Day-2
Series Compensation &
Other FACTS Controllers
Arindam Ghosh
Dept. of Electrical Engineering
Indian Institute of Technology
Kanpur, India
E-mail: aghosh@iitk.ac.in
Series Compensation of
Transmission Systems
A device that is connected in series with the
transmission line is called a series
compensator. In the analysis given below,
we shall investigate the effect of the
series compensator on
• the voltage profile
• the power-angle characteristics
• the stability margin
• the damping of power oscillations
82
Ideal Series Compensator
• The ideal series compensator is represented by
a voltage source that only supplies reactive
power and no real power.
• The location of the series compensator is not
crucial, and it can be placed anywhere along the
transmission line.
83
Voltage Profile
• The series voltage must be injected in such
a way that the series compensator does not
absorb any real power in the steady state.
The injected voltage is then
~
~ µ j 90°
VQ = λI S e
where λ is a proportionality constant.
• The ratio λ/X is called the compensation
level. For example, we call the compensation
level to be 50% when λ = X /2.
84
Voltage Profile
Let us assume
Then
~
~
VS = V∠δ , VR = V∠0°.
~
IS =
~ ~ ~
VS − VR − VQ
jX
V∠δ − V
=
j( X µ λ )
The choice of sign of the injected voltage visà-vis that of the current plays an important
role in the operation of the series
compensator.
85
Mode of Operation
~
~ − j 90°
~ V∠δ − V
Case − 1 : VQ = λI S e
⇒ IS =
j( X − λ )
The above choice corresponds to the voltage
source acting as a pure capacitor. Hence we call
this as the capacitive mode of operation.
~
~ + j 90°
~ V∠δ − V
Case − 2 : VQ = λI S e
⇒ IS =
j( X + λ )
Since the voltage source in this case acts as a
pure inductor, we call this as the inductive
mode of operation.
86
An Example
Consider a system with sending and receiving
end voltages being 1∠30º and 1∠ 0º per unit
respectively, X = 0.5 per unit and λ = 0.15
per unit. Let us assume that the series
compensator operates in the capacitive
mode. Then we have
~
I S = 1.479∠15° per unit
~
VQ = 0.2248∠ − 75° per unit
87
Example (Continued)
The phasor diagram is
given left. Let us now
investigate the
impact of the
location compensator
placement on the
voltage profile. We
shall denote the
voltage on the left of
the compensator by
VQL and on the right
of the compensator
by VQR.
88
Case-1: Compensator
in middle
• The series
compensator is
placed in the
middle.
• The worst voltage
sag occurs at
each side of the
series
compensator
where the voltage
vector aligns with
the current
vector.
89
Case-2: Compensator
before the infinite bus
•
• The series
compensator is
placed just
before the
infinite bus.
• The maximum
voltage rise
occurs just
before the
compensator.
The worst voltage sag still occurs where the
voltage vector aligns with the current vector.
90
Power-Angle Curve
The real and reactive powers at the sending
and receiving end are given by
V sin δ
V (1 − cos δ )
PS + jQS =
+j
X µλ
X µλ
2
2
V sin δ
V (cos δ − 1)
PR + jQR =
+j
X µλ
X µλ
2
2
The real power flow is then
V 2 sin δ
Pe = PS = PR =
X µλ
91
Power-Angle Curve
92
Reactive Power
Since the injected voltage and line current are
in quadrature,
• the real power supplied by the compensator
is zero.
• the reactive power supplied by the
compensator operating in the capacitive
mode is
(
)
2λV
~ ~∗
(cos δ − 1)
QQ = Im VQ I S = j
2
(X − λ )
2
93
Reactive Power
Both the real
power
transfer and
the reactive
injection
requirement
increase with
the increase
in the
compensation
level (λ/X).
94
Alternate Method of
Voltage Injection
• The series compensator injects a voltage that
is in quadrature with the line current.
• So far we have assumed that the injected
voltage
magnitude is proportional to the
magnitude of the line current.
• If we now relax the assumption, then the
magnitude of the injected voltage is given by
~
I S µ j 90°
~
VQ = ~ e
IS
95
Phasor Diagrams
Capacitive Operation
Inductive Operation
Note that it is assumed that
~
~
VR = V∠0°, VS = V∠δ
96
Expression for Power
For the capacitive operation
