IEEE Transactions on Power Delivery, Vol. 12, No. 4, October 1997
1717
B.T. Ooi(SM) M. Kazerani(M) R. Marceau*(M) 2. Wolanski(M)
F.D. Galiana(F) D. McGillis G.
Department of Electrical Engineering
McGill University, *Ecole Polytechnique de Montreal, * *Concoda University
Montreal, Quebec, Canada
Abstmct-Many controllers of Flexible AC Transmission
Systems (FACTS), such as the STATCOM, the Unified
Power Flow Controller (UPFC), the PWM asynchronous dc link, the Thyristor-Controlled Series Capacitor (TCSC) and the PWM Series Static VAR voltage support. Thus, they can be sited at the mid-point of the transmission line, which has been proven by the late
E.W. Kimbark, as the optimum location for shunt capacitor compensation. This paper points out that the ability to double the power transfer of the uncompensated line applies also to the aforementioned FACTS devices. The mid-point siting also facilitates the independent control of reactive power at both ends of the transmission line.
Xeywordr.. Transmission line, Flexible AC Transmission
Systems, FACTS, Thyristor, gate-turn-off thyristors, power electronics, phase shifters, STATCOM, Unified Power Flow
Controller.
I. INTRODUCTION
Through the Flexible AC Transmission Systems (FACTS) initiative of the Electric Power Research Institute (EPRI), the following FACTS devices already exist as prototype installations: the Thyristor-Controlled Series Capacitor
(TCSC) [1,2] and the Static Condenser (STATCOM) [3].
EPRI also sponsors the development of the Unified Power
Flow Controller
Phase Angle Regulator [5]. Looking slightly ahead to the days when the gate-turn-off thyristors (GTOs) will be able to implement the pulse width modulation (PWM) techni- ques, university researchers have proposed a number of
PE-292-PWRD-0-01-1997 A paper recommended and approved by the IEEE Transmission and Distribution Committee of the IEEE
Power Engineering Society for publication in the IEEE Transactions on Power Delivery. Manuscript submitted July 22, 1996; made available for printing January 8, 1997.
PWM-FACTS devices: the PWM-asynchronous dc
[6], the PWM-Shunt SVC [7], [8] and the
PWM-Phase Shifter [9]. Except for the Thyristor-Controlled
Phase Angle Regulator [SI, all the aforementioned FACTS devices have inherent ac voltage support. This may come from the ac capacitance, as in the TCSC [1,2], or from the dc capacitance behind the GTO voltage-source converters, as in the STATCOM [3], the UPFC 141 and all the PWM-
FACTS devices [6-81.
The late E.W. Kimbark pointed out that with shunt capacitor voltage support at the mid-point of the transmis- sion line (which he proved to be the optimum location), it is possible to transmit twice the power of the uncompensated line and to extend the steady-state stability limit from 90"
180" [lo]. This conclusion can be extended to the FACTS devices. When the thermal limit is above twice the transmis- sion power of the line, the FACTS device can provide savings if its cost is below that of a second transmission line.
Another reason for mid-point siting is related to the voltage limit set by equipment and transmission design. As exempiified by the UPFC [4], FACTS devices are increasin- gly expected to control reactive power, which is accomplished by raising or lowering the terminal voltages of the FACTS device. For a given voltage limit, the mid-point siting controls a larger reactive power simply because each side of the FACTS device addresses only half the line impedance and not the full line impedance as in the case of the transmi- ssion line end-siting.
The paper develops the case, in a tutorial form, using a number of phasor diagrams. To keep the presentation simple, a lossless transmission line is assumed. A proof of the steady-state stability limit is given in Appendix A.
It is necessary to point out that real power systems deviate in detail from the simplified assumptions used in the tutorial.
Furthermore, the double power transfer is a consequence of the steady-state stability limit of the radial line between idealized sending-end and receiving-end voltages only. It might not be possible to double the power transfer when non-ideal voltage regulation in the alternators, transient stability, and parallel paths are taken into account. However, the steady-state stability limit points to the opportunity.
11. FACTS DEVICE IN POINT-TO-POINT
TRANSMISSION LINE
0885-8977/97/$10.00 0 1997 IEEE

1718
I
I I
!
I
I
Fig. 1. FACTS device in point-to-point transmission line.
Fig.
