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ïnterline Powen Flow Controller ({PFC}Steady state analysis and devetropment of,
Small sÍgnal model
Victor M. Diez-Valencia
A thesis
submitted in partial fulfilment of the requirements
for the degree of Master of Science
The University of Manitoba
Department of Elechicai and Computer Engineering
Winnipeg, Manitoba, Canada
Februarv 2002
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0-612-76928-3
Canad'ä
THE UNIVERSITY OF MANITOBA
FACULTY OF GRADUATE STUDIES
copyRrc"iiro*ssroN
INTERLINE POWER FLOW CONTROLTER (IPFC)
STEADY STATE ANATYSIS AND DEVELOPMENT OF
SMATL SIGNAL MODEL
VICTOR M. DIEZ-VALENCIA
A ThesisÆracticum submitted to the Faculty of Graduate Studies of The Universify of
Manitoba in partial fulfillment of the requirement of the degree
of
MASTER OF SCIENCE
VICTOR M. DIEZ-VALENCIA @ 2OO2
Permission has been granted to the Library of the University of Manitoba to lend or sellcopies of this thesis/practicum, to the National Library of Canada to microfilm this thesis
and to lend or sell copies of the film, and to University Microfilms Inc. to publish an abstractof this thesisþracticum.
This reproduction or copy of this thesis has been made available by authority of thecopyright owner solely for the purpose of private sfudy and research, and may only be
reproduced and copied as permitted by copyright laws or with express written authorizationfrom the copyright owner.
BY
Acknownedgements
The author expresses his gratitude to his advisors Prof. A.M Gole and prof. Udaya
Annakkage for their valuable support, encouragement and advice throughout this work.
I have had an enorrnous experience from such a wise guidance, learning and enjoying each
and every aspect.
The author wishes to thank Dr. David Jacobson as co-advisor and Manitoba Hydro.
Special thanks to my Power Tower colleagues, vajira, phil, Manish, waruna, Namal,
Pradeep, Ernesto Yazquez, for sure they have been a daily support in Winnipeg, since
simple things are so valuable but your friendship has been a greatgift to me.
I gratefully acknowledge Interconexion Electrica S.A for all the experience and such a
great encouragement offered by Julian cadavid and Jhon Albeiro calderon.
Finally, I am grateful to my parents, sister and Mariana for being always with me during
this time away.
Victor M. Diez-Valencia
'Winnipeg, Feb 2002
rll
Confents
Chapter I -Introduction 7
Chapter 2 - Voltage Sourced Converter (vSC) and F?ICTS topologies tIPrinciples 11
Voltage Sourced Converter Components 12DC Voltage Source I2Power Converter l3
Voltage Sourced Converter Switching and Harmonic Components 15Snubber Circuits 18
Voltage Sourced Converters Topologies lgStatic Synchronous Compensator (STATCOM) I9Static Synchronous Series Compensator (SSSC) 2IUniJied Power Flow Controller (UPFC) 22Interline Power Flow Controller (IPFC) 23Generalized Unífied Power Flow Controller (GUpFe 2a
Chapter 3 - Interline Power Flow Controller (IpFC) 26
Review of Power Transmission 7Main Focus 9
Overview of the report 9
The Interline Power Flow Controller 26
The Static S¡mchronous Series Compensator (SSSC) 28static synchronous series compensator (sssc) without active power source 29Static Synchronous Series Compensator llith Active power Source 30
The UPFC Steady State Model 31Cu'rent through the line when a |IPFC is installed 34
Calailation of the UPFC ratingfor a given angte (õ) 3SActive power injected or absorbed by the UPFC 40Reactive power injected or absorbed by the UpFC 43Active and reactive power at the receiving end 45
1V
chapter 4 - steady state solutionfor IpFC using Matlab 62Newton-RaphsonMethod 62The IPFC Variables 65
Results 71
Chapter 5 - IPFC Small signal stabiliry model 73
Characteristics of an Interline power Flow Controller (IpFC) 4gInterline Power Flow Conrroller (lpFC) constraints 4gThe IPFC case 50P-Q diagram limitation ntled by the first consh.aint 54The impedance efect 58Power transmíssion losses 60
Stability in Power Systems 73
The IPFC Model 74
Linearized IPFC Model 77
The Synchronous Generator model [29] g4
Synchronous Machine Linearized Model g7
The Load Model 90
The system model glState space model neglecting IpFC switching delays 93
Model Results 94
Summary 98
Chapter 6 - Conclusions and Recommendations gg
Conclusions 99
FurtherRecommendations lOf
AppendixA - Nomenclature I0B
Appendix B - Power Flow Program It2
Appendix C - Small Signal Model 120
K-,ist of F ígures
Fig. 2. I Voltage Sourced Conveder block diagram 12
Fig.2.2 Two level six pulse converter voltages 14
Fig.2.3 Th¡ee level bridge 15
Fig.2. 4 Converter arrarigement 6-pulse *4 converters : 24-pulseoperation [6]. t7
Fig. 2. 5 Typical snubber circuit arrangement for GTOs [6] t g
Fíg.2. 6 STATCOM equivalent for steady state Lg
Frg. 2.7 Block diagram of a Static Synchronous Compensator(STATCOM) 20
Fig' 2. 8 Block diagram of a Static Synchronous series compensator(sssc) 21
Fig.2.9 Unified Power Flow Conrroller (I_IpFC) 23
Fig. 2. l0 Interline Power Flow Controller (IpFC) 24Fig.2. ll Generalized Unified Power Flow Controller (GIIpFC) 25Fig. 3. 1 Power converter arrarigement 2lFig.3.2 Interline Power Flow Controller scheme 2gFig. 3. 3 Steady state model for SSSC 29
Fig.3. 4 SSSC effect for a reactive compensation (capacitive orinductive) 30
Fig. 3. 5 Vector diagram for a SSSC with dc side active powersource 31
Fig. 3. 6 System and parameters for the steady state model 32Fig.3.7 Voltage injected by the UPFC, magnitude and angle 33Fig. 3. 8 Line current magnitude (I1) vs. voltage injecred angle 35Fig. 3. 9 UPFCI Reactive Power vs. Active power 3jFig. 3. 10 P-Q Diagram for IIPFC vs. delta angle (ô,, ) for +0.05 p.u 39
Fig. 3. I 1 P-Q diagram for ô,,. : -n/6 surrounded by the dotted
apparentpower 40
Fig.3. 12 Active power (prrocr ) vs. volúage injected angle (ry, ) 4lFig. 3. 13 Voltage injected angle (Vr,,,,o ) for maximum and minimum
active power 43
Fig. 3. l4 Voltage injected angle (V1,,,o,¿2) for maximum and minimumreactive power 44
Fig. 3. 15 P-Q diagram for the receiviag end 46
Vi
Fig. 3. 16 P-Q diagram with respect to the voltage injected (V, and
LVr) 48
Fig. 3. 17 IPFC conformed by two SSSC linked by their dc side 4gFig. 3. 18 The IPFC and its variables 49
Fig. 3. 19 Active power (pr"o ct,p,pFC2) vs. voltage injected angle
(vr, vz) 52
Fig. 3. 20 Limitation imposed by system 2 on system I 53
Fig. 3. 21 First constraint applied to system I 55
Fi9.3.22 First constraint applied to system 2 56
Fi9.3.23 P-Q diagram with active power constraint 5jFig.3.24 Influence of the change in impedance of system 2 on system
159Fig. 3. 25 P-Q diagram for system I after modif,iing the line impedance
in system 2 60
Fig.3.26 Power transmission losses 6lFig. 4. I IPFC steady state va¡iables 66
Fig. 5. I Power system for small signal stability j4Fig. 5.2 IPFC delay 83
Fig. 5.3 Reference frame for d-q transformation 85
Fig. 5. 4 Parameters at the receiving end I 90Fig. 5. 6 Eigenvalues plot for the case considered 97Fig. 5.7 Small signal feasible P-Q region for system I at the receiving
end 98
CHAPT'ER. 1 [nffoduction
1.1 Review of Power Transmission
Since the beginning, Electrical Engineering has taken an active role in the history. The
"battle of the currents" ac vs. dc in late 1800s was an historical moment when Westing-
house and Thomas Edison accompanied by General Electric Company start a fight to
demonstrate the best approach. In 1850s Nicola Tesla enters into the picture with his
invention "The altemating current motor". Later, George V/estinghouse obtained this
patent and was able to solve ali the problems Edison faced by using alternating current and
transformers to step up or step down voltages. The latter made long-distance transmission
feasible.
With the advances made in power electronics towards its application in HVDC, an effi-
cient way to achieve long-distance transmission andlor an alternative to link systems with
different frequencies was achieved. New technologies could be developed using power
electronics but this time applied to ac systems. Thus, it is in 1986 when EPRI (Electric
Power Research Institute) proposed the concept of flexible ac transmission svstem as an
INTRODUCTION
acron)ryn "FACTS" [19]. Later, IEEE (Institute of Electrical and Eiectronics Engineers)
and CIGRÉ (Conferénce internationale des grands réseaux électriques) backed up the pro-
posal which finally made FACTS a new technology. The IEEE definition of FACTS is:
"Alternating current transmission systems incorporating power electronics-based and
other static controllers to enhance controiiability and power transfer capability"
The power electronics devices by that time were not able to withstand the requirements of
HV and EHV levels. Even though developments in sub synchronous resonance damping
(NGH-SSR) using thyristors based devices were already done by this time. In this case the
lack of active cooling was a limitation for tþristors to withstand full load current.
Everything evolves and power electronics has not been an exception. That is why nowa-
days it is possible using FACTS devices to load power transmission lines close to their
thermal limit with accurate flow control. Along these years different projects have been
planned and accomplished successfully. Notable projects are:
- 1993 Thyristor Controlled Series Capacitor (TCSC), 208 Mvar "Line impedance Con-
troller" at C.J.Slatt substation on the Slatt-Buckley 500 kV line in Northern Oregon 1341.
- 1995 Static Synchronous Compensator (STATCOM), +/- i00 Mvar "Voltage Control-
let" at Sullivan substation-Tennessee Valley Authority (TVA) [35].
-1998 Unified Power Flow Controller (UPFC), +l-160 Mvar "All Transmission Parame-
ters Controller" at Inez substation American Electric Power (AEP) [32].
- 2002 Convertible Static Compensator (CSC) 345 kV, +l-200 MVA "Flexible Multi-
functional Compensator" at Marcy substation New York Power Authority (NYPA) tl11.
INTRODUCTION
1) Main Focus
FACTS devices have given rise to many expectations in ac power transmission. Since the
beginning FACTS devices have been considered as an option that not only enhances the
power transferred in power transmission lines but also provides an accurate control to dif-
ferent system parameters. Thus, FACTS are able to improve the power transferred maxi-
mizing the power through the lines [14], the stability margin l23llzíl,reliability [25].
The Interline Power Flow Controller (IPFC) is one of the latest in the family of FACTS
devices and the first to incorporate more than one power transmission line. In order to gain
the maximum benefit in a typical transmission application, it is important to understand
the steady state as well as the dynamic behavior of the device. This thesis therefore
focuses on the development of tools that determine the steady state operating limits of the
IPFC as well as the development of tools for the analysis of the dynamic stability. The
high frequency operation of the semiconductor switches is not considered here as the pri-
mary focus is on the fundamental frequency effects.
1,.3 Overview of the report
Chapter 2 gives a general introduction to power electronics and the voltage sourced con-
verter (VSC), from the simple bridge in power electronics to the harmonics from switch-
ing. The principles of FACTS devices are given starting with the basic configurations such
as Static Synchronous Compensator (STATCOM), Static Synchronous Series Compensa-
tor (SSSC), followed by the combination of these basic configurations to obtain different
INTRODUCTION
FACTS controllers such as Unified Power Flow Controller (UPFC), Interline power Flow
controlier (IPFC) and Generatized unified power Flow controller (GUpFC).
Chapter 3 describes the operating characteristics of the IPFC in the steady state condition.
The analysis begins with the Unified Power Flow Controller (UPFC) applied to a single
transmission line. The resulting equations are then extended to the Interline power Flow
Controller (IPFC) topology which considers series converters (VSC) in two transmission
lines operating under the constraint of zero net generated power in the interconnected dc
link. This chapter studies issues such as the effect of the IPFC on the operation and net-
work parameters, device rating, line impedance effect, and its constraints.
Chapter 4 describes the algorithm used to solve the power flow problem of a simple net-
work which includes the IPFC. The network consists of a pair of power transmission lines
and three generators. The Newton Raphson algorithm is used to calculate the unknown
parameters, and is irnplemented in the Matlabl environment. Our goai is to find out the
magnitude and angle for the injected voltage by each one of the voltage sourced convert-
ers in order to fi;lfilthe constraints imposed by the IPFC.
Chapter 5 explains the procedure to obtain the small signal model for an IPFC. After
developing the IPFC model, this is included in a network with three busbars and two gen-
erators. The eigenvalues from this model are obtained and plotted in order to determine
the system stability for a given operating point.
10
1. MATLAB copyright 1984-2001 The MathWorks, Inc.
CHAPTER 2 Voltage Sourced Converter (VSC)and FACTS topologies
The Voltage Sourced Converter (VSC) is the fundamental building block of a large class
of FACTS devices; inciuding the IPFC. Hence, this chapter begins with the description of
the basic VSC building block, which is then extended to more complex topologies.
2.1 Principles
The Voltage Sourced Converter can produce three alternating voltages from a dc source
by mean of high speed solid state switches. The solid state switches allow us to obtain a
fast and fully controllable amplitude and phase angle in the alternating voltage. From a
fundamental frequency viewpoint, this behaves in a manner similar to a synchronous
machine corurected to an ac network. Like the ac machine, the VSC presents itself as a
voltage behind an inductance. Some of the desirable characteristics of the rotating
machine are: high capacitive output current at iow system voltage levels and an essentially
inductive source impedance that cannot cause harmonic resonance with the network. On
the other hand some drawbacks are: slow response, potential for rotational instability and
11
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES
high maintenance. These latter disadvantages are considerably reduced with the solid state
VSC.
2.2 Voltage Sourced Converter Components
The main components of a voltage sourced converter are depicted in the block diagram
Figure 2. 1. The dc voltage block supplies voltage to a Power Converter which by opera-
tion of its solid state switches (GTOs or IGBTs) generates an ac waveform in the trans-
former side. Through control of the magnitude and phase angle of this ac waveform the
real and reactive po\Ã/er entering the ac network can be precisely controlled.
I2
Inverter
[-l*
{Lfïner
]t_i lt
TransfoDC Voltage powerSorrce Converter
Reciifier
Fig. 2. 1 Voltage Sourced Converter block diagram
2.2.1 DC Voltage Source
The dc voltage source can be one of several alternatives. In the simplest form it is simply a
charged capacitor. If energy exchange is important, abattery or even another converter
can be used. Other arangements include a voltage source generated by connecting a
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGTES
power conditioner to a superconducting inductive coil (Superconducting Magnetic Energy
Storage "SMES").
2.2.2 Power Converter
This is basically an arangement of semiconductor devices called a bridge. The simplest
bridge types are two-level bridges and three level bridges. In the two level bridge, the ac
voltage is generated by alternately connecting the output terminal to one of two dc levels
(+Vdc and -Vdc). This is accomplished through the appropriate turn on and turn off of the
switches, each one of which consists of a controllable element (GTO or IGBT) in anti par-
allel with a diode. The dc voltage is supplied by a capacitor, battery or another power con-
verter and it is ideally constant. If the capacitor is used, it should be small because of
economic reasons. On the other hand the use of a small capacitor may generate a larger
ripple in the dc-Link voltage during steady-state operation and even greater impact on the
dc voltage as a consequence ofsystem disturbances.
In the three level bridge topology, the dc capacitor is divided into two equal sections with
equal voltage across each section. The semiconductor valves connect each phase to three
different voltage levels (+Vdc, Zero (Midpoint) and -Vdc). For this arrangement six
diodes are required as in Figure 2. 3. In this anangement, the zero voltage is obtained if
the two switches in the middle are tumed on. +Vdc is applied to the output if the two
upper switches are tumed on while -Vdc is obtained if the lower two switches are fired.
One of the advantages of this technique is the possibilify to set the pulse width (see o
13
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES
Figure 2.3.) in such away that a selected harmonic can be eliminated. Converters with
more than three levels are not a common option in the market due to their complexity.
I4
1rc
Fig.2.2 Two level six pulse converter voltages
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES 15
+Vdc
0
-vdc
Fig. 2. 3 Three level bridge
2.3 Voltage Sourced Converter Switching and Harmonic Components
In high power applications the switching frequency can be classified as follows. Funda-
mental frequency switching (60 Hz) is considered as low frequency. Medium switching
frequency can be defined between 5 to 9 times fundamental frequency (300 Hzto 540H2).
High frequency is above 15 times the fundamental frequency (> 900 Hz) [16].
VOLTAGE SOURCED CONVERTER (VSC) AND FÀCTS TOPOLOGIES
An important task of the switching technique is the reduction of harmonic conrenr gener-
ated by the converter. Many techniques have been developed throughout the years, from
the basic six pulse, to Pulse Width Modulation (PWM), and Optimal Pulse Width Modula-
tion (OPWM) where the switching angles are calculated in order to avoid certain harmon-
ics components [1]. Aithough Pulse Width Modulation (PWM) creates high switching
losses in the power semiconductors, the continuing improvements in semiconductor tech-
nology still makes this technology potentially attractive.
If the switches are operated at fundamental frequency, with 120o phase shifting, to con-
nect the dc supply sequentially to the outputs, then a balanced set of three square waves
(Va, Vb and Vc) is obtained as shown in Figure 2. 2.In a three phase system with delta
connected converters (isolated neutral), the triplen order harmonics 3rd, 9th, 15th, etc, will
be only of zero sequence and therefore they will not flow in the line currents, unless the
supply voltage or the converter become unbalanced. As a result, triplen harmonics wiil
include positive and negative sequence components that will flow into the system.
The output voltage waveform of the six-pulse converter contains harmonic components of
frequency (6k X 1) 'f and its input current has harmonic components 6k .f and k : l,
2,3,...The large amount of harmonic content in the output voltage makes it an inefficient
device for high power applications. Nevertheless using the principie of harmonic neutral-
ization using n basic six pulse converters operated with certain phase shifting between
them allows us to obtain an overall P : ínmulti pulse arrangement shown inFigsre2.4.
