in matrix form gives:
2 32 32 3
ÀTv Ysc p
Ysc p 0 0
Ip Vp
6 Is 7 6 76 Vs 7
ÀUv Ysc s
0 Ysc s 0
6 7¼6 76 7:
4 It 5 4 54 Vt 5
ÀWv Ysc t
0 0 Ysc t
à à Ã
ÀTv Ysc p ÀUv Ysc s ÀWv Ysc t Tv Ysc p þ Uv Ysc s þ Wv Ysc t þ Y0
2 2 2
0 V0
Equation (3.86) represents the transformer shown in Figure 3.12. However, it is possible
to ¬nd a reduced equivalent matrix that still models the transformer correctly while retaining
only the external buses p, s, and t. This is done by means of Gaussian elimination:
2 3 2 Ã
Uv Ysc p Ysc s þ Wv Ysc p Ysc t þ Ysc p Y0 ÀTv Uv Ysc p Ysc s
2 2
6 7 16 Ã
4 Is 5 ¼ 4 ÀTv Uv Ysc p Ysc s Tv Ysc s Ysc p þ Wv Ysc s Ysc t þ Ysc s Y0
2 2
à Ã
ÀTv Wv Ysc p Ysc t ÀUv Wv Ysc s Ysc t
32 3
ÀTv Wv Ysc p Ysc t Vp
76 7
ÀUv Wv Ysc s Ysc t 54 V s 5; °3:87Þ
Tv Ysc t Ysc p þ Uv Ysc t Ysc s þ Ysc t Y0
2 2

D ¼ Tv Ysc p þ Uv Ysc s þ Wv Ysc t þ Y0 :
2 2 2

The nodal admittance representation of a two-winding transformer can be easily obtained
by introducing simplifying assumptions in Equation (3.87). For instance, when the tertiary
winding does not exist, the row and column corresponding to this bus become redundant and
they are removed from Equation (3.87). Moreover, the tap ratios Wv and Wi become zero.
Hence, the nodal admittance matrix equation representing the two-winding transformer is
arrived at:
! ! !
1 Uv Ysc p Ysc s þ Ysc p Y0 ÀTv Uv Ysc p Ysc s
Ip Vp
¼ ; °3:88Þ
ÀTv Uv Ysc p Ysc s Tv Ysc s Ysc p þ Ysc s Y0 Vs
D 2
D ¼ Tv Ysc p þ Uv Ysc s þ Y0 :
2 2

It must be noted that owing to the ¬‚exibility of the two-winding transformer model in
Equation (3.88), it is possible to assemble a transformer model that represents the
transformer circuit shown in Figure 3.12 by using three of these two-winding transformer
models. An example of how this can be achieved is shown elsewhere (see Acha, Ambriz-
Perez, and Fuerte-Esquivel, 2000).
Transformer models with more constrained tapping arrangements can also be derived
from Equation (3.88). For instance, take the case of the tap-changing transformer shown in
Figure 3.11, represented by Equation (3.77). Such a representation can be derived from
Equation (3.88) by including no magnetising branch, Y0 ¼ 0, and a nominal tapping position
for the secondary winding, Uv ¼ 1. Moreover, the tapping position of the primary winding
is real as opposed to complex, Tv ¼ T, and the short-circuit admittance is assumed to be all
on the primary side, Ysc s ¼ 0 and Ysc p ¼ Ysc . The latter consideration requires application of
L™Hopital differentiation rule with respect to Ysc s .
A further strength of the transformer model in Equation (3.88) is that, owing to the
complex nature of their taps, it represents rather well the system behaviour of a phase-
shifting (PS) two-winding transformer. This is more easily appreciated if it is assumed that
in Equation (3.88) both complex taps have unit magnitudes:
Tv ¼ 1¬t ¼ cos t þ j sin t ;
Uv ¼ 1¬v ¼ cos v þ j sin v :
Ysc p °Ysc s þ Y0 Þ
Ip 1
Ysc p þ Ysc s þ Y0 ÀYsc p °cos t À j sin t Þ ‚ Ysc s °cos u þ j sin u Þ
! !
ÀYsc p °cos t þ j sin t Þ ‚ Ysc s °cos u À j sin u Þ Vp
À Á : °3:90Þ
Ysc s Ysc p þ Y0 Vs
This is a comprehensive model of a PS transformer that yields very ¬‚exible power ¬‚ow and
optimal power ¬‚ow PS models, as will be shown in Chapters 4 and 7, respectively.

