Siemens Energy, Inc.
Power Technology
Issue 111
Phase-shifting Transformer Modeling in PSS®E
Carlos Grande-Moran, Ph.D.
Principal Consultant
carlos.grande@siemens.com
Phase-shifting transformers are often used in power systems to control the active
power flow (MW) in branches in meshed networks or to control the active power flow at
the interface between two large and stiff independent grids. The control of MW flow is achieved by
adjusting the phase angle of the voltages at the phase-shifting transformer terminals. Phase-shifting
transformers are also known as phase angle regulating (PAR) transformers.
Phase-shifting transformers built for transmission grids are generally a three-phase, two-terminal pair
design. The terminal where power is injected into the transformer unit is called the “source terminal” and
the power where load is exiting the transformer unit is called the “load terminal.”
The change in phase angle between the terminal voltages of the transformer unit is carried out by adding
a regulated voltage to the phase-to-neutral voltage at the source terminal. A winding in series with a
network branch is used to insert the regulated voltage that, when added with the appropriate phase to the
source terminal phase-to-neutral voltage, sets up the desired direction of the active power flow between
the transformer terminals. The convention used by manufacturers of these devices regarding an
advanced or retarded phase angle is that an advanced phase angle means that the phase-to-neutral
voltage at the load terminal leads the phase-to-neutral voltage at the source terminal, and a retarded
phase angle means that the phase-to-neutral voltage at the load terminal lags the phase-to-neutral
voltage at the source terminal.
Two phase-shifting transformer designs are the most prevalent in power systems applications: symmetric
phase-shifting transformers and asymmetric phase-shifting transformers. Symmetric phase-shifting
transformers are designed such that the amplitudes of the no-load winding voltages do not change during
the phase shifting operation. The complex transformer voltage ratio for this type of transformer is then
1.0  e  j . The IEEE model for phase-shifting transformers is based on the symmetric phase-shifting
transformer where the no-load phase angle Φ is the angle by which the winding 1 voltage (source side)
leads the winding 2 voltage (load side). Figure 1 below shows a schematic diagram of a symmetric
phase-shifting transformer.
Series Transformer
Excited
Winding
Δ
V1
Source
I1
V2
m
Vm
ΔV
Load
m
I2
Series
Winding

V2
V1  V2
Exciting
Winding
Vm
V1
and
Regulating
Winding
Regulating Transformer
Figure 1 - Symmetric Phase-shifting Transformer
V1
 1.0 e  jφ
V2
Power Technology
March 2012
The asymmetric phase-shifting transformer can add an in-phase and quadrature regulating voltage with a
winding connection angle α to the phase-to-neutral voltage at the source terminal. When the winding
connection angle α is 0º or  180º, the quadrature regulating voltage is zero and the phase-shifting
transformer operates as a conventional voltage or reactive power control transformer. For any winding
connection angle α where the quadrature regulating voltage is not zero, the phase-shifting transformer will
control both active and reactive power flows. A particular case where the winding connection angle α is
 90º is used widely in power systems and is known as a “quadrature booster” transformer. This
transformer type controls mostly active power flow but, because of its asymmetry, it also exerts a small
control action on reactive power flow. Asymmetric phase-shifting transformers change not only the phase
angle between the winding 1 and winding 2 voltages, but also their magnitudes. Thus, the complex
transformer voltage ratio for this type of phase-shifting transformer is t  e  j , where t is the transformer
t
turns ratio, 1 , and t1 and t2 are the winding 1 and winding 2 turns ratios, respectively. Figure 2 below
t2
shows a schematic diagram of an asymmetric phase-shifting transformer operating as a quadrature
booster phase-shifting transformer.
Series
Winding
Series Transformer
Source
V1
V2
I1
In-phase
ΔVd
ΔVq
Load
I2
Δ
ΔVq

Excited
Winding
d
V1
Quadrature
V1  V2 and
q
Exciting
Winding
V2
Regulating Transformer
V1
 t e  jφ
V2
Regulating
Winding
Figure 2 - Asymmetric Phase-shifting Transformer
The no-load complex voltage ratio between winding 1 and winding 2 phase-to-neutral terminal voltages in
phasor notation is given below:
V1  t1   j
    e  t  e j (per unit)
V2  t 2 
Neglecting winding and core losses and using the principle of conservation of energy, the power flowing
into a phase-shifting transformer is equal to the power flowing out of the transformer. Thus, in phasor
notation:
V1 I *1  V2 I *2 (per unit)
and
V1 I2*
=  t  e  j (per unit)
V2 I1*
The asterisk in the equation above represents the complex conjugate operation.
