19
CHAPTER 2
MODELING OF FACTS DEVICES FOR POWER SYSTEM
STEADY STATE OPERATIONS
2.1
INTRODUCTION
The electricity supply industry is undergoing a profound
transformation worldwide. Market forces, scarcer natural resources and an
ever-increasing demand for electricity are some of the drivers responsible for
such an unprecedented change. Against this background of rapid evolution,
the expansion programmes of many utilities are being thwarted by a variety of
well-founded, environmental, land use and regulatory pressures that prevent
the licensing and building of new transmission lines and electricity generating
units. An in-depth analysis of the options available for maximizing existing
transmission assets, with high levels of reliability and stability, has pointed in
the direction of power electronics. There is general agreement that novel
power electronics equipment and techniques are potential substitutes for
conventional solutions, which are normally based on electromechanical
technologies that have slow response times and high maintenance costs. The
power electronics options we have, are HVDC transmission systems and
FACTS (Acha et al 2004).
Until recently, active and reactive power control in AC
transmission networks was exercised by carefully adjusting transmission line
impedances, as well as regulating terminal voltages by generator excitation
control and by transformer tap changes. At times, series and shunt
20
impedances were employed to effectively change line impedances. FACTS
technology is most interesting for transmission planners because it opens up
new opportunities for controlling power and enhancing the usable capacity of
present, as well as new and upgraded, lines. The possibility that current
through a line can be controlled at a reasonable cost enables a large potential
of increasing the capacity of existing lines with large conductors and use of
one of the FACTS controllers, to enable corresponding power to flow through
such lines under normal and contingency conditions. By providing added
flexibility, FACTS controllers can enable a line to carry power closer to its
thermal rating. It must be emphasized that FACTS is an enabling technology,
not a one to one substitute for mechanical switches. In its most general
expression, the FACTS concept is based on the substantial incorporation of
power electronic devices and methods into the high-voltage side of the
network, to make it electronically controllable.
Comprehensive modeling of most popular FACTS devices namely
SVC, TCSC, STATCOM and UPFC for power flow studies are carried out.
The effectiveness of modeling and convergence is tested on a five bus system
without any FACTS devices and further analyzed it with different FACTS
devices. The Newton-Raphson method is used to solve the nonlinear power
flow equation. The work is also extended for IEEE 30 bus system.
2.2
POWER FLOW IN A MESHED SYSTEM
To understand the free flow of power, consider a very simplified
case in which generators at two different sites are sending power to a load
centre through a network consisting of three lines in a meshed connection
(Figure 2.1).
21
Figure 2.1
Power flow in a mesh network: (a) system diagram
(b)
system
diagram
with
Thyristor-controlled
series
capacitor in line AC (c) system diagram with Thyristorcontrolled series reactor in line BC
Let the lines AB, BC and AC have continuous ratings of 1000 MW,
1250 MW and 2000 MW. If one of these generators is generating 2000 MW
and the other generates 1000 MW, then a total of 3000 MW would be
delivered to the load centre. For the impedances shown, the three lines would
carry 600, 1600 and 1400 MW respectively, as shown in Figure 2.1(a). Such
a situation would overload line BC and therefore generation would have to be
decreased at B and increased at A, in order to meet the load without
overloading line BC. Power, in short, flows in accordance with transmission
line impedances (which are 90% inductive). If however a capacitor whose
reactance is -j5
at the synchronous frequency is inserted in one line (Figure
2.1(b)), it reduces the line impedance from 10
to 5 , so that power flow
22
through the lines AB, BC and AC will be 250, 1250 and 1750 MW
respectively (Hingorani and Gyugyi 2000).
A series capacitor in a line may lead to sub-synchronous resonance
(typically at 10 to 50 Hz for a 60 Hz system). If such resonance persists it
will soon damage the turbine and shaft. If all or part of the series capacitor is
thyristor controlled it can be modulated to rapidly damp, any sub synchronous
resonance condition and also low frequency oscillations in power flow. This
enhances easy operation within steady state conditions thus improving
stability of the network. Similar results can be obtained by increasing the
impedance of one of the lines in the same meshed configuration by inserting a
+j7 reactor (inductor) in series with line BC (Figure 2.1(c)).
2.3
BENEFITS OF FACTS DEVICES
Edris et al (1997) proposed terms and definitions for Flexible AC
Transmission System (FACTS) as: Alternating-current transmission systems
incorporating power electronics-based and other static controllers to enhance
controllability and increase power transfer capability.
The main objective of FACTS devices is to replace the existing
slow acting mechanical controls required to react to the changing system
conditions by rather fast acting electronic controls. These devices can
dynamically control line impedance, line voltage, active power flow and
reactive power and when storage becomes economically viable; they can
supply and absorb active power as well. All this can be done in high speed.
The implementation of FACTS devices requires technology for high power
electronics with its real time operating control. With the use of fast acting
controls, the power system security margins could be enhanced by utilizing
the full capacity of transmission lines maintaining the quality and reliability
of power supply. Generation patterns that lead to heavy line flows result in
23
higher losses leading to reduced security and stability margin are
economically undesirable.
Further, transmission constraints make certain
combinations of generation and demand unviable, due to the potential line
outages. In such situations, FACTS devices may be used to improve system
performance by controlling the power flows in transmission lines. By using
reliable, high speed power electronic controllers, technology offers utilities
the following opportunities for increased efficiency:
Greater control of power, so that it flows on the prescribed
transmission routes.
Loading of transmission line to levels nearer their thermal
limits.
Greater ability to transfer power between controlled areas so as
to have reduced generation reserve margin.
