UPFC Ideal Transformer Model
Seungwon An
Thomas W. Gedra
Student Member, IEEE
School of Electrical and Computer Engineering
Oklahoma State University Stillwater, Oklahoma 74078
Email: aseung@okstate.edu
Member, IEEE
School of Electrical and Computer Engineering
Oklahoma State University Stillwater, Oklahoma 74078
Email: gedra@okstate.edu
Abstract— This paper presents a unified power flow controller
(UPFC) model, which consists of an ideal transformer with a
complex turns ratio and a variable shunt admittance. In this
model, the UPFC control variables do not depend explicitly on
UPFC input and output currents and voltages. Accordingly, this
model does not require adding extra busses for UPFC input and
output. This model is easily combined with transmission line
models using ABCD two-port representations, which can then
be converted to Y-parameter representations. The complex turns
ratio and the variable admittance can be mapped to actual UPFC
control variables, such as UPFC injected series and shunt voltages
and currents or other equivalent representations.
I. I NTRODUCTION
The unified power flow controller (UPFC) is one of the
most promising devices in the FACTS family [1], [2], [3],
[4]. It has the capability to control voltage magnitude and
phase angle, and can also independently provide (positive or
negative) reactive power injections. Therefore the UPFC can
provide voltage support, control of real power flow, and other
functions. The UPFC injection model [3] and the uncoupled
model [5] have been proposed for steady-state power flow
analysis. These models can be easily incorporated into steadystate loadflow or optimal power flow problems. However,
in these models four UPFC control variables depend on
the UPFC input and output currents and voltages and both
models require adding two additional busses to the loadflow
or OPF (optimal power flow) problem formulation as well
as a constraint on real power conservation, thereby reducing
the degrees of freedom to three. Therefore, these models do
not seem to be suitable for efficient steady-state power flow
analysis with multiple UPFCs.
Since electricity prices at the UPFC input and output busses,
which are the dual variables (or prices) associated with real
power injections, are uninformative when no power is bought
or sold at these busses, and so their addition to the problem
only serves to increase the size of the problem.
This paper presents a mathematical model of a steady-state
UPFC, which consists of an ideal transformer with a complex
turns ratio and a variable shunt admittance. In this model,
UPFC control variables do not depend on UPFC input and
output voltages and currents, and the UPFC power flow equations are self-contained. An ABCD two-port representation is
used to combine transmission lines cascaded with a UPFC.
This paper also introduces basic UPFC operating principles
[4], the UPFC injection model [3], and the uncoupled model
[5] to provide the background for the development of the
UPFC ideal transformer model.
II. O PERATING P RINCIPLES
A UPFC consists of a shunt transformer and a series
transformer, power electronic switching devices and a DC link,
as shown in figure 1 [4]. Inverter 1 is functionally a static
VAR compensator assuming that inverter 2 is not connected.
It injects reactive power in the form of current at the shunt
transformer, and the current phasor I~T is perpendicular to the
~I .
input voltage V
Inverter 2 alone represents the so-called advanced controllable series compensator (ACSC) assuming that inverter 1 is
not connected. It injects reactive power by adding voltage at
~T is perpendicthe series transformer. The injected voltage V
~
ular to the receiving end current Io .
Now if we connect inverter 1 to inverter 2 through the DC
link, inverter 1 can provide real power to inverter 2. Therefore
the UPFC can independently control real and reactive power
injections through the series transformer, but the real power
injected at the series transformer is provided by the shunt
transformer through the DC link. Inverter 1 must provide
the real power used by inverter 2 via the DC link, but can
also independently inject reactive power (positive or negative)
through the shunt transformer. In summary, note that the UPFC
conserves real power but can still generate (or sink) reactive
power at either transformer or both.
III. E XISTING UPFC M ODELS
The UPFC injection [3] and uncoupled [5] models can both
be easily incorporated in loadflow and optimal power flow, but
both models require adding two additional busses for UPFC
Fig. 1.
General UPFC scheme [4]
Fig. 2.
A series-connected VSI as a voltage source and a reactance
Fig. 3.
input and output voltages and currents, and, in these models,
UPFC behavior is dependent not only on the UPFC control
variables but explicitly on the input and output voltages and
currents. The real power balance equation must be included as
a constraint, so, although these models seem to be suitable for
general purpose UPFC studies in a large power system, there
is still room for improvement. These two additional busses for
each UPFC only serve to make the problem larger.
Injection model for a series connected VSI
2 need to satisfy the following equality condition, assuming
that inverter losses are neglected:
P1 = P2 .
