. 2
( 17)


Figure 2.8 Thyristor-controlled series capacitor (TCSC) electric circuit. Reproduced with
permission from C.R. Fuerte-Esquivel, E. Acha, and H. Ambriz-Perez, ˜A Thyristor Controlled Series
Compensator Model for the Power Flow Solution of Practical Power Networks™, IEEE Trans. Power
Systems 15(1) 58“64, # 2000 IEEE

topology of a TCR in parallel with a capacitor branch, just before the thyristor ¬res on. The
thyristor is represented as an ideal switch, and the contribution of the external network is
assumed to be in the form of a sinusoidal current source. The current pulse through the
thyristor, which exhibits a degree of asymmetry right up to the point when the steady-state
is reached, is shown schematically in Figure 2.9. The time reference, termed the ˜original
time reference™ (OR), is taken at the positive-going zero-crossing of the voltage across

π ’ σ a3 π + σ a4




2 π ’ σ a5 2 π + σ a6
’σ a σ a2

Figure 2.9 Thyristor-controlled series capacitor (TCSC) asymmetrical thyristor current. Note: AR,
auxiliary time reference; OR, original time reference. Reproduced with permission from C.R. Fuerte-
Esquivel, E. Acha, and H. Ambriz-Perez, ˜A Thyristor Controlled Series Compensator Model for the
Power Flow Solution of Practical Power Networks™, IEEE Trans. Power Systems 15(1) 58“64, # 2000

the inductive reactance of the TCSC. It is useful at this stage to introduce an ˜auxiliary time
reference™ (AR) in addition to the OR, which is taken at a time when the thyristor starts to
Expressing the line current given in the circuit of Figure 2.8, iline ¼ cos !t, in terms of the
auxiliary reference plane (AR),

iline ¼ cos°!t À a Þ ¼ cos !t cos a þ sin !t sin a ; °2:21Þ

where a , equal to  À , is the ¬ring advance angle, and is the ¬ring angle with the
capacitor voltage positive-going, zero-crossing as reference.
Applying Kirchhoff™s current law to the circuit of Figure 2.8, we obtain
iline ¼ ithy þ icap : °2:22Þ
During the conduction period the voltage across the TCSC inductive and capacitive
reactances have equal values,
d ithy 1 þ
¼ icap dt þ Vcap ; °2:23Þ
dt C
where Vcap is the voltage across the capacitor when the thyristor turns on.
Expressing Equations (2.21)“(2.23) in the Laplace domain, we obtain
Iline ¼ cos a 2 þ sin a 2 ; °2:24Þ
s þ !2 s þ !2
Iline ¼ Ithy þ Icap ; °2:25Þ
Icap ¼ s2 LC Ithy À CVcap ; °2:26Þ
where s is the Laplace operator.
Substituting Equations (2.24) and (2.26) into Equation (2.25), we obtain the current
through the thyristor in the Laplace domain:
!2 CVcap
s 1 0
Ithy ¼ !2 cos°a Þ þ !0 2 ! sin°a Þ þ 2 :
°s2 þ !2 Þ°s2 þ !2 Þ °s þ !2 Þ°s2 þ !2 Þ s þ !2
0 0 0
The corresponding expression in the time domain is readily established from the above
ithy ¼ A cos°!t À a Þ À A cos a cos !0 t À B sin a sin !0 t þ DVcap sin !0 t; °2:28Þ
A¼ ; °2:29Þ
2 À !2

!0 !
B¼ 2 ; °2:30Þ
!0 À ! 2
D ¼ !0 C; °2:31Þ
!2 ¼ : °2:32Þ
In order to make Equation (2.28) valid for the range [Àa , a2 ] in Figure 2.9, it is
necessary to shift the equation to the original time reference, OR, by adding a /! to the time
variable, to give
h a  i  a 
ithy ¼ A cos ! t þ À a À A cos a cos !0 t þ
! !
a   a 
À B sin a sin !0 t þ þ DVcap sin !o t þ : °2:33Þ
! !

After some arduous algebra, we have,
ithy ¼ A cos !t þ °ÀA cos a cos $a À B sin a sin $a þ D Vcap sin $a Þ cos !0 t
þ °A cos a sin $a À B sin a cos $a þ D Vcap cos $a Þ sin !0 t; °2:34Þ
$¼ : °2:35Þ
Equation (2.34) is valid in the range Àa < !t < a2 , and contains the transient and
steady-state components. One further consideration is added to this result to yield the
desired expression for the thyristor current in steady-state, which is reached when the
current pulse in Figure 2.9 becomes symmetrical (i.e. a ¼ a2 ). Such a condition takes
place when the capacitor voltage, Vcap , reaches such a level that the coef¬cient of the
sinusoidal term, sin !0 t, takes a value of zero. At this point in time the capacitor voltage
Vcap ¼ sin°a Þ À cos°a Þ tan°$a Þ: °2:36Þ
The expression for the steady-state thyristor current is obtained by substituting Equa-
tion (2.36) into Equation (2.34), to give
cos a
ithy ¼ A cos°!tÞ À A °2:37Þ
cos°$a Þ
With reference to Figure 2.9, when the steady-state is reached
a ¼ a2 ¼ a3 ¼ a4 ¼ a5 ¼ a6 : °2:38Þ
A similar equation to Equation (2.37), valid for the interval °p À a Þ < !t < °p þ a Þ,
may be obtained by assuming that a second ¬ring pulse, in Figure 2.9, takes place  radians
just after the ¬rst pulse, producing a current ¬‚ow through the thyristor with opposite polarity
to the current given by Equation (2.37):
cos°a Þ
ithy ¼ A cos°!tÞ þ A cos½$°!t À pފ: °2:39Þ
cos°$a Þ
For completeness, in the interval a < !t < °p À a Þ:
ithy ¼ 0: °2:40Þ
Expressions for the voltage across the TCSC capacitor during the conduction period v on
are obtained by substituting Equations (2.37) and (2.39) into:
d ithy
v on ¼ L : °2:41Þ
The combined solution of Equations (2.37), (2.39), and (2.41) gives the voltages across the
capacitor in the intervals Àa < !t < a , and °p À a Þ < !t < °p þ a Þ:
$ XL cos°a Þ
v on ¼ ÀA XL sin°!tÞ þ A sin°$a Þ; °2:42Þ
cos°$a Þ
$ XL cos °a Þ
v cap ¼ ÀA XL sin°!tÞ À A sin½$°!t À pފ; °2:43Þ
cos°$a Þ
where XL is the inductive reactance de¬ned by the product !L.

