88481823 improving voltage stability in power systems using modal analysis

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CHAPTER 1

INTRODUCTION

The increasing number of power system blackouts in many countries in recent years, is a

major source of concern. Power engineers are very interested in preventing blackouts and

ensuring that a constant and reliable electricity supply is available to all customers. Incipient

voltage instability, which may result from continues load growth or system contingencies, is

essentially a local phenomenon. However, sequences of events accompanying voltage

instability may have disastrous effects, including a resultant low-voltage profile in a

significant area of the power network, known as the voltage collapse phenomenon. Severe

instances of voltage collapse, including the August 2003 blackout in North - Eastern U.S.A

and Canada, have highlighted the importance of constantly maintaining an acceptable level

of voltage stability. The design and analysis of accurate methods to evaluate the voltage

stability of a power system and predict incipient voltage instability, are therefore of special

interest in the field of power system protection and planning. In planning and operating

power systems, the analysis of voltage stability for a given system state involves the

examination of two aspects:

a) Proximity: how close is the system to voltage instability?

Distance to instability may be measured in terms of physical quantities, such as load level,

active power flow through a critical interface and reactive power reserve.

b) Mechanism: how and when voltage instability occurs, what are the key contributing

factors, what are the voltage-weak points, and what areas arc involved? What measures are

most effective in improving voltage stability?

Proximity gives a measure of voltage security whereas mechanism provides information

useful in determining system modifications or operating strategies which could be used to

prevent voltage instability.

[2]

CHAPTER 2

VOLTAGE STABILITY

The voltage stability of a power system refers to its ability to properly maintain steady,

acceptable voltage levels at all buses in the network at all times, even after being subjected to

a disturbance or contingency. A power system may enter a condition of voltage instability

when the system is subjected to a steady increase in load demand or a change in operating

conditions, or a disturbance (loss of generation in an area, loss of major transformer or major

transmission line). This causes an increased demand in reactive power. Voltage instability is

characterized by gradually decreasing voltage levels at one or more nodes in the power

system. Both static and dynamic approaches are used to analyze the problem of voltage

stability. Dynamic analysis provides the most accurate indication of the time responses of the

system.

Voltage stability is indeed a dynamic phenomenon and can be studied using extended

transient/midterm stability simulations. However, such simulations do not readily provide

sensitivity information or the degree of stability. They are time consuming in terms of CPU

and engineering required for analysis of results. Therefore, the application of dynamic

simulations is limited to investigation of specific voltage collapse situations, including fast or

transient voltage collapse and for coordination of protection and controls. Voltage stability

analysis often requires examination of a wide range of system conditions and a large number

of contingency scenarios. For such applications, the approach based on steady state analysis

is more attractive and if used properly, can provide much insight into the voltage/reactive

power problem.

2.1 Reasons of Voltage Collapse

Voltage collapse is a process in which, the appearance of sequential events together with the

voltage instability in a large area of system can lead to the case of unacceptable low voltage

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condition in the network, if no preventive action is committed. Occurrence of a disturbance

or load increasing can leads to excessive demand of reactive power. Therefore, system will

show voltage instability. If additional resources provide sufficient reactive power support, the

system will be established in a stable voltage level. However, sometimes there are not

sufficient reactive power resources and the excessive demand of reactive power can leads to

voltage collapse.

Voltage collapse can be initiated due to small changes of system conditions (e.g. load

increasing) as well as large disturbances (e.g. line outage or generation unit outage). Under

these conditions, shunt FACTS devices such as SVC and STATCOM can improve the

system security with fast and controlled injection of reactive power to the system. However,

when the voltage collapse is due to excessive load increasing, FACTS devices cannot prevent

the voltage collapse and only postpone it until they reach to their maximum limits. Under

these situations, the only way to prevent the voltage collapse is load curtailment or load

shedding. So, reactive power control using FACTS devices is more effective in large

disturbances and contingencies should be considered in voltage stability analysis.

So the principle causes of voltage instability are:

The load on the transmission lines is too high.