The complex power supplied by inverter 2 is calculated as
∗
~T · I~IO
S̄2 = V
,
Ã
= (Vp + jVq ) ejθI
A. Injection Model
We will first review modeling the UPFC as a seriesconnected voltage source inverter (VSI) [3]. Then, inverter 1
is incorporated into the model of the series connected VSI
for a complete UPFC model [3]. Suppose a series connected
voltage source is located between busses I and O in a power
system. The series voltage source inverter can be modelled
with an ideal injection voltage VT in series with a reactance
XS . Figure 2 shows a representation of a series connected
~T consists of in-phase component
VSI. The injected voltage V
Vp and quadrature component Vq with respect to the UPFC
~I . Then, it can be written by
input voltage V
~T = (Vp + jVq ) ejθI ,
V
(1)
where θI is the UPFC input voltage phase angle. The injection
model [3] is obtained by transforming the voltage source VT
in series with XS to an equivalent current source in parallel
with the admittance corresponding to XS . The current source
can be obtained by
~T = bS (Vq − jVp ) ejθI ,
I~S = −jbS V
where
bS =
(2)
1
.
XS
~I · (−I~S )∗ = −bS VI (Vq + jVp ) ,
S̄I = V
~O · I~∗ = bS VO (Vq + jVp ) ej(θO −θI ) ,
S̄O = V
S
(3)
(4)
and the real and reactive powers can be obtained by
¡ ¢
¡ ¢
QI = Imag S̄I ,
PI = Real S̄I ,
¡ ¢
¡ ¢
PO = Real S̄O ,
QO = Imag S̄O .
From equations (3, 4), the injection model of a series connected VSI can be seen as two dependent injections as shown
in figure 3.
Now, let us consider shunt connected inverter 1. It must
provide any real power which is injected to the network via
the series connected voltage source VT . Thus, inverters 1 and
~O
−V
(5)
!∗
.
Then, the real and reactive powers supplied by inverter 2 are
obtained by
P2 = −bS Vq VI
+ bS VO (Vq cos(θI − θO ) + Vp sin(θI − θO )) , (6)
¢
¡
Q2 = bS Vp VI + Vp2 + Vq2 − bS VO Vp cos(θI − θO )
+ bS VO Vq sin(θI − θO ).
(7)
Now, let us consider the capability of the reactive power
support in inverter 1. The reactive power injected by inverter 1
is independently controlled by the UPFC, and can be modelled
as a separate controllable shunt reactive source. We assume
that inverter 1 has the maximum VA rating Smax1 . Thus, the
reactive power injection by inverter 1 need to satisfy the
following inequality condition:
¡ 2
¢
Q21 (Vp , Vq ) ≤ Smax
− P12 (Vp , Vq ) .
(8)
1
Then, the complex power injections at the UPFC input and
output become
S̄I
S̄O
Then, the complex powers injected into each bus become
(VI + Vp + jVq ) e
jXS
jθI
=
=
=
=
PI + jQI ,
−P1 − jQ1 − bS VI (Vq + jVp ),
PO + jQO ,
bS VO (Vq + jVp ) ej(θO −θI ) .
(9)
(10)
Equations (6, 9, 10) can be used to describe the UPFC
operation as equality constraints in the OPF.
B. Uncoupled Model
Figure 4 shows a basic UPFC model, where the UPFC is
located at distance d from bus i. Each part of the transmission
line can be represented as an equivalent Π circuit.
~T can be resolved into in-phase
The injected series voltage V
component Vp and quadrature component Vq with respect to
I~o , and written by
~T = (Vp + jVq ) ejδO ,
V
(11)
where δO is the UPFC output current phase angle. The current
IT injected by the shunt transformer contains a real component
Ip , which is in phase or in opposite phase with the input
voltage. It also has a reactive component Iq , which is in
quadrature with the input voltage. Then the injected current
I~T can be written by
I~T = (Ip + jIq ) ejθI ,
(12)
where θI is the UPFC input voltage phase angle. The magnitudes of the injected voltage VT and current IT are limited by
the maximum voltage and current ratings of the inverters and
their associated transformers.
The UPFC input-output voltage and current can be represented by
~o = V
~I + V
~T = VI ejθI + Vp ejδo + jVq ejδo ,
V
I~o = I~I − I~T = II ejδI − Ip ejθI − jIq ejθI ,
Fig. 5.
(13)
(14)
where δI is the UPFC input current phase angle. Then, the
injected complex power into the series transformer can be
resolved into the real and reactive power in simple form as
~T · I~o∗ = Vp · Io + jVq · Io .