When the thyristor is not conducting, the circuit in Figure 2.8 reduces to a capacitor in
series with a direct current (DC) voltage source, which represents the capacitor voltage at the
time of thyristor commutation,
1 !t
cos°!tÞ dt þ Vcap- off ;
on -
v cap ¼ °2:44Þ
where Vcap- off is the voltage across the capacitor just at the time when the thyristor is turned
on -

off (i.e. a /!). This value is readily obtained from Equation (2.42), to be
Vcap--off ¼ ÀA XL sin°a Þ þ A$ XL cos°a Þ tan°$a Þ: °2:45Þ

Substitution of Equation (2.45) into Equation (2.44) enables the solution of v off in the
intervals Àa < !t < a , and °p À a Þ < !t < °p þ a Þ:
v off ¼ XC ½sin°!tÞ À sin a Š À AXL ½sin a À $ cos a tan $a Š; °2:46Þ

v off ¼ XC ½sin°!tÞ þ sin a Š þ AXL °sin a À $ cos a tan $a Þ; °2:47Þ

where XC is the inductive reactance de¬ned by 1/!C.
Typical TCSC voltage and current waveforms are shown in Figures 2.10(a) and 2.10(b).
They correspond to the TCSC installed at the Kayenta substation (Christl et al., 1992), with
the thyristors ¬red at angles of 150 and having an inductive reactance of 2.6  and an
capacitive reactance of 15 , at a base frequency of 60 Hz. Thyristor-controlled series capacitor fundamental
frequency impedance

As illustrated by the TCSC waveforms shown in Figures 2.10(a) and 2.10(b), the inductor
current is nonsinusoidal but periodic, and it is said to contain harmonic distortion. If the
interest is the study of the TCSC at only the fundamental frequency then it becomes
necessary to apply Fourier analysis to a full period of the inductor current, say Equations
(2.37), (2.39), and (2.40), in order to obtain its expression at the fundamental frequency.
With reference to Figure 2.10(b), it is clear that the TCSC thyristor current has even and
quarter-wave symmetry. Hence, the fundamental frequency component can be obtained by
solving Equation (2.37) only:
4 a cos°a Þ
Ithy°1Þ ¼ A cos°! tÞ À A cos°$ ! tÞ cos°! tÞd! t
p0 cos°! a Þ
! !
2a þ sin°2a Þ 4A cos2 °a Þ $ tan°k a Þ À tan°a Þ
¼A À :
p $2 À 1 p
The thyristor current at the fundamental frequency may be expressed as
ithy°1Þ ¼ Ithy°1Þ cos°! tÞ: °2:49Þ
By recognising that the fundamental frequency voltage across the TCSC module,
VTCSC(1), equals the fundamental frequency voltage across the capacitor, and that the ideal
current source representing the external circuit is taken to be sinusoidal, an expression for


Voltage and current magnitude (p.u.)




0 200
100 300 400
ω t (electrical deg)


Voltage and current magnitude (p.u.)




0 200
100 300 400
ω t (electrical deg)

Figure 2.10 Voltage and current waveforms in (a) thyristor-controlled series capacitor the (TCSC)
capacitor and (b) the TCSC inductor

the TCSC fundamental frequency impedance may be determined:
VTCSC°1Þ Àj XC Icap°1Þ
ZTCSC°1Þ ¼ ¼ : °2:50Þ
Iline Iline
Moreover, the TCSC contains no resistance and the line current splits between the currents
¬‚owing in the capacitive and inductive branches:
h i
Àj XC °cos !t À Ithy°1Þ cos !tÞ 0
XTCSC°1Þ ¼ ¼ ÀjXC 1 þ I thy°1Þ ; °2:51Þ
cos !t

where XTCSC°1Þ is the TCSC equivalent reactance at the fundamental frequency, and I 0thy°1Þ
has the form of Ithy°1Þ in Equation (2.48) but is a dimensionless parameter as it has been
divided by a unitary current.
The TCSC equivalent reactance is as a function of its capacitive and inductive parameters,
and the ¬ring angle:
XTCSC°1Þ ¼ ÀXC þ C1 f2°p À Þ þ sin½2°p À ފg
þ C2 cos2 °p À Þf$ tan½$°p À ފ À tan°p À Þg; °2:52Þ
XLC ¼ ; °2:53Þ
C1 ¼ ; °2:54Þ
4 X2
C2 ¼ À LC : °2:55Þ
XL p
The poles of Equation (2.52) are:
°2n À 1Þ°LCÞ1=2 !
¼pÀ ; n ¼ 1; 2; 3 . . . : °2:56Þ


Equivalent reactance (W )






90 100 110 120 130 140 150 170

Firing angle (deg)

Thyristor-controlled series capacitor (TCSC) fundamental frequency impedance
Figure 2.11

The TCSC capacitive and inductive reactance values should be chosen carefully in order
to ensure that just one resonant point is present in the range of p=2 to p. Figure 2.11 shows
the TCSC fundamental frequency reactance, as a function of the ¬ring angle, for the TCSC
installed at the Kayenta substation (Christl et al., 1992).
For the purpose of power ¬‚ow studies, the TCSC may be adequately represented by the
equivalent reactance in Equation (2.52), which enables a straightforward representation of
the TCSC in the form of a nodal transfer admittance matrix. This is derived with reference


Ik Im



Figure 2.12 Single-phase thyristor-controlled series capacitor (TCSC) comprising an equivalent
capacitor and a thyristor-controlled reactor (TCR) in parallel. Reproduced with permission from C.R.
Fuerte-Esquivel, E. Acha, and H. Ambriz-Perez, ˜A Thyristor Controlled Series Compensator Model
for the Power Flow Solution of Practical Power Networks™, IEEE Trans. Power Systems 15(1) 58“64,
# 2000 IEEE

to the equivalent circuit in Figure 2.12, where it is assumed that the TCSC is connected
between buses k and m.
The transfer admittance matrix relates the nodal currents injections, Ik and Im, to the nodal
voltages, Vk and Vm, via the variable TCSC reactance shown in the equivalent circuit of
Figure 2.12:
232 32 3
1 1
6 Ik 7 6 jXTCSC V
jXTCSC 76 k 7
6 7¼6 76 7: °2:57Þ
454 1 54 5
Im Vm