The voltage source is too far from the load centre.

The source voltages are very low.

There is insufficient load reactive compensation.

2.2 Analysis and Methods of Prevention of Voltage Instability

A number of special algorithms have been proposed in the literature for voltage stability

analysis using the static approach. In general, these have not found widespread practical

application and utilities tend to depend largely on conventional power flow programs to

determine voltage collapse levels of various points in a network. However, this approach is

laborious and does not provide sensitivity information useful in making design decisions.

[4]

Some utilities use Q-V curves at a small number of load buses to determine the proximity to

voltage collapse and to establish system design criteria based on Q and V margins

determined from the curves. One problem with the Q-V curve method is that it is generally

not known apriori at which buses the curves should be generated. In producing a Q-V curve,

the system in the neighborhood of the bus is unduly stressed and results may be misleading.

In addition, by focusing on a small number of buses, system-wide problems may not be

readily recognized.

An approach using V-Q sensitivity and piecewise linear power flow analysis to find the

margin, measured in terms of total load growth, between a given operating condition and the

voltage collapse point is already described. There has been some indication that the linear

power flow solution may not be sufficiently accurate as the collapse point is approached.

Also, V-Q sensitivity information could be misleading when applied to a large system having

more than one area with voltage stability problems.

Most of the approaches proposed to date use conventional power flow models to represent

the system steady state. This may not always be appropriate, especially as the system

approaches critical condition. There is a need to consider more detailed steady state models

for key system components such as generators, SVCs, induction motors and voltage

dependent static loads. Load characteristics in particular could be critical and expanded sub-

transmission representation in the voltage collapse areas may be necessary.

There is a need for analytical tools capable of predicting voltage collapse in complex

networks, accurately quantifying stability margins and power transfer limits, identifying

voltage-weak points and areas susceptible to voltage instability, and identifying key

contributing factors and sensitivities that provide insight into system characteristics to assist

in developing remedial actions.

Modal analysis approach with the objective of meeting the above requirements is used

instead of the conventional methods. It involves the computation of a small number of

eigenvalues and the associated eigenvectors of a reduced Jacobian matrix which retains the

[5]

Q-V relationships in the network. However, by using the reduced Jacobian instead of the

system state matrix, the focus is on voltage and reactive power characteristics. The

eigenvalues of the Jacobian identify different modes through which the system could become

voltage unstable. The magnitude of the eigenvalues provides a relative measure of proximity

to instability. The eigenvectors, on the other hand, provide information related to the

mechanism of loss of voltage stability. Fast analytical algorithms for selective computation

of a specified number of the smallest eigenvalues make the approach suitable for the analysis

of large complex power systems.

2.3 Characteristics of Reactive Compensating Devices

There are different types of reactive compensating devices. How these devices influence

voltage stability are described below.

(a)Shunt capacitors

By far the most inexpensive means of providing reactive power and voltage support is the

use of shunt capacitors. They can be effectively used up to a certain point to exceed the

voltage stability limits by correcting the receiving end power factors. They can also be used

to free up “spinning reactive reserve” in generators and thereby help prevent voltage collapse

in many situations.

Shunt capacitors, however, have a number of inherent limitations from the viewpoint of

voltage stability and control:

In heavily shunt capacitor compensated systems, the voltage regulations tend to be

poor.

Beyond a certain level of compensation, stable operation is unattainable with shunt

capacitors.

The reactive power generated by shunt capacitors is proportional to the square of the

voltage; during system conditions of low voltage the var support drops, thus

compounding the problem.

[6]

(b) Regulated shunt compensation

A static var system (SVS) of finite size will regulate up to its maximum capacitive output.

There is no voltage control on instability problems within the regulating range. When pushed

to the limit, an SVS becomes a simple capacitor. The possibility of this leading to voltage

instability must be recognized.

A synchronous condenser, unlike an SVS, has an internal voltage source. It continues to

supply reactive power down to relatively low voltages and contributes to a more stable

voltage performance.