Prevention of cascading outages by limiting the effects of faults
and equipment failure.
Damping of power system oscillations, which could damage
equipment failure.
Improvement in steady state stability limit and transient stability
margin.
2.4
PRINCIPLE OF OPERATION
This section describes briefly the principle of operation and control
of SVC, TCSC, STATCOM and UPFC (Mohan Mathur and Rajiv Varma
2002).
24
2.4.1
Static Var Compensator
The Static Var Compensators are the most widely installed FACTS
equipment at this point in time. They mimic the working principles of a
variable shunt susceptance and use fast thyristor controllers with settling
times of only a few fundamental frequency periods. From the operational
point of view, the SVC adjusts its value automatically in response to changes
in the operating conditions of the network. By suitable control of its
equivalent susceptance, it is possible to regulate the voltage magnitude at the
SVC point of connection, thus enhancing significantly the performance of the
power system.
The majority of Static Var Compensators have similar controllable
elements. The most common ones are,
Thyristor-Controlled Reactor (TCR)
Thyristor-Switched Capacitor (TSC)
Thyristor-Switched Reactor (TSR)
In the case of the TCR a fixed reactor, typically is connected in
series with a bidirectional thyristor valve. The fundamental frequency current
is varied by phase control of the thyristor valve. A TSC comprises of a
capacitor in series with a bidirectional thyristor valve and a damping reactor.
The function of the thyristor switch is to connect or disconnect the capacitor
for an integral number of half-cycles of the applied voltage. A practical
configuration of a TCR-TSC SVC system is shown in Figure 2.2. The
equivalent susceptance Beq is determined by the firing angle
The variation of Beq as a function of
of the thyristor.
is as shown in Figure 2.3.
25
HV bus
V
PT
Controller
Filter
Figure 2.2 A practical example of TCR-TSC SVC
Beq
unavailable
Bmax
0o
90o
res
Bmin
unavailable
Figure 2.3 Variation of Beq as a function of
2.4.1.1
for SVC
V-I Characteristics of the SVC
The steady-state and dynamic characteristics of SVC describe the
variation of SVC bus voltage with SVC current (Figure 2.4).
26
Vsvc
Over current limit
Over load range
Steady-State Characteristics
Bmin
V1
Vref
Dynamic Characteristics
V2
Bmax
ICr
Linear Range of Control
Capacitive
0
Inductive
ILr
Isvc
Figure 2.4 V-I characteristics of SVC
Dynamic Characteristics
Reference voltage, Vref : This is the voltage at the terminals of the
SVC during the floating condition, that is, when the SVC is neither absorbing
nor generating any reactive power.
Linear range of SVC control: This is the control range over which
SVC terminal voltage varies linearly with the SVC current or reactive power
as it is varied over its entire capacitive to inductive range.
Slope or current droop: The slope or droop of the V-I
characteristics is defined as the ratio of voltage-magnitude change to currentmagnitude change over the linear-controlled range of the compensator.
Overload Range: When the SVC traverses outside the linearcontrollable range on the inductive side, the SVC enters the overload zone,
where it behaves like a fixed inductor.
27
Over current Limit: To prevent the thyristor valves from being
subjected to excessive thermal stress, the maximum inductive current in the
overload range is constrained to a constant value by an additional control
action.
Steady-State Characteristics
The steady-state characteristic of the SVC has a dead band. In the
absence of this dead band, in the steady state the SVC will tend to drift toward
its reactive power limits to provide voltage regulation. It is not desirable to
leave the SVC with very little reactive-power margin for future voltage
control or stabilization excursions in the event of system disturbance.
2.4.2
Thyristor Controlled Series Capacitor
TCSC is a capacitive reactance compensator, which consists of a
series capacitor bank shunted by a thyristor-controlled reactor in order to
provide a smoothly variable series capacitive reactance. The basic TCSC
module comprises a series capacitor C, in parallel with a thyristor-controlled
reactor Ls as shown in Figure 2.5. However, a practical TCSC module also
includes protective equipment normally installed with series capacitors. An
actual TCSC system usually comprises a cascaded combination of many such
TCSC modules, together with a fixed-series capacitor CF. This fixed-series
capacitor is provided primarily to minimize costs.
Figure 2.5 A basic TCSC module
28
2.4.2.1
Basic Operation of TCSC
A TCSC is series-controlled capacitive reactance that can provide
continuous control of power on the ac line over a wide range. From the
system viewpoint, the principle of variable-series compensation is simply to
increase the fundamental frequency voltage across a fixed capacitor (FC) in a
series-compensated line through appropriate variation of the firing angle .
A simple understanding of TCSC functioning can be obtained by
analyzing the behavior of a variable inductor connected in parallel with a FC
(Figure 2.6). The impedance of FC is given by -j(1/ C).
Figure 2.6 A variable inductor connected in shunt with an FC
If
C - (1/ L)
0 or in other words, L
(1/ C), the reactance of
the FC is less than that of the parallel-connected variable reactor and that
this combination provides a variable-capacitive reactance are both implied.
If
C - (1/ L) = 0, a resonance develops that results in an infinite-capacitive
impedance – an obviously unacceptable condition.
If, however,
C - (1/ L)
0, the LC combination provides
inductance above the value of the fixed inductor. In the variable-capacitance
mode of the TCSC, as the inductive reactance of the variable inductor is
increased, the equivalent capacitance reactance is gradually decreased.
29
2.4.3
Static Synchronous Compensator
A static synchronous generator operated as a shunt-connected Static
Var Compensator whose capacitive or inductive output current can be
controlled independent of the ac system voltage.
