19

CHAPTER 2

MODELING OF FACTS DEVICES FOR POWER SYSTEM

STEADY STATE OPERATIONS

2.1 INTRODUCTION

The electricity supply industry is undergoing a profound

transformation worldwide. Market forces, scarcer natural resources and an

ever-increasing demand for electricity are some of the drivers responsible for

such an unprecedented change. Against this background of rapid evolution,

the expansion programmes of many utilities are being thwarted by a variety of

well-founded, environmental, land use and regulatory pressures that prevent

the licensing and building of new transmission lines and electricity generating

units. An in-depth analysis of the options available for maximizing existing

transmission assets, with high levels of reliability and stability, has pointed in

the direction of power electronics. There is general agreement that novel

power electronics equipment and techniques are potential substitutes for

conventional solutions, which are normally based on electromechanical

technologies that have slow response times and high maintenance costs. The

power electronics options we have, are HVDC transmission systems and

FACTS (Acha et al 2004).

Until recently, active and reactive power control in AC

transmission networks was exercised by carefully adjusting transmission line

impedances, as well as regulating terminal voltages by generator excitation

control and by transformer tap changes. At times, series and shunt

20

impedances were employed to effectively change line impedances. FACTS

technology is most interesting for transmission planners because it opens up

new opportunities for controlling power and enhancing the usable capacity of

present, as well as new and upgraded, lines. The possibility that current

through a line can be controlled at a reasonable cost enables a large potential

of increasing the capacity of existing lines with large conductors and use of

one of the FACTS controllers, to enable corresponding power to flow through

such lines under normal and contingency conditions. By providing added

flexibility, FACTS controllers can enable a line to carry power closer to its

thermal rating. It must be emphasized that FACTS is an enabling technology,

not a one to one substitute for mechanical switches. In its most general

expression, the FACTS concept is based on the substantial incorporation of

power electronic devices and methods into the high-voltage side of the

network, to make it electronically controllable.

Comprehensive modeling of most popular FACTS devices namely

SVC, TCSC, STATCOM and UPFC for power flow studies are carried out.

The effectiveness of modeling and convergence is tested on a five bus system

without any FACTS devices and further analyzed it with different FACTS

devices. The Newton-Raphson method is used to solve the nonlinear power

flow equation. The work is also extended for IEEE 30 bus system.

2.2 POWER FLOW IN A MESHED SYSTEM

To understand the free flow of power, consider a very simplified

case in which generators at two different sites are sending power to a load

centre through a network consisting of three lines in a meshed connection

(Figure 2.1).

21

Figure 2.1 Power flow in a mesh network: (a) system diagram

(b) system diagram with Thyristor-controlled series

capacitor in line AC (c) system diagram with Thyristor-

controlled series reactor in line BC

Let the lines AB, BC and AC have continuous ratings of 1000 MW,

1250 MW and 2000 MW. If one of these generators is generating 2000 MW

and the other generates 1000 MW, then a total of 3000 MW would be

delivered to the load centre. For the impedances shown, the three lines would

carry 600, 1600 and 1400 MW respectively, as shown in Figure 2.1(a). Such

a situation would overload line BC and therefore generation would have to be

decreased at B and increased at A, in order to meet the load without

overloading line BC. Power, in short, flows in accordance with transmission

line impedances (which are 90% inductive). If however a capacitor whose

reactance is -j5 at the synchronous frequency is inserted in one line (Figure

2.1(b)), it reduces the line impedance from 10 to 5 , so that power flow

22

through the lines AB, BC and AC will be 250, 1250 and 1750 MW

respectively (Hingorani and Gyugyi 2000).

A series capacitor in a line may lead to sub-synchronous resonance

(typically at 10 to 50 Hz for a 60 Hz system). If such resonance persists it

will soon damage the turbine and shaft. If all or part of the series capacitor is

thyristor controlled it can be modulated to rapidly damp, any sub synchronous

resonance condition and also low frequency oscillations in power flow. This

enhances easy operation within steady state conditions thus improving

stability of the network. Similar results can be obtained by increasing the

impedance of one of the lines in the same meshed configuration by inserting a

+j7 reactor (inductor) in series with line BC (Figure 2.1(c)).

2.3 BENEFITS OF FACTS DEVICES

Edris et al (1997) proposed terms and definitions for Flexible AC

Transmission System (FACTS) as: Alternating-current transmission systems

incorporating power electronics-based and other static controllers to enhance

controllability and increase power transfer capability.

The main objective of FACTS devices is to replace the existing

slow acting mechanical controls required to react to the changing system

conditions by rather fast acting electronic controls. These devices can

dynamically control line impedance, line voltage, active power flow and

reactive power and when storage becomes economically viable; they can

supply and absorb active power as well. All this can be done in high speed.

The implementation of FACTS devices requires technology for high power

electronics with its real time operating control. With the use of fast acting

controls, the power system security margins could be enhanced by utilizing

the full capacity of transmission lines maintaining the quality and reliability

of power supply. Generation patterns that lead to heavy line flows result in

23

higher losses leading to reduced security and stability margin are

economically undesirable. Further, transmission constraints make certain

combinations of generation and demand unviable, due to the potential line

outages. In such situations, FACTS devices may be used to improve system

performance by controlling the power flows in transmission lines. By using

reliable, high speed power electronic controllers, technology offers utilities

the following opportunities for increased efficiency:

Greater control of power, so that it flows on the prescribed

transmission routes.