In three-phase TCSC installations, three independent modules, possibly of the form
shown in Figure 2.6, may be used, one for each phase. For modelling and simulation
purposes, it is assumed that no electromagnetic couplings exist between the TCSC units
making up the three-phase module. This enables a straightforward extension of the single-
phase TCSC transfer admittance, given by Equation (2.57), to model the three-phase TCSC:
2 3 2 32 3
1 1
6 ITCSC ak 7 6 À jXTCSC1 0 0 0 0 76 VTCSC ak 7
6 76 76 7
6 76 76 7
6 76 76 7
1 1
6 ITCSC bk 7 6 76 VTCSC bk 7
0 0 0 0
6 76 76 7
6 76 76 7
6 76 76 7
1 1
6 ITCSC c k 7 6 76 VTCSC c k 7
0 0 0 0
6 76 jXTCSC 3 76 7
6 76 76 7
6 7 76 7
6 76 76 7
1 1
6 ITCSCa m 7 6 76 VTCSC am 7
0 0 0 0
6 7 6 jXTCSC 1 76 7
6 76 76 7
6 76 76 7
6 ITCSCb m 7 6 76 VTCSC bm 7
1 1
6 76 76 7
0 0 0 0
6 76 76 7
6 76 76 7
4I 54 1 54 VTCSC c m 5
TCSCc m 0 0 0 0


where subscripts 1, 2, and 3 are used to indicate that the three single-phase units may take
different values owing to either different design parameters or unequal thyristor ¬ring
If the three single-phase TCSC units have identical reactance values, say XTCSC, then it is
possible to transform the TCSC phase domain model into the sequence domain frame of
reference. Owing to the decoupled nature of Equation (2.58), the positive, negative, and zero
sequence models are identical and have the same form as Equation (2.57), the representation
of the single-phase TCSC.


Modern power system controllers based on power electronic converters are capable of
generating reactive power with no need for large reactive energy storage elements, such as
in SVC systems. This is achieved by making the currents circulate through the phases of the
AC system with the assistance of fast switching devices (Hingorani and Gyugyi, 2000).
The semiconductor devices employed in the new generation of power electronic
converters are of the fully controlled type, such as the insulated gate bipolar transistor
(IGBT) and the gate turn-off thyristor (GTO). Their respective circuit symbols are shown in
Figure 2.13 (Mohan, Undeland, and Robbins, 1995).

Cathode (K) Collector (C)

Gate (G)

Gate (G)

Emitter ( E)
Anode (A)

(a) (b)

Figure 2.13 Circuit symbols for: (a) gate turn-off thyristor and (b) insulated gate bipolar transistor.
Reproduced by permission of John Wiley & Sons Inc. from N. Mohan, T.M. Undeland, and
W.P. Robbins, 1995, Power Electronics: Converter Applications and Design, 2nd edn

The GTO is a more advanced version of the conventional thyristor, with a similar
switched-on characteristic but with the ability to switch off at a time different from when the
forward current falls naturally below the holding current level. Such added functionality has
enabled new application areas in industry to be developed, even at bulk power transmission
where nowadays it is possible to redirect active power at the megawatt level. However, there
is room for improvement in GTO construction and design, where still large negative pulses
are required to turn them off. At present, the maximum switching frequency attainable is in
the order of 1 kHz (Mohan, Undeland, and Robbins, 1995).
The IGBT is one of the most well-developed members of the family of power transistors.
It is the most popular device used in the area of AC and DC motor drives, reaching power

levels of a few hundred kilowatts. Power converters aimed at power systems applications are
beginning to make use of IGBTs owing to their increasing power-handling capability and
relatively low conduction losses. Further progress is expected in IGBT and GTO technology
and applications (Hingorani, 1998).
In DC“AC converters that use fully controlled semiconductors rather than conventional
thyristors, the DC input can be either a voltage source (typically a capacitor) or a current
source (typically a voltage source in series with an inductor). With reference to this basic
operational principle, converters can be classi¬ed as either voltage source converters (VSCs)
or current source converters. For economic and performance reasons, most reactive power
controllers are based on the VSC topology. The availability of modern semiconductors with
relatively high voltage and current ratings, such as GTOs or IGBTs, has made the concepts
of reactive compensation based on switching converters a certainty, even for substantial
high-power applications.
A number of power system controllers that use VSCs as their basic building block are in
operation in various parts of the world. The most popular are: STATCOMs, solid-state series
controllers (SSSCs), the UPFC, and the HVDC-VSC (IEEE/CIGRE, 1995).

2.4.1 The Voltage Source Converter

There are several VSC topologies currently in use in actual power system operation and
some others that hold great potential, including: the single-phase full bridge (H-bridge); the
conventional three-phase, two-level converter; and the three-phase, three-level converter
based on the neutral-point-clamped converter. Other VSC topologies are based on
combinations of the neutral-point-clamped topology and multilevel-based systems.
Common aims of these topologies are: to minimise the operating frequency of the
semiconductors inside the VSC and to produce a high-quality sinusoidal voltage waveform
with minimum or no ¬ltering requirements. By way of example, the topology of a
conventional two-level VSC using IGBT switches is illustrated in Figure 2.14.

+ +
T c+
Ta+ T b+
D a+ D b+ D c+
’ Va
Ta’ Tb’ Tc’ c
D a’ D b’ D c’
’ ’ Vc

Figure 2.14 Topology of a three-phase, two-level voltage source converter (VSC) using insulated
gate bipolar transistors

The VSC shown in Figure 2.14 comprises six IGBTs, with two IGBTs placed on each leg.
Moreover, each IGBT is provided with a diode connected antiparallel to make provisions for
possible voltage reversals due to external circuit conditions. Two equally sized capacitors
are placed on the DC side to provide a source of reactive power.
Although not shown in the circuit of Figure 2.14, the switching control module is an
integral component of the VSC (Mohan, Undeland, and Robbins, 1995). Its task is to control
the switching sequence of the various semiconductor devices in the VSC, aiming at
producing an output voltage waveform, that is as near to a sinusoidal waveform as possible,
with high power controllability and minimum switching loss.
Current VSC switching strategies aimed at utility applications may be classi¬ed into two
main categories (Raju, Venkata, and Sastry, 1997):
 Fundamental frequency switching: the switching of each semiconductor device is
limited to one turn-on and one turn-off per power cycle. The basic VSC topology
shown in Figure 2.14, with fundamental frequency switching, yields a quasi-square-wave
output, which has an unacceptable high harmonic content. It is current practice to use
several six-pulse VSCs, arranged to form a multipulse structure, to achieve better
waveform quality and higher power ratings (Hingorani and Gyugyi, 2000).
 Pulse-width modulation (PWM): this control technique enables the switches to be turned
on and off at a rate considerably higher than the fundamental frequency. The output
waveform is chopped and the width of the resulting pulses is modulated. Undesirable
harmonics in the output waveform are shifted to the higher frequencies, and ¬ltering
requirements are much reduced. Over the years, various PWM control techniques have
been published, but the sinusoidal PWM scheme remains one of the most popular owing
to its simplicity and effectiveness (Mohan, Undeland, and Robbins, 1995).
From the viewpoint of utility applications, both switching techniques are far from perfect.
The fundamental frequency switching technique requires complex transformer arrange-
ments to achieve an acceptable level of waveform distortion. Such a drawback is offset by
its high semiconductor switch utilization and low switching losses; and it is, at present,
the switching technique used in high-voltage, high-power applications. The PWM technique
incurs high switching loss, but it is envisaged that future semiconductor devices will reduce
this by a signi¬cant margin, making PWM the universally preferred switching technique,
even for high-voltage and extra-high-voltage transmission applications. Pulse-width modulation control