ST = V
| {z }
| {z }
PT
The in-phase voltage Vp is associated with a real power supply
and the quadrature voltage Vq with an inductive or capacitive
reactance in series with the transmission line.
Since the real power PT (which may be negative) is
provided by the current Ip in the shunt transformer, we can
derive the following relationship:
(17)
(18)
and the real and reactive powers can be obtained by
¡ ¢
¡ ¢
PI = Real S̄I ,
QI = Imag S̄I ,
¡ ¢
¡ ¢
PO = Real S̄O ,
QO = Imag S̄O .
Equations (13, 14, 16) can be used to describe the UPFC
operation as equality constraints in the OPF.
Since the UPFC conserves real power and provides reactive
power, it can be modelled with an ideal transformer and a
shunt branch, as shown in figure 6. The advantage of this
model is that the ideal transformer turns ratio T̄ and the
variable shunt admittance ρ are independent variables, which
are not associated with the UPFC input-output voltages and
currents. We define the variable T̄ as
T̄ = T ejφ ,
T =
φ =
transformer voltage magnitude turns ratio (real),
phase shifting angle.
The UPFC can be expressed as the ABCD matrix by using its
input-output voltage and current relationship as follows:
#
"
#
"
~o
~I
V
V
= ABCDU ·
,
(20)
I~I
I~o
where
"
ABCDU =
T̄
0
j T̄ ρ
1
T̄ ∗
#
.
Note that equation (20) is not bilateral if T̄ is a complex
number (i.e. φ 6= 0). Now, we will show that this ideal
transformer model represents the UPFC by comparing the
complex power injections at the UPFC input and output. Using
equation (20), the complex power injection at the UPFC input
can be obtained by
=
~I I~I∗ ,
V
¶∗
µ
~o j T̄ ρV
~o + 1 I~o ,
T̄ V
T̄ ∗
~o I~o∗ − j|T̄ |2 · |V
~o |2 ρ,
V
=
~o |2 ρ,
S̄O − j|T̄ |2 · |V
=
=
Basic UPFC model in a transmission line.
(19)
where
S̄I
Fig. 4.
UPFC ideal transformer model
IV. I DEAL T RANSFORMER M ODEL
(16)
This model requires extra two busses to specify the UPFC
input and output, as shown in figure 5. We assume that the
extra two busses are not connected with each other. The
complex powers injected into the UPFC input and output
busses are
~I · (−I~I )∗ = −VI II ej(θI −δI ) ,
S̄I = V
~o · I~o∗ = Vo Io ej(θO −δO ) ,
S̄O = V
Fig. 6.
(15)
QT
Vp · Io − VI · Ip = 0.
Uncoupled UPFC model in a transmission line.
(21)
V. UPFC IN A T RANSMISSION L INE
Fig. 7.
Simplified UPFC circuit
and the real and reactive power injections can be obtained by
¡ ¢
¡ ¢
PI = Real S̄I ,
QI = Imag S̄I ,
¡ ¢
¡ ¢
PO = Real S̄O ,
QO = Imag S̄O .
Thus, we can derive the following relationship between the
UPFC input and output:
PI
QO
=
=
A two-port ABCD matrix is the most convenient method to
represent cascaded networks [6]. Let us divide a transmission
between busses i and k with a UPFC into three cascaded
networks, a UPFC input transmission, a UPFC, and a UPFC
output transmission, as shown in figure 8. The UPFC input
transmission, and the UPFC output transmission are easily
expressed by the two-port ABCD matrix since the transmission
lines are expressed in Π equivalent circuits. We call ABCDi
and ABCDk as the ABCD matrices for each transmission line,
and defined by
"
#
"
#
Ai Bi
Ak Bk
ABCDi =
and ABCDk =
,
Ci D i
Ck Dk
where each element is defined by
PO ,
~o |2 ρ.
QI + |T̄ |2 · |V
(22)
(23)
Equations (22, 23) suggest that the ideal transformer model
conserves real power and generates reactive power, which
verify the validity of this UPFC model. It is important to
note that the ideal transformer does not generate real and
reactive power, and the reactive power is generated by the
shunt admittance only.
Figure 7 shows a simplified UPFC circuit. To obtain how
much real and reactive power is injected in the series and shunt
transformers, we will map the complex turns ratio T̄ in the
ideal transformer and the shunt admittance ρ to the injected
~T and current I~T . Since the UPFC input voltage and
voltage V
current are expressed as
Ã
!