Figure 2.7
The STATCOM operating principle diagram (a) power
circuit (b) equivalent circuit and (c) power exchange
A STATCOM is a controlled reactive-power source. It provides
the desired reactive-power generation and absorption entirely by means of
electronic processing of voltage and current waveforms in a voltage-source
converter (VSC). A single-line STATCOM is shown in Figure 2.7(a), where
a VSC is connected to a utility bus through magnetic coupling. In the Figure
2.7(b) STATCOM is seen as an adjustable voltage source behind a reactance
meaning that capacitor banks and shunt reactors are not needed for reactive
power generation and absorption, thereby giving STATCOM a compact
design.
30
Figure 2.8
The power exchange between the STATCOM and the AC
system
The exchange of reactive power between the converter and the ac
system can be controlled by varying the amplitude of the 3-phase output
voltage, Es of the converter, as illustrated in Figure 2.7(c). If the amplitude of
the output voltage is increased above that of the utility bus voltage, Et, then a
current flows through the reactance from the converter to the ac system and
the converter generates capacitive-reactive power for the ac system. If the
amplitude of the output voltage is decreased below the utility bus voltage,
then the current flows from the ac system to the converter and the converter
absorbs inductive-reactive power from the ac system. If the output voltage
equals the ac system voltage, the reactive-power exchange is zero, in which
case the STATCOM is said to be in floating state.
The reactive and the real-power exchange between the STATCOM
and the ac system can be controlled independently of each other. Any
combination of real power generation or absorption is achievable if the
31
STATCOM is equipped with an energy-storage device of suitable capacity as
depicted in Figure 2.8. With this capability, extremely effective control
strategies for the modulation of reactive and real-power can be achieved.
Furthermore, a STATCOM does the following:
1. It occupies a small footprint, for it replaces passive banks of
circuit elements by compact electronic converters;
2. It offers modular, factory-built equipment, thereby reducing
site work and commissioning time;
3. It uses encapsulated electronic converters, thereby minimizing
its environmental impact.
2.4.4
Unified Power Flow Controller
A combination of a static synchronous compensator (STATCOM)
and a static synchronous series compensator (SSSC) which are coupled via a
common dc link, to allow bi-directional flow of real power between the series
output terminals of the SSSC and the shunt operated terminals of STATCOM,
and are controlled to provide concurrent real and reactive series line
compensation without an external electric energy source. The UPFC, by
means of angularly unconstrained series voltage injection, is able to control,
concurrently or selectively, the transmission line voltage, impedance and
angle or alternatively, the real and reactive power flow in the line. The UPFC
may also provide independently controllable shunt-reactive compensation.
The UPFC which is one of the most promising device in the FACTS concept,
has been researched and put into practical use (Schauder 1998).
32
Figure 2.9
The implementation of UPFC with two back-to-back VSCs
with a common dc-terminal capacitor
The UPFC consists of two voltage source converters. The dc
voltage for both converters is provided by a common capacitor bank (dc link)
(Figure 2.9).
Converter 2 provides the main function of the UPFC by
injecting an ac voltage with controllable magnitude Vpq, in series with the
transmission line via a series transformer which can be varied from 0 to
Vpqmax and phase angle of Vpq can be independently varied from 0o to 360o.
The basic function of converter 1 is to supply or absorb the real power
demand of converter 2 which it derives from the transmission line itself.
Although the reactive power is internally generated / absorbed by the series
converter, the real power generation /absorption is made feasible by the dcenergy-storage device i.e., the capacitor. It can also generate or absorb
controllable reactive power and provide independent shunt reactive
compensation for the line. Converter 2 supplies or absorbs locally the required
reactive power and exchanges the active power as a result of the series
injection voltage. Thus the net real power drawn from the ac system is equal
to the loss of the two converters and their coupling transformers. In addition
33
the shunt converter functions like a STATCOM and independently regulates
the terminal voltage of the interconnected bus.
2.5
POWER FLOW STUDIES
Planning the operation of power systems under existing conditions,
its improvement and also its future expansion require the load flow studies,
short circuit studies and stability studies. The satisfactory operation of the
system depends upon knowing the effects of interconnections, new loads, new
generating stations and new transmission lines before they are installed. They
also help to determine the best size and favourable locations for the power
capacitors both for the improvement of the power factor and also raising the
bus voltages of the electrical network. They help us to determine the best
locality as well as optimal capacity of the proposed generating stations, sub
stations or new lines.
Through the load flow studies we can obtain the voltage
magnitudes and angles at each bus in the steady state.
This is rather
important, as the magnitudes of the bus voltages are required to be held within
a specified limit. Once the bus voltage magnitudes and their angles are
computed using the load flow, the real and reactive power flow through each
line can be computed. Also based on the difference between power flow in
the sending and receiving ends, the losses in a particular line can also be
computed.
2.5.1
The Newton Raphson Algorithm
A popular approach to assess the steady state operation of a power
system is to write equations stipulating that at a given bus, the generation and
powers exchanged through the transmission elements connecting to the bus
must add up to zero. This applies to both active and reactive power. These
34
equations are termed ‘mismatches power equations’ and at bus k, they take
the following form:
Pk = PGk – PLk – Pkcal = Pksch - Pkcal = 0
(2.1)
Qk = QGk – QLk – Qkcal = Qksch – Qkcal = 0
(2.2)
P, Q - mismatch active and reactive powers
PG , QG - active and reactive powers injected by the generator
PL , QL - active and reactive powers drawn by the load at the bus
Under normal circumstances the customer has control over these
variables and in the power flow formulation they are assumed to be known
variables. In principle, at least, the generation and the load at bus may be
measured by the electric utility and in the parlance of power system
engineers, their net values are known as the scheduled active and reactive
powers as given in equations (2.3) and (2.4):
Pksch = PGk - PLk
(2.3)
Qksch = QGk - QLk
(2.4)
The transmitted active and reactive powers, Pkcal and Qkcal, are
functions of nodal voltages and network impedances and are computed using
power flow equations. Provided the nodal voltages throughout the power
network are known to a good degree of accuracy then the transmitted powers
are easily and accurately calculated.