The harmonic frequencies present in this P-pulse arrangement arc (Pk ! I) .f for output
T6
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES t1
voltage and Pk.f for input cunent l2l.
can be implemented by a vanety of circuit
L3,4,51.
Multi pulse (harmonic neutralized) converters
arrangements using di fferent magnetic devices
6-pulse convertorBndge
7.50
ú-pulse convertorBndge
37.50
6-pr,rlse convertorBridge
I {O
ú-pulse convertorBndge
22.5"
Fig.2.4 converter arrangement 6-pulse *4 converters:24-pulse operation 16].
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGTES
2.3.1 Snubber Circuits
GTO devices are available in a wide range of voltage and current ratings including, in par-
ticular, the current turn off capability. A widely used GTO has a peak voltage rating of 4.5
kV and a peak turn off current of 4 kA. As with thyristors, it is important to protect the
individual GTO devices against both forward and reverse overvoltage and against exces-
sive rates ofchange ofinrush current and ofvoltage atturn off. A typical snubber circuit is
shown in Fisure 2. 5.
L - dl/dt Limitine inductorJtct - ulscnafge KeslsEorus Dd - Discharge DiodeGTO - Mau dei¡rce GTODm - Marn DrodeCs - Snubber CapacitorIJS - ÞflUoner ljlodeRs - Snubber Resistor
Fig.2. 5 Typical snubber circuit affangement for GTOs [6]
For the GTO to tum off safely at a high current, the snubber capacitor must have a high
value. The energy stored in this capacitor must be dissipated after every switching. If a
smaller capacitor is used, the switching losses are substantially reduced but the safe tum
off current is also reduced. Energy stored in the di/dt limiting inductor of the snubber cir-
cuit at turn off is dissipated via the discharge resistor and diode. Some of the dvldt limiting
18
VOLTAGE SOURCED CONVERTER(VSC) AND FACTS TOPOLOGIES
inductor of the snubber circuit at turn off is dissipated via the discharge resistor and diode.
Some of the dv/dt and di/dt circuit energy recovered by additional circuits. The added
complexity and cost of these energy recovery techniques must be weighed against factors
such as losses, and switching techniques [6].
2.4 Voltage Sourced Converters Topologies
The voltage sourced converters is the genesis of many controllers, nevertheless the main
topologies are the Static Synchronous Compensator or STATCOM and the Static Syn-
chronous Series Compensator or SSSC.
2.4,1 Static Synchronous Compensator (STATCOM)
The STATCOM is a voltage sourced converter that converts a dc voltage into a three-
phase output voltage at fundamental frequency, a coupling transformer and a dc source
(capacitor, SMES, BESS, etc). The steady state operation is similar to that of a rotating
synchronous compensator but without inertia, so its response is basically instantaneous
and it does not significantly alter the existing system impedance. It is an advance over the
Static Var Compensator (SVC).
t9
VC VAi----------+-*-Þ lnducfiveoperahon
l.qla
+laI va vcL----*--ir-+ UaFaclhve operafton
STATCOM
Ia =Is- It
jxt
Fig.2. 6 STATCOM equivaient for steady state
voLTAcE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES 20
r+<t+r
Va
Couplng Transforrner
Vc
Power Converler
DC Voltage Sorxce
Fig.2. 7 Block diagram of a Static slmchronous compensator (STATCOM)
The arrangement in Figure 2. 7. shows a typical STATCOM, in steady state the power
exchange between the device and the ac system is mainly reactive. The power exchange is
controlled by regulating the amplitude of the STATCOM output voitage. If the per-unit
amplitude of the STATCOM output voltage (Vc) is larger than the per-unit amplitude of
the ac system (Va), the device generates reactive power (capacitive). On the other hand ifthe STATCOM output voltage is lower than that of the ac system, the device will absorb
reactive power (inductive). This operation is clear from the phasor diagram in Figure 2. 6.
Finally if both ac system and STATCOM have the same voltage, there is no power
exchange. The current from the STATCOM is 90' shifted with respect to the ac system
voltage, and it can be leading (generates reactive power) or lagging (absorbs reactive
power) [4].
In this case a capacitor is used to supply dc voltage to the converter. However the con-
verter keeps the capacitor charged to the levels set. In the steady operation the phase angle
between Va and Vc is kept at 0'. By marginally shifting it from 0o, active power can be
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES
made to flow into or out of the VSC, thereby charging or discharging the dc capacitor.
This mechanism is used to keep the capacitor voltage constant if needed. In fact, in the
steady state, the output voltage of the inverter slightly lags the ac system voltage, so that a
small amount of real power from the system flows into the VSC to compensate for intemal
losses and thus, keeping the capacitor voltage constant.
2.4.2 Static Synchronous Series Compensator (SSSC)
When the voltage sourced converter is connected through a coupling transformer in series
with the power transmission line, a new device called Static Synchronous Series Compen-
sator (SSSC) results. The SSSC is shown in Figure 2. 8.
AC System:------------1-¡..Àl--
Couplrrg Transforr¡er
Fower Converier
Dü lloltage Source
Fig. 2. I Block diagram of a Static Synchronous Series Compensator (SSSC)
The Static Synchronous Series Compensator injects a voltage into the power transmission
line in quadrature to the line current, thus emulating an inductive or a capacitive reactance
in series with the transmission line. The power flow on the transmission line can then be
affected throueh the control ofthis series reactance.
2T
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGTES
Up to this point the devices were capable of supplying or absorbing reactive power in the
power transmission system. However a new family of devices is obtained by combination
of the STATCOM and/or SSSCs. For instance the Unified Power Flow Controller (UPFC)
combines a STATCOM and a SSSC. The Interline Power Flow Controller (IPFC) consists
of two or more SSSCs. The Generalized Unified Power Flow Controiler (GUPFC) uses a
STATCOM and two or more SSSCs. The common feature of the devices mentioned is the
possibility to exchange active power between their shunt and series components or
between series components like in the IPFC.
Uniike the STATCOM and SSSC devices which are only capable of providing reactive
power, the UPFC, IPFC and GUPFC have additional degrees of freedom as they allow for
some real power exchange as well. These additional degrees of freedom are expected to
result in greater flexibility and even allow for a better control of system stability through
their rapid control actions l7l.
2.4.3 Unified Power Flow Controller (UPFC)
The SSSC is coupied to a STATCOM in a back-to-back anangement. Thus, the shunt and
series devices are able to exchange active power. This arrangement is called Unified
Power Flow Controller (UPFC) and it is illustrated in Figure 2.9.lnthis arrangement, the
series device injects a voltage in series with the line but this time without any restriction
other than the device rating. In a widely used control concept, the shunt part is operated at
unity power factor and is used to maintain constant voltage on the dc capacitor through the
control of the active power. However one additional degree of freedom is available and
may be used to generate or absorb reactive power at the shunt bus. The first large scale
22
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES
practical demonstration of the UPFC was installed atlnez Substation 138 kV located in
eastern Kentucky in 1998. This project was a collaborative effort between American Elec-
tric Power (AEP), the Westinghouse Electric Corporation and the Electric Power Research
Institute (EPRI). The UPFC comprises two +160MVA Voltage Sourced Converters. The
Power Converter of the three level type as shown in Figure 2.3.Each of the turn off capa-
ble valves in the converter counts of eight or nine 4000 A, 4.5 kV, GTo-Thfistors con-
nected in series.
Fig. 2. 9 Unified Power Flow Controller (UPFC)
2.4.4 lnterline Power Flow Controller (IPFC)
The IPFC topology is shown in Figure 2. 10. A series VSC is inserted into each of several
transmission iine, with all VSCs sharing a coÍrmon dc link. In the IPFC both VSCs
exchange active power with their power transmission lines. However the active power
23
Line impedance
rf<tL+>r
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGIES 1ALA
injected by one power converter to its
losses) to the active power taken away
power transmission line must be equal (neglecting
from the other power transmission line.
The IPFC makes it possible to equalize real
mission lines. It allows us to transfer oower
to underloaded one.
and reactive power flow between power trans_
from the overloaded power transmission lines
Fig.2. 10 Interline Power Flow Controller (IPFC)
2.4.5 Generalized unified Fower Flow controller (GUpF,c)
The GUPFC overcomes one of the limitations of the IPFC, which enforces the net active
power through the dc link to be zero. This constraint generates restrictions for the power
system as we will see later. The addition of the STATCOM to the arrangement allows us
VOLTAGE SOURCED CONVERTER (VSC) AND FACTS TOPOLOGTES
to absorb any active power surplus in the dc link or to provide any def,rcit of active power
present in the dc link. Thus, this arrangement would be equivaient to an independent
UPFC in each line. This device can control simultaneously five power system quantities,
for instance bus voltage at a substation and the real and reactive power flows on two
power transmission lines [18].
Fig.2.11 Generalized Unified Power Flow Controller (GUPFC)
25
CÍTAPTER.3 Interline Fower Flow Conû"oller(PFC)
The Interline Power Flow Controller better known as IPFC is one of the latest in the fam-
ily of FACTS controllers. Although the basic principle of this device is simple, its opera-
tion when it is included in a power system is not well understood. The first device of its
kind will be in serviceby 2002 for New York Power Authority (NYPA) in Marcy 345 kV
substation.
3.1 The Interline Power Flow Controller
The Interline Power Flow Controller provides a flexible pou/er flow confrol scheme for a
multi-line power transmission system, where two or more Static Synchronous Series
Compensators (SSSC) are linked together by their dc side.
From the network point of view, the IPFC consists of two or more power transmission
iines with a common source and different end for each one of them as shown in
Fisure 3. i.
26
INTERLINE POWER FLOW CONTROLLER (IPFC) 27
Fig. 3. I Power converter arrangement
The IPFC scheme is basically a back-to-back dc to ac converter with a cofirmon dc voltage
link' The output of both converters is coupled in series with the power transmission line,
while the dc side allows active power flow from one line to the other or vice versa. A more
detailed schematic diagram is shown in Figure 3.2. Adding the active power exchange
between both converters gives one additional degree of freedom to the system, this means
that another variable can be controiled in this network.
The operation of the IPFC is highly affected by the line current because of its series con-
figuration. Therefore parameters such as the phase difference (ô) between sending end
and receiving end voltages as well as line impedance influence its operation and of course
the device rating, this will be explained in Section3.3.2.
In this section, we investigate the operating regimes of the IPFC and other related devices.
INTERLINE POWER FLOW CONTROLLER (IPFC) 28
Power Converter 1
Fig. 3. 2 Interline Power Flow Controller scheme
3.2 The Static Synchronous Series Compensator (SSSC)
The Static Synchronous Series Compensator (SSSC) is a subset component of the IpFC,
therefore its study has to be extensive and take into account different aspects such as rat-
ing, constraints, and so on.
Under steady state, the SSSC model is considered as a voltage source in series with the
line as seen in Figure 3. 3. The SSSC is assumed to be very close to the source and there-
fore the interconnecting impedance has been ignored.
INTERLINE POWER FLOW CONTROLLER IIPFC) 29
JJÈU
Fig. 3. 3 Steady state model for SSSC
The basic SSSC as it was shown in Chapter 2, is able to inject or absorb reactive power
only (neglecting losses). However if an active po\iler source (battery, power converter
-back-to-back arrangement, large capacitor, superconducting inductor, etc.) is coupled to
the SSSC's dc side, active and reactive power can be exchanged with the network.
3.2,1 Static Synchronous Series Compensator (SSSC) without active po\üer source
For a SSSC without an active power source, the voltage injected (Ll/) in series with the
line has to be 90" leading or lagging the line current (neglecting losses). Thus, only reac-
tive power is injected or absorbed by the SSSC in the steady state. When the voltage
injected leads the line current by 90", the SSSC will perform an action similar to an induc-
tor in series with the power transmission line. The opposite action, similar to a capacitor
will be performed if the voltage is injected 90" lagging the line current. This modus oper-
andi under steady state conditions is shown using phasors in Figure 3. 4 where a voltage
phasor (LV) is injected in series with the voltage source phasor (Zs) and in quadrature
with respect to the line current phasor (Iline).
INTERLINE POWER FLOW CONTROLLER (IPFC)
When the SSSC injects a voltage emulating a capacitor in series with the line, the line cur-
rent is boosted, while the inductor emulation has the opposite effect. At the same time the
electrical length of the power transmission line will be changed stretching the elec1rical
length when the inductor is emulated or shrinking the electrical length when a capacitor is
emulated in a manner similar to that of a series compensation (series capacitors).
30
""1t,-'\
\tnJ (ÅV) emulates an inductor
Ifs
:ìt..
\ ,,,\
Vurj (¿i4 emulates a capacitor
Fig. 3. 4 SSSC effect for a reactive compensation (capacitive or inductive)
3.2.2 static synchronous series compensator with Active power source
On the other hand, If an active power source is coupled to the dc side of the SSSC, the
voltage constraint for LV to be injected at 90o leading or lagging the line current is not a
limitation anymore. Therefore active power as well as reactive power can be supplied or
absorbed by the device.
Illustrated by phasors in Figure 3.5, avoltage phasor (LV) is injected in series with the
voltage source (Vs). Due to the active power source being coupled to the dc side of the
power converter, the phasor (LV) can be injected at any angle within the range 0o to 360o.
INTERLINE POWER FLOW CONTROLLER (IPFC)
Thus a complete circle is generated, and any point within the circle representation is feasi-
ble for operation.
Vrnj (¡fr| is not conskained
Fig. 3. 5 Vector diagram for a SSSC with dc side active power source
3.3 The UPFC Steady State Model
The series part of the Unified Power Flow Controller (UPFC) is identical in operation to
the SSSC with active power source described above. The shunt part provides the dc source
and is also capable of providing reactive power at the shunt connection bus.
For steady state the UPFC model can be assumed to be a sinusoidal voltage source in
series with the line, which can supply a voltage at any angle with respect to the line cur-
rent. Thus, the system consists of a single power transmission line, a couple of sources at
both ends that represent networks equivalent, the sinusoidal voltage source in series with
the power transmission line as the UPFC model.
3l
INTERLINE POWER FLOW CONTROLLER (IPFC)
The elements are shown in Figure 3. 6 and they form the basis of the system to be exam-
ined from different points of view throughout this study. The parameters listed beiow are
given in per-unit for phasors' magnitude and radians for the phasors, angle.
Vslõ, : 1/.0
Vt,Zõ1,: lZ-0.05
LVrl,qr, : -0.05 l+n/2 -+ 0.05 l+n/ 2
Zline : Rt+ jxt: 0.01 +j0.06
ylinel:+_:G1-rjB1ZlineI
kVbase:230 kVMVAbase: 100 MVA
Where
R : Line Resistance
X : LineReactance
G : Line Conductance
B : Line Susceptance
vs / os V lr/ otrIri Lr
Fig. 3. 6 System and parameters for the steady state model
The convention used for the voltage injected phasor in this study allows the voltage mag-
nitude (LV,) to take negative or positive values within the range
(LVt : -0.05 + 0.05 ). This allows us to emulate an inductor in the first case and a
J¿
INTERLINE POWER FLOW CONTROLLER flPFC) aaJJ
capacitor in the second case. Besides the angle (r¡r1) varies in the range
tive and negative magnitude of LVr.
for posi-
For instance when for both magnitudes of A,V, (positive and negative), the angle (ry1)
varies within the range -i to i, a complete circle is drawn. Reflecting this convention, cir-LL
cles of different magnitudes are drawn in Figure 3. 7. These voltage magnitudes will be
used to simulate the voltage supplied by the UPFC.
Fig.3. 7 Voltage injected by the UPFC, magnitude and angle
We first analyze the operation of a UPFC. This analysis can be extended later to the IPFC
by the application of suitabie constraints.
-rtor22
INTERLINE POWER FLOW CONTROLLER (IPFC) 5+
In order to study the UPFC behavior, the following assumptions are made:
' The source voltage ( Zs )will be the reference.
' Losses neglected in the UPFC.
UPFC represented by a purely sinusoidal voltage source.
Harmonics from switchins are not taken into account.
' Analisis is performed for set of given voltage magnitudes and angles.
3.3.1 Current through the line when a UPFC is installed
Using Kirchhoff s Voltage Law to the circuit depicted in Figure 3. 6 and the values given
above. we obtain.
Vs lõ,+6Vrlv, - Vt,lõt,- I(l)"1)(& + jxt) : 0 (Eq 3.1)
1117"1 : (Vs Iõ,XA,V, lV t - V t,Zõ r,.)(& + jxt) (Eq 3.2)
(Eq 3.3)T ./\ _ (l Z0+ A,Vllty; 1l-0.05)rtL'\t - OJl_/.0J6
Plotting the current magnitude (It) with respect to voltage angle (V, ), for various values
of lA (l , different contours for the current magnitude can be observed in Figure 3. 8. In
this plot the current magnitude varies with respect to two different parameters, first the
voltage injected magnitude that corresponds to each one of the contours and the voltage
injected angle which varies along the x-axis. The range of operation for the device current
in this specific case takes place inside the outer contours.
INTERLINE POWER FLOW CONTROLLER (IPFC) 35
t.i' I i ì .rt/' i ì ;
:i:;i.i i i i '..i
i ì i r'.Ìili\lii I i ¡\-il:l\
iij"'-1.6 -1.0? -0.53 0 û.53 1.û? 1.6
Voltage rnjected angle - vr (rad)
Fig. 3. I Line current magnitude (Ii) vs. voltage injected angle ì{l
3.3.2 Calculation of the UPFC rating for a given angle (ô )
We will first develop the relationship between the active and reactive power of the UPFC
device and the injected voltage, from which the rating of the UPFC can be determined.
The apparent power of a UPFC can be calculated as the voltage injected by the series
device times the line current conjugated (see Equation 3.4).
In the following analysis assumes that the delta angles are fixed at both ends. The relation
between the rating and delta angie will be discussed later in this section.
1.3
/- L.)
Ð
ùûfú
c0F
o 0.6
J
n2
Su"n., : LVtlVr. Qt/),)* (Eq 3.4)
TNTERLTNE PO\ryER FLOW CONTROLLER (IPFC)
The apparent power in its rectangular form is given by Equation 3.5, where the real part
corresponds to the active power (Prr,,rr) and the imaginary part corresponds to the reac-
tive power (Qynprr).