The basic PWM control method can be explained with reference to Figure 2.15, in which a
sinusoidal fundamental frequency signal is compared with a high-frequency triangular
signal, producing a square-wave signal, which serves the purpose of controlling the ¬ring of
the individual valves of a given converter topology, such as the one shown in Figure 2.14.
The sinusoidal and triangular signals, and their associated frequencies, are termed reference
and carrier signals and frequencies, respectively. By varying the amplitude of the sinusoidal
signal against the ¬xed amplitude of the carrier signal, which is normally kept at 1 p.u., the
amplitude of the fundamental component of the resulting control signal varies linearly.
In Figures 2.15(a)“2.15(c), the carrier frequency fs is taken to be 9 times the desired
frequency f1.

v tri
v control

(a) (1 /f s)
vAo v Ao




vcontrol > v tri
v control < v tri


(VˆAo)h ’1
() VDC
1.2 2

m a = 0.8, m f = 9





1 2 mf 3 mf

(m f +2) (3 m f +2)
(2m f +1)

Harmonics h of f1


Figure 2.15 Operation of a pulse-width modulator: (a) comparison of a sinusoidal fundamental
frequency with a high-frequency triangular signal; (b) resulting train of square-wave signals; (c)
harmonic voltage spectrum. Reproduced by permission of John Wiley & Sons Inc. from N. Mohan,
T.M. Undeland, and W.P Robbins, 1995, Power Electronics: Converter Applications and Design,
2nd edn

The width of the square wave is modulated in a sinusoidal manner, and the fundamental
and harmonic components can be determined by means of Fourier analysis. To determine
the magnitude and frequency of the resulting fundamental and harmonic terms, it is useful to
use the concept of amplitude modulation ratio, ma, and frequency modulation ratio, mf:
ma ¼ ; °2:59Þ
mf ¼ ; °2:60Þ
^ ^
where Vcontrol is the peak amplitude of the sinusoidal (control) signal and Vtri is the peak
amplitude of the triangular (carrier) signal, which, for most practical purposes, is kept constant.
With reference to the ˜one-leg™ converter shown in Figure 2.16, corresponding to one leg
of the three-phase converter of Figure 2.14, the switches Taþ and TaÀ are controlled by
straightforward comparison of v control and v tri , resulting in the following output voltages:
2 V DC when Taþ is on in response to v control > v tri ;
v ao ¼ °2:61Þ
À 1 VDC when TaÀ is on in response to v control < v tri :

The output voltage v ao ¬‚uctuates between À VDC/2 and VDC/2, as TaÀ and Taþ are never off
simultaneously, and is independent of the direction of io.


D a+
T a+
+ +
D a’
T a’ van



Figure 2.16 ˜One leg™ voltage source converter (VSC). Reproduced by permission of John Wiley &
Sons Inc. from N. Mohan, T.M. Undeland, and W.P. Robbins, 1995, Power Electronics: Converter
Applications and Designs, 2nd edn

The voltage v ao and its fundamental frequency component are shown in Figure 2.15(b),
for the case of mf ¼ 9 and ma ¼ 0:8. The corresponding harmonic voltage spectrum, in
normalised form, is shown in Figure 2.15(c). This is a case of linear voltage control, where
ma < 1, but this is not the only possibility. Two other forms of voltage control exist, namely,
overmodulation and square-wave modulation. The former takes place in the region 1 <
ma < 3:24 and the latter applies when ma > 3:24 (Mohan, Undeland, and Robbins, 1995).
Only the case of linear voltage control (ma < 1) is of interest in this section. The peak
amplitude of the fundamental frequency component is ma multiplied by VDC/2, and the

harmonics appear as sidebands, centred around the switching frequency and its multiples,
following a well-de¬ned pattern given by:
fh ¼ ° mf Æ Þf1 : °2:62Þ
Harmonic terms exist only for odd values of with even values of . Conversely, even
values of combine with odd values of . Moreover, the harmonic mf should be an odd
integer in order to prevent the appearance of even harmonic terms in v ao . Principles of voltage source converter operation

The interaction between the VSC and the power system may be explained in simple terms,
by considering a VSC connected to the AC mains through a loss-less reactor, as illustrated in
the single-line diagram shown in Figure 2.17(a). The premise is that the amplitude and the


γ Vs
Vs ∠0° δ vR
E vR = V ∠ δvR
Ic ∠ γ vR

+ ∆Vx
δ vR

(a) (c)

Figure 2.17 Basic operation of a voltage source converter (VSC): (a) VSC connected to a system
bus. Space vector representation for (b) lagging operation and (c) leading operation

phase angle of the voltage drop, ÁVx, across the reactor, Xl, can be controlled, de¬ning
the amount and direction of active and reactive power ¬‚ows through Xl. The voltage at the
supply bus is taken to be sinusoidal, of value Vs ¬0 (1), and the fundamental frequency
component of the SVC AC voltage is taken to be VvR ¬vR . The positive sequence
fundamental frequency vector representation is shown in Figures 2.17(b) and 2.17(c) for
leading and lagging VAR compensation, respectively.
According to Figure 2.17, for both leading and lagging VAR, the active and the reactive
powers can be expressed as
Vs VvR >
P¼ sin vR ; =
Vs2 Vs VvR >
cos vR : ;
Q¼ À
Xl Xl

Note on notation: V¬ is a single complex number having a magnitude of V and a phase angle .

With reference to Figure 2.17 and Equation (2.63), the following observations are
 The VSC output voltage VvR lags the AC voltage source Vs by an angle vR , and the input
current lags the voltage drop across the reactor ÁVx by p=2.
 The active power ¬‚ow between the AC source and the VSC is controlled by the phase
angle vR . Active power ¬‚ows into the VSC from the AC source for lagging vR °vR > 0Þ
and ¬‚ows out of the VSC from the AC source for leading vR °vR < 0Þ.
 The reactive power ¬‚ow is determined mainly by the magnitude of the voltage source, Vs,
and the VSC output fundamental voltage, VvR . For VvR > Vs, the VSC generates reactive
power and consumes reactive power when VvR < Vs .
The DC capacitor voltage VDC is controlled by adjusting the active power ¬‚ow that goes
into the VSC. During normal operation, a small amount of active power must ¬‚ow into the
VSC to compensate for the power losses inside the VSC, and vR is kept slightly larger than
0 (lagging).
The various power system controllers that use the VSC as their basic building block are
addressed below with reference to key steady-state operational characteristics and their
impact on system voltage and power ¬‚ow control.