Loading of transmission line to levels nearer their thermal

limits.

Greater ability to transfer power between controlled areas so as

to have reduced generation reserve margin.

Prevention of cascading outages by limiting the effects of faults

and equipment failure.

Damping of power system oscillations, which could damage

equipment failure.

Improvement in steady state stability limit and transient stability

margin.

2.4 PRINCIPLE OF OPERATION

This section describes briefly the principle of operation and control

of SVC, TCSC, STATCOM and UPFC (Mohan Mathur and Rajiv Varma

2002).

24

2.4.1 Static Var Compensator

The Static Var Compensators are the most widely installed FACTS

equipment at this point in time. They mimic the working principles of a

variable shunt susceptance and use fast thyristor controllers with settling

times of only a few fundamental frequency periods. From the operational

point of view, the SVC adjusts its value automatically in response to changes

in the operating conditions of the network. By suitable control of its

equivalent susceptance, it is possible to regulate the voltage magnitude at the

SVC point of connection, thus enhancing significantly the performance of the

power system.

The majority of Static Var Compensators have similar controllable

elements. The most common ones are,

Thyristor-Controlled Reactor (TCR)

Thyristor-Switched Capacitor (TSC)

Thyristor-Switched Reactor (TSR)

In the case of the TCR a fixed reactor, typically is connected in

series with a bidirectional thyristor valve. The fundamental frequency current

is varied by phase control of the thyristor valve. A TSC comprises of a

capacitor in series with a bidirectional thyristor valve and a damping reactor.

The function of the thyristor switch is to connect or disconnect the capacitor

for an integral number of half-cycles of the applied voltage. A practical

configuration of a TCR-TSC SVC system is shown in Figure 2.2. The

equivalent susceptance Beq is determined by the firing angle of the thyristor.

The variation of Beq as a function of is as shown in Figure 2.3.

25

Controller

V HV bus

PT

Filter

Figure 2.2 A practical example of TCR-TSC SVC

Bmaxunavailable

unavailableBmin

res 90o0o

Beq

Figure 2.3 Variation of Beq as a function of for SVC

2.4.1.1 V-I Characteristics of the SVC

The steady-state and dynamic characteristics of SVC describe the

variation of SVC bus voltage with SVC current (Figure 2.4).

26

0 ILrICrInductiveCapacitive

Over current limit

Over load range

Dynamic Characteristics

Steady-State Characteristics

Linear Range of Control

Isvc

Vsvc

Bmax

Bmin

Vref

V2

V1

Figure 2.4 V-I characteristics of SVC

Dynamic Characteristics

Reference voltage, Vref : This is the voltage at the terminals of the

SVC during the floating condition, that is, when the SVC is neither absorbing

nor generating any reactive power.

Linear range of SVC control: This is the control range over which

SVC terminal voltage varies linearly with the SVC current or reactive power

as it is varied over its entire capacitive to inductive range.

Slope or current droop: The slope or droop of the V-I

characteristics is defined as the ratio of voltage-magnitude change to current-

magnitude change over the linear-controlled range of the compensator.

Overload Range: When the SVC traverses outside the linear-

controllable range on the inductive side, the SVC enters the overload zone,

where it behaves like a fixed inductor.

27

Over current Limit: To prevent the thyristor valves from being

subjected to excessive thermal stress, the maximum inductive current in the

overload range is constrained to a constant value by an additional control

action.

Steady-State Characteristics

The steady-state characteristic of the SVC has a dead band. In the

absence of this dead band, in the steady state the SVC will tend to drift toward

its reactive power limits to provide voltage regulation. It is not desirable to

leave the SVC with very little reactive-power margin for future voltage

control or stabilization excursions in the event of system disturbance.

2.4.2 Thyristor Controlled Series Capacitor

TCSC is a capacitive reactance compensator, which consists of a

series capacitor bank shunted by a thyristor-controlled reactor in order to

provide a smoothly variable series capacitive reactance. The basic TCSC

module comprises a series capacitor C, in parallel with a thyristor-controlled

reactor Ls as shown in Figure 2.5. However, a practical TCSC module also

includes protective equipment normally installed with series capacitors. An

actual TCSC system usually comprises a cascaded combination of many such

TCSC modules, together with a fixed-series capacitor CF. This fixed-series

capacitor is provided primarily to minimize costs.

Figure 2.5 A basic TCSC module

28

2.4.2.1 Basic Operation of TCSC

A TCSC is series-controlled capacitive reactance that can provide

continuous control of power on the ac line over a wide range. From the

system viewpoint, the principle of variable-series compensation is simply to

increase the fundamental frequency voltage across a fixed capacitor (FC) in a

series-compensated line through appropriate variation of the firing angle .

A simple understanding of TCSC functioning can be obtained by

analyzing the behavior of a variable inductor connected in parallel with a FC

(Figure 2.6). The impedance of FC is given by -j(1/ C).

Figure 2.6 A variable inductor connected in shunt with an FC

If C - (1/ L) 0 or in other words, L (1/ C), the reactance of

the FC is less than that of the parallel-connected variable reactor and that

this combination provides a variable-capacitive reactance are both implied.

If C - (1/ L) = 0, a resonance develops that results in an infinite-capacitive

impedance – an obviously unacceptable condition.