3.3.4 Three-phase Transformers

Based on nodal analysis, quite general models for multiwindings, multiphase transformers
can be derived (Chen and Dillon, 1974; Laughton, 1968). The essence of the method is to
transform the short-circuit parameters of the transformer windings, suitably arranged in a

matrix of primitive parameters Y , into nodal parameters Y . This is done with the help
of appropriate connectivity matrices, namely, C and C . The connectivity matrices relate
the voltages and currents in the unconnected transformer windings to the phase voltages and
currents when the three-phase transformer is actually connected.
The primitive and nodal parameters are related by the following matrix expression:

Y ¼ C Y C : °3:91Þ

The primitive parameters of three identical single-phase transformers, for which the
terminals between transformers are not connected in any way but contain off-nominal
tapping facilities on the primary winding, have the following arrangement:
! ! ! 9
ÀTv Ysc
I1 V1 >
¼ ; >
à >
ÀTv Ysc 2
Tv Ysc >
I2 V2
2 3 2 32 3 >
ÀTv Ysc >
Ysc 0 0 0 0
I1 V1 >
6 I 7 6 ÀT Ã Ysc 76 V 7 >
6 27 6 v 76 2 7 >
Tv Ysc 0 0 0 0
676 76 7
! ! ! =
6 I3 7 6 0 76 V3 7
ÀTv Ysc ÀTv Ysc
Ysc 0 Ysc 0 0
I3 V3
) 6 7¼6 76 7 ;
¼ 6I 7 6 0 0 76 V4 7 >
à Ã
ÀTv Ysc ÀTv Ysc
2 2
76 7 >
6 47 6
Tv Ysc 0 Tv Ysc 0
I4 V4
76 7 > >
ÀTv Ysc 54 V5 5 >
4 I5 5 4 0 >
0 0 0 Ysc >
V6 > >
ÀTv Ysc >
0 0 0 0 Tv Ysc
I6 >
! ! ! >
ÀTv Ysc >
I5 V5 >
¼ :
ÀTv Ysc 2
Tv Ysc
I6 V6
In general, these matrix equations may be expressed in compact form:

I ¼Y V : °3:93Þ

The three single-phase transformers, when suitably connected, electrically speaking, may
serve the purpose of transforming three-phase voltages and currents. The assembly is termed
a ˜three-phase bank™. Each single-phase unit in the bank is closely associated with one phase
of the three-phase system. Depending on the electrical connection and operating conditions,
there may be currents from more than one phase circulating in one single-phase unit at any
one time, but there are not ¬‚ux interactions between windings of different units.
Quite a different situation prevails in multilimb transformers, where all windings of the
three-phase unit are magnetically coupled. The primitive admittance matrix equation of
the two-winding, three-phase transformers is a full matrix, and up to 21 short-circuit tests
may be required to de¬ne fully this primitive admittance matrix. In the remainder of this
chapter only the three-phase bank will be addressed.
The three most popular three-phase transformer connections found in high-voltage
transmission are addressed below, namely the star“star, delta“delta, and star“delta. To
determine their nodal admittance matrix models, one requires information of the matrix of
primitive parameters, Y , and the relevant connectivity matrices, C and C . Star“star connection

The three-phase connection is shown in Figure 3.13 when the windings are connected in star“
star con¬guration, with both star points grounded through admittances, YN and Yn, respectively.

IB Ysc
Ysc Ysc

Star“star connection
Figure 3.13

The transformation matrix, which relates the voltages existing in the unconnected
transformer to the voltages in the connected three-phase transformer shown in Figure 3.13,
is given explicitly in Equation (3.94):
2 32 32 3
0 À1 VA
V1 1 0 0 0 0 0
6 V2 7 6 0 À1 76 VB 7
0 0 1 0 0 0
676 76 7
6 V3 7 6 0 0 76 V C 7
0 À1
1 0 0 0
676 76 7
6 V4 7 6 0 À1 76 Va 7
0 0 0 1 0 0
6 7¼6 76 7: °3:94Þ
6 V5 7 6 0 0 76 V b 7
0 À1
0 1 0 0
676 76 7
6 V6 7 6 0 À1 76 Vc 7
0 0 0 0 1 0
676 76 7
4 V7 5 4 0 0 54 V N 5
0 0 0 0 0 1
V8 0 0 0 0 0 0 0 1 Vn

In compact form, we have,
V ¼ C V : °3:95Þ

The nodal matrix representation of this transformer connection is obtained by substituting
Equations (3.92) and (3.94) into Equation (3.91):
2 32 32 3
ÀTv Ysc ÀYsc
Ysc 0 0 0 0 Tv Ysc
6 IB 7 6 0 76 VB 7
ÀTv Ysc ÀYsc
Ysc 0 0 0 Tv Ysc
676 76 7
6 IC 7 6 0 76 VC 7
ÀTv Ysc ÀYsc
0 Ysc 0 0 Tv Ysc
676 76 7
6 Ia 7 6 ÀTv Ysc ÀTv Ysc 76 Va 7
à Ã
2 2
0 0 Tv Ysc 0 0 Tv Ysc
6 7¼6 76 7
6 Ib 7 6 0 ÀTv Ysc 76 Vb 7
à Ã
ÀTv Ysc 2 2
0 0 Tv Ysc 0 Tv Ysc
676 76 7
6 Ic 7 6 0 ÀTv Ysc 76 Vc 7
à Ã
ÀTv Ysc 2 2
0 0 0 Tv Ysc Tv Ysc
676 76 7
4 IN 5 4 ÀYsc À3Tv Ysc 54 VN 5
ÀYsc ÀYsc Tv Ysc 3Ysc þ YN
Tv Ysc Tv Ysc
à à à Ã
Tv Ysc ÀTv Ysc ÀTv Ysc ÀTv Ysc À3Tv Ysc 3Tv Ysc þ Yn
2 2 2 2
Tv Ysc Tv Ysc
In Vn