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The no-load phasor diagram for an asymmetric phase-shifting transformer is shown in Figure 3 below.
The following parameters are used in the modeling of asymmetric phase-shifting transformers:
 Phase shift angle range [Φmax,Φmin] in degrees
 Number of tap positions

 Nominal tap position: 1.0  e  j0 (per unit on bus voltage base)
 Transformer leakage impedance at nominal tap position in per unit on windings 1 and 2 voltage base
and on either system apparent power base or winding 1-2 apparent power base
 Winding connection angle α in degrees
Applying the Law of Sines to the triangle in the phasor diagram in Figure 3, it is possible to obtain the
magnitude of the regulating voltage ΔV in the series winding for a phase shift angle Φ and winding
connection angle α. The expression for the magnitude of the regulating voltage is then:
V 
sin 
; Φmin ≤ Φ ≤ Φmax and α≠ Φ (per unit)
sin     
and the off-nominal transformer turns ratio, t, for a winding angle α≠0º is given by:
t
sin 
; Φmin ≤ Φ ≤ Φmax and α≠ Φ (per unit)
sin     
The equations above become simpler when applied to the quadrature booster phase-shifting transformer
(α=90º). The new expressions for ΔV and t are then:
V  tan  ; Φmin ≤ Φ ≤ Φmax (per unit)
and
t
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1
; Φmin ≤ Φ ≤ Φmax (per unit)
cos 
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March 2012
  0
  0
0
Smx
0
ΔVqmx
Vmx

ΔVdmx
Sneutral
Vmn
  90
Smn
  90
e j
t e j
 mx   mn
max min
  180
Figure 3 - Asymmetric Phase-shifting Transformer General Model
Phase-shifting Transformer Representation in PSS®E
Phase-shifting transformers are modeled in PSS®E as two-winding transformers. Users have multiple
choices to enter the model data in PSS®E. The choices of data codes for entering each winding’s leakage
impedance and magnetization branch are the same as those for conventional voltage control
transformers. These parameters can be entered in per unit or can be from data collected in the short
circuit and no-load tests. Only winding 1 is allowed to have an under-load tap changer; winding 2 has an
off-load tap changer. Tap position and maximum and minimum phase-shift angles are specified in
electrical degrees. The transformer control band is specified in MW.
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March 2012
The circuit model used in PSS®E for phase-shifting transformers is shown in Figure 4 below.
e j j
eii
t1e j : t 2
Source
Side
Zno  Rno  jX no
t 22 Z no
Load
Side
e j
t
t 1 j
e
t2
Figure 4 - Standard PSS®E Phase-shifting Transformer Circuit Model
The model is made up of an ideal two-winding transformer with a complex transformer ratio, t  e  j : 1 , in
series with the positive sequence phase-shifting transformer’s leakage impedance. Winding 1 is
associated with the source side (“from” bus) of the phase-shifting transformer and winding 2 with the
terminal of the ideal two-winding transformer on the load side (“to” bus) of the phase-shifting transformer.
The positive sequence magnetizing admittance of the phase-shifting transformer is connected as a shunt
to the winding 1 terminal. Voltage at the winding 2 terminal of the two-winding ideal transformer is not
available to the user. Only during no-load conditions is the winding 2 terminal voltage ( e j' ) of the ideal
transformer equal to the load terminal voltage of the phase-shifting transformer. And so, the phase shift
angle range used in the model is the no-load phase shift angle range which relates the source terminal
voltage ( ei ) to the winding 2 terminal voltage ( e j' ) of the ideal transformer.