If, however, C - (1/ L) 0, the LC combination provides

inductance above the value of the fixed inductor. In the variable-capacitance

mode of the TCSC, as the inductive reactance of the variable inductor is

increased, the equivalent capacitance reactance is gradually decreased.

29

2.4.3 Static Synchronous Compensator

A static synchronous generator operated as a shunt-connected Static

Var Compensator whose capacitive or inductive output current can be

controlled independent of the ac system voltage.

Figure 2.7 The STATCOM operating principle diagram (a) power

circuit (b) equivalent circuit and (c) power exchange

A STATCOM is a controlled reactive-power source. It provides

the desired reactive-power generation and absorption entirely by means of

electronic processing of voltage and current waveforms in a voltage-source

converter (VSC). A single-line STATCOM is shown in Figure 2.7(a), where

a VSC is connected to a utility bus through magnetic coupling. In the Figure

2.7(b) STATCOM is seen as an adjustable voltage source behind a reactance

meaning that capacitor banks and shunt reactors are not needed for reactive

power generation and absorption, thereby giving STATCOM a compact

design.

30

Figure 2.8 The power exchange between the STATCOM and the AC

system

The exchange of reactive power between the converter and the ac

system can be controlled by varying the amplitude of the 3-phase output

voltage, Es of the converter, as illustrated in Figure 2.7(c). If the amplitude of

the output voltage is increased above that of the utility bus voltage, Et, then a

current flows through the reactance from the converter to the ac system and

the converter generates capacitive-reactive power for the ac system. If the

amplitude of the output voltage is decreased below the utility bus voltage,

then the current flows from the ac system to the converter and the converter

absorbs inductive-reactive power from the ac system. If the output voltage

equals the ac system voltage, the reactive-power exchange is zero, in which

case the STATCOM is said to be in floating state.

The reactive and the real-power exchange between the STATCOM

and the ac system can be controlled independently of each other. Any

combination of real power generation or absorption is achievable if the

31

STATCOM is equipped with an energy-storage device of suitable capacity as

depicted in Figure 2.8. With this capability, extremely effective control

strategies for the modulation of reactive and real-power can be achieved.

Furthermore, a STATCOM does the following:

1. It occupies a small footprint, for it replaces passive banks of

circuit elements by compact electronic converters;

2. It offers modular, factory-built equipment, thereby reducing

site work and commissioning time;

3. It uses encapsulated electronic converters, thereby minimizing

its environmental impact.

2.4.4 Unified Power Flow Controller

A combination of a static synchronous compensator (STATCOM)

and a static synchronous series compensator (SSSC) which are coupled via a

common dc link, to allow bi-directional flow of real power between the series

output terminals of the SSSC and the shunt operated terminals of STATCOM,

and are controlled to provide concurrent real and reactive series line

compensation without an external electric energy source. The UPFC, by

means of angularly unconstrained series voltage injection, is able to control,

concurrently or selectively, the transmission line voltage, impedance and

angle or alternatively, the real and reactive power flow in the line. The UPFC

may also provide independently controllable shunt-reactive compensation.

The UPFC which is one of the most promising device in the FACTS concept,

has been researched and put into practical use (Schauder 1998).

32

Figure 2.9 The implementation of UPFC with two back-to-back VSCs

with a common dc-terminal capacitor

The UPFC consists of two voltage source converters. The dc

voltage for both converters is provided by a common capacitor bank (dc link)

(Figure 2.9). Converter 2 provides the main function of the UPFC by

injecting an ac voltage with controllable magnitude Vpq, in series with the

transmission line via a series transformer which can be varied from 0 to

Vpqmax and phase angle of Vpq can be independently varied from 0o to 360o.

The basic function of converter 1 is to supply or absorb the real power

demand of converter 2 which it derives from the transmission line itself.

Although the reactive power is internally generated / absorbed by the series

converter, the real power generation /absorption is made feasible by the dc-

energy-storage device i.e., the capacitor. It can also generate or absorb

controllable reactive power and provide independent shunt reactive

compensation for the line. Converter 2 supplies or absorbs locally the required

reactive power and exchanges the active power as a result of the series

injection voltage. Thus the net real power drawn from the ac system is equal

to the loss of the two converters and their coupling transformers. In addition

33

the shunt converter functions like a STATCOM and independently regulates

the terminal voltage of the interconnected bus.

2.5 POWER FLOW STUDIES

Planning the operation of power systems under existing conditions,

its improvement and also its future expansion require the load flow studies,

short circuit studies and stability studies. The satisfactory operation of the

system depends upon knowing the effects of interconnections, new loads, new

generating stations and new transmission lines before they are installed. They

also help to determine the best size and favourable locations for the power

capacitors both for the improvement of the power factor and also raising the

bus voltages of the electrical network. They help us to determine the best

locality as well as optimal capacity of the proposed generating stations, sub

stations or new lines.

Through the load flow studies we can obtain the voltage

magnitudes and angles at each bus in the steady state. This is rather

important, as the magnitudes of the bus voltages are required to be held within

a specified limit. Once the bus voltage magnitudes and their angles are

computed using the load flow, the real and reactive power flow through each

line can be computed. Also based on the difference between power flow in

the sending and receiving ends, the losses in a particular line can also be

computed.