If both star points N and n are solidly grounded then the nodal voltages VN and Vn become
zero. Hence, the rows and columns corresponding to bus N and bus n become redundant and

are deleted from matrix Equation (3.96):
232 32 3
ÀTv Ysc
Ysc 0 0 0 0
6 IB 7 6 0 7 6 VB 7
ÀTv Ysc
Ysc 0 0 0 76 7
6 IC 7 6 0 ÀTv Ysc 76 VC 7
0 Ysc 0 0
6 7¼6 76 7: °3:97Þ
6 Ia 7 6 ÀT Ysc 0 7 6 Va 7
à 2
0 0 Tv Ysc 0
676v 76 7
4 Ib 5 4 0 0 5 4 Vb 5
ÀTv Ysc 2
0 0 Tv Ysc
ÀTv Ysc 2
0 0 0 0 Tv Ysc
Ic Vc Delta“delta connection

This transformer connection is shown in Figure 3.14. In the delta connection the following
relationships exist between the voltages and currents in the connected and unconnected
V ¼ p¬¬¬ C V ; °3:98Þ
3I ¼ C I : °3:99Þ

IB Ysc Ysc Ysc Ysc
IC Ysc

Delta“delta connection
Figure 3.14

The relevant connectivity matrices for this transformer connection are set up and, upon
substitution in Equation (3.91), the following nodal admittance matrix is arrived at:
23 2 32 3
ÀYsc ÀYsc À2Tv Ysc
2Ysc Tv Ysc Tv Ysc
6 IB 7 6 ÀYsc Tv Ysc 76 VB 7
ÀYsc À2Tv Ysc
2Ysc Tv Ysc
67 6 76 7
6 I 7 1 6 ÀY À2Tv Ysc 76 VC 7
6 C7 6 76 7
2Ysc Tv Ysc Tv Ysc
6 7¼ 6 76 7:
à à Ã
6 Ia 7 3 6 À2Tv Ysc ÀTv Ysc ÀTv Ysc 76 Va 7
2 2 2
Tv Ysc Tv Ysc 2Tv Ysc
67 6Ã 76 7
4 Ib 5 4 T Ysc ÀTv Ysc 54 Vb 5
à Ã
À2Tv Ysc ÀTv Ysc
2 2 2
Tv Ysc 2Tv Ysc
à à Ã
À2Tv Ysc ÀTv Ysc ÀTv Ysc
2 2 2
Tv Ysc Tv Ysc 2Tv Ysc
Ic Vc
°3:100Þ Star“delta connection

This transformer connection is shown in Figure 3.15 for the case when the star point is
solidly grounded. Following a similar procedure to that used to derive the nodal admittance

IB Ysc Ysc
VB Ysc

Star“delta connection
Figure 3.15

matrices of the star“star and delta“delta connections, the nodal matrix representation of this
transformer connection is:
p¬¬¬ p¬¬¬
2 32 32 3
ÀTv Ysc
Ysc 0 0 3 Tv Ysc 3 0
p¬¬¬ p¬¬¬ 76 7
ÀTv Ysc 3
6 IB 7 6 Tv Ysc 3 76 VB 7
0 Ysc 0 0
676 p¬¬¬ 76 7
6I 7 6 ÀTv Ysc 3 76 VC 7
6 C7 6 76 7
0 0 Ysc Tv Ysc 3 0
6 7¼6 p¬¬¬ p¬¬¬  76 7:
6 Ia 7 6 ÀTv Ysc 3 ÀTv Ysc 3 76 Va 7
à Ã
ÀTv Ysc 3
2 2 2
0 Tv Ysc 3 2Tv Ysc 3
676  76 7
6 I 7 6 à p¬¬¬ p¬¬¬   76 7
4 b 5 4 Tv Ysc 3 ÀTv Ysc 3 ÀTv Ysc 3 ÀTv Ysc 3 54 Vb 5
2 2 2
0 2Tv Ysc 3
p¬¬¬ p¬¬¬   
à Ã
Tv Ysc 3 ÀTv Ysc 3 ÀTv Ysc 3 ÀTv Ysc 3
2 2 2
Ic Vc
0 2Tv Ysc 3