2.4.2 The Static Compensator

The STATCOM consists of one VSC and its associated shunt-connected transformer. It is
the static counterpart of the rotating synchronous condenser but it generates or absorbs
reactive power at a faster rate because no moving parts are involved. In principle, it
performs the same voltage regulation function as the SVC but in a more robust manner
because, unlike the SVC, its operation is not impaired by the presence of low voltages
(IEEE/CIGRE, 1995).

Bus k Bus k
+ VvR δ vR ’ YvR
E vR
Ik γ k
Vk θ k
I vR


Figure 2.18 Static compensator (STATCOM) system: (a) voltage source converter (VSC) connected
to the AC network via a shunt-connected transformer; (b) shunt solid-state voltage source

A schematic representation of the STATCOM and its equivalent circuit are shown in
Figures 2.18(a) and 2.18(b), respectively. The equivalent circuit corresponds to the Thevenin
equivalent as seen from bus k, with the voltage source EvR being the fundamental frequency
component of the VSC output voltage, resulting from the product of VDC and ma.
In steady-state fundamental frequency studies the STATCOM may be represented in the
same way as a synchronous condenser, which in most cases is the model of a synchronous

generator with zero active power generation. A more ¬‚exible model may be realised by
representing the STATCOM as a variable voltage source EvR , for which the magnitude and
phase angle may be adjusted, using a suitable iterative algorithm, to satisfy a speci¬ed
voltage magnitude at the point of connection with the AC network. The shunt voltage source
of the three-phase STATCOM may be represented by:
EvR ¼ VvR °cos vR þ j sin vR Þ; °2:64Þ
where  indicates phase quantities, a, b, and c.

The voltage magnitude, VvR , is given maximum and minimum limits, which are a

function of the STATCOM capacitor rating. However, vR may take any value between 0 and
2p radians.
With reference to the equivalent circuit shown in Figure 2.18(b), and assuming three-
phase parameters, the following transfer admittance equation can be written:
½Ik Š ¼ ½YvR ÀYvR Š ; °2:65Þ
Ik ¼ ½Ik ¬
k Ik ¬
k Ik ¬
k Št ; °2:66Þ
a a b b c c

‚a Ãt
Vk ¼ Vk ¬a Vk ¬b Vk ¬c ; °2:67Þ
b c
k k k

EvR ¼ ½VvR k ¬vRk VvR k ¬vRk VvR k ¬vRk Št ; °2:68Þ
a a b b c c

2a 3
YvR k 0 0
6 7
YvR ¼ 4 0 0 5: °2:69Þ
YvR k
0 0 YvR k

2.4.3 The Solid State Series Compensator

For the purpose of steady-state operation, the SSSC performs a similar function to the static
phase shifter; it injects voltage in quadrature with one of the line end voltages in order to
regulate active power ¬‚ow. However, the SSSC is a far more versatile controller than the
phase shifter because it does not draw reactive power from the AC system; it has its own
reactive power provisions in the form of a DC capacitor. This characteristic makes the SSSC
capable of regulating not only active but also reactive power ¬‚ow or nodal voltage
magnitude. This functionality is addressed further in Section 2.5. A schematic
representation of the SSSC and its equivalent circuit are shown in Figures 2.19(a) and
2.19(b), respectively.
The series voltage source of the three-phase SSSC may be represented by
EcR ¼ VcR °cos cR þ j sin cR Þ; °2:70Þ

where  indicates phase quantities, a, b, and c.
The magnitude and phase angle of the SSSC model are adjusted by using any suitable
iterative algorithm to satisfy a speci¬ed active and reactive power ¬‚ow across the SSSC.
Similar to the STATCOM, maximum and minimum limits will exist for the voltage

Bus k + VcR δ cR ’ Bus m
Bus m
Bus k
Y cR
Ik γ k Im γ m


Vk θ k Vm θ m
E cR
I cR

(a) (b)

Figure 2.19 Solid state series compensator (SSSC) system: (a) voltage source converter (VSC)
connected to the AC network using a series transformer and (b) series solid state voltage source

magnitude VcR, which are a function of the SSSC capacitor rating; the voltage phase angle
cR can take any value between 0 and 2 radians. The control capabilities of the SSSC
are addressed in Section 2.5.
Based on the equivalent circuit shown in Figure 2.19(b), and assuming three-phase
parameters, the following transfer admittance equation can be written:
2 3
! ! Vk
Vm 5:
¼ °2:71Þ
Im YcR
In addition to parameters used in Equations (2.66)“(2.69) the following quantities are
Im ¬
m Št ;
Im ¼ ½Im ¬
m Im ¬
m °2:72Þ
a a b b c c

Vm ¼ ½ Vm ¬a Vm ¬c Š ; °2:73Þ
Vm ¬b
a b c
m m m

EcR ¼ ½VcR ¬cR VcR ¬cR VcR ¬cR Št ; °2:74Þ
a a b b c c

2a 3
YcR k 0 0
6 7
YcR ¼ 4 0 0 5: °2:75Þ
YcR k
0 0 YcR k

2.4.4 The Uni¬ed Power Flow Controller

The UPFC may be seen to consist of two VSCs sharing a common capacitor on their DC side
and a uni¬ed control system. A simpli¬ed schematic representation of the UPFC is given in
Figure 2.20(a), together with its equivalent circuit, in Figure 2.20(b) (Nabavi-Niaki and
Iravani, 1996).
The UPFC allows simultaneous control of active power ¬‚ow, reactive power ¬‚ow, and
voltage magnitude at the UPFC terminals. Alternatively, the controller may be set to control
one or more of these parameters in any combination or to control none of them (Fuerte-
Esquivel, Acha, and Ambriz-Perez, 2000b).