For power flow calculations the user must set: (1) the control mode in the transformer data record to
either MW control (COD1=3) or asymmetric MW control (COD1=5), (2) the transformer upper RMA1 and
lower RMI1 tap limits to the phase-shifting transformer angle limits in electrical degrees, (3) the
transformer loading limits in MVA, (4) the winding connection angle α in electrical degrees, and (5) the
upper VMA1 and lower VMI1 control band limits to be enforced by the phase-shifting transformer in MW.
The MW control mode (COD1=3) is used for symmetric phase-shifting transformers and the asymmetric
MW control mode (COD1=5) is used for asymmetric phase-shifting transformers. A negative control mode
(COD1= -3 or -5) indicates that the phase shift angle is held fixed in the load flow solution.
The winding connection angle default value is 0º. The use of this default value in the load flow
computations with asymmetric control mode (COD1=5) results in PSS®E handling this phase-shifting
transformer as a symmetric phase shifting transformer. Quadrature booster transformers are specified by
selecting the asymmetric control mode and a winding connection angle of ± 90º. Selection of winding
connection angles in the range 0º < α < 360º, and an asymmetric control mode, are used to specify
asymmetric phase-shifting transformers where both in-phase and quadrature regulating voltage
components are considered.
The phase-shift angle adjustment is continuous, and if the regulated active power flow of at least one of
the phase-shifting transformers falls outside its scheduled MW control band, all phase-shifting
transformers and bus voltage phase angles are adjusted simultaneously. An excessively narrow control
band may cause non-convergence of the power flow solution. Thus, a reasonable control band must be
chosen in proportion to the active power target flow. For example, if the MW target flow is about 500 MW,
a control band of  5 MW is a reasonable choice. The upper limit of the control band will be 505 MW and
the lower limit will be 495 MW.
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For short circuit calculations the phase-shifting transformer model for the negative sequence network will
employ the same leakage impedance used in the load flow calculations, but the sign of the phase shift
angle will be reversed. Hence, if the phase shift angle is +Φ in the positive sequence network, then this
angle will be -Φ in the negative sequence network. This change in sign causes the shunt and series
components of the negative sequence pi-equivalent network model to be different from those used in the
positive sequence network. However, if the classical short circuit solution is selected in the PSS®E short
circuit module, both the positive and negative sequence representations of a phase-shifting transformer
will be identical, because all phase-shift angles of transformer units are set to 0º.
The zero sequence network representation for phase-shifting transformers is dependent on the winding
connections of the series and regulating (shunt) transformers, their core design, and whether the
regulating transformer has any delta-connected tertiary winding. The connection code CC=9 can be used
to represent phase-shifting transformers in the zero sequence network. Winding 1 is the regulating
transformer and winding 2 is the series transformer. For the T-equivalent network used for CC=9, the
impedance connected to the winding 1 terminals represents the zero sequence impedance of the
regulating transformer as seen from its input terminals; the impedance connected to winding 2 terminals
represents the zero sequence impedance of the series transformer as seen from its output terminals; and
the shunt branch represents the impedance of a tertiary winding or the stray air path of the zero sequence
flux through the tank walls.
As an example of data preparation for a symmetrical phase-shifting transformer, consider a three phase,
two-winding transformer, 230/230 kV, 700 MVA OA rating, and nominal winding phase-to-phase voltage
of 230 kV. Its positive sequence leakage impedance at nominal tap is 0.0 +j0.1257 per unit on 700 MVA
and 230/230 kV base. The total number of taps is 49 and the no-load phase shift angle range is  32º.
The minimum, nominal and maximum tap positions are 1, 25 and 49, respectively.
The PSS®E data record for this transformer unit is shown in Figure 5 below. Note that the winding
connection angle is set to 0º and the control mode is set to COD1=3.
Figure 5 - PSS®E Data Record for the Symmetrical Phase-shifting Transformer
If the phase-shifting transformer data record shown in Figure 5 above were to belong to an asymmetric
quadrature booster phase-shifting transformer, the only change required to be implemented would be to
change the winding connection angle to 90º. Figure 6 below shows the data record for the quadrature
booster phase-shifting transformer. Note that for a phase-shift angle Φ of 15º, the required value for
winding 1 turns ratio, t1, is 1.0353 per unit on a 230 kV bus voltage base.
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Power Technology
Figure 6 - PSS®E Data Record for the Asymmetrical Phase-shifting Transformer
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