3.3.5 Sequence Domain Parameters

Transformer parameters are also amenable to representation in the frame of reference of the
sequences (Chen and Dillon, 1974). The matrix of symmetrical components and its inverse,
given in Equations (3.57), are used to such effect. This requires that the order of all matrices
involved in the exercise be a multiple of three. This characteristic is met by matrices
representing the star“star connected transformer with both star points solidly grounded, the
delta“delta transformer, and the star“delta transformer with the star point solidly grounded.
It should be noted that the symmetrical components transform given in Equations (3.57)
cannot directly be applied to cases of star-connected windings, where one or two star points
are not grounded or are grounded through earthing impedances. In such cases, Kron™s
reductions are applied ¬rst to ¬nd out reduced equivalent representations which are a
function only of phase terminals. This follows the spirit of the procedure presented in
Section 3.2.2 for the elimination of transmission line ground wires.
A generic, compact representation of Equations (3.97), (3.100), and (3.101) correspond-
ing to the star“star, delta“delta, and star“delta connections may be expressed as:
! ! !
¼ ; °3:102Þ
Iabc YIV Vabc

where the order of matrices YI , YII , YIII , YIV is 3 ‚ 3 and suitable for direct treatment by
the matrix of symmetrical components, to enable representation in the frame of reference of
the sequences. This is achieved by applying the following symmetrical component
Yi ¼ TÀ1 YI TS ; > >
Yii ¼ TS YII TS ; =
Yiii ¼ TÀ1 YIII TS ; >
Yiv ¼ TS YIV TS :

Table 3.1 shows matrices Yi , Yii , Yiii , and Yiv in explicit form, for the star“star, delta“delta,
and star“delta transformer connections.

Table 3.1 Transformer sequence domain admittances
Matrix type Star“star Delta“delta Star“delta
2 3 2 3 2 3
Ysc 0 0 00 0 Ysc 0 0
40 05 4 0 Ysc 05 40 05
Ysc Ysc
0 0 Ysc 00 Ysc 0 0 Ysc
2 3 2 3 2 3
Tv Ysc 0 0 0 0 0 0 0 0
40 05 40 05 40 5
Tv Ysc ¬30
Tv Ysc Tv Ysc 0
Tv Ysc ¬ À30
0 0 Tv Ysc 0 0 Tv Ysc 0 0
2 3 2 3 2 3
Tv Ysc 0 0 0 0 0 0 0 0
40 05 4 0 Tv Ysc 05 4 0 Tv Ysc ¬ À 30 5
à à Ã
Tv Ysc 0
à Ã
Tv Ysc ¬30
0 0 Tv Ysc 0 0 Tv Ysc 0 0
2 3 2 3 2 3
2 0 0 0 0 0 0
Tv Ysc 0 0
40 05 4 0 Tv Ysc 05 4 0 Tv Ysc 05
2 2
Yiv Tv Ysc
2 2
2 0 0 Tv Ysc 0 0 Tv Ysc
0 0 Tv Ysc

The sequence domain representation of a transformer, in compact form, is:

!" # !
°iÞ °iiÞ
I012p V012p
¼ ; °3:104Þ
°iiiÞ °ivÞ
I012s V012s

where the subscripts 0, 1, and 2 refer to zero, positive, and negative sequence quantities,
respectively. It has been emphasised in various points.
Careful examination of the sequence domain parameters indicates that three independent
transfer admittance matrix equations, leading to three independent circuits, are generated for
a three-phase transformer. This is more easily realised if the transformer taps are taken to be

real as opposed to complex, yielding symmetrical matrix equations and, hence, reciprocal
circuits. The star“star, delta“delta, and star“delta connections share the same positive and
negative sequence equivalent circuits, given in Figure 3.16.

Ip Is

Vp Vs

Figure 3.16 Positive and negative sequence equivalent circuit for the star“star, delta“delta, and star“
delta connections

In contrast, the zero sequence equivalent circuits for the three connections differ from one
another. The equivalent circuits are shown in Figures 3.17(a), 3.17(b), and 3.17(c) for the
star“star, delta“delta, and star“delta connections, respectively.

I0p I0s Ysc

(1’T)Ysc V0p
V0p V0s
V0p V0s

(a) (b)

Zero sequence equivalent circuits for: (a) star“star, (b) delta“delta, and (c) star“delta
Figure 3.17

It should be noted that for the star“delta transformer connection the primary and
secondary terminals of the zero sequence equivalent circuit are not electrically connected.
However, the primary terminal contains an admittance Ysc connected between this terminal
and the reference. It is also interesting to note that the positive and negative transfer
admittances contain an asymmetrical phase shift of 30 between the primary and secondary
terminals giving rise to nonreciprocal equivalent circuits. The asymmetrical phase shift is
entirely attributable to the star“delta connection and it is present even when no taps are
availabe in the transformer. It is common practice in application studies, such as positive
sequence power ¬‚ow and sequence domain-based fault levels to ignore the phase shift
during the calculations and then to account for it during the analysis of results.