1 shows a two-port black box representation of a
FACTS device in a transmission line modelled by an inductive reactance X
=
XI
+
X2 between the sending-end and the receiving-end voltages, E, and E,. The line resistance is assumed to be negligible. Irrespective of the technological realization of the FACTS device, it is its terminal voltages,
VI and V, at the two ports, which interact with the transmis- sion line sections through Kirchhoffs Voltage Laws,
= jX1I1 + VI (1)
V,
= jX21, + E,.
The power transferred is still based on the formula
(2) at the sending-end section, and
Fig. 2. (a) FACTS device - STATCOM. (b) Voltage and current phasor diagram. at the receiving-end section. The symbol 6 represents voltage angle. Since the FACTS device is neither a source nor a sink of power, the real power which enters one port must exit through the other port:
Re(VII;) = Re(V,I;).
The asterisk symbol operation.
* denotes the complex conjugate
III. MID-POINT VOLTAGE SUPPORT
(5) posite phasor diagram for (1) and (2) is shown in Fig. 2@)
[10,11]. The internal voltage E3 of the STATCOM supports
V, = V,. The currents I, and I, are perpendicular to the phasors representing the voltage drops across the half-line sections. The FACTS device must provide for their dif- ference I3 (capacitor current)
= I, - I,. From [IO], the conclusions relevant to this paper is that the steady - state stability limit is 8 = 8 / 2 = (8,-81) = (8,-br) = (6,-83/2 =
90: Since X,
= X, = X/2, from (3) and (4), the steady-state power limit is 2/X or
that of the uncompensated line.
A proof of the steady-state stability limit is given in [12]. The
VAr rating of the STATCOM is Q = P tan(b/4). It is assumed that the FACTS device is located at the mid- point of the line, i.e., XI =
&
= X/2 and the device can provide 1.0
I
V,
I
=
I
V,
I
= 1.0 pu. B. Back-to-Back dc Link [6]
A. STATCOM [3,7]
Starting from the familiar grounds of the STATCOM, ch is modelled by the voltage E3 in Fig. 2(a), the com-
Fig. 3(a) shows the two transmission line sections con- nected together through a asynchronous dc
The ideal voltage sources E3 and E, represent the voltages of the rectifier and the inverter which are voltage source

I I I
Fig. 3. (a) Back-to-back PWM asynchronous dc link. (b) Voltage and current phasor diagram.
1719 rating can be reduced
the shunt-series voltage-source arrangement consisting
& and E4, as shown in Fig. 4(a).
can be realized again by the rectifier/iverter of the back-to-back dc
in the unified power flow controller
(UPFC) [4,9], with the difference that one converter is connected in shunt and the other in series.
As illustrated in Fig. 4(b), the voltage support is provided by the shunt voltage source
5
= Vl and the series voltage source E4 = Vl
-
V2.
The phase-shifter capability comes mainly through the series voltage E4. But since I,, which flows through it, is not in quadrature, the real power Re(E4123 must find a passage through the dc link to the shunt voltage source E3 to be returned as Re(E313?.
Except for the phase-shift range and the internal realizati- ons for Vl and V2,
Figs. 3 and
4 are similar. Thus, one reaches the same conclusion that mid-point sited UPFC can transmit twice the maximum power of the uncompensated line.
1
The VA rating is determined by
I I I
.When
I
13,"
I
I I I
1.0 pu and
I
13,"
E4,"
I
I
+
<
1.0 pu, the UPFC is more economical than the back-to-back dc
converters joined back-to-back through a dc link. The magnitudes of E3 and E4 are independently controllable and the angles of E3 and E4 have each a 360" range. Thus E3 and
E4 can be controlled to equal the phasors of VI and V, in
Fig. 2(b) so that its function is the same as that of the
STATCOM. With the mid-point voltage support, it also can transmit twice the power of the uncompensated line.
However, the back-to-back dc link has phase shifting capability as well, so that the power transmitted is indepen- dent of the voltage angle 8. Thus, as illustrated in Fig. 3@), where 8 is a large obtuse angle, the same power transfer of
Fig. 2(b) can be obtained by rotating the congruent triangles by y , where y=8-28. It is a matter of opening the angle between E3 and E4 so that they coincide with the new angles of Vl and V,.
The ac power from the sending-end side, Re(E311?, is rectified to dc power and thereafter, the dc power is inverted as ac power, Re(E41i), on the receiving-end side thus satisfying (5). The voltages E3 and E4 are supported from an internal stabilized regulated dc voltage v d c across a dc capacitor [13,14].