S,r¡oc, : Puppct tjQrroct (Eq 3.5)
The exact expressions for Pyn¡6' , and Quon ' in Equation 3.5 can be readily shown to be:
36
P rn,,r, : (LVr. Zscos(ry,) + Lt/t, - LV,. 2,,. . cos(ry, - ô,,))G,
+ (Ln. Zs . sin(ry,) - AV, . Vr,.. sin(y, - ô,"))8,
Qurrrt : (LVr. Zs . sin(ryt) - AV, . Vr,. sin(r.¡r, - ô,.))G,
- (LVr. Zs . cos(ry,) + AVt2 - LVt ' 2,,.. cos(y, - ô,,))8,
(Eq 3.6)
(Eq 3.7)
Here, Zs is the cornmon sending end voltage, Vr,. is the voltage at the receiving end, A,V,
is the voltage in series, Ylinel:G, +jB, is the line admittance (see Figure 3. 6). The
apparent po\ryer handled by the UPFCT is thus:
sunoc, : JPT;,* tur*, (Eq 3.8)
The rating of the UPFCT is then the peak value of Sypp¿1 . Figure 3. 9 illustrates the P upoc,,
vs. Quro6 graph for the model parameters as in Section 3.3 with an angle
ô : -0.05 rad : 2.9" which corresponds to a power flow of 80 MW without compensa-
tion.
Note: Figure 3. 9 is a plot of the active and reactive power injected by the UPFC, not the
power at the receiving end.
TNTERLTNE POWER FLOW CONTROLLER (rpFC) anJI
? oo8g
Èí u.u ¡o
cY o.06?
o
6 U.U)ótr
B o.o+:o(do nn?,
fJ n nreE.Þ
o.ote
- 0.ûül
MVAbase: 100
kVbase :230
- *:û.06 -0.049-0.036-0.024-0.012 0 0.012 0.024 0.036 0.û48 û0ó
UPFCl ,A-ctive Power Puprcr fu.*)
Fig.3. 9 UPFCr Reactive Power vs. Active power
In actual operation the delta angle varies according to demand, season, and usage whether
normal or emergency, etc. Therefore a relation between the rating of the device and angle
(ô) is necessary; the UPFC should be rated large enough for the worst case, as shown
below.
The rating of the device can be obtained from the equations above (Equation 3.6 and
Equation 3.7), with the following assumptions:
" The line impedance is known, this means the candidate line for installation of the
UPFC has been chosen.
" The voltage source is an infinite busbar, therefore it can supply any amount of power
required keeping the voltage fixed.
" The voltage injected will be provided as given in Figure 3. 7.
INTERLINE POWER FLOW CONTROLLER (IPFC)
' The delta angle (ð1,) will vary from 0 to -i radians or -30", which means that all the
power goes from the source to the receiving end.
" The UPFC can fulfill any requirement, this means it has enough power and can inject or
absorb active Dower.
Taking into account the above considerations, the P-Q diagram with respect to the delta
angle at the receiving end (ô',.) is presented as a 3-D plot shown in Figure 3. 10. This plot
shows a cone shape where at larger delta angles (ô,") the overali device rating is
increased. Thus, for every angle delta there is a corresponding conic section.
Following the procedure used in Section 3.3.2 to obtain the device rating for a specific
case, we can calculate the device rating for a range for delta angle of 0 to -] which is typ-
ical for transmission lines. From Figure 3. 10 it can be seen that the largest *nur"n, power
in the UPFC results for the largest operating vaiue of delta, which in our case is -f; or 30".
Figure 3. l l illustrates the P-Q diagram for this case and is obtained considering the cross
section of the cone in Fizure 3. 10.
Hence, a circle of radius 0.47 p.u is drawn surrounding the P-Q diagram where the UPFC
operates at delta equal to f;
. Thus an UPFC with an apparent power of 0.41p.u can with-
stand the steady state operation within the range -å to 2 radians. A view from the cone's
base is shown in Figure 3. 11, where the dotted circle represents the UPFC apparent power
and the other circle the apparent power for delta equal to T .
38
INTERLINE POWER FLOW CONTROLLER ûPFC) 39
61¡ (rad)
"I:,:I
Ïl::I
-u.)
-0.4
-0.4.
_o.2
g
Þ
.E
d
U
Þ
ô1¡=0c
(Eq 3.s)
(Eq 3.10)
(Eq 3.11)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
UPFC I Active power Puprcr (p.u)
Fig. 3. 10 P-Q Diagram for UPFC vs. delta angle (ô,,.) for LV, : +0.05 p.u
In Figure 3. 10 the operating point for an angle ô,, : 0o results to be a point with active
and reactive power components. In spite of the fact that there is no angle difference
between sending and receiving end of the power transmission line, the voltage injected
makes the receiving end to see a difference in angle. Thus, the apparent power for angle
ôr, : 0o can be demonstrated to be:
LVt : LVZr4rl
, _ Vs l0 + LVIV t + Vt,lO L--¿ r,,:ffi but I-s:L-t,
LVl,qt,' Zlinel l0
And the apparent power is given by Equation3.4, then we have:
0.41 .. ......,
Srrnc, : d,Vlr4r, . (=+!+ L-).\Zlinel l0/ (Eq 3.12)
INTERLINE POWER FLOW CONTROLLER IIPFC) 40
d Lf taÐuPF('t - zli""l
The largest value of S,,,o.. : {p'* d is as shown by the
which is the radius of circle C. In this case S,,,,, : 0.047 p.u,
UPFC.
(Eq 3.13)
vector S,o, in Figure3. 11
that must be the rating of the
0.5
0.4
0.3
rJ.2
0.1
û
-û.1
-0.2
-0.3
-frÁ
-n5-0.5 -0.4 -û.3 -0.2 -0.1 0 0.1 0.2 û.3 0.4 0,5
UPFC 1 Active Power Pupncr fu.r)
Fig. 3. 11 P-Q diagram for ô1,. : -|rad surrounded by the dotted apparent power
3.3.3 Active power injected or absorbed by the UPFC
The UPFC is able to inject or absorb active power allowing us to generate a fictitious pos-
itive or negative resistance, which is a very attractive feature in power flow control, tran-
sient stability and power oscillation damping t8lt9lt10l. Therefore its operating
characteristics are potentially superior to other options such as series compensation or
/a
;lJÊr
uÞo
Ê.Ð
(úU
¡T
Ê.ll
INTERLINE POWER FLOW CONTROLLER (IPFC)
Thyristor Controlled Series Capacitor (TCSC), which can only insert purely imaginary
impedances.
The relation between the active power (Pur,,rr) and the voltage injected angle (v,) is
very important, since it helps us to see how much active power we are injecting or absorb-
ing in the system with respect to the voltage supplied. From the typical case given in Sec-
tion 3.3 we obtain the results shown in Figure 3. 12, below.
+l
0.05
û.039
0.028
0.û16
t.005
0.00625
-0.018
-u.029
-0.04-1,6 -L.2 -û 3 -0.4 0 û.4 0.8
Voltage nlected angtes ryt (rad)
Fig. 3. 12 Active power (P rnorr) vs. voltage injected angle (ry1 )
Each contour in Figure 3. 12 corresponds to a constant voltage injected magnitude (LVr),
^t)&
tr
oB
trÐ
o4[JH
1.61.1
while on the x-axis the angle (y1) varies from -l to
with respect to the active po\ /er axis means that the
i.The lack of symmetry in this plot
UPFC is able to inject more active
INTERLINE POWER FLOW CONTROLLER ûPFC) A1AL
power into the power transmission line than it is able to absorb, for this particular value
of ô.
By taking the derivative of the active power given in Equation 3.6 with respect to the
injected angle (ryt ) and setting it equal to zero gives us the angle at which the active
power can be maximum. In our case, the maximum occurs fot V, : Vtu,o,p as in
Equation 3.14.
For any given value of A,Vt, the maximum real power injected occurs for an angle V tntaxp
given by Equation3.l4. Figure 3. 13 shows the real power vs. the injected angle ry1 for
three different values of LV, (-0.05 p.u, 0 p.u and 0.05 p.u); and they all peak for the same
Vr : Vlr,¿.r¿. Iû our casg 'Vtntaxp : 0.14 rad.
(Eq 3.1a)
Thus setting LV.t: 0.05 and replacitrg Vr,,o'p in the active power expression Equation 3.6,
we obtain the maximum active power injected to the transmission line.
on the other hand setting LVt: -0.05 and Vr : Vlr,,," in the active power expression
Equation 3.6, we obtain the maximum active power that can be taken away from the trans-
missionline. Thus, wehave Purnct60.0s : 0.0477 p.u and Prrrcrg_0.0s : -0.0342p.u.
INTERLINE POWER FLOW CONTROLLER ûPFC) 43
0.05
0.039
;̂- n n'l?F.
Ê' 0.016
oÞ
E rr.oojü
:l 0.0062J
õk -n nlR
-0.029
-fì fl4' ' ' ' I ' ' t- - -1 .6 -1 .2 -0.8 -0.4 0 0.4 0.8 1.1 1 .6
Voltage üJectect argles url (rad)
Fig. 3. 13 Voltage injected angle (Vr,,o,¿) for maximum and minimum active power
3.3.4 Reactive power injected or absorbed by the UPFC
As well as the active power, the reactive power will change according to the voltage
injected (angle and magnitude). Thus, similar to active power, the reactive power injected
or absorbed can be plotted with respect to the voltage angle injected (ry, ) for the parame-
ters given in Section 3.3. Ax expression for the voltage injected angle (V, ) at which the
reactive power is maximum is illustrated in Equation 3.15. The angle Vr for maximum
reactive power is denoted by V t,,o,o .
Vi*p
I:,/: ,/'t/
. (Vs.G.-Vr,.Gr. cos(Er,) + Vt,.. B, . sin(ô{ (Eq3.1s)Vln,oro : u,*\
,r/
INTERLINE POWER FLOW CONTROLLER ûPFC) 44
The results from the example presented in Section 3.3 are as follows: the injected angle at
which the reactive power is maximum is -i.4 radians. The maximum reactive power has a
voitage injected magnitude of -0.05 while the minimum reactive power has a voltage
injected of magnitude equal to 0.03 as shown in Figure 3. L4. Thus, replacing vr,,,,e in
the reactive power expression Equation 3.7, we obtain the maximum reactive power
injected to the line for a voltage injected magnitude of -0.05 and the maximum reactive
power taken away from the line for a voltage injected magnitude of 0.03. These values are
Qun,,rt6-0.0s : 0.0814 p.u and Quoorrgo¡3 : -0.0100 p.u.
0.081
û.063
0.t44
û.025
0.0û625
-t.012
-û.031
-0.05-1.O - r.¿ -0.8 -u.4 0 0.4 û.8 1.2
Voltage m1ected anges ryt (rad)
Fig.3.14 Voltageinjectedangle(Vt,,o,q)formaximumandminimumreactivepower
9)
CT
0Fo
EÐ
O(d0
tr
=
t.b
Vtro e
INTERLINE POWER FLOW CONTROLLER IIPFC)
3.3.5 Active and reactive power at the receiving end
The Flexible AC Transmission System or FACTS devices according to many experts has
been described as a short to midterm solution for bottlenecks or even considered as one
alternative for the future system enlargement [3]. In this section it will be shown how the
active and reactive po\ /er are influenced by the installation of an UPFC. power enhance-
ment, an important advantage of FACTS, will be addressed in this section ll4l.
In previous sections, we were concerned with the rating of the UPFC converter. Hence, all
P-Q diagrams were drawn showing only the powers injected/absorbed by the UpFC. In the
treatment in this and subsequent sections, we are investigating the impact of the UpFC on
the power flow in the power transmission line. Thus, the diagrams such as Figure 3. 15,
shows the active and reactive power at the receiving end. These should not be confused
with the active and reactive power of the converter module.
The active and reactive power at the receiving end of our system 1 in Section 3.3 is
described bv:
45
Pr, : Gr(Vr,. Zs . cos(ô t,) + Vt,.. LVr.cos(ô,, - Vr) - Vt,z)
+.8,(lV,,l .lZsl . sin(ô, ,) + lV,,l .lLV,l. sin(ô1, - V,))
Qr, : Gr(Vr,. Zs . sin(ô t,) t Vt,. LV, . sin(ô,, - Vr))
-Br(Vr, ' Zs . cos(ô t,) t Vt,. LV, . cos(ô,, - Vz) + Vr,')
(Eq 3.16)
(Eq 3.17)
INTERLINE POWER FLOW CONTROLLER (IPFC)
The parameters given in Section 3.3 are replaced in Equation 3.16 and Equation 3.17. We
can plot the relation between active power and reactive power at the receiving end as illus-
trated in Figure 3. 15.
The operating point without compensation (without UPFC) corresponds to an active
power of 0.805 p.u and reactive power for -0.155 p.u, which is a capacitive power factor
by convention. The point without compensation corresponds to the center of the circle
drawn in Figure 3. 15. Each one of the contours corresponds to a different voltage injected
magnitude. The closest contour to the centre has a voltage injected magnitude of +0.01
p.u and the outer contour which includes all the other a voltage injected contours has a
magnitude of +0.05 p.u.
46
0.?
4.4'¡
f̂:.
a'ú n nr<
Ooûl ^^uÞ
Ê -û.43o
H -0.ø¡ÐÉ
Operabng pornt
\flTNOUI U¡'f U
-l 1 L-u.1 û.13 0.35 0.58 0.8 1,tz 1.25 1 .4't
,A-ctive Power Received - Ptr (p u)
Fig. 3. 15 P-Q diagram for the receiving end
TNTERLTNE POWER FLOW CONTROLLER (IPFC)
The addition of the UPFC for this specific case allows us to reach I.62 p.u in active power,
which means that our active power is 100% more than it would be with no compensation.
Similarly, the reactive power without compensation is -0.155 p.u, but with the addition of
the UPFC, the reactive power can vary between 0.66 p.u and -0.91 for voltage injected
magnitudes of 0.05 p.u and -0.05 p.u, respectively.
Another feature that can be added to the P-Q diagram at the receiving end is the voltage
injection angle (V' ). Itr Figure 3. 16, additional lines representing constant voltage injec-
tion angles are drawn for vr = 0 and vr : Ìt radians.
The vr = i line divides the P-Q diagram into two halves, the upper side for positive volt-
age injected magnitudes (ÂZt > 0 ) and the lower side for negative voltage injected magni-
tudes (A V, < 0).
From this graph we can obtain the operating point in the P-Q diagram corresponding to a
given injection LV,l,y* as indicated by the arbitrary vector shown in Figure 3. 16. The
active and reactive power at the receiving end are (P ¡:1.23 p.u and Qr,:T.0, which cor-
respond to a voltage injected of magnitude A,V, : 0.03 and angle v, : I in our particular
case. In similar manner any voltage injected has a corresponding operating point.
4l
INTERLINE POWER FLOW CONTROLLER (IPFC) 48
0.7
.+ n¿?
- n 1{
;.fr o.ozsoo@* -n')ÐÞoF- -t.43ùÞ.
u
0
ii
1l
-0.88
-1 1L---û1 û.13 0.35 0.58 0.8 1.02 1.25 1.41
Active Power Received - ptr (p.u)
Fig.3. 16 P-Q diagram with respect to the voltage injected (ry1 and AZ, )
3.4 Characteristics of an Interline Power Flow Controller (IPFC)
In previous sections the focus was on a UPFC. This approach was adopted to explain the
impact of single device in a power transmission system and to understand this basic unit of
the IPFC. The IPFC consists of at least two systems (see Figure 3. l7) subject to additional
constraints imposed by the nature of the IPFC.
Fig. 3. 17 IPFC conformed by two SSSC linked by their dc side
INTERLINE POWER FLOW CONTROLLER ûPFC)
3.4.1 Interline Power Flow Controller (IPFC) constraints
In the IPFC, we connect the converters in the two lines through the dc link, as shown in
Figure 3. 17. Note that only real power can flow through the link.
The net active power has to be zero (neglecting losses). This means that the active power
supplied by one of the power transmission lines is the same active power that the other
power transmission line absorbs. Failure to do this would either overcharge or discharge
the dc capacitors [11].
49
Pvsc,,* Pvscz : 0
Re((AV,l,tt,. (Itl\t)*) + ne((A[ lyr. QrlX)*) : 0
LVt . It. cos(y, - À,) + L[/2. 12. cos(ry, - Àr) : 0
(Eq 3.18)
(Eq 3.1e)
(Eq 3.20)
Vlr/ örr
Vs/ õs
Fig. 3. 18 The IPFC and its variables
The second constraint is due to the dependency of the active power supplied by the VSC
on the line current. The maximum amount of active power that can be transferred to the
INTERLINE POWER FLOW CONTROLLER ûPFC)
other line is also constrained by the current in the line. As a consequence of this limitation,
a region of the P-Q diagram will not be feasible for operation. Thus, the maximum active
power transferred through the dc link will be constrained by the power converter with the
lowest current.
3.4.2 The trPFC case
Consider a typical IPFC application in Figure 3. 18, where the two power transmission
lines terminate in the same ac network X at one end, and have separate ac networks
(Y and Z) attheir remote ends. The IPFC is located at the coÍr.mon end as illustrated.
The operation of the IPFC will be explored with special attention to its constraints.
The system parameters are iisted below:
V,ZE, : I/.0
Vt,lEt,: ll-0.05
A,Vrlry, : -0.051+n/2 -+ 0.051+n/2
Zlinel : Rr+ jXt : 0.01 +j0.06
ylinei:+:Gt+jBlZlinel
Vz,lEz, : Il-0.03
A,V2lty2 : -0.05 /+n/2 --> 0.051+n/2
ZIjne2 : Rr+ jXr: 0.01 +i0.06
yline2: + : Gr+¡ï,Zline2
50
INTERLINE POWER FLOW CONTROLLER (IPFC)
Taking into account constraints identified in Section 3.4.I. we propose an operating mode
such that one of the voltage sources in series (UPFC model in steady state) will operate to
control P and Q in its line to their desired values (Master). The other (Slave) will thus be
forced to fulfil the active power constraint of Equation 3.18.
Piotting the active power in the VSCs for each one of the lines with voltage injected mag-
nitudes of +0.05 p.u and voltage injected angles varying from -i to I, we obtain
Figure 3. 19. Active power contours are sketched as a solid line for system 1 and addition
signs (+) for system 2. Equation 3.14 allows us to calculate the voltage injected angle for
maximum active power. Thus,
Vtntaxr, : 0.I4 rad:8.02o
'Vznaxp : 0.15 rad:8.6o
Assuming independent VSCs in each line, i.e., removing the active power constraint for
the time being. Once again, the reader is reminded that Figure 3. 19 refers to the power in
the VSC device, not to be confused with the power in the line.