In this situation, the corresponding
mismatch powers are zero for any practical purposes and the power balance at
each bus is satisfied. However, if the nodal voltages are not known precisely
then the calculated transmitted powers will have only approximated values
and the corresponding mismatch powers are not zero. The power flow
solution takes the approach of successively correcting the calculated nodal
35
voltages and hence, the calculated transmitted powers until values accurate
enough are arrived at, enabling the mismatch powers to be zero or fairly close
to zero. In modern power flow programs, it is normal to all mismatch
equations to satisfy a tolerance as tight as 1e-12 before the iterative solution
can be considered successful.
In order to develop suitable power flow equations, it is necessary to
find relationships between injected bus currents and bus voltages. Based on
the Figure 2.10 the injected complex current at bus k, denoted by Ik , may be
expressed in terms of the complex bus voltages Ek as follows:
Ik = 1*( Ek – Em ) / Zkm = ykm * (Ek – Em )
(2.5a)
Similarly for bus m,
Im = 1*(Em – Ek ) / Zmk = ymk * (Em – Ek )
(2.5b)
Figure 2.10 Equivalent impedance
The above equations can be written in matrix form as,
Ik
Im
Ykk
Ymk
Ykm
Ymm
Ek
Em
(2.6)
The bus admittances and voltages can be expressed in more explicit form:
Yij = Gij + jBij
(2.7)
Ei = Vie = Vi * ( cos i + jsin i)
(2.8)
36
The complex power injected at bus k consists of an active and
reactive component and is expressed as a function of the nodal voltage and
the injected current at the bus as given in equation (2.9).
Sk = Pk + jQk = EkIk*
(2.9)
Sk = Ek ( YkkEk + YkmEm)*
where Ik* is the complex conjugate of the current injected at the bus k.
The expressions for Pkcal and Qkcal are as follows:
Pkcal = Vk2Gkk + VkVm[Gkmcos(
m)
k
Qkcal = -Vk2Bkk + VkVm[Gkmsin(
+ Bkmsin(
m)
k
m)]
k
- Bkmcos(
(2.10)
m)]
k
(2.11)
For specified levels of power generation and power load at bus k and
according to equations (2.1) and (2.2), the mismatch equations may be written
down as follows:
Pk=PGk – PLk –{ Vk2Gkk + VkVm[Gkmcos(
Qk=QGk – QLk –{-Vk2Bkk + VkVm[Gkmsin(
m)
k
k
+ Bkmsin(
m)
- Bkmcos(
k
k
m)]}
= 0 (2.12)
m)]}
= 0 (2.13)
Similar equations may be obtained for bus m simply by exchanging subscripts
k and m in equations (2.12) and (2.13).
It should be remarked that equations (2.10) and (2.11) represent
only the powers injected at bus k through the ith transmission element, i.e.,
Pki
cal
and Qki
cal
. However a practical power system will consists of many
buses and transmission elements. This calls for equations (2.10) and (2.11) in
more general terms, with the net power flow injected at bus k expressed as the
summation of the powers flowing at each one of the transmission elements
terminating at this bus.
37
The generic active and reactive powers injected at bus k are as follows:
where Pki
Pkcal =
Pki cal
(2.14)
Qkcal =
Qki cal
(2.15)
cal
and Qki
cal
are computed using equations (2.10) and (2.11)
respectively.
As an extension, the generic power mismatch equations at bus k are as
follows:
Pk = PGk – PLk –
Pki cal = 0
Qk = QGk – QLk –
(2.16)
Qki cal = 0
(2.17)
In large-scale power flow studies, the Newton Raphson has proved
most successful owing to its strong convergence characteristics. The power
flow Newton Raphson algorithm is expressed by the following relationship.
P
=Q
P/
Q/
P /( v / v)
Q /( v / v)
(2.18)
( v / v)
It may be pointed out that the correction terms Vm are divided by
Vm to compensate for the fact that jacobian terms ( Pm Vm)Vm and
Qm Vm)Vm are multiplied by Vm. It is shown in the directive terms that this
artifice yields useful simplifying calculations.
Consider the lst element connected between buses k and m in Figure
2.10, for which self and mutual Jacobian terms are given below:
For k
m
Pk ,l
m, l
= Vk Vm [G km sin(
k
m
)
Bkm cos(
k
m
)]
(2.19)
38
Pk ,l
Vm , l V m , l
Qk ,l
= Vk Vm [G km cos(
k
m
) B km sin(
k
m
)]
Pk ,l
=
Vm ,l Vm ,l
m, l
Q k ,l
Vm , l V m , l
=
Pk ,l
(2.20)
(2.21)
(2.22)
m,l
for k = m,
Pk ,l
= Qkcal V k2 Bkk
(2.23)
k, l
Pk ,l
Vk ,l Vk ,l
Qk ,l
Pkcal
Vk2 G kk
(2.24)
Pkcal Vk2 Gkk
(2.25)
Qkcal
(2.26)
k ,l
Qk , l
V k , l Vk ,l
Vk2 Bkk
The mutual elements remain the same whether we have one
transmission line or n transmission lines terminating at the bus k.