The regulated dc voltage v d c provides the steady-state stabfity to the limit 8 = 8 ,
-
= 82
- tir = 90; A proof is given in Appendix A. The comprehensive 360" phase-shift range of the back-to-back dc
has an expensive price tag as the MVA ratings must be twice the full line ratings:
1
V1,max
1 I
I1,max
I
+
1
V2,max
I I
I2,max
I
*
C.
(UPFC) [4,9]
The concept of unified power flow controller [4] has added the dimension of reactive power control to FACTS devices.
! !
When the phase-shift requirement is modest, the MVA
Fig. 4. (a) Unified Power Flow Controller. (b) Voltage and current phasor diagram.

1720
The black box of
Fig.
or decreasing the m udes of Vl and V,. Thus, all the examples treated in Figs. 2, 3 and 4 are capable of reactive power control, except that the STATCOM does not have independent control of the reactive power at both ends of the transmission line, since V, = V, = E3.
V transm~ssi~n is already close to the limit set by equipment and transmission design. The overvoltage margin is in the range of q = 0.05 or 0.10 pu.
A. Back-to-Back ak Link [ti] and UPFC [4,9]
For instance, Fig. 5
illustrates the zero reactive power are in phase with E, and E, respec-
I
V,
I are increased to (1.0
+ q) pu. s of (1) and (2) are represented by right-angle triangles. From the Pythagorean theorem, it can be shown that the real power at this voltage limit is:
I i i
I
‘
’
,I
,
1) Numerical Example: As a base reference, consider an uncompensated line in which X = 0.3 pu and the transmitted powerPS = lEsl IE,I sin(8,-83/X = 1pufor IE,I
= IE,I
= 1 pu and 8 ,
-
8, = 17.451 h mid-point siting and zero reactive power operat
11s of Fig. 5, P, = 2.13 pu for q = 0.05 and P, = 3.05 pu for q = 0.10. P, is computed from (6) and in Fig. 5, the angle 8 = cos”[l/(l+q)].
B. Series SVC [1,2,8]
The capacitor voltages of the TCSC [1,2] and the P W M -
SVC voltages [8] are represented by the voltage source E4 in the black box of Fig. 6(a). For series connection, 1, = I,.
The voltage constraint its
I
VI
1 and
I
V,
I to (1.0
+ q) pu.
I
I j
_ _ - - - - - . - _ _ _ _
Fig. 5. Power transfer at zero reactive power at both ends.
0
Fig. 6. (a) FACTS device
- series SVC. (b) Phasor diagram for end-point siting. (c) Phasor diagram for mid-point siting.
1) End-Point Siting: For a siting at the receiving end, E, =
V2 in the phasor diagram of Fig. 6(b). The sending-end voltage E, is 8 degrees ahead of E,. The compensating voltage E4 is added to V, to form VI = V, + E,. The side cd represents the jXIIl voltage of the transmission line. The

power transmission is:
P, =
1% I
111
I
6/2*
C. Practical Implications
(7)
1721
The readers’ attention is called to the last paragraph of the Introduction cautioning the reader that doubling the power transfer might not be fidly realizable from practical considerations.
2) Mid-Point Siting: For a mid-point siting, the phasor diagram is shown in Fig. 6(c). The voltage V2 is ob, and the side ab represents the voltage jX212. The side bc represents the compensating voltage E& As in Fig. 6@), cd is the jXIIl voltage. Both VI and V2 operate at the voltage limit (1.0
+ q) PUS
In comparing Figs. 6(b) and (c), one sees that the voltage triangles
are identical
both diagrams. The side cd represents the same reactance voltages. However, since in the mid-point siting, Xl = X2 = X/2,
I
Ill is twice as large as the end-point siting where X. Applying
(3, power transmissible at the voltage limit is doubled for the mid-point siting.
In order to avoid the crowding
the many symbols
Figs. 6@) and 6(c), the horizontal phasors have been enlarged, thus sacrificing accuracy for clarity of exposition.
Therefore, the large magnitude of E4 in Fig. 6(c) is for illustration purpose only and does not imply that it must exceed 1.0 pu.