~T
V
~I = V
~o + V
~T = V
~o 1 +
~o T ∠φ,
V
=V
(24)
~o
V
µ
¶
1
I~I = I~o + I~T = I~o +
(25)
∠φ − 1 I~o + I~ρ ,
T
Yii
1
Yii
,
Bi =
, Ci = Yii (2 +
),
YiI
YiI
YiI
Ykk
1
Ykk
Ak = Dk = 1 +
, Bk =
, Ck = Ykk (2 +
).
Yok
Yok
Yok
Ai = Di = 1 +
Now, the three cascaded networks are combined to obtain
"
#
"
#
~i
~k
V
V
= ABCDi · ABCDU · ABCDk
,
I~i
−I~k
"
#"
#
~k
Aik Bik
V
=
,
(30)
Cik Dik
−I~k
where
~T and current I~T can be obtained by
the injected voltage V
~T = (T ∠φ − 1) V
~o ,
V
µ
¶
1
I~T =
∠φ − 1 I~o + I~ρ .
T
(26)
(27)
Then, the power flows through each inverter can be obtained
by
~I I~T∗ ,
S̄1 = V
Aik
=
Bik
=
Cik
=
Dik
=
1
B i Ck ,
T̄ ∗
1
T̄ Ai Bk + j T̄ Bi Bk ρ + ∗ Bi Dk ,
T̄
1
T̄ Ci Ak + j T̄ Di Ak ρ + ∗ Di Ck ,
T̄
1
T̄ Ci Bk + j T̄ Di Bk ρ + ∗ Di Dk .
T̄
T̄ Ai Ak + j T̄ Bi Ak ρ +
By rearranging equation (30) to solve for Ii and Ik , we have
"
#
"
#
~i
V
I~i
= Ȳbusik
,
(31)
~
I~k
V
k
where
·µ
¶
1
∠φ − 1 I~o + I~ρ
T
~ o |2 ,
= (1 − T ∠φ) S̄o − jρ|T̄ |2 |V
~o T ∠φ
=V
¸∗
Ȳbusik =
,
(28)
~T I~o∗ ,
S̄2 = V
Dik
Bik
− B1ik
Cik −
Aik Dik
Bik
Aik
Bik
#
.
If the phase shifting angle φ is zero, then
Aik Dik
1
= −Cik +
,
Bik
Bik
~o I~o∗ ,
= (T ∠φ − 1) V
= (T ∠φ − 1) S̄o .
"
(29)
We can also see from equations (28, 29) that the UPFC
conserves real power and can generate reactive power.
and hence Ȳbusik becomes a symmetrical matrix, which implies that the transmission line between busses i and k is
bilateral and consists of only passive components.
R EFERENCES
Fig. 8.
Cascaded transmission line with a UPFC
VI. C ONCLUSIONS
Since the UPFC is embedded in a transmission line by
using ABCD matrix, and the UPFC control variables T, φ
and ρ are independent of UPFC input and output voltages,
this model can reduce the size of OPF problem. Therefore,
this model seems most suitable for UPFC sensitivity analysis,
which may require multiple UPFCs in several transmission
lines. Currently, we are developing the first- and second-order
sensitivities of the UPFC using this model [5], [7].
[1] Colin Schauder. The unified power flow controller - a concept becomes
reality. In IEE Colloquium on Flexible AC Transmission Systems-The
FACTS(Ref. No. 1998/500), pages 7/1–7/6, November 1998.
[2] L. Gyugyi, C. D. Schauder, S. L. Williams, T. R. Rietman, D. R.
Torgerson, and A. Edris. The unified power flow controller: A new
approach to power transmission control. IEEE Transactions on Power
Delivery, 10(2):1085–1097, April 1995.
[3] M. Noroozian, L. Ängquist, M. Ghandhari, and G. Andersson. Use of upfc
for optimal power flow control. IEEE Transactions on Power Delivery,
12(4):1629–1634, October 1997.
[4] L. Gyugyi. Unified power-flow control concept for flexible ac transmission systems. In IEE Proceedings C. on Generation, Transmission and
Distribution, volume 139(4), pages 323–331, July 1992.
[5] Seungwon An and Thomas W. Gedra. Estimation of upfc value using
sensitivity analysis. In Proceeding of the 2002 Midwest Symposium on
Circuits and Systems, Oklahoma State University, Tulsa, OK, August
2002.
[6] J. Duncan Glover and Mulukutla Sarma. Power System Analysis &
Design. PWS Publishing Company, second edition, 1994.
[7] Parnjit Damrongkulkamjorn, Prakash K. Arcot, Peter Dcouto, and
Thomas W. Gedra. A screening technique for optimally locating phase
shifters in power systems. In Proceedings of the IEEE/PES Transmission
and Distribution Conference, pages 233–238, April 1994.