2.5.2
The Sample Five Bus System
First we have considered the five bus system as a case study shown
in Figure 2.11 (Stagg and Abiad 1968). The input data for the considered
system are given in Table 2.1 for the bus and Table 2.2 for transmission line.
39
Figure 2.11 The five-bus network
Table 2.1 Input Bus data (p.u.) for the system under study
Bus
Bus code
Impedance
Line charging
No.
(k-m)
(R+jX)
admittance
1
1-2
0.02+j0.06
0+j0.06
2
1-3
0.08+j0.24
0+j0.05
3
2-3
0.06+j0.18
0+j0.04
4
2-4
0.06+j0.18
0+j0.04
5
2-5
0.04+j0.12
0+j0.03
6
3-4
0.01+j0.03
0+j0.02
7
4-5
0.08+j0.24
0+j0.05
40
Table 2.2 Input Transmission line data (p.u.) for the system under study
Bus
No.
Type
1
Generation
Load
Voltage
P
Q
P
Q
|v|
slack
0
0
-
-
1.06
0
2
P-V
0.4
0.3
0.2
0.1
1
0
3
P-Q
-
-
0.45
0.15
1
0
4
P-Q
-
-
0.4
0.05
1
0
5
P-Q
-
-
0.6
0.1
1
0
Assuming base quantities of 100 MVA and 100 KV.
The power flow result for the above system without any FACTS
devices is mentioned in Table 2.3. All the nodal voltages are achieved to be
within acceptable voltage magnitude limits.
Table 2.3 Bus voltage of system under study without FACTS devices
Parameter
BUS 1
BUS 2
BUS 3
BUS 4
BUS 5
|V| (p.u)
1.06
1
0.987
0.984
0.972
0
-2.06
-4.64
-4.96
-5.77
(deg)
2.6
FACTS CONTROLLERS MODELING
This chapter explains the power flow modeling of different FACTS
devices we have adopted in our project. The unified approach that combines
the state variables describing controllable equipment with those describing the
network in a single frame of reference for iterative solutions using the
Newton-Raphson algorithm is followed, which retains its quadratic
convergence characteristics (Acha et al 2004).
41
2.6.1
Power Flow Model of SVC
Two models are presented in this category, namely, the variable
shunt susceptance model and firing-angle model.
2.6.1.1
Variable susceptance model
In practice the SVC can be seen as an adjustable reactance with
either firing angle limits or reactance limits. The equivalent circuit shown
below in Figure 2.12 is used to derive the SVC nonlinear power equations and
the liberalized equations required by the Newton’s method.
Figure 2.12 SVC -Variable shunt susceptance
With reference to Figure 2.12, the current drawn by the SVC is
given by equation (2.27),
ISVC = jBSVCVk
(2.27)
The reactive power drawn by the SVC, which is also the reactive power
injected at bus k, is given by equation (2.28),
QSVC = Qk = -Vk2BSVC
(2.28)
42
At the end of each iteration, the variable shunt susceptance B is
updated as given in equation (2.29),
(i )
SVC
BSVC
BSVC
( i 1)
SVC
B
B
(i)
( i 1)
BSVC
(2.29)
The changing susceptance represents the total SVC susceptance necessary to
maintain the nodal voltages at specified value.
2.6.1.2
Firing angle model
An alternative SVC model, which circumvents the additional
iterative process, consists in handling the TCR firing angle
as a state
variable in the power flow formulation. The positive sequence susceptance of
the SVC, is given by equation (2.30).
Vk2
{X L
XC XL
Qk
XC
[2(
SVC
) sin(2
SVC
)]}
(2.30)
From the above equation, the linearised SVC equation is given as follows:
Pk
(i)
Qk
=
0
0
0
2V 2
[cos(2
XL
(i)
SVC
k
) 1]
At the end of iteration (i), the variable firing angle
(i)
(2.31)
SVC
SVC
is updated as given in
equation (2.32).
(i)
SVC
2.6.2
(i-1)
=
SVC
+
(i)
SVC
(2.32)
Power Flow Model of TCSC
Two alternative power flow models to assess the impact of TCSC
equipment in network wide applications are presented in this section. The
simpler TCSC model exploits the concept of a variable series reactance. The
43
series reactance is adjusted automatically, within limits, to satisfy a specified
amount of active power flows through it. The more advanced model uses
directly the TCSC reactance-firing angle characteristics, given in the form of
a non-linear relation. The TCSC firing angle is chosen to be the state variable
in the Newton-Raphson power flow solution.
2.6.2.1
Variable series reactance power flow model
The TCSC power flow model presented in this section is based on
the simple concept of a variable series reactance, the value of which is
adjusted automatically to constrain the power flow across the branch to a
specified value. The amount of reactance is determined efficiently using
Newton’s method. The changing reactance XTCSC, shown in Figure 2.13(a)
and 2.13(b), represents the equivalent reactance of all the series-connected
modules making up the TCSC, when operating either in the inductive or in the
capacitive regions.
Figure 2.13 TCSC equivalent circuits (a) Inductor (b) Capacitive
operative regions
The transfer admittance matrix of the variable series compensator is
given by
Ik
Im
=
jB kk
jBkm
Vk
jB mk
jBmm
Vm
(2.33)
44
For inductive operation, we have
Bkk = Bmm = -
Bkm = Bmk =
1
(2.34)
X TCSC
1
(2.35)
X TCSC
For capacitive operation the signs are reversed.