Bus m
Bus k

Vk IcR
Shunt Series
EvR +
I vR

δ vR δ cR


Bus m
Bus k
I1 + E cR ’ Im

I vR

{ — —=
Y vR
Re E vR I vR + EcR I m 0
Vk Vm



Figure 2.20 Uni¬ed power ¬‚ow controller (UPFC) system: (a) two back-to-back voltage source
converters (VSCs), with one VSC connected to the AC network using a shunt transformer and the
second VSC connected to the AC network using a series transformer; (b) equivalent circuit based on
solid-state voltage sources. Redrawn, with permission, from A. Nabavi-Niaki and M.R. Iravani,
˜Steady-state and Dynamic Models of Uni¬ed Power Flow Controller (UPFC) for Power System
Studies™, IEEE Trans. Power Systems 11(4) 1937“1943, # 1996 IEEE

The active power demanded by the series converter is drawn by the shunt converter from
the AC network and supplied to bus m through the DC link. The output voltage of the series
converter is added to the nodal voltage, at say bus k, to boost the nodal voltage at bus m. The
voltage magnitude of the output voltage VcR provides voltage regulation, and the phase
angle cR determines the mode of power ¬‚ow control (Hingorani and Gyugyi, 2000).
In addition to providing a supporting role in the active power exchange that takes place
between the series converter and the AC system, the shunt converter may also generate or
absorb reactive power in order to provide independent voltage magnitude regulation at its
point of connection with the AC system.

The UPFC equivalent circuit shown in Figure 2.20(b) consists of a shunt-connected
voltage source, a series-connected voltage source, and an active power constraint equation,
which links the two voltage sources. The two voltage sources are connected to the AC system
through inductive reactances representing the VSC transformers. In a three-phase UPFC,
suitable expressions for the two voltage sources and constraint equation would be:
EvR ¼ VvR °cos vR þ j sin vR Þ; °2:76Þ
EcR ¼ VcR °cos cR þ j sin cR Þ; °2:77Þ
 Ã  Ã
RefÀEvR IvR þ EvR Im g ¼ 0: °2:78Þ
where  indicates phase quantities, a, b, and c.
Similar to the shunt and series voltage sources used to represent the STATCOM and the
SSSC, respectively, the voltage sources used in the UPFC application would also have limits.
Based on the equivalent circuit shown in Figure 2.20(b), and assuming three-phase
parameters, the following transfer admittance equation can be written:
2 3
! !
°YcR þ YvR Þ ÀYcR ÀYcR ÀYvR 6 Vm 7
Ik 6 7
¼ 4 EcR 5; °2:79Þ
Im YcR YcR 0
where all the parameters have been de¬ned in Equations (2.66)“(2.69) and (2.72)“(2.75).

2.4.5 The High-voltage Direct-current Based on
Voltage Source Converters

The HVDC-VSC comprises two VSCs, one operating as a recti¬er and the other as an
inverter. The two converters are connected either back-to-back or joined together by a DC
cable, depending on the application. Its main function is to transmit constant DC power from
the recti¬er to the inverter station, with high controllability. A schematic representation of
the HVDC-VSC and its equivalent circuit are shown in Figures 2.21(a) and 2.21(b),
One VSC controls DC voltage and the other the transmission of active power through the
DC link. Assuming loss-less converters, the active power ¬‚ow entering the DC system must
equal the active power reaching the AC system at the inverter end minus the transmission
losses in the DC cable. During normal operation, both converters have independent reactive
power control (Asplund, 2000).
The HVDC-VSC system is suitably represented by two shunt-connected voltage sources
linked together by an active power constraint equation. Each voltage source is connected to
the AC system by means of its transformer reactance. Suitable expressions for the three-
phase voltage sources and the linking power equation are:
EvR1 ¼ VvR1 cos vR1 þ j sin vR1 ; °2:80Þ
EvR2 ¼ VvR2 cos vR2 þ j sin vR2 ; °2:81Þ
 Ã  Ã
Ref À EvR1 IvR1 þ EvR2 Im g ¼ 0; °2:82Þ
where  indicates phase quantities, a, b, and c.

Rectifier Inverter
Bus k station station Bus m
E vR1 E vR 2
ma I vR 2
I vR1
Vk Vm

VvR1 δ vR 1 VvR 2 δ vR 2


Bus m
Bus k
Ik I vR1 I vR2

{ }
— —
Re EvR1 I k + EvR 2 Im = 0
Y vR1 Y vR2

+ +
E vR 2
E vR 1

’ ’


Figure 2.21 High-voltage direct-current based on voltage source converter (HVDC-VSC) system:
(a) the VSC at the sending end performs the role of recti¬er, and the VSC at the receiving end
performs the role of inverter; (b) equivalent circuit

In this application, the two shunt voltage sources used to represent the recti¬er and
inverter stations have the following voltage magnitudes and phase angles limits:
VvR min 1 < VvR1 < VvR max 1 ;

0 < vR1 < 2;
VvR min 2 < VvR2 < VvR max 2 ;

0 < vR2 < 2p:

Based on the equivalent circuit shown in Figure 2.21(b), and assuming three-phase
parameters, the following transfer admittance equation can be written:
2 3
! !
6 EvR1 7
YvR1 ÀYvR1
Ik 0 0 6 7;
¼ °2:83Þ
YvR2 ÀYvR2 4 Vm 5
Im 0 0

where all the parameters have been de¬ned in Equations (2.66)“(2.69) and (2.72)“(2.75).


To a greater or lesser extent, the three ˜series™ VSC-based controllers, namely the SSSC, the
UPFC, and the HVDC-VSC, share similar power system control capabilities. They are able
to regulate either nodal voltage magnitude or injection of reactive power at one of its
terminals, and active power ¬‚ow through the controller. The UPFC and the HVDC-VSC
employ two converters and are able to regulate nodal voltage magnitude with one of them
and reactive power injection with the other. From the perspective of fundamental frequency
power system studies, there is little difference between the control ¬‚exibility afforded by
the three controllers, except that the UPFC and HVDC-SVC do it more robustly than
does the SSSC. The individual control functions are illustrated in Figure 2.22, with
reference to the operating regions of the SSSC.
The equivalent circuit of the SSSC shown Figure 2.19(b) is used as the basis for the
analysis. The voltage magnitude of Vm jm can be controlled at a speci¬ed value by injecting
an in-phase or antiphase voltage increment ÁVcR jcr ¼ m , as illustrated in Figure 2.22(a).
Notice that for the purpose of drawing the phasor diagrams in Figure 2.22, the phase angle
m is taken to have a value of 0 . Series reactive compensation can be achieved by injecting
a complex voltage, ÁVcR jcR ¼
m Æ 90 , which is in quadrature with the line current,
Im j
m , as illustrated in Figure 2.22(b). Pure phase-angle control is also possible, as shown in
Figure 2.22(c), by injecting an angular quantity, 1jÆcR , to the otherwise unaffected voltage,
Vm jm . Furthermore, all three functions can be applied simultaneously by injecting an
incremental complex voltage ÁVcR jcR to Vm jm , as shown in Figure 2.22(d), a
characteristic that adds unrivalled ¬‚exibility in power system operation.