(c) Series capacitors

Series capacitors are self regulating. The reactive power supplied by series capacitors is

proportional to square of the line current and is independent of the bus voltages. This has a

favorable effect on voltage stability.

Series capacitors are ideally suited for effectively shortening long lines. Unlike shunt

capacitors, series capacitors reduce both the characteristics impendence (Zc) and the electrical

length of the line. As a result, both voltage regulation and stability are significantly

improved.

[7]

CHAPTER 3

DEFINING FACTS DEVICES

FACTS, an acronym which stands for Flexible AC Transmission System, is an evolving

technology-based solution envisioned to help the utility industry to deal with changes in the

power delivery business. The term FACTS refers to alternating current transmission systems

incorporating power electronic-based and other static controllers to enhance controllability

and increase power transfer capability. Technology concepts were conceived in the 1980’s

and projects sponsored by the Electric Power Research Institute (EPRI) demonstrated many

of these concepts with laboratory scale circuits.

The concept of Flexible AC Transmission Systems (FACTS) was first defined by Hingorani,

N.G. in 1988. Up to now, lots of advanced FACTS devices have been put forward due to the

rapid development of the modem power electronics technology. These FACTS devices have

a large potential ability to make power systems operate in a more flexible, secure and

economic way. Moreover, these FACTS devices can also make the power systems operate in

a more sophisticated way. A good coordination and adaptation is needed to fully exploit the

new characteristics of FACTS. Presently, the studies on FACTS are mainly focused on

FACTS devices developments and their impacts on the power system, such as power flow

modulation and control, transient stability enhancement, small-disturbance stability

improvement and oscillation damping. It is also significant to study the impact of the FACTS

devices on improving performance of power systems such as optimization related software

algorithms in modem Energy Management System (EMS).

3.1 Facts Controller Applications

The simplest way to identify the potential roles to be played by FACTS Controllers is to

examine their functions as they relate to conventional equipment. The definition of FACTS

systems incorporates both power electronic-based and other static controllers to enhance

[8]

controllability and increase power transfer capability. One of the system planners’ tasks is to

determine which combinations of controllers provide both the capacity to supply the reactive

power, dynamic reserve and continuous regulation needed for the application. Table 1 lists

the main functions that can be performed by FACTS Controllers and show both FACTS and

other conventional equipment that performs these functions.

Table 1- System Control Functions

Function Non FACTS Control Methods FACTS Controllers

Voltage

Control

Electric generators

Synchronous Condensers

Conventional Transformer tap-changer

Conventional Shunt Capacitor/Reactor

Conventional Series Capacitor/Reactor

Static Var Compensator (SVC)

Static Synchronous Compensator

(STATCOM)

Unified Power Flow Controller (UPFC)

Superconducting Energy Storage

(SMES)

Battery Energy Storage System (BESS)

Convertible Static Compensator (CSC)

Active

and

Reactive

Power

Flow

Control

Generator schedules

Transmission line switching

Phase Angle Regulator (PAR)

Series Capacitor (switched or fixed)

High Voltage Direct Current

Transmission (HVDC)

Interphase Power Controller (IPC)

Thyristor controlled Series Capacitor

(TCSC)

Thyristor Controlled Series Reactor

(TCSR)

Thyristor Controlled Phase Shifting

Transformer (TCPST)

UPFC

Static Synchronous Series Compensator

(SSSC)

Interline Power Flow Controller (IPFC)

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Transient

Stability

Braking Resistor

Excitation Enhancement

Special Protection Systems

Independent Pole Tripping

Fast Relay Schemes

Fast Valving

Line Sectioning

HVDC

Thyristor Controlled Braking Resistor

(TCBR)

SVC, STATCOM, TCSC, TCPST,

UPFC

BESS, SMES, SSSC, CSC, IPFC

Dynamic

Stability

Power System Stabilizer

HVDC

TCSC, SVC, STATCOM, UPFC, SSSC,

TCPST, BESS, SMES, SSSC,CSC,

IPFC

Short

Circuit

Current

Limiting

Switched series reactors

Open circuit breaker arrangements

Thyristor switched series reactor, TCSC,

IPC, SSSC, UPFC; These are secondary

functions of these controllers and their

effectiveness may be limited.