The active and reactive power equations at bus k are as follows:
Pk
VkVm Bkm sin(
V k2 Bkk
Qk
k
m
(2.36)
)
Vk Vm Bkm cos(
k
m
)
(2.37)
For the power equations at bus m, the subscripts k and m are exchanged in
equations (2.36) and (2.37).
The state variable XTCSC of the series controller is updated at the
end of each iterative step as given in equation (2.38).
X
2.6.2.2
(i )
TCSC
X
( i 1)
TCSC
X TCSC
X TCSC
(i )
( i 1)
X TCSC
(2.38)
Firing angle power flow model
The model presented above in section 2.6.2.1 uses the concept of an
equivalent series reactance to represent the TCSC. Once the value of
reactance is determined using Newton’s method then the associated firing
angle
TCSC
can be calculated.
However, such calculations involve an
iterative solution since the TCSC reactance and the firing angle are nonlinearly related. One way to avoid the additional iterative process is to use the
alternative TCSC power flow model presented in this section.
45
Figure 2.14 TCSC compensator firing angle modules
The fundamental frequency equivalent reactance XTCSC of the
TCSC module shown in Figure 2.14 is given by equation (2.39).
XTCSC
XC C1{2(
) sin[2(
)]} C2 cos2 (
){ tan[ (
)] tan(
)}
(2.39)
where
C1
C2
X LC
XC
X LC
(2.40)
2
4 X LC
XL
(2.41)
XC XL
XC X L
XC
XL
(2.42)
1/ 2
(2.43)
The TCSC active and reactive power equations at bus k are as follows:
Pk
Qk
VkVm Bkm sin(
Vk2 Bkk
k
m
(2.44)
)
Vk Vm B km sin(
k
m
)
(2.45)
46
where
Bkk
Bkm
(2.46)
BTCSC
For equations at bus m, exchange subscript k and m in equation (2.44) and
(2.45).
2.6.3
Power Flow Model of STATCOM
The Static Synchronous Compensator is represented by a
synchronous voltage source with minimum and maximum voltage magnitude
limits. The bus at which STATCOM is connected is represented as a PV bus,
which may change to a PQ bus in the events of limits being violated. In such
case, the generated or absorbed reactive power would correspond to the
violated limit. The power flow equations for the STATCOM are derived
below from the first principles and assuming the following voltage source
representation:
Figure 2.15 STATCOM equivalent circuit
Based on the shunt connection shown in Figure 2.15, the following
equation can be written.
EvR
VvR (cos
SvR
*
VvR I vR
vR
j sin
vR
)
VvRYvR* (VvR* Vk* )
(2.47)
(2.48)
47
The following are the active and reactive power equations for the
converter at bus k:
PvR
QvR
Pk
Qk
2.6.4
VvR2 GvR VvRVk [GvR cos(
vR
VvR2 BvR VvRVk [GvR sin(
Vk2GvR VkVvR [GvR cos(
vR
k
Vk2 BvR VkVvR [GvR sin(
k
k
vR
k
) BvR sin(
) BvR cos(
) BvR sin(
vR
vR
)]
vR
k
) BvR cos(
k
k
vR
k
(2.49)
)]
)]
vR
)]
(2.50)
(2.51)
(2.52)
Power Flow Model of UPFC
The equivalent circuit consists of two coordinated synchronous
voltage sources should represent the UPFC adequately for the purpose of
fundamental frequency steady state analysis. Such an equivalent circuit is
shown in Figure 2.16.
Figure 2.16 UPFC equivalent circuits
48
The UPFC voltage sources are as follows:
EvR
VvR (cos
vR
j sin
vR
)
(2.53)
EcR
VcR (cos
cR
j sin
cR
)
(2.54)
where VvR and
vR
are the controllable magnitude (VvRmin
phase angle (0
VvR VvRmax) and
) of the voltage source representing the shunt
vR
converter. The magnitude VcR and phase angle
cR
of the voltage source
representing the series converter are controlled between limits (VcRmin
VcRmax) and (0
cR
VcR
), respectively.
The phase angle of the series injected voltage determines the mode
of power flow control. If
cR
is in phase with the nodal voltage angle
UPFC regulates the terminal voltage.
If
cR
is in quadrature with
controls active power flow, acting as a phase shifter. If
cR
k,
the
k,
it
is in quadrature
with line current angle then it controls active power flow, acting as a variable
series compensator.
At any other value of
cR,
the UPFC operates as a
combination of voltage regulator, variable series compensator and phase
shifter. The magnitude of the series injected voltage determines the amount
of power flow to be controlled.