2.5.1 The Newton Raphson Algorithm

A popular approach to assess the steady state operation of a power

system is to write equations stipulating that at a given bus, the generation and

powers exchanged through the transmission elements connecting to the bus

must add up to zero. This applies to both active and reactive power. These

34

equations are termed ‘mismatches power equations’ and at bus k, they take

the following form:

Pk = PGk – PLk – Pkcal = Pk

sch - Pkcal = 0 (2.1)

Qk = QGk – QLk – Qkcal = Qk

sch – Qkcal = 0 (2.2)

P, Q - mismatch active and reactive powers

PG , QG - active and reactive powers injected by the generator

PL , QL - active and reactive powers drawn by the load at the bus

Under normal circumstances the customer has control over these

variables and in the power flow formulation they are assumed to be known

variables. In principle, at least, the generation and the load at bus may be

measured by the electric utility and in the parlance of power system

engineers, their net values are known as the scheduled active and reactive

powers as given in equations (2.3) and (2.4):

Pksch = PGk - PLk (2.3)

Qksch = QGk - QLk (2.4)

The transmitted active and reactive powers, Pkcal and Qk

cal, are

functions of nodal voltages and network impedances and are computed using

power flow equations. Provided the nodal voltages throughout the power

network are known to a good degree of accuracy then the transmitted powers

are easily and accurately calculated. In this situation, the corresponding

mismatch powers are zero for any practical purposes and the power balance at

each bus is satisfied. However, if the nodal voltages are not known precisely

then the calculated transmitted powers will have only approximated values

and the corresponding mismatch powers are not zero. The power flow

solution takes the approach of successively correcting the calculated nodal

35

voltages and hence, the calculated transmitted powers until values accurate

enough are arrived at, enabling the mismatch powers to be zero or fairly close

to zero. In modern power flow programs, it is normal to all mismatch

equations to satisfy a tolerance as tight as 1e-12 before the iterative solution

can be considered successful.

In order to develop suitable power flow equations, it is necessary to

find relationships between injected bus currents and bus voltages. Based on

the Figure 2.10 the injected complex current at bus k, denoted by Ik , may be

expressed in terms of the complex bus voltages Ek as follows:

Ik = 1*( Ek – Em ) / Zkm = ykm * (Ek – Em ) (2.5a)

Similarly for bus m,

Im = 1*(Em – Ek ) / Zmk = ymk * (Em – Ek ) (2.5b)

Figure 2.10 Equivalent impedance

The above equations can be written in matrix form as,

m

k

mmmk

kmkk

m

k

EE

YYYY

II

(2.6)

The bus admittances and voltages can be expressed in more explicit form:

Yij = Gij + jBij (2.7)

Ei = Vie = Vi * ( cos i + jsin i) (2.8)

36

The complex power injected at bus k consists of an active and

reactive component and is expressed as a function of the nodal voltage and

the injected current at the bus as given in equation (2.9).

Sk = Pk + jQk = EkIk* (2.9)

Sk = Ek ( YkkEk + YkmEm)*

where Ik* is the complex conjugate of the current injected at the bus k.

The expressions for Pkcal and Qk

cal are as follows:

Pkcal = Vk

2Gkk + VkVm[Gkmcos( k m) + Bkmsin( k m)] (2.10)

Qkcal = -Vk

2Bkk + VkVm[Gkmsin( k m) - Bkmcos( k m)] (2.11)

For specified levels of power generation and power load at bus k and

according to equations (2.1) and (2.2), the mismatch equations may be written

down as follows:

Pk=PGk – PLk –{ Vk2Gkk + VkVm[Gkmcos( k m) + Bkmsin( k m)]} = 0 (2.12)

Qk=QGk – QLk –{-Vk2Bkk + VkVm[Gkmsin( k m) - Bkmcos( k m)]} = 0 (2.13)

Similar equations may be obtained for bus m simply by exchanging subscripts

k and m in equations (2.12) and (2.13).

It should be remarked that equations (2.10) and (2.11) represent

only the powers injected at bus k through the ith transmission element, i.e.,

Pki cal and Qk

i cal. However a practical power system will consists of many

buses and transmission elements. This calls for equations (2.10) and (2.11) in

more general terms, with the net power flow injected at bus k expressed as the

summation of the powers flowing at each one of the transmission elements

terminating at this bus.

37

The generic active and reactive powers injected at bus k are as follows:

Pkcal = Pk

i cal (2.14)

Qkcal = Qk

i cal (2.15)

where Pki cal and Qk

i cal are computed using equations (2.10) and (2.11)

respectively.

As an extension, the generic power mismatch equations at bus k are as

follows:

Pk = PGk – PLk – Pki cal = 0 (2.16)

Qk = QGk – QLk – Qki cal = 0 (2.17)

In large-scale power flow studies, the Newton Raphson has proved

most successful owing to its strong convergence characteristics. The powerflow Newton Raphson algorithm is expressed by the following relationship.

QP

= -)//(/)//(/

vvQQvvPP

)/( vv (2.18)

It may be pointed out that the correction terms Vm are divided by

Vm to compensate for the fact that jacobian terms ( Pm Vm)Vm and

Qm Vm)Vm are multiplied by Vm. It is shown in the directive terms that this

artifice yields useful simplifying calculations.