The maximum and minimum active power in the VSC are obtained when ry, is replaced
in Equation 3.6 by the voltage injected angle for maximum active power (v t,,o,p o.V z,,o,r)
and the voltage injected magnitude for +0.05 p.u.
5l
TNTERLTNE POWER FLOW CONTROLLER (rpFC) 52
0.05
û.0436
0.03?1
û.030?
0.024?
û.01?9
0.û114
0.0
-û.ü014
-tl.û079
-0.0143
-0.020?
-0.02?1
-0.0336
-nn4'-.t .Ð-L.J ¡_1 l+0.91-0.6tr0.46-0.23 0 0.23 0.4ó 0.69 0.91 1.14 1.31 1.6
Voltage fiJected angles vt,rpz (rad)
Fig.3. 19 Active power (Prrrr, Prrcr) vs. voltage injected angle (V,, Vr)
Hence, the maximum active power that both systems can inject into their respective sys-
tems is:
Prsc,rgo.o, : 0.0477 p.u
Pnsczgo.os: 0.0313 p.u
The maximum active power that either of the systems can supply to the other system is:
P rsc, g-0.0, : -0.0342 p.u
[email protected]: - 0.0178 p.u
These values show the maximum achievable active power supplied or absorbed by the
VSCs. However the first constraint for the IPFC (Prsc't * P vscz : 0 ¡ imposes additional
limitations since the active power that the VSCr is able to provide to the power transmis-
,+
,3/.¡+
NUß
Ê"
I?
tr
ütsorO
O<c
rìlr')Þ
INTERLINE POWER FLOW CONTROLLER ûPFC)
sion line I ts 0.0417 p.u. The maximum active power that the VSCz is able to take from
the power transmission line 2 in order to supply it to the system 1 is -0.0178 p.u (negative
sign means that the active power is going out of system 2). Therefore system I will be lim-
ited by system 2 given its lack of active power.
The active power that the VSCr can supply to power transmission line 1 is given by the
positive half of the active power plot of Figure 3. 20. The maximum amount of active
power available from system 2 is drawn as a line of constant active power at 0.0178 p.u in
Figure 3.20. Thus, system 2 irnposes a constraint on system 1, which is unable to operate
above the active power line. The impact on the P-Q diagram at the receiving end is shown
in Fizure 3.2L
53
/}UØÞ
Er.
I(J
tr
UBo
E"0Þ
q
0.05
0.t4
0.03
B.t2
t.01
0
-0.01
-0.02
-0.03
-0.04
-0.05 -1.05 -0.52 0 0.52 r.05
Voltage ,nJected anges vt,r¡rz (rad)
-1.5J
Fig. 3. 20 Limitation imposed by system 2 on system i
INTERLINE POWER FLOW CONTROLLER ûPFC)
Simiiar to system 1, the active power supplied by the VSCz to its power transmission line
is shown in Figure 3. 20. The maximum amount of active power available from system 1
is drawn as a dashed line of constant active power at 0.0342 p.u in Figure 3 . 20.
System 1 is able to supply as much active power as required by system 2, which means
that system 2 will not be constrained. As a result, the fulI P-Q diagram at the receiving end
for system 2 is feasible.
In terms of our first constraint, the system 2willbe able to operate throughout the entire P-
Q diagram at the receiving end. On the other hand, system i in the P-Q diagram shows one
non feasible region of operation.
3.4.3 P-Q diagram limitation ruled by the first constraint
Demonstrated in the previous section, the lack of active power in one system constrains
the other system according to the amount of active power available. This limitation has
repercussions for the P-Q diagram atthe receiving end.
For system 1 to operate without limitation, system 2 has to supply the minimum amount of
active power of 0.0477 p.u. However system 2 can supply a maximum of 0.0179 p.u
which generates an active power constraint, on the other hand system 2 can absorbs from
system 1 a maximum amount of active power of 0.0314 p.u but system I is able to supply
0.0343 p.u. The lack of active power is 0.0298 p.u in the first part and 0.003 p.u which
means that some regions in the P-Q diagram will not be feasible for operation as noted in
Figure 3.21.
54
INTERLINE POWER FLOW CONTROLLER ûPFC)
The line that divides the non feasible region (solid color in Figure 3.21) from the feasible
region is a line of constant active power supplied by the VSC. The family of lines at con-
stant active power supplied by the VSC in the P-Q diagram are called "Voltage Compen-
sation Lines" 112]. These lines were straight lines in the reference because the resistance
was neglected.
0?
u.53
0.36
0.19
0.02
-0.15
-0.32
-0.49
-0.66
-û.83
-1-0.1 0.08 0.26 0.4 t.62 0.8 0.98 1.16 I .34 1.52
Active Power Received - Plr (p.u)
Fig.3. 21 First constraint applied to system I
For system 2 to operate without any restriction, system I must supply active power of
0.0313 p.u. However in this case, system 1 is able to supply up to 0.0342 p.u which means
that system I can fulfil completely the active power requirement of system 2. Hence the
complete P-Q diagram for system 2 is feasible for operation as illustrated in Figure 3.22.
55
,+tr"
.Éo'õOÐ
oPÕ
E.o
o(úù
t.?
INTERLINE POWER FLOW CONTROLLER (IPFC) 56
0.8
t.62
0.44
8.26
0.08
-0.1
-t1.28
-0.46
-t1.64
-0.82
--t-0.4 -0.22 -0.04 0.14 0.32 0.5 0.68 û.86 I.04 1t22
Active Power Received - p2r fu u)
Fig. 3. 22 First constraint applied to system 2
The procedure illustrated as a flow chart in Figure 3. 23 helps us to draw the P-Q diagram
at the receiving end considering the active power limitation. The input phasors of this pro-
cedure arc Vs , V1,., Zlinel , and the phasor A Z which varies in both magnitude and angle.
Equation 3.2 ís used to calculate the line current with respect to the ^tr/.
Thus, for each
voltage injected phasor a different current value results, and implies a coffesponding
active and reactive power in the VSCI. The apparent power in the VSCI is calculated as
the product between the phasor LV and the current conjugated (l+ ). tn a similar manner,
the active and reactive po\ /er supplied by the source can be calculated since the source is
an infinite busbar and the reference for the systom.
;N
'ìJoD
Ð
0Fo
Itr
ù
o(d0
É,
1n
INTERLINE POWER FLOW CONTROLLER ûPFC) 57
Input Variables
Vs, Vr,, Zlinel
V t =-n12,-t c/2+0.0I,...n12
ItSrsc-¡:AVr.hx
SvscrPvsc¡*J?vsct
S.r,ce: VS.I *
r.oMin P<Pz, cr < Max
yes
P, : llrl2 .R,
g, : llrl2 'xt
Ptr: Pror,tulPvsa-P,
Qtr: Qror,"n-lQvsu-Q,
ne Plot Pr. vs Qt,
y Min P and Max Pare the limits imposedby the VSC from onesystem on the other
Fig. 3. 23 P-Q diagram with active power constraint
INTERLINE POWER FLOW CONTROLLER ûPFC)
Therefore using a power balance equation, adding the active power supplied by the source
to the active power supplied by the VSCI and substracting the resistance losses in the
power transmission line gives the active power at the receiving end. A similar procedure is
followed for the reactive po\¡/er calculation.
The calculated active power that the VSCr provides can be compared with the maximum
active power that the other system is able to supply. Thus, we will find that the P-Q dia-
gram at the receiving end for system 1 wilt be constrained if the active power is not suffi-
cient to fulfil active power requirement in system l.
The result of this procedure will provide us with pairs of points with P and Q values,
which will be drawn according to the limit imposed by the other system. Each one of these
pairs of points has an active and reactive power at the receiving end that corresponds to
different voltage inj ected phasors.
The result is a P-Q diagram with a line described by a constant active power supplied by
the VSC as illustrated previously in Figure 3.21.
3.4.4 The impedance effect
Here we investigate the effect on the characteristic of different line impedances. For
example, if the system 2 impedance is reduced by 25% from
Zline2 : Rr* jXr: 0.01 + j0.06 to Zline2 : Rr+ jXr: 0.0075 +j0.045. It is
expected that the current will be increased. As a result of this change we wiil be able to
supply more active power from system 2 by mean of VSCz. Therefore we will obtain a
s8
INTERLINE POWER FLOW CONTROLLER ûPFC)
larger feasible area for operation at the receiving end of system 1 as illustrated in plot (A)
Figure 3. 24. At the same time system 2 has been enhanced and this time the system I is
not able to fuIfil the active power requirement of system 2 creating a non feasible area as
shown in the plot (B) Figure 3.24.
59
0.?
0.53
0.36
0.19
0.02
-0.15
-0.32
-0.49
-0.66
-0.83
t;l
Fig. 3. 24 Influence of the change in impedance of system 2 on system 1
The maximum active power that VSCz is now able to supply is 0.0238 p.u compared with
its original value of 0.0178 p.u. Its active power is increased as a consequence of the
reduction in line impedance. If we increase the active po\rver available from VSCz at the
same time we are able to supply more active power to system 1. In our case the impedance
effect enlarges the feasible region in the P-Q diagram at the receiving end for system 1. In
Figure 3.25 the highlighted band is the additional region for system 1 that can be feasible
'-0 I 0.08 0.26 0.44 4.62 0.8 0.98 t.t6 1.34 | 52
Active Powr Received - Ptr (P u)
(A)
-036 -A)2 0.t2 0i6 0.6 0.84 1.08 1.32 r.56
Active Power Received - F2r (p.u)
(B)
0e1l
I
066 |
I
I o.ar IYIc/ o.rr
IEI.R -o.o?5
|õlql! -032 I
Èlolù -¡st I
9lß -0.81
I
"l-r.06
I
I
-t ? L
-U
INTERLINE POWER FLOW CONTROLLER (IPFC)
for operation given the reduction in impedance. The highlighted band's edges correspond
to constant active power lines in VSCI.
60
^
cr.Ëoooo
oFoÊ.0
o(dü
0.1
0.53
0.36
0.19
0.02
-0.1J
-t.32
-0.49
-0.66
-0.83
-I
Enhanced area wrth 25%less impedance in powertransmission line 2
-0.1 0.08 0.26 8.4 t.62 0.8 0.98 1.16 r.34
.A.ctive Power Received - Ptr fu.u)
t.52 1.,
Fig. 3. 25 P-Q diagram for system 1 after modifying the line impedance in system 2
3.4.5 Power transmission losses
The power transmission losses due to Joule effect are another important phenomena to
consider, they are given by equation Equation3.21. Where they are proportional to the
current square and the line resistance. FACTS devices such as the IPFC or UPFC tend to
boost the line current which at the same time will increase the power transmission losses.
Losses : lI1' .R (F.q3.21)
INTERLINE POWER FLOW CONTROLLER (IPFC)
As an example considering the system 1 without any compensation, has losses for
0.007 p.u. However taking into account a voltage injected magnitude of L\ : *0.05
and the voltage injected angle ry' varying from 0 to !,we obtain Figure 3.26.In this plot
is illustrated the increment in power transmission losses reaching 0.027 p.u as the maxi-
mum.
6l
0.0261
fi o.ozzzs0g 0.0184ço'*
0.0145
Ë o.orna0ta
F o.oooa15o
0.0029
Fig. 3. 26 Power transmission losses
_ñññt L i , r '
| | r
- --- 0 0.2 8.4 0.6 0.8 t 1.2 1.4 1.6
Voltage rnjected angle r¡.rl
I These velues correspond io a diflerent wltaç injected magnitude
CT{APTER 4 Steady State Solution for PFCusing Vlatlab
In Chapter 3, we looked at operating limits for the IPFC using the simple IPFC equations.
However, for more detaiied analysis, one may have to compute solutions for IPFC operat-
ing points which may not be reducible to a closed mathematical form. Hence, the objec-
tive of this chapter is to develop a computer program to determine the steady state
conditions of the IPFC for a given operating condition. This is essentially a load flow pro-
gram with an IPFC. The Newton Raphson method will be used to solve the load flow
problem and the method is briefly explained in this chapter. MATLABI is used to develop
the program due to its simplicity.
4.1 Newton-RaphsonMethod
Taylor's series expansion for a function of two or more variables is the basis for the New-
ton-Raphson method of solving the power-flow problem. Our study of the method begins
1. MATLAB copyright 1984-2001 The MathWorks, Inc
62
STEADY STATE SOLUTION FOR IPFC USING MATLAB
by a discussion of the solution of a problem involving only two equations and two vari-
ables.
Let us consider the equation of a function h, of two variables x, and x2 equal to a con-
stant å' expressed as
OJ
fr(xr, xz, u) : hr(xv xz, u) - br : 0
and a second equation involving another function h, suchthat
(Eq 4.1)
(Eq 4.3)
.fr(xr, xz, u) : hz(xs xz, u) - bz : 0 (El4.2)
Where ór is also a constant. The symbol z represents an independent control, which is
considered constant in our case. The functions f, and f, are introduced for convenience to
allow us to discuss the differences between calculated values of å, and h. and their
respective specified values b1 and b2.
For a specified value of uletus estimate the solution of these equation to be x!0) and x!o) .
The zero superscripts indicate that these values are initial estimates and not the actual
solutions x,* and x2*. We designate the corrections Ax!0) and Ax!O) as the values to be
added to x!o) and x!o) to yield the correct solutions x1x and x2x. So, we can write
fr(x,*,xz*,u) : fr(*\o) +axlo),x!o)+Âx!o),u)-bt : 0
fr(rr*, xz*, u) : fr(*\o) + axÍO), xf) + Lxf), u) - b1 : 0 ftqa.4)
STEADY STÄTE SOLUTION FOR IPFC USING MATLAB
Our problem now is to solve for Âx!o) and Ax!O), which we do by expanding the
Equation 4.3 and Equation 4.4 in Taylor's series about the assumed solution to give
fr(xr*, xz*, u) : ¡rçx\o), xf) (Eq a.5)
fr(xr*, xz*, u) : rt(r\o), xf), u) * n*\o'ff,1"' + n*:!'ffi|"' * .. : 0 (Eqa.6)
: þ -r,' \o', *f', ul : lø, - ,r,ç*10', *f' , ulLo -,t(r!o', *f', u)l lu, - t ,ç*',0', *f' , ù)
Where the partial derivatives of order greater than 1 in the series of terms of the expansion
have not been listed. The term ôf,/ ôx1¡tol indicates that the partial derivative is evaluated
for the estimated values of xlo) and xlo) . Other such terms are evaluated similarlv.
If we neglect the partial derivatives of order greater than 1, we can rewrite Equation 4.5
and Equation 4.6 in matrix form. We then have
(Eq 4.7)
Jtol
Where the square matrix of partial derivatives is called the Jacobian J or in this case J(o) f6
indicate that the initial estimates x!o) and x!o) have been used to compute the numerical
values of the partialderivatives. We note that f1(*\o',r\",2) is the calculated value of/
based on the estimated values of x!o) and x!o), but this calculated vaiue is not the zero
value specified by Equation4.l unless our estimated values x!o) and xf) are correct. As
I u¡, ôf,f . -l
lô*, ¿,,,ll¡'10'lt-|llôf, ôfrl I¡"lo'l
L 'J
| ôx, ôx,l
STEADY STATE SOLUTION FOR IPFC USING MATLAB
r,,,þ'l'i : þ¡''lL¡'!'l ln/,')
We repeat the process until the corrections become so small in magnitude that
a chosen precision index e > 0; that is, until lAx,l and lÀr2l are both less than
65
before, we designate the specific value ofl minus the calculated value of/ as the mis-
match A,{o) and define the mismatch nS) similarly. We then have the linear system of
mismatch equations
(Eq 4.8)
By solving the mismatch equations, either by triangular factonzation of the Jacobian or
(for very small problems) by finding its inverse, v/e can determine Ax!O) and L*f) . How-
ever, since we truncated the series expansion, these values added to our initial guess do
not determine for us the correct solution and we must try againby assuming new estimates
x!t) and x!t), where
rÍ') : x!o)+ Àx!o) and *L') : xf) + tx'f) (Eq a.e)
they satis$r
e 11 51.
4.2 The IPFC Variables
We start the IPFC with steady state variables as illustrated in Figure 4. 1. This set of vari-
ables will be used to develop a MATLAB program that considers the same assumptions
made in the previous chapter such as.
The source voltage will be the reference.
Losses in the power converters are neglected.
STEÀDY STATE SOLUTION FOR IPFC USING MATLAB 66
0
ø
The VSC is represented by a purely sinusoidal voltage source.
Harmonics from switching are not taken into account.
Infinite sources at both ends with fixed delta ansle.
Vlr/ örr
Fig. 4. 1 IPFC steady state variables
The program consists basically of two parts. The first one coffesponds to the system 1 and
the second one to system 2.
The first part calculates the voltage injected phasor (LVr) using as input variables the
voltage source phasor (Vs), the voltage phasor at receiving end (v,,),line impedance
(Zlinel) and the active and reactive power demanded by the load. Equation 4.10 and
Equation 4.11 which are restatements of Equation 3.16 and Equation 3.17 in the manner
of Equation 4.i.
The voltage injected phasor (LVt) is calculated using Equation4.l0, Equation4.li and
the Newton Raphson algorithm explained previously.
Vs / Òs
I n¡ f-I -./
STEADY STATE SOLUTION FOR IPFC USING MATLAB
fr: Gr(Vr,.Vs. cos(ð,,.) I Vr,.. AZ, . cos(ôr¡-Vr¡-Vr,2¡+ Pr,.
Br(Vr,. Zs . sin(ð t,) * Vt,.' LV, . sin(õ,,. - Vr)) (Eq a'10)
f, : G,(V,,' Vs. sin(ô,,.) t Vt,.AZ, . sin(ô,,-Vr)) t Qr,
_-Br(Vr,. Zs . cos(ô t,) i Vt,. LV, ' cos(ô1, - V,) + Vr,.') (Eq a'11)
Using this equations we obtain the Jacobian ( J )
I ar, ôf,fr,ãfr
J:lonlt ôYtl rEq4.12l
| ôf, ôf, I
|
-
-llôLV, ôy,)
The Newton Raphson algorithm uses the Jacobian noted in Equation{.l2 to solve
Equation 4.10 and Equation 4.L1 in an iterative way. The result of this algorithm will give
us the values of A,V, and ry, for the input phasors and the load demand.