Based on the equivalent circuit shown in Figure 2.16 and equations
(2.53) and (2.54), the active and reactive power equations at bus k are as
follows:
Pk
Vk2Gkk VkVm[Gkm cos(
k
m
) Bkm sin(
VkVcR [Gkm cos(
k
cR
) Bkm sin(
k
cR
)]
VkVvR [GvR cos(
k
vR
) BvR sin(
k
vR
)]
k
m
)]
(2.55)
49
Vk2 Bkk VkVm [Gkm sin(
Qk
k
m
) Bkm cos(
VkVcR [Gkm sin(
k
cR
) Bkm cos(
k
cR
)]
VkVvR [GvR sin(
k
vR
) BvR cos(
k
vR
)]
k
m
)]
(2.56)
At bus m
Pm
Vm2Gmm VmVk [Gmk cos(
VmVcR [Gmm cos(
m
cR
m
VmVcR [Gmm sin(
m
) Bmk sin(
) Bmm sin(
Vm2 Bmm VmVk [Gmk sin(
Qm
k
m
m
k
Bmm cos(
cR )
cR
m
k
)]
(2.57)
)]
) Bmk cos(
)]
m
k
cR
k
(2.58)
cR )]
m
Series converter
PcR
VcR2 Gmm VcRVk [Gkm cos(
VcRVm [Gmm cos(
QcR
cR
m
cR
) Bmm sin(
VcR2 Bmm VcRVk [Gkm sin(
VcRVm [Gmm sin(
cR
k
cR
) Bkm sin(
cR
k
Bmm cos(
m)
m
(2.59)
)]
) Bkm cos(
cR
)]
cR
k
)]
(2.60)
m )]
Shunt converter
PvR
QvR
VvR2 GvR VvRVk [GvR cos(
VvR2 BvR VvRVk [GvR sin(
vR
vR
k
k
) BvR sin(
) BvR cos(
vR
vR
k
k
)]
)]
(2.61)
(2.62)
Assuming lossless converter values, the active power supplied to
the shunt converter, PvR, equals the active power demanded by the series
converter, PcR; that is,
PvR
PcR
0
(2.63)
50
Further more, if the coupling transformers are assumed to contain
no resistance then the active power at bus k matches the active power at bus
m. Accordingly,
PvR
PcR
Pk
Pm
0
(2.64)
The UPFC power equations are combined with those of the AC
network.
2.7
CASE STUDIES WITH FACTS CONTROLLERS
The five bus network is modified to include different FACTS
devices and examine the voltage control capabilities and power flow control
capabilities of the same.
2.7.1
Power Flow Study with SVC
The five bus network is modified to examine the voltage control
capabilities of different SVC models. One SVC is included in the bus 3
(Figure 2.17) to maintain the nodal voltage at 1 p.u.
Figure 2.17 Study system with SVC
51
The SVC inductive and capacitive reactances are taken to be 0.288
p.u. and 1.07 p.u. respectively. The SVC firing angle is set initially at 1400, a
value that lies on the capacitive region. Convergence is achieved in 5
iterations, satisfying a prespecified tolerance of 1e-12 for all variables. The
result for the voltage magnitude and phase angle obtained for both variable
susceptance and firing angle model is shown in Table 2.4. Due to the
inclusion of SVC in the third bus, its voltage is maintained at 1 p.u. The SVC
susceptance values and firing angle values are shown in Table 2.5 for each
step of the iterative process.
Table 2.4 Bus voltage of modified network (with SVC)
Type of model Parameter BUS 1
SVC
|V| (p.u)
1.06
(Varying
0
(deg)
Susceptance)
SVC
|V| (p.u)
1.06
(Varying Firing
0
(deg)
Angle)
BUS 2
1
BUS 3
1
BUS 4
0.9944
BUS 5
0.9752
-2.0532 -4.8378 -5.1071 -5.7972
1
1
0.9944
0.9752
-2.0533 -4.8379 -5.1073 -5.7975
Table 2.5 SVC state variables
Iteration
1
2
3
4
5
2.7.2
Susceptance model Firing angle model
Bsvc (p.u)
Bsvc (p.u)
SVC
0.02
0.4788
140
0.1
0.1038
130.2
0.168
0.1096
131.3
0.2048
0.2048
132.5
0.2048
0.2048
132.5
Power Flow Study with TCSC
The original five-bus network is modified to include one TCSC
(Figure 2.18) to compensate the transmission line connected between bus 3
52
and 4. The TCSC should maintain the real power flow in transmission line 6
as 21 MW.
Figure 2.18 Study system with TCSC
The starting value of TCSC is set at 50 percent of the value of the
transmission inductive reactance i.e. 0.015 p.u. for varying reactance model.
Convergence is achieved in 6 iteration to power mismatch tolerance of 1e-12.
The TCSC upholds the target value of 21 MW, which is achieved with 70
percent series capacitive compensation. The initial value of firing angle is set
at 1450. Convergence is obtained in 6 iterations, with final value of
at
148.660, with TCSC upholding the target value of 21 MW in line 6. The result
for the voltage magnitude and phase angle obtained for the above system for
both variable reactance and firing angle model is shown in Table 2.6.
53
Table 2.6 Bus voltage of modified network (with TCSC)
Type of model
TCSC
|V| (p.u)
(Varying
Reactance)
(deg)
TCSC
|V| (p.u)
(Varying Firing
Angle)
2.7.3
Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5
(deg)
1.06
0
1
2.0378 4.7239 4.8145 5.7012
1.06
0
0.9869 0.9846 0.9719
1
0.987 0.9845 0.9719
2.0377 4.7263 4.8116 5.7004
Power Flow Study with STATCOM
The STATCOM is included in bus 3 (Figure 2.19) of the sample
system to maintain the nodal voltage at 1 p.u.
Figure 2.19 Study system with STATCOM
The power flow result indicates that the STATCOM generates 20.5
MVAR in order to keep the voltage magnitude at 1 p.u. at bus3. The source
54
impedance is ZvR = 0.1 p.u., the STATCOM parameters associated with this
amount of reactive power generation are VvR = 1.0205 p.u. and
vR
= -4.830.
Use of STATCOM results in improved network voltage profile as shown in
Table 2.7.
Table 2.7 Bus voltages of STATCOM upgraded network
Type of model Parameter BUS 1 BUS 2
BUS 3
BUS 4
BUS 5
|V| (p.u)
1
0.9944
0.9752
STATCOM
2.7.4
(deg)
1.06
1
0
-2.04
-4.7526 -4.821 -5.8259
Power Flow Study with UPFC
The original five-bus network is modified to include one UPFC to
compensate the transmission line linking bus 3 and bus 4 (Figure 2.20).