Consider the lst element connected between buses k and m in Figure

2.10, for which self and mutual Jacobian terms are given below:

For k m

lm,

,lkP= )]cos()sin([ mkkmmkkmmk BGVV (2.19)

38

lmlm

lk

VVP

,,

, = )]sin()cos([ mkkmmkkmmk BGVV (2.20)

lm,

,lkQ=

lmlm

lk

VVP

,,

, (2.21)

lmlm

lk

VVQ

,,

, =lm,

,lkP (2.22)

for k = m,

lk,

,lkP= kkk

calk BVQ 2 (2.23)

kkkcal

klklk

lk GVPVV

P 2

,,

, (2.24)

kkkcal

klk

lk GVPQ 2

,

, (2.25)

kkkcalk

lklk

lk BVQVV

Q 2

,,

, (2.26)

The mutual elements remain the same whether we have one

transmission line or n transmission lines terminating at the bus k.

2.5.2 The Sample Five Bus System

First we have considered the five bus system as a case study shown

in Figure 2.11 (Stagg and Abiad 1968). The input data for the considered

system are given in Table 2.1 for the bus and Table 2.2 for transmission line.

39

Figure 2.11 The five-bus network

Table 2.1 Input Bus data (p.u.) for the system under study

BusNo.

Bus code(k-m)

Impedance(R+jX)

Line chargingadmittance

1 1-2 0.02+j0.06 0+j0.06

2 1-3 0.08+j0.24 0+j0.05

3 2-3 0.06+j0.18 0+j0.04

4 2-4 0.06+j0.18 0+j0.04

5 2-5 0.04+j0.12 0+j0.03

6 3-4 0.01+j0.03 0+j0.02

7 4-5 0.08+j0.24 0+j0.05

40

Table 2.2 Input Transmission line data (p.u.) for the system under study

BusNo.

TypeGeneration Load Voltage

P Q P Q |v|

1 slack 0 0 - - 1.06 0

2 P-V 0.4 0.3 0.2 0.1 1 03 P-Q - - 0.45 0.15 1 0

4 P-Q - - 0.4 0.05 1 0

5 P-Q - - 0.6 0.1 1 0

Assuming base quantities of 100 MVA and 100 KV.

The power flow result for the above system without any FACTS

devices is mentioned in Table 2.3. All the nodal voltages are achieved to be

within acceptable voltage magnitude limits.

Table 2.3 Bus voltage of system under study without FACTS devices

Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5|V| (p.u) 1.06 1 0.987 0.984 0.972

(deg) 0 -2.06 -4.64 -4.96 -5.77

2.6 FACTS CONTROLLERS MODELING

This chapter explains the power flow modeling of different FACTS

devices we have adopted in our project. The unified approach that combines

the state variables describing controllable equipment with those describing the

network in a single frame of reference for iterative solutions using the

Newton-Raphson algorithm is followed, which retains its quadratic

convergence characteristics (Acha et al 2004).

41

2.6.1 Power Flow Model of SVC

Two models are presented in this category, namely, the variable

shunt susceptance model and firing-angle model.

2.6.1.1 Variable susceptance model

In practice the SVC can be seen as an adjustable reactance with

either firing angle limits or reactance limits. The equivalent circuit shown

below in Figure 2.12 is used to derive the SVC nonlinear power equations and

the liberalized equations required by the Newton’s method.

Figure 2.12 SVC -Variable shunt susceptance

With reference to Figure 2.12, the current drawn by the SVC is

given by equation (2.27),

ISVC = jBSVCVk (2.27)

The reactive power drawn by the SVC, which is also the reactive power

injected at bus k, is given by equation (2.28),

QSVC = Qk = -Vk2BSVC (2.28)

42

At the end of each iteration, the variable shunt susceptance B isupdated as given in equation (2.29),

( ) ( 1)i iSVC SVCB B

( )i

SVC

SVC

BB

( 1)iSVCB (2.29)

The changing susceptance represents the total SVC susceptance necessary to

maintain the nodal voltages at specified value.

2.6.1.2 Firing angle model

An alternative SVC model, which circumvents the additionaliterative process, consists in handling the TCR firing angle as a state

variable in the power flow formulation. The positive sequence susceptance ofthe SVC, is given by equation (2.30).

2

{k Ck L

C L

V XQ XX X

[2( ) sin(2 )]}SVC SVC (2.30)

From the above equation, the linearised SVC equation is given as follows:

k

k

PQ

(i) = 2

0 020 [cos(2 ) 1]SVC

L

VX

(i) k

SVC

(i) (2.31)

At the end of iteration (i), the variable firing angle SVC is updated as given in

equation (2.32).

SVC (i)

= SVC(i-1) + SVC

(i) (2.32)

2.6.2 Power Flow Model of TCSC

Two alternative power flow models to assess the impact of TCSCequipment in network wide applications are presented in this section. The

simpler TCSC model exploits the concept of a variable series reactance. The

43

series reactance is adjusted automatically, within limits, to satisfy a specifiedamount of active power flows through it. The more advanced model usesdirectly the TCSC reactance-firing angle characteristics, given in the form ofa non-linear relation. The TCSC firing angle is chosen to be the state variable

in the Newton-Raphson power flow solution.

2.6.2.1 Variable series reactance power flow model

The TCSC power flow model presented in this section is based onthe simple concept of a variable series reactance, the value of which is

adjusted automatically to constrain the power flow across the branch to aspecified value. The amount of reactance is determined efficiently usingNewton’s method. The changing reactance XTCSC, shown in Figure 2.13(a)and 2.13(b), represents the equivalent reactance of all the series-connected

modules making up the TCSC, when operating either in the inductive or in thecapacitive regions.