~
~
X I S = VQ + 2V sin (δ 2 )
For the inductive operation
~
~
X I S = − VQ + 2V sin (δ 2 )
The power flow is then given by
2
V
V ~
Pe =
sin δ ± VQ cos(δ 2 )
X
X
97
Power-Angle Curve
98
Series Compensator
Modes of Operation
With appropriate control, the series
compensator operates in two modes:
• Constant Reactance
~
~ µ j 90°
VQ = λI S e
• Constant Voltage
~
I S µ j 90°
~
VQ = ~ e
IS
99
Comparison Between the
Modes of Operation
• The two curves match
at π/2.
• The maximum power
for constant voltage
case occurs earlier
than π/2.
• The power transfer for
constant voltage case
for δ = 0 is greater
than zero.
100
Comparison Between the
Modes of Operation
• The increase in line
current in either
case is monotonic.
• However the rate of
rise in the constant
voltage mode is lower
than constant
reactance mode.
• Constant voltage is
the more desirable
mode of operation
101
Power Flow Control and
Power Swing Damping
Let us consider an example to illustrate
• the power flow control and
• The power swing damping
capabilities of ideal series compensator.
The system contains
• a double circuit transmission line
• one of the two lines compensated by an
ideal series compensator.
102
An Example
~
~
VS = VR = 1.0 pu, X = 0.5 pu, δ 0 = 30°
Constant reactance mode with λ = 0.15 pu
Pe1 = 1.0 pu, Pe 2 = 1.43 pu, Pm 0 = 2.43 pu
103
Example (Continued)
For any increase or decrease in the power
flow, the series compensator can be
controlled in one of the following two
modes.
• Regulating Control: Channeling the increase
(or decrease) in power through line-1. In
this case the series compensator holds the
power flow over line-2 constant.
• Tracking Control: Channeling the increase
(or decrease) in power through line-2. In
this case the series compensator helps in
holding constant the power flow over line-1.
104
Example (Continued)
• The input to the control system is the power
flow over line-2 (Pe2).
• Pe2 is compared with the reference value Pref
and the error is passed through a PI controller.
• In addition, a damping controller is also added
to the feedback loop.
• The output of the controller is the
compensation level CI (=λ/X).
d∆δ
C I = K P Pref − Pe 2 + K I ∫ Pref − Pe 2 dt + C P
dt
The controller parameters are
(
)
(
)
K P = 0.1, K I = 1 and C P = 75
105
Example:
Regulating Control
• The system is in the nominal steady state with
Pm0 = 2.43 per unit.
• The mechanical power input is suddenly raised
by 10% (i.e., 0.243 per unit).
• It is expected that the series compensator
will hold the power through line-2 constant at
Pe2 such that entire power increase is
channeled through line-1.
• We then expect that the power Pe1 will
increase to 1.243 per unit and the load angle to
go up to 0.67 rad.
• The compensation level will then change to 13%.
106
Example:
Regulating Control
107
Example:
Tracking Control
•
•
•
•
•
The system is in the nominal steady state with
Pm0 = 2.43 per unit.
The mechanical power input is suddenly raised
by 25% (i.e., 0.6 per unti).
It is expected that the series compensator
will make the entire power increase to flow
through line-2.
Then both Pe1 and load angle are maintained
constant at their nominal values.
The power, Pe2, through line-2 will then
increase to about 2.04 per unit and the
compensation level will change to 51%.
108
Example:
Tracking Control
109
Practical Series Compensator
• The series compensator structure assumed
throughout this section is essentially that of
a static synchronous series compensator or
SSSC . Like in the case of STATCOM, the
SSSC includes an SVS (supplied by a dc
capacitor) and a coupling transformer.
• A thyristor controlled series compensator
or TCSC is an older thyristor and passive
element based devices that controls the
fundamental reactance. We shall discuss it
first.
110
Thyristor Controlled Series
Compensator (TCSC)
Equivalent Circuit
Voltage-Current
Waveforms
111
TCSC - Circuit Equations
The TCSC voltage and currents are combination
of two piecewise linear models. Let the
system state vector be xT = [vc iP]. Then when
the thyristor is on