UPFC should maintain real and reactive power flowing towards bus 4 at 40
MW and 2 MVAR, respectively. The UPFC shunt converter is set to regulate
the nodal voltage magnitude at bus 3 at 1 p.u.
Figure 2.20 Study system with UPFC
55
The starting values of the UPFC voltage sources are taken to be VcR
=0.04 p.u.,
cR
= 87.130 , VvR = 1 p.u. and
vR
= 00. The source impedances
have values of ZcR = ZvR = 0.1 p.u. . Convergence is obtained in five iterations
to a power mismatch tolerance of 1e-12. The UPFC upheld its target values
and the bus voltage are given in Table 2.8.
Table 2.8 Bus voltages of UPFC upgraded network
Type of model Parameter BUS 1
UPFC
2.8
|V| (p.u)
(deg)
BUS 2
BUS 3
BUS 4
BUS 5
1
1
0.9917
0.9745
1.06
0
-1.7691 -6.016 -3.1905 -4.9737
RESULTS AND DISCUSSION
The power flow for the five bus system was analyzed without and
with FACTS devices performing the Newton-Raphson Method. The
consolidated comparative bus voltage results have been represented in
Table 2.9. The largest power flow takes place in the transmission line
connecting the two generator buses: 89.3 MW and 74.02 MVAR leave bus 1
and 86.8 MW and 72.9 MVAR arrive at bus2. The operating conditions
demand a large amount of reactive power generation by the generator
connected at bus1 (i.e., 90.82 MVAR). This amount is well in excess of the
reactive power drawn by the system loads (i.e., 40 MVAR). The generator at
bus 2 draws the excess of reactive power in the network (i.e., 61.59 MVAR).
This amount includes the net reactive power produced by several transmission
lines, which is addressed by different FACTS devices. Thus SVC upholds its
target value and as expected identical power flows and bus voltages are
obtained for both Shunt variable susceptance model and Firing angle power
flow models.
56
57
The TCSC Variable series compensator model is used to maintain
active power flowing from the extra fictious bus 6 towards bus 3 at 21 MW.
The starting value of the TCSC is set at 50% of the value of transmission line
inductive reactance. The TCSC upholds the target value of 21 MW, which is
achieved with 70% series compensation of the transmission line 6. In the case
of the firing angle model the initial value of firing angle is set at 145º and the
TCSC upholds the target value of 21 MW.
The power flow result indicates that the STATCOM generates 20.5
MVAR in order to keep the voltage magnitude at 1 p.u at bus 3. Use of
STATCOM results in an improved network voltage profile, except at bus 5,
which is too far away from bus 3 to benefit from the influence of STATCOM.
The original five-bus network is modified to include one UPFC to
compensate the transmission line linking bus 3 and bus 4. The UPFC is used
to maintain active and reactive powers leaving UPFC, towards bus 4, at 40
MW and 2 MVAR, respectively. Moreover the UPFC shunt converter is set
to regulate the nodal voltage magnitude at bus 3 at 1 p.u. There is a 32%
increase of active power flowing towards bus 3. The increase is in response
to the large amount of active power demanded by the UPFC series converter.
Thus from the above analysis we find that within the framework of traditional
power transmission concepts, the UPFC is able to control, simultaneously or
selectively, all the parameters affecting power flow in the transmission line
(voltage, impedance and phase angle) and this unique capability is signified
by the adjective ‘Unified’ in its name.
The power flow study is then extended for IEEE 30 bus system
whose single line circuit diagram is shown in Figure 2.21. Voltage at bus 12
for the base case (without any FACTS devices) is found to be |V| = 0.94773 &
= - 18.080. The SVC is connected at bus 12, which bring the bus voltage to
1 p.u. and convergence is obtained within 4 iterations. The impact of different
58
59
FACTS devices on the neighbouring buses is given in Table 2.10. The power
flow in the line between bus 13-bus 12 is found to be 16.9 MW and 38.4
MVAR. The SVC is able to control the reactive power flow of the line 13-12
to the specified control reference 0.0 p.u., while the active power flow is
almost unchanged. By driving the reactive power flow on the line to zero
using SVC, the un-used (available) transmission line capacity can be
increased. It can be seen that the base case reactive power of line 13-12 is
38.4 MVAR, so the reactive power control by the SVC is significant. Similar
result is also obtained when SVC is replaced by STATCOM at bus 12.
Desired real power flow is also obtained when TCSC is connected in the line
connecting bus 12 and bus 15. The UPFC is installed between bus 12 and the
sending end of the transmission line connecting bus 12 – bus 15. In the
simulation, the bus voltage control reference is |V12 | = 1 p.u. Convergence is
achieved within 6 iterations with specified real power and reactive power in
line connecting bus 12 – bus 15. The detail results show the validity of
different types of model proposed for FACTS devices with the model
proposed by Zhang et al (2006).
2
1
18
15
14
19
3
28
8
4
5
7
6
11
9
13
12
16
17
10
20
23
26
25
27
21
22
29
24
30
Figure 2.21 IEEE 30 Bus system
60
2.9
SUMMARY
The non-linear power flow equations of the various FACTS
controllers have been linearised and included in Newton Raphson power flow
algorithm. The state variables corresponding to the controllable devices have
been combined simultaneously with the state variables of the network in a
single frame of reference for unified, iterative solutions. The effectiveness of
modeling and convergence is tested and results are analyzed.
The next chapter discusses about dynamic modeling of Unified
Power Flow Controller (UPFC) and proposed suitable controller, for damping
the electromechanical oscillations in a power system.