VI. APPENDIX A. Steady-State Stability Limit
Foliowing the method in [E!], the stability analysis assumes
to
the infinite bus. The dynamic equation
the swing of
o,Ms(d28a3
/ dt2 = P,
-
P,, where P,=turbine power, P,=powertransmitted byalternator, wo = synchronous angular frequency and M,= inertia of alternator. Linearizat- ion about the equilibrium yields:
Steady-state stability is assured when the synchronizing power
(A-2)
ForXl = X , = X / Z a n d / E , / =
1.0 pu, (3) and (4) yield:
=
AP, =
- AB,)
= /V,l =
(A-3)
It is assumed that protection against over-voltages is provided by devices such as zinc-oxide nonlinear resistors.
Fail-safe must be assured under worst case scenarios.
In the case of Fig.
when
I
I
=
I
E,
I
= l + q , the zero reactive power condition cannot be satisfied. I, and I2 will have to take up angles which bisect 8. As the FACTS device can accommodate these adjustments, failure is not expected when ] E s /
= l + q .
On the other hand, in the events when in Fig. 6(c),
I
VI
I
E,
I
=
I
E,
I
= 1+ q
I
=
I
V,
I will exceed the voltage limit. The series SVC must have the feedback control which reduces the magnitude of the compensating voltage E4
that
coin- cides with d and c coincides with a. where
A C =
-
Ab,) (A-4)
6 = ( 6 -
61,) = (6% ) (A-5) where B,, 81,, 6, and 6 , are the equilibrium voltage angles. S i c e the FACTS device is neither a source nor a
of real power, (AP, - APr) at most will alter the dc link voltage
AV,,. As the dc voltage regulation feedback keeps
AVd, = 0, it follows that:
V. CONCLUSIONS
ApS = AP,. (A-6)
The study has shown that FACTS devices derive maximum benefit from their stabilized voltage support when sited at the mid-point of the transmission line. The power transfer capacity is increased to twice that of the uncompensated line.
The mid-point siting is also most effective in reactive power control. The transmission line must be operating below the thermal limit and the transient stability limit. The FACTS device must be costed against the savings derived from not building a second line as well as the economic value provid- ed by the phase shifting and VAR controllability in terms of the increased trammission capability of the network.
Substituting (A-3) and (A-4) in (A-6) and since Ab, = 0 (E, infinite bus),
A 6 1
+
Ah62
= A6,. (A-7)
From (A-3),
When the FACTS device swings as a single unit, AB1 =

1722
A 8 2 so that (1 - A8,/AsS) = 0.5 > 0. Thus, steady-state stability is determined by cos0 which is positive for < 0
< 90; The steady-state stability limit is 8 = 90" and from
(3), the maximum power is Ps = 2/X pu. When feedbacks mod+ the responses of A 8 1 and A 8 2 , as long as b 8 1 / A 8 ,
>
0 and A8,/h8, > 0, (1 -
A 8 1 / A 8 , )
> 0 so that the limit remains to be 8 = 90:
Boon-Teck Ooi (S'69, M'71, SM'85) received the B.Eng. Hons. degree from the University of Adelaide, Australia, the S.M. degree from the Mas- sachusetts Institute of Technology, Cambridge, and the Ph.D. degree from
McGill University,Montreal,Quebec, Canada, in 1962, 1967, and 1970, respectively. He is presently a Professor with the Department of Electrical
Engineering, McGill University. His research interests are in the areas of linear induction motors, electrodynamic magnetic levitation with supercon- ducting magnets, subsynchronous resonance phenomena, stability of long- distance power transmission, PWM rectifiers, HVdc and FACTS.
VII. REFERENCES t11
J.Urbanek, RJ. Piwko,E.V.Larsen, B.L.Damky, B.C.Furuma- su, W. Mittlestadt and J.D. Eden, "Thyristor-Controlled Series
Compensation Prototype Installation at the Slatt 500 kV
Substation", IEEE Trans. Power Delivery, vol. 8, no. 3, July 1993,
Mehrdad Kazerani (S'88, M'96) received the B.Sc. degree from Shim
University, Iran, M.Eng. degree from Concordia University, Montreal,
Canada, and Ph.D. degree from McGill University, Montreal, Canada, in
1980, 1990 and 1995, respectively. From 1982 to 1987, he was with the
Energy Ministry, Iran. He is presently a post-doctoral fellow in the
Department of Electrical Engineering, McGill University. His research interests are in the areas of power converters and FACTS.