In general, synchronous machines are grouped into two main types, according to their rotor
structure: round rotor and salient pole machines (Grainger and Stevenson, 1994). Steam
turbine driven generators (turbogenerators) work at high speed and have cylindrical rotors.
The rotor carries a DC excited ¬eld winding. Hydro units work at low speed and have salient
pole rotors. They normally have damper windings in addition to the ¬eld winding. Damper
windings consist of bars placed in slots on the pole faces and connected together at both
ends. Turbogenerators contain no damper windings but the solid steel of the rotor offers a
path for eddy currents, which have similar damping effects.
For simulation purposes, the currents circulating in the solid steel or in the damping
windings can be treated as currents circulating in two closed circuits. Accordingly, a three-
phase synchronous machine may be assumed to have three stator windings and three rotor
windings. This is illustrated in Figure 3.18, where all six windings are magnetically coupled.



q (t)


Figure 3.18 Schematic representation of a three-phase synchronous generator. Redrawn by
permission of the Institution of Electrical Engineers from R.G. Wasley and M.A. Shlash, ˜Steady-
state Phase-variable Model of the Synchronous Machine for Use in 3-phase Load-¬‚ow Studies™,
Proceedings of the IEEE 121(10) 1155“1164 # 1974 IEEE

The relative position of the rotor with respect to the stator is given by the angle  between
the rotor™s direct axis and the stator™s phase a axis, termed the d axis and a axis, respectively.
In the stator, the axis of phases a, b, and c are displaced from each other by 120 electrical
degrees. In the rotor, the d axis is magnetically centred in the north pole of the machine. A

second axis, located 90 electrical degrees behind the d axis is termed the quadrature axis or
q axis.
Three main control systems directly affect the turbine-generator set, namely the boiler™s
¬ring control, the governor control, and the excitation system control. The excitation system
consists of the exciter and absolute value recti¬er (AVR). The latter regulates the generator
terminal voltage by controlling the amount of current supplied to the ¬eld winding by the
exciter. For the purpose of steady-state analysis, it is assumed that the three control systems
act in an idealised manner, enabling the synchronous generator to produce constant power
output, to run at synchronous speed, and to regulate voltage magnitude at the generator™s
terminal with no delay and up to its reactive power design limits.

3.4.1 Machine Voltage Equation

The objective of this section is to derive a steady-state expression for the stator three-phase
voltages and currents of the synchronous generator (Wasley and Shlash, 1974b). The rotor
emfs (electromagnetic forces) and saliency are accounted for in the resulting voltage
equation, which may form the basis for connecting the machine model to a given three-
phase bus of an unbalanced power system representation.
With reference to Figure 3.18, using stator and rotor quantities expressed in frames of
reference attached to their respective physical circuits, namely stator and rotor circuits, the
instantaneous voltages of the machine may be expressed as:
v ¼ Ri þ pLi; °3:105Þ
where R and L are the machine resistance and inductance matrices, respectively, and p is the
time derivative operator.
Furthermore, expanding Equation (3.105) into stator and rotor subsets, we obtain:
! !! !! ! !
Gss Gsr is Lss Lsr pis
vs Rs 0 is
¼ þ !r þ ; °3:106Þ
Gtsr 0 Ltsr Lrr pir
vr 0 Rr ir ir
where G ¼ dL=d; !r , equal to d=dt, is the rotor speed; and  ¼ !r t þ .
The submatrix coef¬cients L, G, and R are:
2 3
Laa0 þ La2 cos°2Þ ÀLab0 À La2 cos°2 þ 60Þ ÀLab0 À La2 cos°2 À 60Þ
Lss ¼ 4 ÀLab0 À La2 cos°2 þ 60Þ Laa0 þ La2 cos°2 þ 120Þ ÀLab0 À La2 cos°2 À 180Þ 5;
ÀLab0 À La2 cos°2 À 60Þ ÀLab0 À La2 cos°2 À 180Þ Laa0 þ La2 cos°2 À 120Þ

2 3
Laf cos°Þ Laf cos°Þ ÀLaf sin°Þ
4 Laf cos° À 120Þ ÀLaf sin° À 120Þ 5;
Laf cos° À 120Þ
Lsr ¼ °3:108Þ
Laf cos° þ 120Þ Laf cos° þ 120Þ ÀLaf sin° þ 120Þ

2 3
Lfd Lmkd 0
4 Lmkd 0 5;
Lrr ¼ °3:109Þ
0 0 Lkq

2 3
À2La2 sin°2Þ 2La2 sin°2 þ 60Þ 2La2 sin°2 À 60Þ
4 2La2 sin°2 þ 60Þ À2La2 sin°2 þ 120Þ 2La2 sin°2 À 180Þ 5;
Gss ¼ °3:110Þ
2La2 sin°2 À 60Þ 2La2 sin°2 À 180Þ À2La2 sin°2 À 120Þ