The voltage injected phasor for system | (LVt) allows us to calculate the line current
using Equation 4.13.
r /.\ _ (Vslõ,+ AVtlr1r-Vt,.lõ1,.)11/-ìv1 :
(R, +/.xJ (Eq4'13)
With the line current and the voltage injected phasors, we are able to calculate the appar-
ent power supplied by the VSCI to its power transmission line. At this point we must fulfil
the first constraint of the IPFC, where the active po\Ã/er in VSCr plus the active power in
VSCz must be zero as illustrated in Equation 4.14.
67
LVt.It.cos(ry1-I,)+ LV2.12 'cos(ry2-Àr) : 0 Eqa.lal
STEADY STATE SOLUTION FOR IPFC USING MATLAB
The Equation 4.14 helps us to calculate the amount of active power demanded by system 1
and therefore how much active power must be provided by system 2.
Another set of equations is used for system 2 in order to obtain the voltage injected phasor
which fuIfils the IPFC requirements. Here Equation 4. 15 is a restatement of Equation 3.6
and Equation 3.18 as in Equation 4.1. Similarly Equation4.16 is obtained for system 2.
68
The voltage injected phasor (LVt) can be calculated using the Newton Raphson method
for Equation4.l5 and Equation4.16. The Jacobian is obtained as illustrated in
Equation 4.17 for system 2.
(Eq 4.171
uation 4.18 is solved so as to obtain the
I ar. ôf^1|
-
-lJ:lô^v' av'l| ôfo ôfol
lo\v 2 o\yzl
Finally, the voltage injected phasor (LVr) in Eq
reactive power at the receiving end of system 2.
f, : Gr(LV.r. Vs. cos(y2) + LV22 - LVr. Zr,. cos(ryr-ôr,)) - Prr.
+ (LV2. Zs . sin(ry r) - LVr. 22, . sin(ryr- õr,))8,
fo : Gr(Vr,. - Vs. cos(ô2.) + V2,..Ä2, . cos(õ z,-Vz) - Vr,t) + Pr,
+ Bz(V2,.. Vs . sin(ô2,) t Vr,.A% . sin(ô2, - Vz))
: Gr(V2,, . Vs . sin(82,.) + Vz, 'Â Z, . sin(ô2" - Vr))
-Br(Vr,. Zs . cos(ô z,) * Vz,- LV, .cos(ô2,. - Vz) + Vr,')
(Eq 4.15)
(Eq 4.16)
Q,,(Eq 4.18)
STEADY STATE SOLUTION FOR IPFC USING MATLAB 69
Inout Variables
Initial Conditions for
Vs, Vt,, Zlinet
Yz,,Zline4Pr
Qt', Pz'
AVr, y r, AVz, y z, Iteratiofl:0,
Iterationl :0, Varcalc: lAVr, ty r]
Calc > 0.001 andIteration< 15
F sysr :7'(Equation 4.1Ot-l
þfz (Equation 4.1 1l
Calculate J (Equation 4.12)
Calc: -J -t* F sysr
Varcalc: Calc + Varcalc
AVr : Varcalc (1)
y r: Varcalc (2)
Iteration: Iteration * 1
STEADY STATE SOLUTION FOR IPFC USING MATLAB 70
¡:(Vs+AVrYr)lZliner,Srscr:ÂVl.h8
Pvscr:-Real {Szscr}
Varcalci : [AVz, r,¡r z]
Calci > 0.001 andIterationl < 15
F sysz :ft'@quation 4.15f
I rr,r @quation 4.161
Calculate J @quation 4.17)
Calc: -J -t* F sysz
Varcalcl : Calc + Varcalcl
AVz: Varcalcl(1)
ry z: Varcalcl(z)
Iterationl : Iterationl * 1
STEADY ST,A.TE SOLUTION FOR IPFC USING M,{TLAB 71
B
1z: (V s+AV z-Y z,) I Zlinez
Svsc,z:LYz.Iz*
Pvscz:-Real {Srsr-z}
Varcalcl:[AVz,yz]
Calculate Qz,in Equation 4.18
Vtr, Pt., Qrr, AVr, Ir
YzrrPzr, Qzr, AVz, Iz
END---- ,..-
4.3 Results
In order to demonstrate the successful solution by our program, we consider the IPFC sys-
tem studied in Chapter 3.
Given the line impedances
Zlinel : 0.01 +70.06 p.u
Zline2 : 0.01 +i0.06 p.u
Ylinel : Yline2ZlineI
STEADY STATE SOLUTION FOR IPFC USING MATLAB
Rated voltages for busbars are
Vs lõ" : 1.l0 p.u
Vt,lõt,: 1l-0.05 p.u
Vz,lõ2,: Il-0.03p.u
The power demanded at busbar I are
P,, : 0.75 p.u
Qr. : 0.1 P.u
Pr, : 0.45 p.u
The solutions given by the program are
LVrl,4tr: 0.01591 l-0.4351 p.u
LVrl'4t, : -0.02951-0.1225 P.u
With the solutions for the voltage injected, we can calculate the reactive power at busbar 2
and line currents
Qr, : -0.572T p.lI1 : 0.75661-0.182 p.u
12 : 0.727810.874 p.u
Note that the angles of phasors are expressed in radians
In this chapter we have presented a method for solving the nonlinear equations of the
IPFC using Newton-Raphson techniques. The numerical examples shown indicate that
using this method, the steady state behavior of the IPFC can be investigated under all con-
ditions.
72
CT{APTER. 5 IPFC Small signal stability modetr
A power system is said to be stable for small disturbances if the system returns to the orig-
inal operating point after being subjected to a small disturbance. A disturbance is consid-
ered to be small if the changes in system variables caused by the disturbance are
sufficiently small to permit the use of models based on first order linear approximation.
The objective of this chapter is to develop a small signal modei for the IPFC, and to use it
to investigate the stability when incorporated into a network with transmission lines and
generators.
5.L Stability Ín Fower Systems
In linearized small signal analysis a system is considered unstable if it moves away from a
given equilibrium point. It is possible that the system is still stable in a large signal sense,
in that it settles down at some other equilibrium state. However smali signal analysis does
give an indication whether large and possible undesirable changes are going to occur.
-aIJ
IPFC SMALL SIGNAL STABILITY MODEL 74
5.2 The IPFC Model
The simple system of three nodes, two transmission lines and an IPFC discussed in previ-
ous chapter is used in this chapter to investigate the small signal stability. The power sys-
tem connected to the receiving ends of the transmission lines are modelled as a
combination of a constant impedance load and a synchronous generator.
The aim of this chapter is to develop a state space model describing the dynamics of two
generators and the IPFC in the form given by equations Equation 5.19 and Equation 5.20.
Lx : ALx+ BLu
Ly = CA,x
(Eq 5.1e)
(Eq 5.20)
Vtr/ ôrr
Vs/ õs
Fig. 5. 1 Power system for small signal stability
It is important to mention that in this chapter the notation for the voltage injected in series
will be changed from ÀV, to Vr,,, to avoid confusion with the use of delta (A) for small
changes. Thus, the voltage and current relations for the power transmission lines are given
bv:
V2inji V2
IPFC SMALL SIGNAL STABILITY MODEL 75
Vs -l Vt¡n¡ - Vt, : Zlinel - 11 (Eq s.21)
Vs i Vr,,, - Vr, : Zline2 . I, @qs.zz)
wnere,
Zs is the sending end (source) voltage
V1,. isthe receiving end voltage at busbar I
V2, is the receiving end voltage at busbar 2
Vr,,,, is the voltage injected in series with the line I called earlier LV1
V2,,, is the voltage injected in series with the line 2 called earlier L,V,t
11 is the line I current
12 is the line 2 current
Zlinel, : Rt + jxt is the line I impedance
Ylinel : =J - : Gr+¡\, isthelineladmitranceZlinel
ZlineL : R, + jX, is the line 2 impedance
Yline2 : + : G2* jB2 is the line 2 admitianceZlineL
Additional subscripts x and / coffespond to the real and imaginary part of each phasor,
with respect to the infinite bus voltage.
The voltage of the infinite busbar is given by,
(Eq 5.23)v, : lr'l: ltllv'À Lol
Other voltages and currents in x-y reference frame are:
IPFC SMALL SIGNAL STABILITY MODEL 76
The injected voltages in the x-y reference frame are defined as follows:
Linearizing Equation 5.28 and Equation 5.29 for small changes, we obtain,
vt,,j : l'r,,,Å : vt¡,,j[""tt,lLVr,,¡À fsinyll
v2,,j : l'"À : vz,njþ"t*'llVr,,¡À [sinyr]
f/Y lr -
f// 2r -
r-
r-t2-
ln,,llVt,À
lr,,llv,,À
[''lL/'J
t'lv,À
(Eq 5.24)
(Eq s.2s)
(Eq s.26)
(Eq s.27)
(Eq 5.28)
(Eq 5.29)
(Eq 5.30)
(Eq 5.31)
lor,,,l : [.ory, -v,,n,siny,l þ n,,r]
lLVr,,¡À lsiny, /',,rcosry,_l L ^V,l
lorr,,l : [.or,y, -v2,,,sintyl!r,dlLVr,,¡À fsinry, V2,,,cosy1) L AVrl
The admittance matrices of the power transmission lines are defined as,
Equation 5.2I canbe expressed in rectangular form as follows:
I*+ jIu : (Gr +,lB')[(Vs,+ jVsr) t (Vr,,¡,t jVt,,¡r) - (Vr,.,+ jVt,.r)] (Eq 5.32)
IPFC SMALL SIGNAL STABILITY MODEL
Separating real and imaginary parts of the above equation and expressing the two resulting
equations in matrix form:
A similar expression can be derived for line 2 and the admittance matrices are defined as:
Þ,;l : [o, -r,.1 (ln'l *1n,,.,1- [n,,llLr,,l Lr, c,l ( [rz'l 1v,,,,À Lr,,))
Ylinei : IZlinel
Yline2ZLineT
Lxtnrc
LIt :
LIz :
(Eq 5.33)
(Eq 5.34)
(Eq 5.35)
(Eq 5.36)
(Eq 5.37)
(Eq 5.38)
(Eq 5.3e)
- [c, -¡JÞ; c'.1
- lc., _.s)- lt, ,S
5.3 Linearized IPFC Model
In this section, linearized model of the IPFC is modelled as a set of differential equations
and a set of algebraic equations in the following form:
: A 1 p ¡r¿A,x t p rc t B I p pçLu ¡ p p'¿
(LVs + AVt¡,1- AZr,.)Ylinel
(LVs + AVz¡,j- A,V2,.)Yhne2
lor,,,ÅLxtprc: I o*,
I
lLv'''À
I or,,,,,")L,'tt¡p¡,ç:
lor,,rllLv''""Å
(Eq 5.40)
IPFC SMALL SIGNAL STABILITY MODEL
where,
Lx,ro¡; is the linearized state vector for the IPFC
Lulp¡,¿ is the linearized control vector for the IPFCNote: Although there are 4 vaiables associated with the IPFC, there are only 3 independent vari-ables.
As the infinite busbar voltage is constant, changes in voltage at the infinite busbar are
zero. Thus, writing Equation 5.33 for small changes and using the relationship in
Equation 5.30 the following relationships is obtained:
78
a1r : þt',:l : - [o, -u'l [on'":l141,J LB, G,)L^vbÀ
Similarly for transmission line 2:
* þ, -B,l [.o'rrr -zri,rsinry'l þn'",r]lLB, G,l lsiny, Vr,,,cosy,) L ¡V, I
-;l Þ r,,f *lo, -tlþo,*, -v,,,,sintyllor,,Å
Gr)VVr,l lB, G2l lsinry2 Vr,,,cosy2)L¡Vrl
(Eq 5.41)
(Eq 5.42)^r, : [^¿']
lot,)
As explained in Chapter 3, the active power supplied by one of the VSCs must be equal to
the active power extracted by the other as shown in Equation 5.43.
Prsct* Pvscz : Puu, : 0 (Eq 5.43)
Where,
Pvsct : G1lVsrVr,,,,rcosr1r, +t,r¡(cosry,)' -V1,,V¡¡,¡così{l + 4,r¡(sinyr¡'1 (Eqs.zl4)
-V1rrV1¡,¡sirv,]+BlIV1,rV1,,rcostyr-lVs,V1ir,7sinqr, -V1,,Vy¡,¡sirV,]
P rscz : GzlVs,V,nrcosVr+ t,,¡(cosryr)' - V2,,V2¡,¡così./2 + ú,,¡(sinyr¡z1 (Eq 5.4s)
-V 2, rV 2,,., sin V, ] + B rlV r,Vy n/ cos V2 + V s,V, nr sinty 2 - V 2 uV2 ¡,¡ sin V, ]
IPFC SMÀLL SIGNAL STABILITY MODEL
In Equation5.4I and Equation5.42 there are four IPFC variables (LVt,,¡, A\r' , LVz,,j,
ÂVr). But as mentioned earlier there are only three independent variables (LVt,,,, AV, ,
LVr.,,¡).4 relationship between small changes of those four variables can be obtained
using the active power constraint (Equation 5.43). Then, one of the variables can be
expressed in terms of the other three variables.
Adding Equation 5.44to Equation 5.45 and making this result equal to zero, we obtain the
IPFC constraint. By taking the derivative of Equation 5.43 we obtain,
From Equation 5.46 we can obtain Âry, in terms of the other variables. Thus, the con-
straint is included in our smali sisnal model.
| 9!tuov,.- * õP n et Lv,..., + .ôP,", ¡y..- ¡ ôP nu, 6v,,.,, t ô P,", ¡y,,.,]
lUrr,.r ôVt,y "/ õVr,, õVz,y ''' ôVt,u¡ "
I
| *g!r,m, +ôP,", aV,,_, I
Âv, : -L ôv t ' ' ôVr,,, '"'' )
(Eq 5.47)
ôP,"t
ãw
IPFC SMALL SIGNAL STABILITY MODEL 80
ôPu", ôPu", ôPn", ôPnu, ôPr",
(Eq s.48)ôVr,,
^ r, ôVr,,,, ,r.r-- Jt linl
ôP,u, õP,n,
ã'v, ãw
-U,ru orr.,ôP nu,;-oVz
AT/ *Àv, : -o
/ t'* ¡vr,.'-%Lvrr,' ôP,", ôP,,o,
ãw av,
ôP,n, ôP,",
- lv' ov,-%Lv,u,ôP,o, ôP,",
ãr,,ôrV,
The variables are labeled Pl to PB to make the equations more compact. Thus
Equation 5.48 can be written as,
(Eq 5.4s)
(Eq 5.50)
A v, : -ffit r,,,-#o r r, r-Ð#o r r,,-#o v r, r-ffiL v r,,,
-DP6 ar,-QP7 tv.,..,DP\ '' DP\
Combining Equation 5.41 and Equation 5.42 we obtain,
[or,"'] [-c, B, o o I [on,J [c, -rJlor',1 : l-r, -o, 0 0 llon,,,l * lB, c, l["o'y, -v,,,,,sintyl [on,,,,:ll^1r,1 l0 0 -G2Arllnttr*l lO 0lfsinry,Z,-,cos'y,lL¡V,l1o,,,) lo o -8,-Gllor,,) lo ol-
[o o.l
*lo o l["otv, -v2,,,sin,4tllonr,ÀlG, -Brl fsinrY, Vr,,,cost4t2) L ¡Vr]ln' G')
Substituting in Equation 5.50 the value of Àry, from Equation 5.49 we obtain,
IPFC SMALL SIGNAL STABILITY MODEL 81
F-ITlA1',1 lG, cosy, - B, sinr.¡r, -lot',| : lr,cosv,
+ G,sinyl -l^1,'l I e
lN,À L e
GrVr,,,,sinty, - BrV,nrcorV,l trlBrVy,,,sinytr * Gr V,l,"orryrllorr,,f * I O I Tr,, I7' llo*,1 l?1r,,,,0tILO]
o l[¡ø,"]il "1o llov,,,l
B, * r ll\V""-lc, *'lÁ;,;)
[-c, B,
* l-u, -o,l" ßILv n
0
0
-Gr+ þ
-Br+' e -
'n:
q:
where,
cf:
þ:
v:
€:
IJ:
(D:I
-|fl t- G 2v2,,,,sinyt, - B rvr,,, cosyr r)DP8'
-P* C G 2v 2,,,, sin,q, - B rv 2,,, co s,4r 2)DP8'
-?4 t- B 2v2,,,,sinty z t G zv 2,,, cos,y 2)DP8'
-o*e B rv r,,, sinrq z * G 2Vr,,,., cosr4r 2)DP8'
-"=ff t- G rv r,,,,sinty, - B rv 2,,, cos,4t 2)DPSI
-PffiC G rv r,,,sinry, - B rv r,,, cosy r)DP8'
-K t- B rVr,,,,sinty z * G zV2, n, cosry 2)DP8'
-Pffjç B 2v2,,,siny z * G zvr,,, costy 2)DP8'
-H f - G 2v2,,,,sinry, - B rV2, u, costy 2)DP8'
-'Æt- G2v r,,,, sin,q, - B rv 2,,, cos t4t 2)DP8'
(Eq 5.s1)
(Eq 5.s2)
(Eq 5.53)
(Eq 5.54)
(Eq s.5s)
(Eq 5.56)
(Eq 5.57)
(Eq 5.58)
(Eq 5.se)
(Eq 5.60)
(Eq 5.61)
IPFC SMALL SIGNAL STABILITY MODEL 82
e : -lH(- Btvr,,,srn\y2 + G2vr,,,costqtr)
" : -#(- BrV2,,,slnì.{2 + GrV,n,cosytr)
e : Grcos{2 --B,sinry, .(-#)G Grvr,,,síntyr- B,vr,,,,costyr)
@ : Brcosv2 + Grsinr¡r2.l-'g^J (- BrV2,,,sinry, + GrV2,,,,cost¡t2)\ DP8l'
(Eq s.62)
(Eq 5.63)
(Eq 5.64)
(Eq 5.6s)
þt',:l [G, "or,y, - B¡ siny¡ - G,v,,,,,sinvr - B1V1,,,cosyr, 0l [n ,, I
lA1'rl : lB, così{r + Gr sinrl¡r - BrV,n,sinVr r Gr Vr,,,cosry, 0 | l^^" '"'l
lor,.l I e n ol lï'llnt,) | e , ;] lav,,,¡)
[-c, B, o o-]þn,.1+ l-a' -G' o o
ll 6v"'l
l" B -G,+þ Bz*tllor,,,lLY n -Bz*t-Gz+vllavr,)
For simplicity Equation 5.66 will be abbreviated as,
Finally combining the state variables associated with two VSCs in Equation 5.51, the lin-
eanzed expression for the IPFC is given by,
(Eq 5.66)
AI,nrc : T,prçAx tp rc * Ylpp¿A,V1p¡r¿ (Eq 5.67)
The relationship between the reference settings of the IPFC and the actual values are mod-
elled by first order transfer functions. In the analysis so far the IPFC modei's response has
IPFC SMALL SIGNAL STABILITY MODEL
been assumed to be instantaneous. The time constants in Fieure 5. 2 therefore can be con-
sidered the controller delay andlor any additional delay in the IPFC itself.
lVri"il
l\|/r'erl
lVzin¡'"rl Vz;n;l
Fig. 5. 2 IPFC delay
83
LVr,,¡: *f- a.vt¡,jt avt¡,¡,,r)Tr'
aV, : *(- o*, + aV,,ø)T2'
Lv),,¡: *(- avz¡,¡-r LVz,,¡,"1T2'
Which can be expressed in matrix form as,
(Eq s.68)
(Eq 5.6s)
(Eq 5.70)
(Eq 5.71)1o,,,,À
loy'l:lLvr,,rl
-1 o oTr
o -l oT2
0 0 -lT3
lor,,Àlo*'l*lLV,,,,À
1ov,,,,,"
I ot';'
lLV,u¡,".
looTr
oloT2
001T3
TPFC SMALL SIGNAL STABILITYMODEL
With this equation we have the IPFC smali signal model given in the form proposed in
Equation 5.19 and Equation 5.20. which correspond to Equation 5.7I and Equation 5.66
respectively. Equation 5.77 canbe abbreviated as,
Ax tpFC : A ¡p¡çLx trrc I B 1p¡çLu ¡¡,¡,ç (Eq 5.721
5.4 The Synchronous Generator model [29]
S¡mchronous generators are modelled ignoring the amortisseur winding but do include
field flux dynamics. The excitation and voltage regulator systems are represented by a
simple first order transfer function.