3.2 Overview of Facts Controllers

The value of FACTS applications lies in the ability of the transmission system to reliably

transmit more power or to transmit power under more severe contingency conditions with the

control equipment in operation. If the value of the added power transfer over time is

compared to the purchase and operational costs of the control equipment, relatively complex

and expensive applications may be justified. Other economic considerations include the

market structure, transmission tariff and identification of winners and losers. Realization of

the value added by a proposed transmission project often requires a coordinated

implementation of conventional transmission equipment, possibly including transmission line

segments, FACTS Controllers, coordinated control algorithms and special operating

procedures.

Commonly used FACTS controllers are:

[10]

1. Static Var Compensator (SVC)

2. Static Synchronous Compensator (STATCOM)

3. Superconducting Magnetic Energy Storage (SMES)

4. Battery Energy Storage System (BESS)

5. Thyristor Controlled Series Capacitor (TCSC)

6. Static Synchronous Series Compensator (SSSC)

7. Unified Power Flow Controller (UPFC)

8. Interphase Power Controller (IPC)

3.3 Static Var Compensator (SVC)

The Static Var Compensator used for transmission system applications is a dynamic source

of leading or lagging reactive power. It is comprised of a combination of reactive branches

connected in shunt to the transmission network through a step up transformer. The SVC is

configured with the number of branches required to meet a utility specification as indicated

in Figure 3.1. This specification includes required inductive compensation and required

capacitive compensation. The sum of inductive and capacitive compensation is the dynamic

range of the SVC. One or more thyristor-controlled reactors may continuously vary reactive

absorption to regulate voltage at the high voltage bus. This variation is accomplished by

phase control of the thyristors, which results in the reactor current waveform containing

harmonic components that vary with control phase angle. A filter branch containing a power

capacitor and one or more tuning reactors or capacitors is included to absorb enough of the

harmonic currents to meet harmonic specifications and provide capacitive compensation. The

thyristor switched capacitor is switched on or off with precise timing to avoid transient inrush

currents.

[11]

Figure 3.1 Circuit diagram of a SVC containing a thyristor controlled reactor, a thyristor

switched capacitor and a double tuned filter

3.4 Static Synchronous Compensator (STATCOM)

The STATCOM shown in Figure 3.2 performs the same voltage regulation and dynamic

control functions as the SVC. However, its hardware configuration and principle of operation

are different. It uses voltage source converter technology that utilizes power electronic

devices (presently gate turn-off thyristors (GTO), GCTs or insulated gate bi-polar transistors

(IGBT)) that have the capability to interrupt current flow in response to a gating command.

Analogous to an ideal electromagnetic generator, the STATCOM can produce a set of three

alternating, almost sinusoidal voltages at the desired fundamental frequency with controllable

magnitude. The angle of the voltage injected by the STATCOM is constrained to be very

nearly in-phase with the transmission network at the point of connection of the coupling

transformer.

[12]

Figure 3.2: STATCOM circuit diagram

When the voltage is higher in magnitude than the system voltage, reactive current with a

phase angle 90 degrees ahead of the voltage phase angle flows through the coupling

transformer. This is analogous to the operation of a shunt capacitor. When the generated

voltage is lower than system voltage, the current phase angle is 90 degrees behind the voltage

phase angle that is analogous to the operation of a shunt reactor. The slight deviation in

voltage phase angle absorbs power needed for the losses in the circuit. For high power

applications a number of six or twelve pulse converters are operated in parallel to meet both

the current rating requirement and the harmonic requirement of the network. Two different

switching patterns, phase displaced converters with electronic devices switched once per

cycle and pulse width modulation, have been used to form the sinusoidal waveform.