0
x&= 
1

L
1
−  1
C x +  i

C L
0   0 

112
TCSC - Circuit Equations
Similarly when the thyristor is off
1


0 0 


x&= 
x
+
C iL

0 0  0 
The waveform of vC is a combination of the
solution of these two state equations and
therefore is not a smooth sinusoidal function.
Therefore both the inductor current and
capacitor voltage are the solutions of two
piece-wise linear models.
113
TCSC - Fundamental
Characteristics
• A TCSC is a parallel combination of a fixed
capacitor and a thyristor controlled reactor.
• Therefore the steady state fundamental
impedance of the TCSC is given by
X TCSC
X C X P (α )
=
X C − X P (α )
• We can therefore see that by varying the
conduction angle, the fundamental frequency
reactance of the TCSC can be made inductive
or capacitive.
114
TCSC - Fundamental
Reactance
The fundamental reactance of the TCSC is given by
X TCSC = β1 ( X C + β 2 ) − β 4 β 5 − X C
β1 =
2(π − α ) + sin 2(π − α )
π
XC X P
, β2 =
, β3 =
XC − X P
XC
XP
4 β 22 cos 2 (π − α )
β 4 = β 3 tan[β 3 (π − α )] − tan (π − α ), β 5 =
π XP
• α is the firing angle
• XC and XP respectively are the reactances of
the capacitor and parallel inductor.
115
TCSC - Fundamental
Reactance Plot
116
TCSC - Fundamental
Impedance Plots for Finite Q
•
•
The fundamental reactance and resistance
change with the quality factor of the coil.
The above design curves can be used for
characterizing the TCSC.
117
TCSC Waveforms for
Finite Q Factor
118
TCSC Control
A TCSC can be controlled in three modes.
Blocking Mode:
• The thyristors are not gated (i.e., the TCR is
blocked).
• The line current passes through the capacitor.
• The TCSC performs the task of a fixed series
capacitor.
Bypass Mode:
• The thyristors are gated for full conduction
of inductor current.
• The TCSC behaves as a parallel of fixed
capacitor and inductor.
119
TCSC Control
Bypass Mode (continued):
• The capacitor voltage is less for a given line
current.
• Therefore this mode is utilized to reduce
capacitor stress during faults.
Vernier Control Mode:
• The conducting angle of the TCSC is
continuously varied to operate in either
capacitive boost or inductive boost modes.
• In this mode the TCSC follows the
fundamental reactance curve shown earlier.
120
Static Synchronous Series
Compensator (SSSC)
• An SSSC contains an
SVS and a coupling
transformer that is
connected in series
with the line.
• The SSSC is operated
such that the injected
voltage is almost in
phase quadrature with
the line current.
121
Equivalent Circuit of SSSC
Compensated System
• In the equivalent circuit of an SSSC
compensated system, the SSSC is represented
by a voltage source and impedance (Lr,Rr).
• The SSSC is connected between buses 1 and 2.
• The pair (L1,R) represent the line and L2
represents a transformer.
122
SSSC Control
123
SSSC Control
• An instantaneous 3-phase set of line voltages
at bus 1 is used to calculate the angle θ that
is phase locked to the phase-a of the line
voltage.
• An instantaneous 3-phase set of measured line
currents is first decomposed into real and
reactive components.
• The amplitude and the relative angle of the
line current θir are then calculated.
• The phase locked angle θp and θir are added to
obtain the angle θl which is the angle of the
line current.
124
SSSC Control
• The are two control loops.
• Since the SSSC voltage must lag the line current
by 90º, a fixed angle equal to -90º is added to θl
to obtain θfref in the main loop.
• In the auxiliary loop, the reactance demand Xtref
is added to Xr of SSSC.
• The sum is multiplied by the magnitude of the
line current and a constant to obtain Vdcref.
• The error between Vdcref and the actual value of
Vdc is passed through a PI controller to obtain θd.
• This quantity is then added to θfref to obtain θf
of the inverter.
125
SSSC Control
• The PI controller retains the charge on the dc
capacitor by injecting a voltage nearly in
quadrature with the line current.
• The real power exchange between the ac system
and SSSC takes place if the injected voltage is
not in quadrature with the line current, which
either charges or discharges the dc capacitor.
• The PI controller then advances or retards the
phase of the injected voltage relative to line
current in order to adjust the power at ac
terminals and keep the dc voltage constant.
126
Comparison Between SSSC and TCSC
127
Comparison (Continued)
128
Comparison (Continued)
129
Other FACTS Controllers
There are many other FACTS controllers that
use the power electronic technology. We
shall briefly discuss the following:
• Interline Power Flow Controller (IPFC)
• Thyristor Controlled Braking Resistor
(TCBR)
• Thyristor Controlled Phase Angle Regulator
(TCPAR)
• Unified Power Flow Controller (UPFC)
130
Interline Power Flow
Controller (IPFC)
An IPFC contains two or more SSSCs that are
connected to a common dc bus to facilitate
real power exchange between them.
131
IPFC (Continued)
• Each individual SSSC can provide controllable
series compensation to the line it is connected.
• In addition, it can also exchange power
between them.
• Example: Assume that Line-1 is lightly loaded
while Line-2 is heavily loaded.
• SSSC-1 then absorbs power to charge the dc
capacitor.
• SSSC-2 is then supplied real power by the dc
capacitor.
• In this way the load sharing between the lines
can be equalized.
132
Thyristor Controlled
Braking Resistor
This is a shunt connected thyristor switched
resistor, which is used for minimizing the
power acceleration during a fault.
• In the schematic
diagram, the
thyristors are
usually blocked.
• They are
switched on when
a fault is
detected.
133
Braking Resistor
For the system shown above, the critical
clearing angle is computed to be 52.24º.
We shall now place a dynamic brake at the
generator terminals.
134
Braking Resistor
We assume that the
braking resistor is
pressed into
service as soon as
the fault occurs
and is removed as
soon as the fault is
cleared. Then
2
′
E
∗
Pe = Re (E ′∠δ )I ≈
10 R
{
}
135
Braking Resistor
It can be seen that the critical clearing angle
increases with the increase in the value of
the resistor.
136
Phase Angle Regulator (PAR)
137
PAR - Operation
•
•
•
The PAR is inserted between the sending end
and transmission line.
It is a sinusoidal voltage source with
controllable amplitude.
For an ideal PAR the angle of the phasor Vσ is
stipulated to vary such that the magnitude of
Vseff remains constant.
138
PAR - Power-Angle Curve
139
PAR - Phase Shifting
Assume that a resistive load is connected at the
output.
140
PAR - Phase Shifting
The voltages obtained at the lower and upper
taps are v1 and v2 respectively.
• At the zero crossing of these voltages, the
switch Sw1 is turned on.
• At α the switch Sw2 is turned on.
• This commutates the current of Sw1 by
forcing a negative anode to cathode voltage
across it, thereby making the voltage across
the load v2.
• The switch Sw2 turns off when the current
through it reverses.
141
PAR - Phase Shifting
The voltage thus obtained contains harmonics.
The fundamental component of the voltage
is given by
~
2
2
−1  a 
V0 = a + b ∠ tan  
b
~
~
~
~
V2 − V1
~ V2 − V1
(cos 2α − 1), b = V1 +
a=
2π
2π
sin 2α 