Figure 2.22 Phasor diagram illustrating the general concept of: (a) magnitude voltage control,
(b) impedance line compensation, (c) phase-angle regulation, and (d) simultaneous control. Redrawn,
with permission, from Institute of Electrical and Electronic Engineers/Conseil International des
Grands Reseaux Electrique, FACTS Overview, # 1995 IEEE


This chapter has presented an overview of the most salient characteristics of the power
electronic equipment currently used in the electricity supply industry for the purpose of
voltage regulation, active and reactive power ¬‚ow control, and power quality enhancement.
The emphasis has been on steady-state operation, and a distinction has been made between
power electronic equipment, which uses conventional power semiconductor devices (i.e.
thyristors) and the new generation of power system controllers, which use fully controllable
semiconductor devices such as GTOs and IGBTs. The latter devices work well with fast
switching control techniques, such as the sinusoidal PWM control scheme, and, from the
power system perspective, operate like voltage sources, having an almost delay-free
response. Equipment based on thyristors have a slower speed of response, greater than
one cycle of the fundamental frequency, and use phase control as opposed to PWM control.
From the power system perspective, thyristor-based controllers behave like controllable
reactances as opposed to voltage sources.
The TCR, SVC, and TCSC belong to the category of thyristor-based equipment. The
STATCOM, SSSC, UPFC, and HVDC-VSC use the VSC as their basic building block. It has
been emphasised that all these power electronic controllers produce harmonic distortion,
which is an undesirable side-effect, as part of their normal operation. The various means of
harmonic cancellation open to system engineers have been mentioned, such as switching
control, multilevel con¬gurations, three-phase connections, and, as a last resort, ¬ltering
equipment. The remit of this book is not power system harmonics; hence, it is assumed that
harmonic distortion is effectively contained at source. The mathematical modelling
conducted for the various power electronic controllers addressed in the chapter re¬‚ect this
fact. The emphasis has been on deriving ¬‚exible models in the form of nodal admittance
matrices that use the frame of reference of the phases, which is a frame of reference closely
associated with the physical structure of the actual power system plant. A major strength of
this frame of reference is that all design and operational imbalances present in the power
system are incorporated quite straightforwardly in the model. Nevertheless, it is
acknowledged that very often it is desirable to reduce the comprehensiveness of the
power system solution and to carry out the study in the frame of reference of the sequences
rather than in that of the phases. This has the advantage of speedier calculations, but key
information becomes unavailable since sequence domain modelling tacitly assumes that
no imbalances are present in the plant being modelled. When such an assumption is
incorporated in the phase domain nodal admittance models, it yields simpler models
expressed in the frame of reference of the sequences.
The phase domain nodal admittance models are used in Chapter 6 as the basis for
developing the power ¬‚ow equations of three-phase power systems. Similarly, the sequence
domain nodal admittance models are used to develop in Chapter 5 the power ¬‚ow equations
of positive sequence power systems.

Acha, E., Madrigal, M., 2001, Power System Harmonics: Computer Modelling and Analysis, John Wiley
& Sons, Chichester.

Anaya-Lara, O., Acha, E., 2002, Modelling and Analysis of Custom Power Systems by PSCAD/
EMTDC™, IEEE Trans. Power Delivery 7(1) 266“272.
Arrillaga, J., 1998, High Voltage Direct Current Transmission, 2nd edn, Institution of Electrical
Engineers, London.
Arrillaga, J., Arnold, C.P., 1990, Computer Analysis of Power Systems, John Wiley & Sons, Chichester.
Asplund, G., 2000, Application of HVDC Light to Power System Enhancement™, IEEE Winter Meeting,
Session on Development and Application of Self-commutated Converters in Power Systems,
Singapore, January 2000.
Christl, N., Hedin R., Sadek, K., Lutzelberger, P., Krause, P.E., McKenna, S.M., Montoya, A.H.,
Torgerson, D., 1992, ˜Advanced Series Compensation (ASC) with Thyristor Controlled Impedance™,
International Conference on Large High Voltage Electric Systems (CIGRE), Paper 14/37/38-05,
Paris, September 1992.
Fuerte-Esquivel, C.R., Acha, E., Ambriz-Perez, H., 2000a, ˜A Thyristor Controlled Series Compensator
Model for the Power Flow Solution of Practical Power Networks™, IEEE Trans. Power Systems 15(1)
Fuerte-Esquivel, C.R., Acha, E., Ambriz-Perez, H., 2000b, ˜A Comprehensive UPFC Model for the
Quadratic Load Flow Solution of Power Networks™, IEEE Trans. Power Systems 15(1) 102“109.
Hingorani, N.G., 1993, ˜Flexible AC Transmission Systems™, IEEE Spectrum 30(4) 41“48.
Hingorani, N.G., 1995, ˜Introducing Custom Power™, IEEE Spectrum 32(6) 41“48.
Hingorani, N.G., 1996, ˜High Voltage DC Transmission: a Power Electronics work-horse™, IEEE
Spectrum, Vol. 33, No. 4, April 1996, pp. 63“72.
Hingorani, N.G., 1998, ˜High Power Electronics and Flexible AC Transmission Systems™, IEEE Power
Engineering Review (July) 3“4.
Hingorani, N.G., Gyugyi, L., 2000 Understanding FACTS Concepts and Technology of Flexible AC
Transmission Systems, Institute of Electrical and Electronic Engineers, New York.
IEEE/CIGRE (Institute of Electrical and Electronic Engineers/Conseil International des
Grands Reseaux Electriques) FACTS Overview, Special Issue, 95TP108, IEEE Service Center,
Piscataway, NJ.
Kinney, S.J., Mittelstadt, W.A., Suhrbier, R.W., 1995, ˜Test Results and Initial Operating Experience for
the BPA 500 kV Thyristor Controlled Series Capacitor: Design, Operation, and Fault Test Results,
Northcon 95™, in IEEE Technical Conference and Workshops Northcon 95, Portland, Oregon, USA,
October 1995, IEEE, New York, pp. 268“273.
Kundur, P., 1994, Power System Stability and Control, McGraw-Hill, New York.
Larsen, E.V., Bowler, C., Damsky, B., Nilsson, S., 1992, ˜Bene¬ts of Thyristor Controlled Series
Compensation™, International Conference on Large High Voltage Electric Systems (CIGRE), Paper
14/37/38-04, Paris, September 1992.
McMurray, W., 1987, ˜Feasibility of GTO Thyristors in a HVDC Transmission System™, EPRI
EL-5332, Project 2443-5, Final Report, August 1987, Electric Power Research Institute.
Miller, T.J.E., 1982, Reactive Power Control in Electric Systems, John Wiley Interscience, Chichester.
Mohan, N., Undeland, T.M., Robbins, W.P., 1995, Power Electronics: Converter Applications and
Design, 2nd edn, John Wiley & Sons,Chichester.
Nabavi-Niaki, A., Iravani, M.R., 1996, ˜Steady-state and Dynamic Models of Uni¬ed Power Flow
Controller (UPFC) for Power System Studies™, IEEE Trans. Power Systems 11(4) 1937“1943.
Raju, N.R., Venkata, S.S., Sastry, V.V., 1997, ˜The Use of Decoupled Converters to Optimise the Power
Electronics of Shunt and Series AC System Controllers™, IEEE Trans. Power Delivery 12(2) 895“900.
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New York.
Wood, A.J., Wollenberg, B., 1984, Power Generation, Operation and Control, 2nd edn, John Wiley &
Sons, Chichester.
Modelling of Conventional
Power Plant