3.5 Thyristor controlled series capacitor (TCSC)

The thyristor controlled series capacitor (TCSC) is placed in series with a transmission line

and is comprised of three parallel branches: a capacitor, a thyristor pair in series with a

reactor (TCR), and a metal oxide varistor (MOV) that is required to protect against

overvoltage conditions (see Figure 3.3). The TCSC can function as a series capacitor if the

thyristors are blocked or as variable impedance when the duty cycle of the thyristors is

[13]

varied. Applications of TCSCs currently in service provide impedance variations to damp

inter-area system oscillations. The most economical installations often contain one segment

of thyristor-controlled capacitors in series with one or more segments of conventionally

switched series capacitors.

Figure 3.3: One Line Diagram of the TCSC

3.6 Static synchronous series compensator (SSSC)

A static synchronous series compensator (SSSC) is connected in series with a transmission

line and is comprised of a voltage source converter operated without an external electric

energy source. (See Figure 3.4) This configuration serves as a series compensator whose

output voltage is in quadrature with, and controlled, independently of the transmission line

current.

Figure 3.4: Circuit diagram for a Static Synchronous Series Compensator

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The purpose of the SSSC is to increase or decrease the overall reactive voltage drop across

the line and thereby control the transmitted real electric power. The SSSC may include

transiently rated energy storage or energy absorbing equipment to enhance the dynamic

behaviour of the power system by additional temporary real power compensation, to increase

or decrease momentarily, the overall real (resistive) voltage drop across the line. This action

controls the reactive power flow on the line.

3.7 Unified Power Flow Controller (UPFC)

The Unified Power Flow Controller (UPFC) provides voltage, and power flow control by

using two high power voltage source converters (VSC) coupled via a dc capacitor link.

Figure 3.5 shows the two interconnected converters. VSC 1 is connected like a STATCOM

and VSC 2 is connected as a SSSC in series with the line. With the dc bus link closed, the

UPFC can simultaneously control both real and reactive power flow in the transmission line

by injecting voltage in any phase angle with respect to the bus voltage with the series

converter. The shunt-connected converter supplies real power required by the series

connected converter. With its remaining capacity the shunt converter can regulate bus

voltage.

The UPFC circuit can be reconfigured by use of external switches and possibly additional

transformers to form STATCOM, SSSC, or coupled SSSC circuits.

Figure 3.5: Circuit Diagram of a Unified Power Flow Controller

[15]

Today, most power systems are operating near their steady-state stability limits, which may

result in voltage instability. Flexible ac transmission system (FACTS) devices are good

choices to improve the stability of power systems. Many studies have been carried out on the

use of FACTS devices for voltage and angle stability problems. Taking advantages of the

FACTS devices depends largely on how these devices are placed in the power system,

namely, on their location and size. In a practical power system, allocation of these devices

depends on a comprehensive analysis of steady-state stability, transient stability, small signal

stability, and voltage stability. Moreover, other practical factors such as cost and installation

conditions also need to be considered. In the literature, a tool has been reported based on the

determination of critical modes, which is known as modal analysis.

Modal analysis has been used to locate static Var compensator (SVC) and other shunt

compensators to avoid voltage instability. The setting of many controllable power system

devices, such as HVDC Links and FACTS devices, are based on the issues unrelated to the

damping of oscillations in the system. For instance, an SVC improves transmission system

voltage, thereby enhancing the maximum power transfer limit; static synchronous series

compensator (SSSC) control reduces the transfer impedance of a long transmission line,

enhancing the maximum power transfer limit. In addition to the primary function, the

supplementary damping control is also added, and how to utilize their control capabilities

effectively as stabilizing aids is becoming very important.

[16]

CHAPTER 4

BUS CLASSIFICATIONS AND EIGEN PROPERTIES OF THE

STATE MATRIX

4.1 Bus Classifications

Each bus is defined by four variables: net active power, net reactive power, voltage

magnitude and voltage phase angle. Since there are only two equations per bus, two out of

the four variables must be specified in each bus in order to have a solvable problem.