2 3
ÀLaf sin°Þ ÀLaf sin°Þ ÀLaf cos°Þ
Gsr ¼ 4 ÀLaf sin° À 120Þ ÀLaf sin° À 120Þ ÀLaf cos° À 120Þ 5; °3:111Þ
ÀLaf sin° þ 120Þ ÀLaf sin° þ 120Þ ÀLaf cos° þ 120Þ

2 3
Ra 0 0
40 0 5;
Rr ¼ °3:112Þ
0 0 Rc
2 3
Rfd 0 0
Rr ¼ 4 0 0 5: °3:113Þ
0 0 Rkq
Since the rotor circuits are represented by a ¬eld winding on the d axis and two short-
circuited damper windings on the d axis and q axis, respectively, the rotor voltage vector
may be written as
v fd
vr ¼ 4 0 5 ; °3:114Þ
where vfd is the applied direct ¬eld voltage.
For the purpose of steady-state analysis, it will be assumed that the applied direct ¬eld
voltage equals the voltage drop across the ¬eld resistance owing to the DC component of the
¬eld current and that additional voltages from Rf if can be neglected. Using such a
simpli¬cation, the relevant part of Equation (3.106) is solved for pir :
pir ¼ ÀLÀ1 ½!Grs is þ Lrs pis Š: °3:115Þ

Assuming the following set of unbalanced stator currents:
2 3
I1 sin°!t þ 1 Þ
is ¼ 4 I2 sin°!t þ 2 Þ 5; °3:116Þ
I3 sin°!t þ 3 Þ
and the fact that the rotor runs at synchronous speed (i.e. !r ¼ !), we have,
2 3
Im cos° m Þ
4 Im cos° m Þ 5;
pir ¼ À!Laf LÀ1 °3:117Þ
m¼1 ÀIm sin° m Þ

¼ 2!t þ  þ 1 ; >
¼ 2!t þ  þ 2 À 120; °3:118Þ
¼ 2!t þ  þ 3 þ 120:

The expression for pir can be further simpli¬ed by substituting the inverse relation of
Equation (3.109) into Equation (3.117):
2 3
X k1 Im cos° m Þ
4 Àk2 Im cos° m Þ 5;
pir ¼ °3:119Þ
k2 Im sin° m Þ

!Laf Lmkd >
k1 ¼ À 1À =
Lfd Lkd
!Laf >
k2 ¼ :
It should be mentioned that the following practical simpli¬cations have been made while
substituting the inverse relation of Equation (3.109) into Equation (3.117): Lfd is much
greater than Lmkd, and Lkq ¼ Lkd.
Equation (3.119) is now integrated:
2 323
k1 Im sin° m Þ
1 X4
5 þ 4 0 5;
ir ¼ Àk2 Im sin° m Þ °3:121Þ
2! m¼1
Àk2 Im cos° m Þ 0

where ifd is the DC component of the ¬eld current.
Inspection of Equation (3.118) and (3.121) reveals that the presence of negative sequence
currents at the machine terminals gives rise to rotor currents of double the supply frequency.
In contrast, positive sequence currents are associated with zero frequency rotor currents,
other than the direct ¬eld current. Also, owing to balanced machine design considerations,
there is no contribution from zero sequence currents:
Substituting Equations (3.119) and (3.121) into Equation (3.106) we obtain a reduced
expression for the stator voltage vector:
2 3 2 32 3 2 32 3
va I1 sin°!t þ 1 Þ À1 À1 I1 cos°!t þ 1 Þ
Ra 0 0 1
676 76 7 6 76 7
4 vb 5 ¼ 4 0 0 54 I2 sin°!t þ 2 Þ 5 þ !Laa0 4 À1 1 À1 54 I2 cos°!t þ 2 Þ 5
vc I3 sin°!t þ 3 Þ À1 À1 I3 cos°!t þ 3 Þ
0 0 Rc 1

2 32 3
cos°!t þ 1 Þ cos°!t þ 2 À 120Þ cos°!t þ 3 þ 120Þ I1
°k1 À 2k2 ÞLaf 6 76 7
þ 4 cos°!t þ 1 þ 120Þ cos°!t þ 2 Þ cos°!t þ 3 À 120Þ 54 I2 5
cos°!t þ 1 À 120Þ cos°!t þ 2 þ 120Þ cos°!t þ 3 Þ I3

2 3
cos°!t À 1 þ 2Þ À cos°!t À 2 þ 2 þ 60Þ À cos°!t À 3 þ 2 À 60Þ
!La2 6 7
À 4 À cos°!t À 1 þ 2 þ 60Þ cos°!t À 2 þ 2 þ 120Þ À cos°!t À 3 þ 2 À 180Þ 5
À cos°!t À 1 þ 2 À 60Þ À cos°!t À 2 þ 2 À 180Þ cos°!t À 3 þ 2 À 120Þ