84
dr¡,dt
9t : co,(o,- l)dt
: + lrMi-vc*lc¡o - vc¡alc¡¿- K¡¡(o, - 1)]lI7 ¡
LE' o, : I l,Er,-8, qi - (xai - x, ¿,)Ia,¿f
dt Tt ¿o,
L,!r, : Llxn,(v,ut¡- lvo,l) - Er,ldr Tu,
(Eq 5.73)
(Eq 5.74)
(Eq 5.75)
(Eq 5.76)
the ter-
(Eqs.77)
Neglecting the stator winding resistances, the voltage behind transient reactance,
minal voltage, and the terminal current are related by,
, [-o- l-*,,
ô, is the generator's rotor angle
rrr, is the synchronous speed
rrr, is the normalized angular speed of the generator
H, is the rotor-turbine inertia constant
'41'"1o )LIo,)
lu',7 : ln",L ol LVo,
IPFC SMALL SIGNAL STABILITY MODEL
T¡a¡ is the mechanical torque of the generator
V6¡,¿ is the d-axis component of the generator terminal voltage
V6¡,,, is the q-axis component of the generator terminal voltage
16,,¿ is the d-axis component of the generator current
16,,, is the q-axis component of the generator current
Kp, is the damping coefficient accounting for mechanical damping losses and the effects ofdamper windings
E ni is the q-axis voltage behind transient reactance
x¿r is the direct axis synchronous reactance ofthe generator
xni is the quadrature axis s¡mchronous reactance ofthe generator
x'¿¡ is the direct axis transient reactance of the generator
T'ao¡ is the direct axis open circuited transient time constant of the armature of the generator
Ep, is the field voltage referred to the armatwe circuit
T r¡ is the time constant of the exciter
Va, is the AVR (Automatic Voltage Regulator) Voltage
S¿¡ is the saturation function of the exciter
K¡, is the exciter constant
T ¿¡ is the time constant of the AVR
Kt, is the AVR's gaín
V,"¡, is the reference voltage of the AVR
2,, is the exciter feedback stabilizer voltage
7s¡ is the time constant of the exciter feedback stabilizer
K9 is the gain of the exciter feedback stabilizer
x axis
85
y axls
Fig. 5.3 Reference frame for d-q transformation
IPFC SMALL SIGNAL STABILITY MODEL 86
I6¡ : Y6¡E'c,-Yc,Vc, (Eq 5.78)
In Equation5.77, voltages and currents are given with respect to d-q axis. Transforming
those variables into the infinite bus reference frame, Equation 5.77 canbe written as:
(Eq 5.7s)
(Eq 5.80)
(Eq 5.8r)
yc : lro", ,ou1: ,",[o
-"f *,
LYo,, Yorl lt olV'u, J
t ^ ^tSo, : lcosÒ'
smÒt I
fsinô, -cosõ/
Yco¡ : (*-f cosô,sinô,tx'd¡ x,t,
Yor, : llcosô,)'* lf sinô,)2Xq¡ ' x'd¡
yG,¡ : - !qcosô,)' - a(sinô,)2X'¿¡ Xq¡
Yoo, : -YGo¡ : (+ -Ð cosô,sinô,
E',G' :t'fu'r;l : þ'ø,cosô,1
L 0I [Enisinôl
The transformations from dq to xy and vice versa are given by,
(Eq 5.82)
(Eq 5.83)
(Eq 5.84)
(Eq 5.85)
(Eq 5.86)
(Eq 5.87)
(Eq s.88)
(Eq 5.8e)
VG,,t¿ : So,Vo,
Iç¡n¿: S¡¡16¡
Vc, : SpVc,o,
16¡ : S¡¡Iç¡¿,
IPFC SMALL SIGNAL STABILITYMODEL 87
The components of voltages can be expressed as vectors as follows:
Vc,q¿ : (Eq 5.e0)
t^ , --utqd (Eq 5.e1)
T/YGi - (Eq 5.e2)
r-rG¡ _ (Eq 5.e3)
5.5 Synchronous Machine Linearized Model
It can be shown that the linearized form of the generator equations given in Equation 5.73
thru Equation 5.76 is given by 1291,
ln"'llVo'À
[*';lVo,u)
lr"llVo,À
fr';lVo,À
I ou,l
l ot'l
ILE'n'lL^Eol
lr"llv,,À
Where,
Lio, : A6,A,x6,+ Bo,A,uç,* Ec¡LVc¡
AIc¡ : S6,Ax6, -YciLVci
Lxc¡ :
(Eq s.ea)
(Eq 5.e5)
(Eq 5.e6)
Lttc¡ : (Eq 5.s7)
IPFC SMALL SIGNAL ST,{BILITY MODEL 88
,^ = lor",)'(rt | |
LLIo'')
vG¡ : lor",flLVo'À
Io cte,, o olÁ^ - lor^ ol,) or, o
I"ut lor, o og, ogrol
þg^ 0 0 aSqa)
(Eq 5.e8)
(Eq 5.e9)
(Eq s.100)
(Eq 5.101)
(Eq 5.102)
(Eq s.103)
(Eq 5.10a)
(Eq s.105)
(Eq s.r06)
(Eq s.r07)
(Eq 5.108)
agn : (Ðo
o3zt : #l(H - L,,) ( z6ocosô, - z6,,sinô,)
-(? - n,ò( zo,'cosô, + zo,,sinô,) l
aSzz
aBzz
_ _Ko,2H,
- - Vo,o
2H¡x'¿¡
agtt: -;e t)( zo,r"orô, - Zo,,sinô,)
oTzz : - xa¡
T dn,x'¿,
gg^o : L
T oo,
I -_açq : -ffilVç,r(V6,rcosô,- Z6,,sinôr) +
V o, ¿(V 6, * cos ô, + Zoo sinô,)
Agoo : -LTn,
(Eq s.10e)
IPFC SMALL SIGNAL STABILITY MODEL 89
Bc¡ =
DLGi -
sc, :
']
,]
lI
,l
1ð
0
0
0
Ku,
Tn,
0
o o^.-ò¿,
o g^,'õ J,
o o,,'ö+.
\I sll
22
42
e
o
e
I
)
0
I2H,Hi
0
0
)¡^,
T.,tJl
!+t
0
o^è'¿
q-òJ
ø,Ò+
IC
Ic
Gid.
jid
tL
f"
;
+
t
(Eq 5.110)
(Eq 5.111)
(Eq s.112)
(Eq 5.11s)
(Eq s.r14)
(Eq 5.11s)
(Eq s.116)
(Eq s.117)
(Eq 5.118)
(Eq 5.11e)
(Eq 5.120)
(Eq s.r2r)
(Eq5.1221
€ szt : #l(? - n,) "o,a, - (Ikn
€ szz : r r-l(? - h,) sinõ, + (b
€ptt : J-(a- 1l cosô,T oo,\x'r, /
€pt.¡ : L(u- r'lsinô,T oo,\x'o, /
egqt : -ffifV6¡rcosõ¡+ Zo,¿sinô,)
€ g qz : -ffirv 6 ¡ n sinõ ¡-v6¡,,cos ô,)
sln
cos
þt,, o sg', o-l
Lsgr' 0 sgæ 0l
Yco,l Vo,rYcu,sgrr : -1o,nsinð, * Ici¿cosô, - Vc,y
sinô,çø,r : J"ö lJ
edi
sgzt : lo,ocosô, * 1c¡¿sinô¡ - Vo,Yo,,t Vc,,Yca,
cosð,.çø^- :'ö¿J
X'a¡
IPFC SMALL STGNAL STABILITY MODEL
5.6 The Load Model
The load is modelled as a constant shunt admittance representing P and Q at the given
voltage.
D +;aì.G]'JYGI
Fig. 5. 4 Parameters at the receiving end 1
Thus, we model the load as,
S¿r : Pu+jQLt (Eq 5.r23)
Referring to Figure 5. 4:
v Sr,*t Lt - lv¿s,,f
Yt; : GLt+ jBr.l
(Eq 5.124)
(Eq s.12s)
(Eq 5.126)
(Eq5.1271
The x-y components of the load voltage and current have the following relationship:
[t,,'l : lc,, -n,llr,,lV'''À lB', G',JIV',À
Vlr/ olr
IPFC SMALL SIGNAL STABILITY MODEL 9r
T -'l
YLt : lcu B"lI8,, G,,I
And combining load models at two receiving ends:
(Eq s.128)
(Eq 5.12e)
(Eq 5.r30)
5.7 The system model
The complete model consists of two generators with their dynamic behavior described by
Equation 5.133 and Equation 5.i34, the IPFC model which is given by Equation 5.66 and
Equation 5.7I, and the load given in Equation 5.130.
For the generators,
Defining
which correspond to
Lxc : A6A,x6+ BoLu6+ EGLVG
LIc : S6Axo -YGLI/G
T I T -IT Il^/,,1 :lY,,0lloz,,llLI,,l | 0 Y,,llLv,,lL..-JL.--JL--)
LIL : YLAVL
þ*"'1 : loo, o I þ"";l * þo, o I fo,"| *luo, .l Þ";ll}to,l L0 Aç2)lLxç2) L0 Bo')êuor) L0 EG,)lLVcr)
(Eq 5.1s1)
(Eq 5.132)
(Eq 5.133)
IPFC SMALL SIGNAL STABILITY MODEL 92
For the IPFC
þt",] : [so, o-l [o"";l -lto, o'] [on"]lnt"l f o snlfa'ol [o r"lltrt"l
(Eq 5.134)
(Eq 5.r35)
(Eq 5.136)
(Eq 5.137)
(Eq 5.138)
(Eq 5.r3e)
(Eq 5.140)
(Eq 5.141)
(Eqs.142l
Litptrc
AItrt,c
A ¡ p ¡r¿Ax ¡ p ¡,ç I B ¡ p pç A.u ¡ p ¡rç
T ¡ p ¡,-s\,x ¡ p F¿ -f Y ¡ p ¡,¿LV ¡ p p¿
By applying Kirchhoff current law to the generator bus, we obtain
a,I¡p¡,¿+ LIG- A,I¡. : 0
56Á16 - Y7LVG* Ttprc\xtprc* Y,nocLV,orc- YTLVL : 0
but,
LVo: A,Vlp¡,ç: LVr: LV
Hence, Equation 5.138 can be re-written as,
S6Áx6 - Y7LV-f Trcnc\xtprc I Y,norLV - YLAV : 0
From Equation 5.140 an expression for LV can be obtained as follows
LVlYrpFc - YG - Y,.f + So\,xc * TtprcLxtprc : 0
A,V : -lY,o,,r- YG- Y,,1-t Solxo_ lY,nrr- YG- Yr1-t T,ror\,x,r,,,
Equation 5.I42 replaces d,V6 inEquation 5.131 obtaining,
Lxc : A6Lx6-r Bc\uc+ EcelYrpoc- Yo- fr1-',So)Âxo
- l-Y,ro, - YG - YLf-t TtpFCLx*rc)
(Eq s.143)
IPFC SMALL SIGNAL STABILITY MODEL 93
Lxc : lAc- Ec(lYtpFC- YG- y¿l-',Sc)lAx6+ B6Lu6
-lEolY,rnc - YG - Y,.l-t T,rnrlLx,nnc)
(Eq s.144)
I ot" I : ltr" -8"(lY,r,,r-YG-)'r]-'so)l -lEolY,nrr-Yo-Yrl'r,rorll o"" Il\i'rnr) | o Atrrc l[Ax'"o.J
* þ" t l[ o""fL 0 Btprc)l\u'rrd
Finally, the equation for the fulI system, including the generators, the IPFC and the load is
given by the Equation 5.144 and Equation 5.135. Note that all state variables have been
combined to form one vector of state variables.
(Eq s.14s)
The model obtained in Equation 5.145 is the desired state space model. This can be used
to determine the system's small signal stability through the calculation of the eigenvalues
of the "A" matrix (see Equation 5.146).
l- , - ----r- --l
A : llAo - E^(LY,rr, - YG - I'rl-tso)l -lEGlYrpFC - Yc - Yrf 'T,rnrfl ,ro u.,rou,
I o A,nrc -l
5.7.1 State space model neglecting IPFC switching delays
The use of power electronics devices make the operation of devices such as the IPFC very
fast compared with mechanical devices. Therefore, for a case where the IPFC response is
almost instantaneous we have that Lx,rr, : Ltttppc. Hence the state space matirx 4,,,,,
will be equal to A1¡.y¡ inEquation 5.146
IPFC SMALL SIGNAL STABILITY MODEL 94
/ -/ - l- Iá(r.r) ninst - LIAG_EcfiyrpFC_yG_ fr]-,So)l
And the control matrix of the state space model (B) is given by,
(Eq 5.147)
(Eq 5.148)
(Eq s.149)
B': llr olr,',,r - Y o - Y,,f' r, n nrtr
8i,,, : lr" ul
a.u : I o"o | ,Eqs.lso)
l\u,rr")
Finally, the equation that describes the system for this particular case is given by,
Lx¡irr, : A¡,rr,Lx6l B,rrr\u (Eq s.151)
5.8 Model Results
We discussed in Chapter 3 that the receiving end power can be controlled by controlling
the voltage injected by the IPFC. It was shown that the range of control is given by the
P-Q diagram illustrated in Figure 3.16. In Section 3.4.3 |t was shown that some parts of
this region are not achievable due to power flow constraints.
In this formulation, the system considered has no Power System Stabilizer (PSS) on its
generators, as the pulpose of this section is to present a method of analysis, not necessarily
to consider the fulI details.
For the IPFC controller we choose typical values for time constants [30]
IPFC SMALL SIGNAL STABILITY MODEL 95
Tt : l0.6ms
Tz : 10.6ms
T3 : I0.6ms
Consider an operating point described by the following data:
Vslõ" : ll0 p.u
Vl,.l\t,.: ll-0.05 p.u
L'V.'ltYt : 0.01591 l-0.4351 P.u
Zlinel : R1-r jXr: 0.0i +j0.06 p.u
Ylinel : +: Gr+jïrp.uZIinel
P,, : 0.75 p.u
Q', : o'1 p.u
V2,lõ2,: Il-0.03p.u
Vt,,¡ltq, : -0.02951-0.1225 p.u
ZIine2 : Rr+ jXr: 0.01 +j0.06 p.uPr, : 0.45 p.u
Qr, : -0.5721 p.u
Scr : 0.4 +70.3 p.u
Scz : 0.4 +70.3 p.u
For the above operating point, the system matrix can be computed using the equation
derived earlier.
The eigenvalues of the system matrix give information on frequencies involved in small
oscillation of this system and also whether these oscillations are suff,rciently damped. The
imaginary part of the eigenvalues give the frequency. The amount of damping is given by
the damping factor (o
IPFC SMALL SIGNAL STABILITY MODEL 96
(Eq 5.152)
Eigenvalues
-0.t7233 + 9.90938i
-0.11233 - 9.90938i
-0.1 1800 + 7.06088i
-0.1 1800 - 7.06088i
-1.33806 + 1.36008i
-1.33806 - 1.36008i
-1.32622 + t.07292i
-L32622 - r.07292i
-94.33962
-94.33962
-94.33962
Damping Factor (( )
0.0174
0.0r74
0.0167
0.0r67
0.70i3
0.7013
0.7774
0.7774
The first result obtained from these eigenvalues is that the system is small signal stable for
the operating point. Because all the eigenvalues are located in the left hand side of the
plane as illustrated in Figure 5. 6. However it can be noted that the damping factors for the
first two pair of eigenvalues are very small.
IPFC SMALL SIGNAL STABILITY MODEL 97
;:1t¡--. 'f' rl
:' :tl' 'llr¡ 0l"-lrI
I
,, ,:6,f^t'I
-1n | |
l.ïsn -9û
Eigenvalues
-60 rJ -40 -30 -2Û -1û 0
Fig. 5. 6 Eigenvalues plot for the case considered
In Chapter 3, it was shown that the receiving end active and reactive power can be
changed by injecting a series voltage. It was also shown that only a certain portion of the
P-Q diagram is feasible from steady state operating point of view.