Buses are classified according to which two out of the four variables are specified:

4.1.1 Load PQ bus:

No generator is connected to the bus; hence the control variables PG and QG are zero.

Furthermore, the active and reactive powers drawn by the load PL and QL are known from

available measurements. In these types of buses the net active power and net reactive power

are specified, and V and θ are computed.

4.1.2 Generator PV bus:

A generating source is connected to the bus; the nodal voltage magnitude V is maintained at

a constant value by adjusting the field the current of the generator and hence it generates or

absorbs reactive power. Moreover, the generated active power PG is also set at a specified

value. The other two quantities θ and QG are computed. Constant voltage operation is

possible only if the generator reactive power design limits are not violated, that is, QG min <

QG < QG max.

4.1.3 Generator PQ bus:

If the generator cannot provide the necessary reactive power support to constrain the voltage

magnitude at the specified value then the reactive power is fixed at the violated limit and the

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voltage magnitude is freed. In this case, the generated active power PG and reactive power QG

are specified, and the nodal voltage magnitude V and phase angle θ are computed.

4.1.4 Slack (swing) bus:

One of the generator buses is chosen to be the slack bus where the nodal voltage magnitude,

Vslack, and phase angle θslack, are specified. There is only one slack bus in power system and

the function of a slack generator is to produce sufficient power to provide for any unmet

system load and for system losses, which are not known in advance of the power flow

calculation. The voltage phase angle at the slack bus θslack is chosen as reference against

which all other voltage phase angles in the system are measured. It is normal to fix its value

to zero.

Generator buses are also called regulated buses or voltage controlled buses.

4.2 Eigen properties of The State Matrix

4.2.1 Eigenvalues

The eigenvalues of a matrix are given by the values of the scalar parameter λ for which exist

non-trivial solutions (i.e. other than Φ = 0) to the equation

A Φ= λ Φ ….…….. (4.1)

Where

A is n x n matrix (real for a physical system such as power system)

Φ is an n x 1 matrix

To find the eigenvalues above equation may be written in the form

(A- λ I) Φ = 0 ….…….. (4.2)

For a non-trivial solution

det (A- λ I) = 0 ….…….. (4.3)

Expansion of the determinant gives the characteristics equation. The n solutions of λ = λ1, λ2,

…… λn are eigenvalues of A.

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The eigenvalues may be real or complex. If A is real, complex eigenvalues always occurs in

conjugate pairs. Similar matrices have identical eigenvalues. A matrix and its transpose have

the same eigenvalues.

4.2.2 Eigenvectors

For any eigenvalue λi, the n-column vector Φi which satisfies (4.1) is called right eigenvector

of A associated with eigenvalue λi. Therefore, right eigenvector is

AΦi= λiΦi i =1,2,…..n ….…….. (4.4)

The eigenvector Φi has the form

Φi =

1i

2i

ni

Since equation (4.2) is homogenous, kΦi (where k is a scalar) is also a solution. Thus, the

eigenvalues are determined only to within a scalar multiplier.

Similarly, the n-row vector Ψi which satisfies

ΨiA = λiΨi i= 1,2,….n; ….…….. (4.5)

is called the left eigenvector associated with the eigenvalue λi.

The left and right eigenvectors corresponding to different eigenvalues are orthogonal. In

other words, if λi is not equal to λj,

ΨjΦi = 0 ….…….. (4.6)

However, in case of eigenvectors corresponding to the same eigenvalues,

ΨiΦi = Ci ….…….. (4.7)

where Ci is a non-zero constant.

Since, as noted above, the eigenvalues are determined only to within a scalar

multiplier, it is a common practice to normalize these vectors so that

Ψi Φi = 1 ….…….. (4.8)

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4.2.3 Eigenvalue sensitivity

Consider equation (4.4) which defines the eigenvalues and eigenvectors:

AΦi= λiΦi

Differentiating with respect to akj (the element of A in kth

row and jth

column) gives

kj kj kj kj

i ii i

ii

AA

a a a a

Premultiplying by Ψi ,and noting that Ψi Φi = 1 and Ψi (A- λiI) = 0, we see that the above

equation simplifies to

kj kj

i iiA

a a

All elements of kj

Aa

are zero, except for the element in the kth

row and jth

column which

is equal to 1. Hence,

kj

ik jii

a

….…….. (4.9)

Thus the sensitivity of the eigenvalue λi to the element akj of the state matrix is equal to the

product of the left eigenvector element Ψik and the right eigenvector element Φji.