[41
E.V. Larsen, KClark, S A . Miske, Jr. and J. Urbanek, "Chara- cteristics and Rating Considerations of Thyristor-Controlled
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April 1994, pp. 992-1000.
C. Schauder, M. Gernhardt, E. Stacey, T. Lemak, L. Gyugyi,
T.W. Cease, A. Edris, "Development of a 2100 MVAR Static
Condenser for Voltage Control of Transmission Systems", IEEE
Trans. Power Delivery, vol. 10, no. 3, July 1995, pp. 1486-1496.
L. Gyugyi, C.D. Schauder, S.L. Williams, T.R Rietman, D.R
Torgerson and A. Edris, "The Unified Power Flow Controller:
Richard Marceau (M'80) received the B.Eng. degree from McGill
University, Montreal, Canada, the M.ScA. degree from Ecole Polytechni- que de Montreal, Canada, and the Ph.D. degree from McGill University, in 1977, 1983, and 1993, respectively. From 1978 to 1982, he worked for
MONENCO in system planning, relaying and power station design. He also worked at Hydro-Quebec from 1982 to 1990. He is currently Assistant
Professor at B o l e Polytechnique de Montreal. His immediate research interests include dynamic security analysis, the use of FACTS devices in power systems, and developing virtual reality operator training systems.
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S. Nyati, M. Eitzmann, J. Kappenman, D. VanHouse, N. Mohan and A. Edris, "Design Issues for a Single Core Transformer
Thyristor Controlled Phase-Angle Regulator", IEEE Trans.
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System", IEEE Trans. PowerDelivery, vol. 6, no. 4, Oct. 1991, pp.
1557-1563.
L.Moran, P.D. Ziogas and C. Joos, "Analysis and Design of a
Synchronous Solidstate VAR Compensator', IEEE Trans.
Industry AppIicm'ons, vol.IA-25,no.4, 1989, pp.39848.
B.T. Ooi, S-Z. Dai and X. Wang, "Solid-state Series Capacitive
Reactance Compensators", IEEE Trans. Power Delivery, vol. 8, no. 2, April 1993, pp. 712-718.
B.T. Ooi, S Z . Dai and F.D. Galiana, "A Solid-state PWM
Phase Shifter", IEEE Trans. Power Delivery, 8, no. 2, April
1993, pp. 573-579.
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Zbigniew Wolanski (M'95) received the M.S. degree in Electrical En- gineering from the Technical University in Zielona Gora, Poland, and the
Ph.D. degree from the Kharkov Polytechnical Institute, in 1974 and 1978, respectively. He is presently an Assistant Professor in the Department of
Electrical Engineering, McGill University, Montreal, Canada. His research interests are in power electronics and linear and non-linear control.
Francisco D. Galiana (F) received the B.Eng. Hons. degree from McGill
University, Montreal, Canada, and the S.M. and Ph.D. degrees from the
Massachusetts Institute of Technology, in 1966,1968 and 1971, respectively.
He spent a few years at the Brown Boveri Research Genter and the
University of Michigan. Since 1977, he has been a Professor with the
Department of Electrical Engineering, McGill University. His research interests are in the areas of power system operation and planning including expert system applications, load flow and optimal load flow analysis, system restoration, power system computer simulation environments, load forecasting and management, symbolic computation, unit commitment and hydro-thermal scheduling.
Donald T. McGillis received the B.Sc. degree from Loyola College,
Montreal, Canada, and the B.Eng. degree from McGill University,
Montreal, Canada, in 1949 and 1951, respectively. Since 1951, he was with
Hydro-Quebec as the head of the Analytical Division, Manager of System
Engineering and Advisor to the Vice-president, System Planning. As a member of Canadian Electrical Association, he served as Chairman of the
Engineering and Operating Division, Consultative Committee on Outage
Statistics, Committee on Static Compensation and Committee on Expert
Systems where he is presently active. He is a lecturer at Concordia
University and an adjunct profemor at McGill University,
Geza Joos (M79, SM'89) received the M.Eng. and Ph.D. degrees from
McGill University, Montreal, P.Q. Canada, in 1974 and 1987, respectively.
From 1975 to 1978, he was with Brown Boveri and from 1978 to 1988 with the B o l e de Technologies Superieure, University of Quebec. Since 1988, he has been with the Department of Electrical and Computer Engineering at Concordia University, Montreal. His research interests are in rotating machines, power converters, and electrical drives.