In this section, the feasibility of operation within the steady state range is investigated
from the small signal stability point of view. The region of interest was divided into a
large number of discrete points and the eigenvalues of the system matrix were evaluated
for each point.
It was found that for the particular system being analyzed, the complete region that is fea-
sible from steady state point of view is also feasible from small signal stability point of
view as shown below. The criterion used is simply that the eigenvalues must be on the left
IPFC SMALL SIGNAL STABILITY MODEL
hand side of the complex plane. One may also specifu a minimum damping requirement in
the above analysis, in which case some parts of the steady state feasible region may not be
feasible from small signal stability point of view.
0.5P-G diagnm SmalfSignal slable received from IPFC at end 1
û 0.2 u.4 t.6 8.8 1 1.2 1.4 1.6 1.8
Active Power (P.u)
Fig. 5. 7 Small signal feasible P-Q region for system I at the receiving end
5.9 Summary
A small signal stabilify model for a simple power system with one IPFC and two synchro-
nous generators was developed from first principles. The eigenvalues of the system matrix
were investigated for the complete "steady state feasible" region of the IPFC for a given
operating condition.
98
n=-.;
ÞtL.
'ç
0¿ -t.5
CHAPTER. 6 Conctrusions and
Recomrnendations
6.1 Conclusions
The main focus in this thesis was to obtain more details about the IPFC operation.
Although its individual components are well understood, its performance once assembled
is more complex. It was necessary to investigate the IPFC behavior using steady state as
well as small signal dynamic analysis.
The IPFC is certainly a very interesting FACTS device because it incorporates two or
more power transmission lines, making it the first "multi-line FACTS device". However
the IPFC series configuration in each line plays a strong role, since any change in the line
current will affect the IPFC performance. In order to gain an understanding of its opera-
tion, parametric studies which include variation in factors such as line impedance, po\Ã/er
available, sending, receiving and injected voltages (magnitudes and angles) and so on
were carried out.
99
CONCLUSIONS AND RECOMMENDATIONS r00
The IPFC can be controlled in various modes with the constraint that (neglecting losses)
the active power into one of the series converters must equal the active power out of the
other. In the operating mode considered in this thesis, the active and reactive power flows
on one of the lines (primary) was completeiy controllable (inside maximum limits), but
only one quantity such as receiving end power or voltage was controllable on the second-
ary line. Indeed it was even possible to control the direction of the power flow in both
lines.
In Chapter 3 it was shown that the receiving end active and reactive power could be
changed by injecting a series voltage and that the range of active and reactive power that
can be graphically represented within a circle in the P-Q plane. However when the con-
straints are considered only a certain portion of this circular region is feasible from the
steady state point of view
The IPFC is the next device in a sequence which starts with fixed series capacitors and
progresses to TCSC and then to SSSC devices. The tools developed here can be extended
by future workers to study the many advantages offered by any VSC based FACTS tech-
nology. These include better control of the power flow so that thermal limits are not
exceeded, better sharing of power between power transmission lines, the possibility of
using modulation signals to improve overall stability and the prevention of network loop
flows.
It is expected that with further accumulated experience, the IPFC and allied FACTS
devices will undergo further penetration into the power networks and with more use,
CONCLUSIONS AND RECOMMENDATIONS
prices are also likely to drop. The experience from prototype installation such as the Inez
UPFC (1998) l32l and the Marcy Convertible Static Compensator (CSC) l33l in New
York is thus invaluable.
The small signal model for the IPFC developed in this study is one of many research
directions in this field. The small signal model allows us to determine if a given operating
state is stable. The small signal model can be used to identi$r the feasible operating region
as a function of the operating condition and the IPFC constraints. Such analysis was con-
ducted in Chapter 5. The region of interest was divided into a large number of discrete
points and the eigenvalues of the system matrix calculated for each operating point. It was
found that the particular system analyzed the complete region feasible from steady state
point of view is also feasible from small signal stability point of view.
Although not done in this thesis, the state space model developed here is readily amenable
to desisn of feedback controllers for the IPFC.
6.2 FurtherRecommendations
The IPFC study is an interesting approach to a new device that it is expected to be in oper-
ation by the year 2002. Recommendations for a more extensive study of the IPFC are out-
lined as the follows:
The mode of operation for the IPFC used in this thesis considered fixed active and reactive
power at the receiving end in the primary line and the active power in the secondary line.
Other possible modes of operation such as active and reactive power in the pnmary line as
101
CONCLUSIONS AND RECOMMENDATIONS r02
well as voltage control on the other line, or voltage control on both lines with active or
reactive power control on the primary line could be investigated and their advantages and
disadvantases catalo zued.
The dynamic model developed only considered the IPFC device, a simple network repre-
sentation with machine dynamics and simplified controls. Additional control equations
depicting the dynamics of feedback control loops could be added to the state formulation
for more comprehensive studies.
In this thesis, the design of controller for IPFC to improve the damping of the power sys-
tem oscillations was not considered. It is recommended that the dynamic model be used to
identify the most suitable feedback control signals for improving a given mode of oscilia-
tion.
'When one or more parallel paths for current exist, it is difficult to control the current in
any given path. The efficacy of the IPFC could also be investigated for the purpose of mit-
igating network loop flows. Such loop flows may lead to bofflenecks and even congestion.
L'Abbate et al in [27] describes a method to prevent loop flows using FACTS devices and
in particular a UPFC to carry out this task. Another approach to reduce or even eliminate
loop flows using FACTS devices is given by Huang in [28] where an Optimal Power
Delivery (OPD) is developed to solve loop flow problems in a grid.
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APPENDTx A þ[omenclafure
Table 4.1 Operators
* Complex conjugate operator
j Complex operator
J Jacobian
* First order derivative
Table 4.2 Parameters
V,, Magnitude of voltage at receving end of line I
ô,, Angle of voltage aT receiving end of line i
I i Magnitude of current in line i
?'.i Angel of current in line i
Vs Magnitude of voltage at sending end
ôs Angle of voltage at the sending end
LVi Viinj Magnitude of voltage injected in series to line i
V¡ Angle of the voltage injected in series to line i
Zlnei Impedance of Line i
Ylinei Admittance of Line i
108
NOMENCLATURE
Table 4.2 Parameters
,R¡ Resistance of line I
Xi Reactance of line i
G i Conductance of line i
B i Susceptance of line i
P,, Active power received at the end of line i from IPFC o UPFC
Q¡, Reactive power received at the end of line I from IPFC o UPFC
S,, Apparent power received at the end of line i from IPFC o UPFC
Supt c,, Apparent power of IJPFC coupled to line I
P gt,trc' Active power of UPFC coupled to line I
Qurrc, Reactive power of IJPFC coupled to line i
Voltage injected angle ry, for maximum active power of UPFC coupled to line I'V inraxP
V ¡ntaxo Voltage injected angle ry, for maximum reactive power of UPFC coupled to line i
ôi Rotor angle of the generator
610 Synchronous speed
(Ð, Normalized angular speed of the generator
H i Rotor-Turbine inertia constant
Tu, Mechanical torque of the generator
Vo, Generator tenninal voltage phasor
VG,, Generator tenninal voltage x axis cornponent
Vo,, Generator terminal voltage y axis component
Vc,na Generator lrå terminal voltage referred to the dq frarne
Vc,¿ d-axis component of the generator terminal voltage
Vo,, q-axis component of the generator terminal voltage
Io, Generator current phasor
IG,, Generator current x axis component
Io,, Generator current y axis component
Ic,ao Generator ¡/rt current referred to the dq frame
109
NOMENCLATURE 110
Table 4.2 Parameters
Ic,¿ d-axis component of the generator current
Io,n q-axis component of the generator current
Damping coefficient accounting for mechanical damping losses and the effects of^D¡ damper windings
E q¡ q-axis voltage behind transient reactance
E q¡ q-axis voltage behind transient reactance
xrt¡ Direct axis synchronous reactance of the generator
x,t¡ Quadrature axis slmchronous reactarce of the generator
x'di Direct axis transient reactance ofthe generator
T,do¡ Direct axis open circuited transient time constant of the armature of the generator
Et, Field voltage referred to the armahrre circuit
T ¡¡¡ Time constant of the exciter
V¿, Automatic Voltage Regulator (A\IR) voltage
S¡;, Saturation function ofthe exciter
Kn, Exciter constant
T ¿¡ AVR time constant
Kn, AVR gain
Vrefi AVR reference voltage
Vr¡ Exciter feedback stabilizer voltage
T s¡ Tilne constant ofthe exciter feedback stabilizer
Ks, Gain of the exciter feedback stabilizer
Yco, Self admittance of the i¡å generator, x axis
Yct , Transfer admittance of the ith generator,from y axis to x axis
YGr, Transfer admittance of the irrt generator, from x axis to y axis
Yo¿, Self adrnittance of the i//t generator, y axis
x State vector
xcl State vector for generator l//'
xtprc State vecto¡ for IPFC
u Control vector
NOMENCLATURE 111
Table 4.2 Parameters
uG¡
,l^]PFC
xtptrc
U]PFC
' iitljx
'iinjy
D'úl
Qc¡
,c^.*ltl
AaGi
eGi
uG¡
.q,
vtGi
/n
A inst
Þ
PSS
AVRSSSC
STATCOM
SVC
TCR
TSC
L]PFC
IPFC
Control vector for generator ith
Systern matrix of the linear state space model of an IPFC
State vector of an IPFC
Control vector of an IPFC
Voltage injected in series to line i, x axis component
Voltage injected in series to line i, y axis cornponent
Active power of Generator i
Reactive power of Generator i
Apparent power of Generator i
Systern matrix of the linear state space model of a generator I
State vector of a generator I
Control vector ofa generator i
Transfonnation matrix between qd reference fralne and xy reference frame for generator i
Adrnittance matrix of generator i
Systern matrix of the linear state space
System rnatrix of the linear state space for instantaneous response
Darnping ratio
Power System Stabilizer
Automatic Voltage Regulator
Static Synchronous Series Compensator
Static Synchronous Compensator
Static Var Compensator
Thyristor Controlled Reactor
Thyristor Switched Capacitor
Unified Power Flow Controller
Interline Power Flow Controller
AFPErntDrx B Fower Flow Program
In this appendix the case is given as well as one case with its solution
VsZõ-: 110
Vlr/.81,.: ll-0.05
LVrlty, : 0.01591 l-0.4351
Zlnel -- \+ jxt: 0.01+70.06
Ylinel:=+:G1+jB7ZlneI
P,' = 0.75
Qr. : 0.1
MVAbase = 100MVA
kVbase:230 kV
Vlr/ ötr
tt2
POWER FLOW PROGRAM 113
V2rZE2,.: ll-0.03
LVrAy, : -0.0295 /.-0.1225
Zlne2 = Rr+ jXr: 0.01 +j0.06Pr, : o'45
Qz, : -0.5121
rt : 0.7440-j0.t313
12 : 0.4669 +i0.5583
The program developed in Matlab, this set of four programs correspond to four functions
that calculate the functions and Jacobians
The Main Program is shown below
o/o Prograrn IPFC power flow
%
ciear all
close all
warning off
fnnnaf ìnno
% Define Value
%
Plr:0.75;
Qlr:0.1;
P2=0.45;
Vs=1;deltas:0;
VlFl;deltal:-0.05;
V2r1;delta2:-0.03;
RLl=0.01;XLl =0.06;
ZLi=RL1+i"XLl;
YLI:llZLl;
G1=eal(Yl.l);
B1:imag(Yll);
RL2=0.01;XL2:0.06:
POWER FLOW PROGRAM TT4
ZL2=P*L2+]*XL2;
YL2=1/ZL2;
GZ:real(YL2);
B2:imag(YL2);
Vsp:pol ar2cart(Vs,deltas);
V I rp:polar2cart(V 1 r,delta I );
%
%o Initial Condition
%
dV 1=0.01;
psi 1:0.01;
dV2:0.001;
psi2:0.01;
Iteration:1;
Iteration 1:1 ;
calc= I ;
calcl:l;
%
var_calc:[dV1;psil];
%
7o Newton Raphson for systern 1
%
while (rnax(abs(calc)) > 0.00 1)&(Iteration < I 5);
delS:systern_set(Gl,V1r,Vs,deltal,dV1,psi1,Bl,Plr,Qlr);
J:Jmatrix(Gl,V I r,deltal,psi 1,8 1,dV I );
calc=-inv(J)*del S;
var_calc=calc*var_calc;
dV 1=var_calc(l );psil =var-calc(2);
Iteration:Ite¡ation+ I ;
end
if(psil > pi/2)
n=ound(psi 1/(pi));
nci I :nci I -n *ni'ye.. re,. .. r.)
dVl:(-1)^n*dVl;
if (psil <pi/2)&(psil > -pil2)
POWERFLOW PROGRAM 115
psi3:psi 1 ;
dV3=dV1;
end
else
dV3:dv1;
psi3:psi 1 ;
end
if(psil < -pi/2)
n:round(psi 1 /(pi));
psi 1:psi I -n*pi;
dV1:(-l)"n*dVl;
if (psiI <pil2)&(psil >-pil2)
psi3:psi 1;
dV3:dvl;
end
else
dV3:dVr;
psi3:psi i;
end
if (rnax(abs(calc)) < 0.001)&(lteration < l5);
dV 1 p:polar2cart(dV3,psi3);
o/o Current Cal culations
¡¡ | =(Vsp+dV Ip_y lrp) I ZLt ;
% Apparent Power supplied by the SSSCI
Ssssc l =dVlp''coni(IL l );
end
Psssc2=-real(Ssssc 1 );
var_calc I :[dV2;psi2]:
%
o/o Newton Raphson for systern 2
%
while (rnax(abs(calcl)) > 0.001)&(Iterationl < l5);
del S=systern_set I (G2,V2r, Vs,delta2,dV2,ps i2,B2,P2r,P sssc2) ;
J 1 = Jm atri x 1 (G 2,Y 2r deltaZ,psi2,B2, dV2,Vs) ;
calcl:-inv(Jl )*delS;
var_calc I =calc I +var_calc I ;
POWERFLOW PROGRAM 116
dV2:var_calc I ( 1 );psi2=var_calc 1 (2);
Iteration I :lteration 1 + I ;
end
if (psi2 > pii2)
n=ound(psi2l(pi));
psi2:psi2-n*pi;
¿y2=(_t)^n*dV2;
if (psi2 <pil?)&(psi?> -pil2)
psi4:psi2;
dY4:dY2;
end
else
psi4=psi2;
dv4:dV2;
end
if (psi2 < -pi/2)
n:round(psi2/(pi));
psi2:psi2-n*pi;
dV2:(-l)"nxdV2;
if (psi2 < pil2)&(psi? > -pil2)
psi41si2;
dY4:dY2;
end
else
psi4=psi2;
dV4=dV2;
end
if (max(abs(calcl)<0.001)&(Iterationl < 15)&((abs(dv3)<0.05)&((abs(dva))<0.05)
dV2p:pola12cart(dV4,psi4) ;
Y 2ryp ol arZ cart(Y 2r delta2) ;
I L2: (V sp+dV 2p -Y 2rp) I ZL2 ;
Ssssc2=dV2p*conj(IL2);
Q2r:(G2) * ((V2r) * (Vs) * sin(deita2)+(V2r) *(dV2)* sin(delta2-psi2)) -
(82)*(V2r)*(Vs)*cos(delta2)+(V2r)*(dV2)*cos(delta2-psi2)-(V2r)"2);
%%
Result(k,l ):P l r;
POWER FLOW PROGRAM 111
%
Result(k,2):Q 1r;
%dv1
Result(k,3)=dV3;
%psi1
Result(k,4):psi3;
o/oLine current
Result(k,5):lL1;
%Apparent power in the UPFCi
Result(k,6):Ssssc I ;
%
Result(k,7)=P2r;
%Q2r
Result(k,8)=Q2r;
%dv2
Result(k,9):dV4;
%psi?
Result(k,10)=psi4;
%Pinjected
Result(k,l I):ILZ;
%Apparent power in UPFC2
Result(k, l2)=Ssssc2;
%
Result(k, 1 3):-Psssc2;
k:k+l;
end
Result.'
Other Functions emnloved
Function to calculate the set of functions for Svstem I
function [out]:systern_set(G l,V 1 r,Vs,delta l,dV l,psi l,B 1,P I r,Q I r)
f(i,l)=(Gl)*(Vlr)*(Vs)*cos(deltal)+(Vlr)*(dVl)xcos(deltal-psil)-(Vlr)^2)+(Bl)x(Vlr)*...
PO\ryERFLOW PROGRAM 118
(Vs)xsin(deltal )+(Vlr)*(dV I )*sin(deltal -psi l))-P lr;
f(2,1)=(Gl )*(Vl r)*(Vs)*sin(deltal )+(V1r)x(dVl)*sin(deltal-psi i))-(B i)*((Vlr)*(Vs)"...
cos(deita I )+(V I r)*(dV I )*cos(deltal -psi 1 )-(V 1r)"2)-Q 1 r;
%
out:f;
refurn
Function to calculate the Jacobian for the first svstem
function fout]:Jrnatrix(G I ,V 1 r,delta 1 ,psi 1 ,B 1 ,dV 1 );
J( l, 1 )=(G I )*(V I r){'cos(deltal -psi I )+(B I )*(V I r)*sin(delta 1 -psi 1 );
i(1,2):(G1)*CV lr)*(dV l)*sin(deltal-psi I )-(B I )*(V1r)x(dVl)*cos(deltal-psi 1);
%
J(2, I )=(c I )*(V 1 r)*sin(delta I -psi I )-(B 1 )*(V1 r)*cos(deltal -psi 1 );
J(2,2)=-(Gl )*(V lr)*(dV 1)*cos(deltal -psi I )-(Bl )*(Vlr)*(dVl)*sin(deltal -psi1);
ouFJ:
return
Function to calculate the set of functions for Svstem 2
fu nction [out]=system_set I (G2,V2r,Vs,delta2,dV2,p si2,B2,P2r,P upfc2)
%
f(i,1):(G2)*(dV2)*(Vs)*cos(psi2)+(dV2)"2-(V2r)*(dY2)*cosþsi2-delta2))+(B2)*((dV2)*...
(Vs)*sin(psi2)-(V2r)*(dV2)*sin(psi2-delta2))-Pupfc2;
f(2,1)=(G2)*(V2r)*(Vs)*cos(delta2)+(V2r)*(dV2)*cos(delta2-psi2)-(V2r)"2)+(82)x...