Figure 2.13 TCSC equivalent circuits (a) Inductor (b) Capacitiveoperative regions

The transfer admittance matrix of the variable series compensator isgiven by

m

k

II

=mmmk

kmkk

jBjBjBjB

m

k

VV

(2.33)

44

For inductive operation, we have

kkB = mmB = -TCSCX1 (2.34)

kmB = mkB =TCSCX1 (2.35)

For capacitive operation the signs are reversed.

The active and reactive power equations at bus k are as follows:

)sin( mkkmmkk BVVP (2.36)

kkkk BVQ 2 )cos( mkkmmk BVV (2.37)

For the power equations at bus m, the subscripts k and m are exchanged in

equations (2.36) and (2.37).

The state variable XTCSC of the series controller is updated at the

end of each iterative step as given in equation (2.38).

)1()(

)1()( iTCSC

i

TCSC

TCSCiTCSC

iTCSC X

XXXX (2.38)

2.6.2.2 Firing angle power flow model

The model presented above in section 2.6.2.1 uses the concept of an

equivalent series reactance to represent the TCSC. Once the value of

reactance is determined using Newton’s method then the associated firing

angle TCSC can be calculated. However, such calculations involve an

iterative solution since the TCSC reactance and the firing angle are non-

linearly related. One way to avoid the additional iterative process is to use the

alternative TCSC power flow model presented in this section.

45

Figure 2.14 TCSC compensator firing angle modules

The fundamental frequency equivalent reactance XTCSC of the

TCSC module shown in Figure 2.14 is given by equation (2.39).

)}tan()](tan[){(cos)]}(2sin[)(2{ 221 CCXX CTCSC

(2.39)

where

LCC XXC1 (2.40)

L

LC

XX

C2

24 (2.41)

LC

LCLC XX

XXX (2.42)

2/1

L

C

XX (2.43)

The TCSC active and reactive power equations at bus k are as follows:

)sin( mkkmmkk BVVP (2.44)

)sin(2mkkmmkkkkk BVVBVQ (2.45)

46

where

TCSCkmkk BBB (2.46)

For equations at bus m, exchange subscript k and m in equation (2.44) and

(2.45).

2.6.3 Power Flow Model of STATCOM

The Static Synchronous Compensator is represented by a

synchronous voltage source with minimum and maximum voltage magnitude

limits. The bus at which STATCOM is connected is represented as a PV bus,

which may change to a PQ bus in the events of limits being violated. In such

case, the generated or absorbed reactive power would correspond to the

violated limit. The power flow equations for the STATCOM are derived

below from the first principles and assuming the following voltage source

representation:

Figure 2.15 STATCOM equivalent circuit

Based on the shunt connection shown in Figure 2.15, the following

equation can be written.

(cos sin )vR vR vR vRE V j (2.47)

* * * *( )vR vR vR vR vR vR kS V I V Y V V (2.48)

47

The following are the active and reactive power equations for the

converter at bus k:

2 [ cos( ) sin( )]vR vR vR vR k vR vR k vR vR kP V G V V G B (2.49)

2 [ sin( ) cos( )]vR vR vR vR k vR vR k vR vR kQ V B V V G B (2.50)

2 [ cos( ) sin( )]k k vR k vR vR k vR vR k vRP V G V V G B (2.51)

2 [ sin( ) cos( )]k k vR k vR vR k vR vR k vRQ V B V V G B (2.52)

2.6.4 Power Flow Model of UPFC

The equivalent circuit consists of two coordinated synchronous

voltage sources should represent the UPFC adequately for the purpose of

fundamental frequency steady state analysis. Such an equivalent circuit is

shown in Figure 2.16.

Figure 2.16 UPFC equivalent circuits

48

The UPFC voltage sources are as follows:

(cos sin )vR vR vR vRE V j (2.53)

(cos sin )cR cR cR cRE V j (2.54)

where VvR and vR are the controllable magnitude (VvRmin VvR VvRmax) and

phase angle (0 vR ) of the voltage source representing the shunt

converter. The magnitude VcR and phase angle cR of the voltage source

representing the series converter are controlled between limits (VcRmin VcR

VcRmax) and (0 cR ), respectively.

The phase angle of the series injected voltage determines the mode

of power flow control. If cR is in phase with the nodal voltage angle k, the

UPFC regulates the terminal voltage. If cR is in quadrature with k, it

controls active power flow, acting as a phase shifter. If cR is in quadrature

with line current angle then it controls active power flow, acting as a variable

series compensator. At any other value of cR, the UPFC operates as a

combination of voltage regulator, variable series compensator and phase

shifter. The magnitude of the series injected voltage determines the amount

of power flow to be controlled.

Based on the equivalent circuit shown in Figure 2.16 and equations

(2.53) and (2.54), the active and reactive power equations at bus k are as

follows:

2 [ cos( ) sin( )]

[ cos( ) sin( )]

[ cos( ) sin( )]

k k kk k m km k m km k m

k cR km k cR km k cR

k vR vR k vR vR k vR

P V G V V G B

V V G B

V V G B

(2.55)

49

2 [ sin( ) cos( )]

[ sin( ) cos( )]

[ sin( ) cos( )]

k k kk k m km k m km k m

k cR km k cR km k cR

k vR vR k vR vR k vR

Q V B V V G B

V V G B

V V G B

(2.56)

At bus m

2 [ cos( ) sin( )]

[ cos( ) sin( )]

m m mm m k mk m k mk m k

m cR mm m cR mm m cR

P V G V V G B

V V G B(2.57)