2 3 2 3
67 6 7
‚ 4 I2 5 À !Laf ifd 4 sin°!t þ  À 120Þ 5:
sin°!t þ  þ 120Þ

The last term in Equation (3.122) may be interpreted as an array of rotor emfs. Moreover,
taking the stator a phase as reference,
232 3
va V1 sin°!tÞ
4 v b 5 ¼ 4 V2 sin°!t þ 2 Þ 5: °3:123Þ
vc V3 sin°!t þ 3 Þ
It is seen that a root mean square (rms) form of Equation (3.122) may be established very
readily. Also, by negating the stator currents to correspond to generator operating
conditions, we have
ES ¼ ½RS þ j°X1 þ X2 ފIS þ jX3 Ià þ VS ; °3:124Þ

2 3
ÀLab0 ÀLab0
4 ÀLab0 ÀLab0 5;
X1 ¼ ! °3:125Þ
ÀLab0 ÀLab0 Laa0
2 3
Laf °k1 À 2k2 Þ 4 1 h
h2 5;
X2 ¼ °3:126Þ
h 1
h2 h 1
2 3
1 h
!La2 e
4 h2 1 5:
X3 ¼ À °3:127Þ
h 1
It is observed that the term Laf °k1 À 2k2 Þ=4 reduces to À!L2 =4Lfd if damper windings are
not present.
As a means of evaluating the reactance elements in Equation (3.124), it is noted that the
usually available dq0 reactances of the machine may be used in the following expressions:

!Laa0 ¼ Xd þ Xq þ X0 ; > >
!Lab0 ¼ Xd þ Xq À 2X0 ;
6 >
1À Á >
!La2 ¼ Xd À Xq :
Equation Á (3.124) includes the effect of machine saliency through matrix X3, where
Xd À Xq expresses the degree of saliency. Notice that if saliency can be ignored (i.e.
Xd ¼ Xq ) matrix X3 plays no role in machine performance. Also, X3 is dependent on
external circuit conditions through the machine angle . Matrix X2 contributes negative
sequence impedance, impairing the balanced behaviour of the machine.


In general, power system loads can be classi¬ed into rotating and static loads (Weedy,
1987). A third category corresponds to power electronic-based loads. Rotating loads consist
mainly of induction and synchronous motors, and their steady-state operation is affected by

voltage and frequency variations in the supply. Power electronic-based loads are also
affected by voltage and frequency variations in the supply. There is general agreement that
such loads are more dif¬cult to operate because, in addition to being susceptible to supply
variations, they inject harmonic current distortion back into the supply point (Acha and
Madrigal, 2001).
Detailed representation of a synchronous motor load in a three-phase power ¬‚ow study
requires use of Equation (3.124), with changed signs to re¬‚ect the motoring action. An
expression of comparable detail can be derived for the induction motor load. However,
owing to the large number and diversity of loads that exist in power networks, it is
preferable to group loads and to treat them as bulk load points. It is only very important
loads that are singled out for detailed representation. It is interesting to note that a group of
rotating loads operating at constant torque may be adequately represented as a static load
that exhibits the characteristic of a constant current sink (Weedy, 1987).
In steady-state applications, most system loads are adequately represented by a three-
phase power sink, which may be connected either in a star or delta con¬guration, depending
on requirements (Chen and Dillon, 1974). Figure 3.19(a) shows the schematic rep-
resentation of a star-connected load with the star point solidly grounded, whereas
Figure 3.19(b) shows a schematic represantation of a delta-connected load.

Va Va

Vb Vb

Vc Vc


(a) (b)

Figure 3.19 System load representation: (a) star-connected load with star point solidly grounded and
(b) delta-connected load

In three-phase power ¬‚ow studies it is normal to represent bulk power load points as
complex powers per phase, on a per-unit basis:
SLa ¼ PLa þ jQLa ; >
SLb ¼ PLb þ jQLb ; °3:129Þ
SLc ¼ PLc þ jQLc :

Re¬nements can be applied to the above equations to make the power characteristic more
responsive to voltage performance:
1 1 >
SLa ¼ PLa þjQLa
Va > >
! >>
! =
1 1
SLb ¼ PLb þjQLb ;
Vb > >
! >>
! >
1 1
SLc ¼ PLc þjQLc
Vc Vc

In Equations (3.130), and take values in the range 0“2 and Va, Vb, and Vc are the per-unit
three-phase nodal voltage magnitudes at the load point. Notice that when ¼ ¼ 0 the
complex power expressions in Equations (3.130) coincide with those in Equations (3.129).
However, if ¼ ¼ 1, Equations (3.130) resemble complex current characteristics more
than complex power characteristics. Also, if ¼ ¼ 2, the complex powers in Equa-
tions (3.130) would behave like complex admittances.
The admittance-like characteristic in Equations (3.130) may be expressed in matrix form
for both kinds of load connections, star and delta, respectively:
2 3
SLa Va 0 0
40 0 5; °3:131Þ
SLb Vb 2
0 0 SLc Vc
2 2 2 2
2 3
SLa Va þ SLb Vb ÀSLb Vb ÀSLa Va
2 2 2 2
16 7
ÀSLb Vb SLb Vb þ SLc Vc ÀSLc Vc 5: °3:132Þ
2 2 2 2
ÀSLa Va ÀSLc Vc SLc Vc þ SLa Va