The conventional elements of an electrical power system are: generators, transformers,
transmission lines, cables, loads, banks of capacitors, nonlinear inductors, and protection
and control equipment. These elements are suitably interconnected to enable the generation
of electricity in suf¬cient quantity to meet system demand at any one point in time. The
operational objective is to transmit the electricity to the load centres at minimum production
cost, maximum reliability, and minimum transmission loss (Elgerd, 1982).
For most practical purposes, the electrical power network may be divided into four
subsystems, namely, generation, transmission, distribution, and utilisation. Transmission
networks operate at high voltages, typically in the range 500“132 kV, although even higher
voltages are used in parts of North America (Weedy, 1987). Conversely, electricity is
produced at relatively low voltages, in the range of 25“11 kV, and step-up transformers are
used at the generator substation to increase the voltage up to transmission levels. In contrast,
step-down transformers are used to reduce the high voltages used in transmission systems to
levels that are appropriate for industrial, commercial, and residential applications. In the
United Kingdom, a typical voltage level used in distribution networks is 33 kV; and
industrial and residential consumers are fed at 11 kV and 415 V or 240 V.
Three-phase synchronous generators are used to produce most of the electricity consumed
worldwide (Grainger and Stevenson, 1994) and, except for a small percentage which is
transported in direct current (DC) form using high-voltage direct-current (HVDC) links,
all electricity is brought to the points of demand using alternating current (AC) three-phase
transmission lines and cables. This point deserves further analysis because quite
often the generating stations are located far away from where the load sites are, and
long-distance transmission becomes necessary (Shlash, 1974). More often than not, long-
distance transmission circuits consist of more than one three-phase circuit, and contain
series and shunt compensation to enable stable operation. Nevertheless, it has long been
recognised that remote generating stations, which are mostly of the hydroelectric type, are
only weakly interconnected and that the nonuniform nature of their rotors (i.e. saliency)
increases the overall system unbalance. It should also be remarked that the windings of

FACTS: Modelling and Simulation in Power Networks.
´ ´
Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho
# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2

three-phase transformers can be connected in a variety of ways to suit speci¬c requirements
and that transformer connections should be modelled explicitly when system imbalances
cannot be ignored in power system studies (Hesse, 1966). The bulk load points associated
with transmission systems may be taken to be highly balanced, but such an assumption is no
longer valid in low-voltage distribution systems, where load points may be highly
unbalanced owing to an abundance of individual single-phase loads within a distribution
load point.
The application tool used to assess the steady-state operation of power systems exhibiting
a considerable degree of geometric unbalance or load unbalance is known as three-phase
power ¬‚ow (Chen and Dillon, 1974; Laughton, 1968; Wasley and Shlash, 1974a). In this
application, all operations are carried out on a per-phase basis, and all power plant
components making up the power system are modelled in the frame of reference of the
phases (Chen et al., 1990; Harker and Arrillaga, 1979). However, if system geometric
imbalances may be taken to be insigni¬cant and system load is balanced then there is much
numerical advantage to be gained by representing all power plant components in the frame of
reference of the sequences as opposed to that of the phases. In this situation, a positive
sequence power ¬‚ow solution can be carried out, as opposed to the full blown three-phase


High-voltage and extra-high-voltage transmission lines consist of a group of phase
conductors, which are responsible for transmitting the electrical energy. All power network
transmission lines are located at a ¬nite distance from the earth™s surface and may use the
ground as a return path. Accordingly, it becomes necessary to take this effect into account
when calculating transmission-line parameters (Anderson, 1973). High-voltage transmission
lines may contain several conductors per phase (bundle conductors) and ground wires, and
distribution lines may include a neutral wire as a return path. Transmission and primary
distribution circuits may be responsible for introducing considerable geometric imbalances,
even at the fundamental frequency, depending on their electrical distance (Acha and
Madrigal, 2001; Arrillaga et al., 1997).

Rseries + jX series
a ,b ,c a ,b ,c

( Gshunt + jBshuntc )
1 a ,b ,c a ,b ,


Transmission line representation in the form of an equivalent p-circuit
Figure 3.1

In power system studies it is current practice to model the inductive and resistive effects
of multiconductor transmission lines as a series impedance matrix, and the capacitive effects
as a shunt admittance matrix. The overall transmission line model can then be represented
by either a nominal p-circuit or an equivalent p-circuit, as shown in Figure 3.1, if the
electrical length of the line is suf¬cient to merit the extra work involved in calculating it.

3.2.1 The Voltage-drop Equation

The phase conductors of a three-phase transmission line, with ground as the return path and
negligible capacitive effects, are illustrated schematically in Figure 3.2. If the circuit
terminal conditions enable current to ¬‚ow in conductors a, b, c, and in the ground return
path, the voltage-drop equation of the transmission line shown in Figure 3.2, at a given
frequency, may be expressed in matrix form as follows:
32 3 2 0 3
RaaÀg þ j!LaaÀg RabÀg þ j!LabÀg RacÀg þ j!LacÀg Va
Va Ia
6 07
4 Vb 5 ¼ 4 RbaÀg þ j!LbaÀg RbbÀg þ j!LbbÀg RbcÀg þ j!LbcÀg 54 Ib 5 þ 4 Vb 5; °3:1Þ
RcaÀg þ j!LcaÀg RcbÀg þ j!LcbÀg RccÀg þ j!LccÀg 0
Vc Ic Vc
the subscript g indicating that the ground return effect has been included.


Lab Lb

Lbc Lc


Lg Ig= Ia+ Ib+ Ic


. 2
( 17)