((V2r) *(Vs)*sin(delta2)+(V2r)*(dV2)*sin(delta2-psi2))-P2r;
Function to calculate the Jacobian for the second svstem
tunction fout]=Jrnatrix I (G2,Y 2r delta2,psi2,B2,dV2,Vs);
J 1 ( I , I )=(G2)*((Vs)*cosþsi2)+2*(dVZ)-(V2r)*cos(psi2-delta2))+(82)*((Vs)*sin(psi2)-(V2r)+sin(psi2-delta2));
Jl ( 1,2)=(G2)*(-(dV2)x(Vs)*sin(psi2)+(dV2)*(V2r)*sin(psi2-delta2)+(82)*((Vs)*(dV2)*cos(psi2)-(dV2)*(V2r)*cos(psi2-delta2));
%
ouFf;
retum
POWER FLOW PROGRAM 119
%
J I (2, 1 ):(G2)*(V2r)*cos(delta2-psi2)+(82)*(V2r)*sin(delta2-psi2);
Jl(2,2):(G2)*(V2r)*(dV2)xsin(delta2-psi2)-(82)*(V2r)*(dV2)*cos(delta2-psi2);
%
out=J I ;
return
Other functions employed
fu nction Iout]:polar2cart(Mag,Angle);
o/o Transfonnation from Polar to cartesian coordinates, this retums a vector in cartesian form
%
x:Mag*cos(Angle);
y:Mag*sin(Angle);
%
ouFx+i*yl
renrm
.dPPENÐrx c Small Signal Model
In this appendix are shown the parameters as well as the program developed for small sig-
nal analysis considering the IPFC and two generators.
Generators Parameters [29]
Table 8.1. Generator and AVR parameters
vlr/ ôrr
Unit H(s)
r 6.40
2 3.01
xd xq¡
(p.u) (p.u)
0.89s8 0.864s
r.3t25 1.2s78
KD edi
(p.u)
0.1 1 98
0.18 r 3
T'doi
(s)
s.90
s.89
-Ä¡
25.0
25.0
T¿(s)
0.s0
0.50
VArr¡, VA*o,
(p u) (p.u)
-4.r2 4.12
-4.r2 4.12
0
0
t20
SMALL SIGNAL MODEL t2r
VslE": ll0
Vt,lEt,.: ll-0.05
LVrlty, : 0.01591 Z-0A351
Zllnel: \+ixt = 0.01+i0.06
Ylinel:+':Gt+illZlinel
Pr,. : 0.75
Qt, : 0.1
V2,/-62,: 1l-0.03
LVrZty, : -0.0295 l-0.1225
Zlne2 : R2+ jX2: 0.01+j0.06P2, : o'45
Q2,. -- -0.572r
scr : 0.4 +j0.3
scz:0.4+j0.3
The main program is shown below
o/o Program IPFC Srnall-Signal
%
clear all
close all
waming off
fnnn¡t lons
SMALL SIGNAL MODEL t22
% Define Value
%
%
% Network Parameters: Voltages' Phasors:
%
% Vs, Vir, V2r, Vlinj, V2inj), Line Impedance's phasors
%
% (zLr,zL2)
%
Vs:1;deltas:0;
IVsx Vsy]:pola12cart(Vs,deltas);
Vsxy:Yt*n¡*Utt'
V lr:1;delta1:-0.05;
[V 1 x V 1 y]=polar2cart(V lr,deltal );
Vlxy=Vix+i*Vly;
V2r:1 ;delta2:-0.03;
[V 2x V2y] 1 olar2 cart(Y 2r delta2) ;
V2xpV2x+i+V2y;
%
%
%
%
IPFC Pararneters:
Vl inj:0.01 591 556733465;
psi 1:-0.435 I 3 10 I 860565;
V2inj=-0.0295 0306447 7 13 ;
psi}: -0. 1225 1 699 121 87 0;
P1r0.75;
Q1t:0.1;
Þ)eO 4\'
Q2r=-0.572r:
Tu:377*0.0106;
Tl2=377a0.0106;
Tl3=377*0.0106;
%
SMALL SIGNAL MODEL t23
Slr:Plrri*Q1r;
S2rP2r+ixQ2r;
SGI=0.4+ix0.3;
sG2:0.4+i*0.3;
%
RLl=0.01;XL1=0.06;
ZL1=RLl+i*XL1;
YLl=llZLr;
G1:real(YLi);
Bl:imag(YLl);
RL2=0.0 I ;XL2:0.06;
ZL2=RL2+1*XL2;
YL2:IIZL2;
G2:real(YL2);
B2:itnag(YL2);
%
o/oLoad
%
Sloadl=S1rrSGl;
Yload I =coni(Sloadl )/V I r"2;
GL1=eal(Yloadl);
BLI=imag(Yloadl );
%
Sload2:S2rrSG2;
Y load2:conj (S load2) I Y 2{2;
GL2=eal(Yload2);
BL2=imag(Yload2);
%
Yioad(1,1)=GLl;
Yload( I ,2)=-BL 1 '
Yload(2,1):BLi;
Yload(2,2):GLl;
%
Yload(3,3)=Ç¡2;
Yload(3,4):-BL2;
Yload(4,3):BL2;
SMALL SIGNAL MODEL r24
Yload( ,4)=GL2;
Yloadful l:full(Yl oad) ;
%
%
%
o/o Generators Pararneters:
%
0/o Generator I tied to line I and AVR I
%
I1=conj(SG1/V I xy);
%
Wol:1;
Hl=6.40*377:
Kdl=0;
Tdo 1 prirne=S. 90* 37'l ;
Kal=25;
Ta1=0.5*377:
Xdl=0.8958;
Xdlprime0. 1198;
Xq1:0.8645;
%
Vtl:Vl xy+i*Xq I *(l l);
deltal r:atan(imag(Vt I )/real(Vt I ));
V I d:V I x{'sin(deltal r)-V 1 yxcos(deltal r);
V I q=y 1**.or(deltaIr)+VI y*sin(deltalr);
%
Gamma 1 =angle(V 1 xy)-angle(I 1 )+delta I r;
%
Ig I q:abs(I 1 )*cos(Garnma 1 );
Ig I d=abs(l 1)*sin(Garnrnal );
%
Ygl a=(( 1/Xd lprirne)-( 1/Xq I ))*cos(deltal r)*sin(deltal r);
Ygl b:( I /Xq 1 )*(cos(deltal r))^2+( I D(dl prime)*(sin(deltal r))"2;
Yg I c:-( 1/Xd lprirne)*(cos(deltal r))^2-( lr(ql )*(sin(deltalr))"2;
SMALL SIGNAL MODEL t25
Ygld:-Ygla;
%
o/o Generator 2 tied to line 2 and AVR 2
%
I2:conj(SG2/V2xy);
%
Wo2=l;
H2:3.0t*377;
Kd2=0;
Tdo2prim*5.89*377;
Ka2=25;
Ta2:0.5*377;
Xd2=1.3125;
Xd2prirne0. 1813;
Xq2:1.2578;
%
Yt2--Y2xy+i*Xq2á'(12);
delta2r:atan(imag(Vt2)/real(Vt2)) ;
V2d=V2x*sin(delt¿r)-V2y*cos(delta2r);
V2q=y2**"or(delta2r)+V2y*sin(delta2r);
%
Gamnr a2:ang I e(V2xy) -an gl e (12)+ ¿s1¡^2r'
Ig2q=abs(I2)*cos(Gamrna2);
I92d=abs(I2) * sin(Garnrna2) ;
%
Y g2æ((UXd2prirne)-( I /Xq2))*cos(delta2r)*sin(delta2r);
Y 92È (l lXq2) x (cos(delta2r))^2+( I D(d2prirne) * (sin(delta2r))"2;
Y g2c:-(1 lXd2prirne)*(cos(d elta2r))"2-(l lXq2)*(sin(delta2r))"2;
Yg?d:-Ys2a;
%
% Admitance Matrix for Generator 1 and 2, the elements of this lnafrices are:
%o Y gl a, Yg 1 b, Yg 1 c, Y gl d, Y g2a, Y gZb, Y g2c, Y g2d
%
YG(1,1):Yela;
YG(1,2):Yglb;
SM.A.LL SIGN.A.L MODEL r26
YG(2,1)=Yglc;
YG(2,2)=Ygld;
%
YG(3,3)=Yg2a;
YG(3,4)=Ye2b;
YG(4,3):Yg2c;
YG(4,4)=Yg2d;
%
YGturi:tu11(YG);
%
SG( l, I ):-lg I q*sin(delta lr)+lg I d*cos(deltal r)-V ly*Ygl a+V I x*Yg I b;
SG( 1,3):sin(deltal r)/Xd iprime;
SG(2, I )=Ig 1 q*cos(deltalr)+Ig I d*sin(deltal r)-V lyxYg I c+V I x*Yg1 d;
SG(2,3):-cos(delta I r)/Xd I prime;
SG(3,5 )=-l 92q* sin(delta2r)+Ig2d*cos(delta2 r) -Y 2y*Y g2a+V2x *Yg2b;
SG(3,7)=sin(delta2r)/Xd2prime;
sG(3,8)=0;
SG(4,5):Ig2q* cos(delta2r)+I 92 d*sin(delta2 r) -Y 2y*Y g2c+Y 2x*Y g\d;
SG(4,7)=-ç651delta2r)/Xd2prime;
sG(4,8)=0,
%
SGturl=tulr(SG);
%
DPl:[-Gl *Vl inj*cos(psi1)-81 *Vlinjxsin(psi1)];
DP2=[-c 1*Vlinj*sin(psil)+B l *Vl inj*cosþsil )];
DP 3 =l-cz*V 2inj * cos(psi2)-82''V2inj *sin(psi2)l ;
DP 4:l-G2*V 2inj *si n(psi2)+B2*V2inj *cos(psi2)l;
DP5=Gl x[Vsx*cos(psi 1)+2*Vlinj*(cos(psi 1))"2-Vlx*cos(psil)+2*Vlinj*(sin(psil))^2-...
VI y*sin(psi 1)l+B I xIVsx*sin(psi I)-VIx*sin(psi I)+V1y*cos(psi I)];
DP6=c1*[-Vlinjr'Vsx*sin(psi I )+Vl inj*V I x*sin(psil)-VlinjxVly*cos(psi 1)]+B 1*þVlinj*...
V1y*sin(psil)+Vsx*Vl inj*cos(psi 1)-Vl x*V 1inj*cos(psi l )l;
DP7=G2+[Vsx*cos(psi2)+2*V2inj*(cos(psi2))^2-Y2x*cosþsi2)+2xV2injx(sin(psi2))^2-...
V2y*sin(psì2)l+B2xIVsx*sin(psi2)-V2x*sin(psi2)+V2y*cos(psi2)] ;
DP8=G2*[-V2inj*Vsxxsin(psi2)+V2inj*V2x*sinþsi2)-V2inj*V2y*cos(psi2)]+82*[-V2inj*...
V2y*sin(psi2)+Vsx*V2inj *cosþsi2)-V2x*V2inj *cosþsi2)l;
%
SMALL SIGNAL MODEL r27
Z I =G2 xcos(psi2)-B28sin(psi2);
22:82¿' c o s(p si 2 )+ G2't' t; n lOttr r'
%
W I :-G2*V2inj xsin(psi2)-82*V2inj *cos(psi2);
W 2=-82*Y 21nj*sin(psi2)+G2*V2inj *cos(psi2);
o/o
o% Admitance nratrices
%
Y(1,1):-G1;
Y(1,2):Bl;
Y(2,1)=-81;
Y(2,2):-Gr;
%
Y(3,1)=-DPl *W1/DP8;
Y(3,2):-DP2*W1/DP8;
Y(4, r ):-DP 1 *W2/DP8;
Y(4,2)=-9P2*W2¡DP8;
%
Y(3,3)=-62-PP3*W i/DP8;
Y(3,4)=82-DP4"W I /DP8 ;
Y(4,3):-B2-DP3 *W2/DP8 ;
v (4,4):-G2-DP4*W2IDP8;
%
Ytull:tulr(Y);
%
T( 1, I )=6 1 x.osþsi I )-B I *sin(psi I );
T(1,2):-Gl *V1inj*sin(psi1)-81 *Vlinjxcos(psil);
T(1,3):0;
T(2,1 )=91 x.os(psil)+G1*sin(psil)'
T(2,2)=-B 1*V I inj*sin(psi l)+G1*V I inj*cos(psil);
r(2,3)=0;
T(3, 1 )=(-DPs*W l/DP8);
T(3,2)=(-DP6*W 1 /DP8);
T(3,3)=Z l+(-DP7*W 1/DP8);
T(4, I ):(-DP5*W2/DP8);
T(4,2)=(DP6*W2/DP8);
SMALL SIGNAL MODEL t28
T (4,3):22+ (-DP 7 *W2lDP 8) ;
%
Yrotal:Yfu li-YGfull-Yloadfull ;
Yinl-inv(Ytotal);
M=-Yinv*SGfull;
N=Yinv*T;
%
%Differential Equations the Cenerators
A(1,1):0;
A(1,2):Wo1;
A(i,3):0;
A(1,4):0;
A(2,1)=(11(2*Hl)*(Vl d/Xdlprirne-lglq)*(Vly*cos(deltalr)-V1x*sin(deltalr))-(Vl q/Xq1+Igl d)*...
(V I x*cos(deltalr)+V 1 yxsin(delta I r)));
A(2,2):-K|U(2*H1);
A(2,3):-V 1 d/(2*H I *Xd1 prirne);
A(2,4)=0;
A(3,1)=( l/Tdolprirne)*(Xd1/Xdlprirne-l)*(V1y*cos(deltalr)-V1x*sin(deltalr));
A(3,2):0;
A(3,3):-Xd I /(Tdo 1 prime+Xd 1 prirne);
A(3,4)=1/Tdolprirne;
A(4,1):-(Ka1/(Tai "V1r))*(Vl q*(V1y+cos(deltalr)-Vlx{'sin(deltalr))+Vld*(V 1x*cos(deltalr)+...
V 1y*sin(deltal r)));
A$,2):0;
A(4,3):0;
A(4,4)=-ttTat'
%
A(5,5)=0;
A(5,6):Wo2;
A(5,7)=o;
A(5,8):0;
SMALL SIGNAL MODEL r29
A(6,5):(l(2*H2)*((V2d/Xd2prime-Ig2q)*(V2y*cos(delta2r)-V2xxsin(delta2r))-(Y2qlXq2+lgZdy6...
(V2x*cos(delta2r)+V2y*sin(delta2r)));
A(6,6)=-Kd2t(2*H2);
A(6,7 )= -v zdl (2 +H2 *Xd2prime) ;
A(6,8):0;
A(7,5):( 1 /Tdo2prime)*(Xd2lXd2prime- 1 )*(V2y*cos(delta2r)-V2x*sin(delta2r));
A(7,6)=0;
A(7,7): -Xd2 I (Tdo2prirne*Xd2prime) ;
A(7,8):i/Tdo2prime;
A(8,5)=-(Ka2l(Ta2*V2r))*p2q*(V2y*cos(delta2r)-V2x*sin(delta2r))+V2d*(V2x*cos(delta2r)+...
V2y*sin(delta2r))l;
A(8,6):o;
A(8,7):0;
A(8,8)=-1lTa2;
%
Afull=full(A);
%
E( 1,1):0;
E(i,2):0;
E(2,1):(1(2*H1)*(V 1d/Xdlprime-Iglq)*cos(deltalr)-(V1q/Xq1+Ig1d)+sin(deltalr));
E(2,2)=(ll(2*Hl)+(V td/Xdlprime-lg1q)+sin(delta 1r)+(V 1q/Xql+lgld)*cos(deltalr));
E(3, I ):(1(Tdo 1 prirne)){'(Xd1/Xd lprime- 1)*cos(deltalr);
E(3,2)=(1(Tdolprirne))*(Xd1/Xd lprime- I )*sin(deltalr);
E(a, I )=-[Kal /(Ta1 *V 1r)]*(Vl q*cos(deltalr)+Vld*sin(deltalr));
E@,2)=-lKal/(Tal *V I r)l*(V I q*sin(delta lr)-V I d*cos(deltair));
%
E(5,3)=0;
E(5,4):0;
E(6,3)=(1(2*H2))'+((V2dlxd2prirne-Ig2q)*cos(delta2r)-(V2qlXq2+lg2d)*sin(delta2r));
E(6,4):(11(z*HZ))*((VZd/Xd2prirne-Ig2q)*sin(delta2r)+(V2qlXq2+1g2d)*cos(delta2r));
E(7,3)=( I(Tdo2prirne))*(Xd2lXd2prirne- I )*cos(delta2r);
E(7,4):( i (Tdo2prime)) *(Xd 2lXd2prime' 1)*sin(delta2r) ;
E(8,3):-[Ka2l(Ta2*V2r)] *(V2q*cos(delta2r)+V2d*sin(delta2r));
E(8,4): -lKa2l(Ta2*V2r)l *(V2q* sin(delta2r)-V2d*cos(delta2r));
%
Etul=tul1(E);
SMALL SIGNAL MODEL 130
%
B(r,l):0;
B(1,2)=0;
B(2,1):1/(2*H I );
B(2,2)=o;
B(3,1)=0;
B(3,2):0;
B(4,1)=0;
B(4,2)=çu1¡^¡u1'
%
B(s,3)=0;
B(s,a)=0;
B(6,3):U(2*H2);
B(6,4):0;
B(7,3):o;
B(7,4)=0;
B(8,3)=0;
B(8,4)--Ka2lTa2'
%
Bturl=tull(B);
EM:EfuII*M;
EN:Etu11*N;
Xcen:Afull+EM;
XIPFC:EN;
%
%
% Differential Equations for the IPFC
%
F(l,t¡=-1¡11'
F(2,2)=-VTr2;
F(3,3)=-1lTI3;
Fturl=tul1(F);
l3iSMALL SIGNAL MODEL
Xttotal:IXGen,-XIPFC;zeros(3,8),Ffu1i] ;
ei gA:ei g(Xtto tal)* 37 7
figure
plot(real(eigA),imag(eigA),'.');
end
Other Functions employed
function Ix,y]lolar2cart(Mag,Angle);
o/o Transfonnation from Polar to cartesian coordinates, this retums a vector in cartesian fonn
%
x=Mag*cos(Angle);
5Mag*sin(Angle);
%
return