2 [ sin( ) cos( )]

[ sin( ) cos( )]

m m mm m k mk m k mk m k

m cR mm m cR mm m cR

Q V B V V G B

V V G B(2.58)

Series converter

2 [ cos( ) sin( )]

[ cos( ) sin( )]

cR cR mm cR k km cR k km cR k

cR m mm cR m mm cR m

P V G V V G B

V V G B(2.59)

2 [ sin( ) cos( )]

[ sin( ) cos( )]

cR cR mm cR k km cR k km cR k

cR m mm cR m mm cR m

Q V B V V G B

V V G B(2.60)

Shunt converter

2 [ cos( ) sin( )]vR vR vR vR k vR vR k vR vR kP V G V V G B (2.61)

2 [ sin( ) cos( )]vR vR vR vR k vR vR k vR vR kQ V B V V G B (2.62)

Assuming lossless converter values, the active power supplied to

the shunt converter, PvR, equals the active power demanded by the series

converter, PcR; that is,

0vR cRP P (2.63)

50

Further more, if the coupling transformers are assumed to contain

no resistance then the active power at bus k matches the active power at bus

m. Accordingly,

0vR cR k mP P P P (2.64)

The UPFC power equations are combined with those of the AC

network.

2.7 CASE STUDIES WITH FACTS CONTROLLERS

The five bus network is modified to include different FACTS

devices and examine the voltage control capabilities and power flow control

capabilities of the same.

2.7.1 Power Flow Study with SVC

The five bus network is modified to examine the voltage control

capabilities of different SVC models. One SVC is included in the bus 3

(Figure 2.17) to maintain the nodal voltage at 1 p.u.

Figure 2.17 Study system with SVC

51

The SVC inductive and capacitive reactances are taken to be 0.288

p.u. and 1.07 p.u. respectively. The SVC firing angle is set initially at 1400, a

value that lies on the capacitive region. Convergence is achieved in 5

iterations, satisfying a prespecified tolerance of 1e-12 for all variables. The

result for the voltage magnitude and phase angle obtained for both variable

susceptance and firing angle model is shown in Table 2.4. Due to the

inclusion of SVC in the third bus, its voltage is maintained at 1 p.u. The SVC

susceptance values and firing angle values are shown in Table 2.5 for each

step of the iterative process.

Table 2.4 Bus voltage of modified network (with SVC)

Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5SVC(VaryingSusceptance)

|V| (p.u) 1.06 1 1 0.9944 0.9752

(deg) 0 -2.0532 -4.8378 -5.1071 -5.7972

SVC (Varying FiringAngle)

|V| (p.u) 1.06 1 1 0.9944 0.9752

(deg) 0 -2.0533 -4.8379 -5.1073 -5.7975

Table 2.5 SVC state variables

IterationSusceptance model Firing angle model

Bsvc (p.u) Bsvc (p.u) SVC

1 0.02 0.4788 1402 0.1 0.1038 130.23 0.168 0.1096 131.34 0.2048 0.2048 132.55 0.2048 0.2048 132.5

2.7.2 Power Flow Study with TCSC

The original five-bus network is modified to include one TCSC

(Figure 2.18) to compensate the transmission line connected between bus 3

52

and 4. The TCSC should maintain the real power flow in transmission line 6

as 21 MW.

Figure 2.18 Study system with TCSC

The starting value of TCSC is set at 50 percent of the value of the

transmission inductive reactance i.e. 0.015 p.u. for varying reactance model.

Convergence is achieved in 6 iteration to power mismatch tolerance of 1e-12.

The TCSC upholds the target value of 21 MW, which is achieved with 70

percent series capacitive compensation. The initial value of firing angle is set

at 1450. Convergence is obtained in 6 iterations, with final value of at

148.660, with TCSC upholding the target value of 21 MW in line 6. The result

for the voltage magnitude and phase angle obtained for the above system for

both variable reactance and firing angle model is shown in Table 2.6.

53

Table 2.6 Bus voltage of modified network (with TCSC)

Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5TCSC(VaryingReactance)

|V| (p.u) 1.06 1 0.9869 0.9846 0.9719

(deg) 0-

2.0378-

4.7239-

4.8145-

5.7012

TCSC (Varying FiringAngle)

|V| (p.u) 1.06 1 0.987 0.9845 0.9719

(deg) 0-

2.0377-

4.7263-

4.8116-

5.7004

2.7.3 Power Flow Study with STATCOM

The STATCOM is included in bus 3 (Figure 2.19) of the sample

system to maintain the nodal voltage at 1 p.u.

Figure 2.19 Study system with STATCOM

The power flow result indicates that the STATCOM generates 20.5

MVAR in order to keep the voltage magnitude at 1 p.u. at bus3. The source

54

impedance is ZvR = 0.1 p.u., the STATCOM parameters associated with this

amount of reactive power generation are VvR = 1.0205 p.u. and vR = -4.830.

Use of STATCOM results in improved network voltage profile as shown in

Table 2.7.

Table 2.7 Bus voltages of STATCOM upgraded network

Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5

STATCOM|V| (p.u) 1.06 1 1 0.9944 0.9752

(deg) 0 -2.04 -4.7526 -4.821 -5.8259

2.7.4 Power Flow Study with UPFC

The original five-bus network is modified to include one UPFC to

compensate the transmission line linking bus 3 and bus 4 (Figure 2.20).