Moreover, if it is assumed that the load powers and voltage magnitudes are taken to be
balanced, SLa ¼ SLb ¼ SLc ¼ SL, and Va ¼ Vb ¼ Vc ¼ V, then application of the following
symmetrical component operation, Y012 ¼ TÀ1 Yabc TS , leads to the load model representa-
tion for zero, positive, and negative (0, 1, 2) sequences:
2 2 3
SL V 0 0

40 05
2; °3:133Þ
SL V 2
0 0 SL V
2 3
0 0 0
40 05
2: °3:134Þ
0 0 SL V

Notice that no zero sequence loads exist for the case of a three-phase delta-connected load,
only positive and negative sequences.
As an extension of the above result, the positive, negative, and zero sequence expression
of a star-connected load with its star point solidly grounded may be expressed as
! !
1 1
0 0 0
SL °1Þ ¼ SL °2Þ ¼ SL °0Þ ¼ PL þjQL ; °3:135Þ

whereas for the case of a delta-connected load we have
! !
1 1
0 0 0
SL °1Þ ¼ SL °2Þ ¼ PL þjQL ; SL °0Þ ¼ 0: °3:136Þ

It should be remarked that the exponents and are not con¬ned to integer values and that
a wide range of load characteristics can be achieved by judicious selection of and ,
depending on the group of loads present in the study.
Also, a three-phase delta connected load can always be transformed into an equivalent
star circuit by using a delta“star transformation. However, notice that the transform-
ation will generate an extra bus in the form of the star point, which yields no physical


The chapter has addressed the mathematical modelling of the most common elements found
in conventional electrical power systems, namely, transmission lines, transformers,
generators, loads, and shunt and series passive compensation. The tools and methods
covered in the book are limited to fundamental frequency steady-state phenomena, and the
modelling approach followed in this chapter re¬‚ects this fact. Notwithstanding this, the
overall modelling philosophy is quite general in the sense that all plant component models
are formulated in the frame of reference of the phases, which is closely associated with the
physical structure of the equipment and its actual steady-state electrical operation. It is
shown throughout the chapter that simpler models do exist to represent a given plant
component but that these models are based on the assumption of perfect geometric balance
conditions. These models are realised with the help of the symmetrical component
transform, leading to plant component representation in the frame of reference of the
Multiphase transmission line parameters are calculated with great accuracy, incorporating
all key effects that affect fundamental frequency operation such as geometric imbalances,
ground return loops, and even long-line effects. Practical transmission lines include several
conductors per phase and ground wires as well as more than one three-phase circuit sharing
the same right of way, giving rise to a large number of electromagnetically coupled
conductors. The chapter has presented a methodology for handling all these effects in a
systematic and ef¬cient manner. A comprehensive computer program in Matlab1 has been
written to calculate multiconductor transmission line parameters.
Three-phase power transformers have been modelled in the frame of reference of the
phases, with particular reference to complex off-nominal tapping positions. This caters for
the possibility of the transformer acting as a tap changer or as a phase shifter. The most
popular transformer connections used in high-voltage transmission have been addressed
and, under the assumption of perfect geometric conditions, transformer models in the frame
of reference of the sequences have been derived. The thrust of these models is fundamental
frequency, steady-state operation, and there is little loss of accuracy in representing the
three-phase transformer as a three-phase bank of transformers. A detailed model of the
synchronous generator, based on its physical windings arrangement, has been presented.

The effects of saliency and the generator load angle are explicitly represented in the model.
This model also serves the purpose of representing a synchronous motor, by suitable
modi¬cation of signs to conform to motoring action. Static loads suitable for bulk load
representation have also received attention.
The models of conventional power plant components developed in this chapter interface
quite naturally with the models of FACTS components developed in Chapter 2. Together,
they provide a very sophisticated tool with which to represent power system networks
containing a vast array of power electronic controllers of various kinds. These are the
power systems that may be in operation tomorrow. Two different modelling ¬‚avours
emerge from this modelling exercise, the frame of reference of the phases and the frame of
reference of the sequences, each one having its own time and space. Chapters 4, 5, and 7,
dealing with positive sequence power ¬‚ow and optimal power ¬‚ow, use the positive
sequence models derived in this and Chapter 2. Chapter 6 covers the topic of three-phase
power ¬‚ow and uses the comprehensive models developed in the frame of reference of
the phases.

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