π − α +

2 

We can therefore vary the fundamental voltage
by varying the delay angle α.
142
Unified Power Flow
Controller (UPFC)
A UPFC contains a
shunt SVS and a
series SVS that
are connected to
a common dc bus
such that real
power exchange
can take place
between them.
143
UPFC - Equivalent Circuit
The term Ppq
indicates
real power
exchange
between
the shunt
and series
branches.
144
UPFC - Phasor Diagrams
(a) Voltage Regulation, (b) Line impedance compensation,
(c) Phase shifting and (d) simultaneous control.
145
UPFC - Real & Reactive Power
Let
~
~
~
VS = V∠0°, VR = V∠ − δ , V pq = V pq ∠ − ρ
Then for the uncompensated system (Vpq = 0)
we have
2
V
sin δ
P = P0 =
X
2
V
(cos δ − 1)
QR = −Q0 =
X
146
UPFC - Real & Reactive Power
For the compensated system
V − V∠ − δ + V pq ∠ − ρ 
P − jQR = V∠ − δ 

jX


∗
Solving we get
P = P0 −
VV pq
QR = Q0 −
sin (δ − ρ )
X
VV pq
X
cos(δ − ρ )
147
UPFC - Real & Reactive Power
Then
(P − P0 ) + (QR − Q0 )
2
2
 VV pq 

= 
 X 
2
• This is the equation of circle with its center
at P0,Q0 and radius of VVpq/X.
• Suppose V2/X = 1.0 and Vpq = 0.5V.
• Then VVpq/X = 0.5.
148
Real Versus Reactive Power
Even though the
uncompensated
system cannot
transfer any
power in this
case, the UPFC
is capable of
transmitting
power in either
direction.
149
Real Versus Reactive Power
150
Real vs Reactive Power
• The figures show that the UPFC has the
unique capability to control independently
the real and reactive power flow at any
transmission angle.
• It is however assumed here that the
sending and receiving end voltages are
provided by independent power systems
which are able to supply and absorb real
power without any internal angular change.
• In practice the situation will be different
depending on the change in load angle.
151