UPFC should maintain real and reactive power flowing towards bus 4 at 40

MW and 2 MVAR, respectively. The UPFC shunt converter is set to regulate

the nodal voltage magnitude at bus 3 at 1 p.u.

Figure 2.20 Study system with UPFC

55

The starting values of the UPFC voltage sources are taken to be VcR

=0.04 p.u., cR = 87.130 , VvR = 1 p.u. and vR = 00. The source impedances

have values of ZcR = ZvR = 0.1 p.u. . Convergence is obtained in five iterations

to a power mismatch tolerance of 1e-12. The UPFC upheld its target values

and the bus voltage are given in Table 2.8.

Table 2.8 Bus voltages of UPFC upgraded network

Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5

UPFC|V| (p.u) 1.06 1 1 0.9917 0.9745

(deg) 0 -1.7691 -6.016 -3.1905 -4.9737

2.8 RESULTS AND DISCUSSION

The power flow for the five bus system was analyzed without and

with FACTS devices performing the Newton-Raphson Method. The

consolidated comparative bus voltage results have been represented in

Table 2.9. The largest power flow takes place in the transmission line

connecting the two generator buses: 89.3 MW and 74.02 MVAR leave bus 1

and 86.8 MW and 72.9 MVAR arrive at bus2. The operating conditions

demand a large amount of reactive power generation by the generator

connected at bus1 (i.e., 90.82 MVAR). This amount is well in excess of the

reactive power drawn by the system loads (i.e., 40 MVAR). The generator at

bus 2 draws the excess of reactive power in the network (i.e., 61.59 MVAR).

This amount includes the net reactive power produced by several transmission

lines, which is addressed by different FACTS devices. Thus SVC upholds its

target value and as expected identical power flows and bus voltages are

obtained for both Shunt variable susceptance model and Firing angle power

flow models.

56

57

The TCSC Variable series compensator model is used to maintain

active power flowing from the extra fictious bus 6 towards bus 3 at 21 MW.

The starting value of the TCSC is set at 50% of the value of transmission line

inductive reactance. The TCSC upholds the target value of 21 MW, which is

achieved with 70% series compensation of the transmission line 6. In the case

of the firing angle model the initial value of firing angle is set at 145º and the

TCSC upholds the target value of 21 MW.

The power flow result indicates that the STATCOM generates 20.5

MVAR in order to keep the voltage magnitude at 1 p.u at bus 3. Use of

STATCOM results in an improved network voltage profile, except at bus 5,

which is too far away from bus 3 to benefit from the influence of STATCOM.

The original five-bus network is modified to include one UPFC to

compensate the transmission line linking bus 3 and bus 4. The UPFC is used

to maintain active and reactive powers leaving UPFC, towards bus 4, at 40

MW and 2 MVAR, respectively. Moreover the UPFC shunt converter is set

to regulate the nodal voltage magnitude at bus 3 at 1 p.u. There is a 32%

increase of active power flowing towards bus 3. The increase is in response

to the large amount of active power demanded by the UPFC series converter.

Thus from the above analysis we find that within the framework of traditional

power transmission concepts, the UPFC is able to control, simultaneously or

selectively, all the parameters affecting power flow in the transmission line

(voltage, impedance and phase angle) and this unique capability is signified

by the adjective ‘Unified’ in its name.

The power flow study is then extended for IEEE 30 bus system

whose single line circuit diagram is shown in Figure 2.21. Voltage at bus 12

for the base case (without any FACTS devices) is found to be |V| = 0.94773 &

= - 18.080. The SVC is connected at bus 12, which bring the bus voltage to

1 p.u. and convergence is obtained within 4 iterations. The impact of different

58

59

FACTS devices on the neighbouring buses is given in Table 2.10. The power

flow in the line between bus 13-bus 12 is found to be 16.9 MW and 38.4

MVAR. The SVC is able to control the reactive power flow of the line 13-12

to the specified control reference 0.0 p.u., while the active power flow is

almost unchanged. By driving the reactive power flow on the line to zero

using SVC, the un-used (available) transmission line capacity can be

increased. It can be seen that the base case reactive power of line 13-12 is

38.4 MVAR, so the reactive power control by the SVC is significant. Similar

result is also obtained when SVC is replaced by STATCOM at bus 12.

Desired real power flow is also obtained when TCSC is connected in the line

connecting bus 12 and bus 15. The UPFC is installed between bus 12 and the

sending end of the transmission line connecting bus 12 – bus 15. In the

simulation, the bus voltage control reference is |V12 | = 1 p.u. Convergence is

achieved within 6 iterations with specified real power and reactive power in

line connecting bus 12 – bus 15. The detail results show the validity of

different types of model proposed for FACTS devices with the model

proposed by Zhang et al (2006).

26

3029

22 2125

27

24

23

1518

14

13

20

1716

12

10

43

21

7 5

9 11

28

8 6

19

Figure 2.21 IEEE 30 Bus system

60

2.9 SUMMARY

The non-linear power flow equations of the various FACTS

controllers have been linearised and included in Newton Raphson power flow

algorithm. The state variables corresponding to the controllable devices have

been combined simultaneously with the state variables of the network in a

single frame of reference for unified, iterative solutions. The effectiveness of

modeling and convergence is tested and results are analyzed.

The next chapter discusses about dynamic modeling of Unified

Power Flow Controller (UPFC) and proposed suitable controller, for damping

the electromechanical oscillations in a power system.