4.2.4 Participation factor

One problem in using right and left eigenvectors individually for identifying the relation

between the states and the modes is that the elements of the eigenvectors are dependent on

units and scaling associated with the the state variables. As a solution to this problem, a

matrix called the participation matrix (P), which combines the right and left eigenvectors as

follows is a measure of the association between the state variables and the modes.

1 2 nP p p p

With

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1i 1 1

2i 2 2

i

ni

p

pp

p

i i

i i

ni in

Where

Φki = the element on the kth

row and ith

column of the modal matrix Φ

= kth

entry of the right eigenvector Φi.

Ψik = the element on the ith

row and kth

column of the modal matrix Ψ

=kth entry of the left eigenvector Ψi.

The element pki = Φik Ψki is termed the Participation factor. It is a measure of the relative

participation of kth

state variable in the ith

mode, and vice versa.

Since Φki measures of the activity of xk in the ith

mode and Ψik weighs the contribution of

this activity to the mode, the product pki measures the net participation. The effect of

multiplying the elements of the left and right eigenvectors is also to make pki dimensionless

(i.e. independent of the choice of the units).

In view of the eigenvector normalization, the sum of the participation factors associated

with any mode n

ki

i=1

p or with any state variable

n

ki

k=1

p is equal to 1.

From equation 9, we see that participation factor pki is actually equal to the sensitivity of the

eigenvalue λi to the diagonal element akk of the state matrix A

iki

kk

p =a

….…….. (4.10)

[21]

CHAPTER 5

MODAL ANALYSIS FOR VOLTAGE STABILITY

EVALUATION

A system is voltage stable at a given operating condition if for every bus in the system, bus

voltage magnitude increases as reactive power injection at the same bus is increased. A

system is voltage unstable if for at least one bus in the system bus voltage magnitude

decreases as the reactive power injection at the same bus is increased. In other words, a

system is voltage stable if V-Q sensitivity is positive for every bus and unstable if V-Q

sensitivity is negative for at least one bus.

Modal analysis is a method for voltage stability evaluation. In this method, stability analysis

is done by computing eigenvalues and right and left eigenvectors of a Jacobian matrix which

obtained from the power flow equations. Assume that a power system is located at a primary

operating point. In this operating point, the relations between main power system quantities

(voltage magnitude, voltage angle, injected active power and injected reactive power) can be

expressed by power flow equations as follows:

5.1 Reduced Jacobian Matrix

The linearized steady state system power voltage equations are given by.

Pθ PV

Qθ QV

P J J θ

Q J J V

…………(5.1)

Where,

∆P = incremental change in bus real power.

∆Q= incremental change in bus reactive power injection.

∆ = incremental change in bus voltage angle.

∆V = incremental change in bus voltage magnitude.

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If the conventional power flow model is used for voltage stability analysis, the Jacobian

matrix in (5.1) is the same as the Jacobian matrix used when the power flow equations are

solved using the Newton-Raphson technique.

System voltage stability is affected by both P and Q. However, at each operating point we

keep P constant and evaluate voltage stability by considering the incremental relationship

between Q and V. This is analogous to the Q-V curve approach. Although incremental

changes in P are neglected in the formulation, the effects of changes in system load or power

transfer levels are taken into account by studying the incremental relationship between Q and

V at different operating conditions.

1QV Qθ Pθ PVQ J -J .J .J . V ..……….(5.2)

Rearrange (5.2), then

1R V=J Q ….……..(5.3)

where,

1R QV Qθ Pθ PVJ J -J .J .J ….……..(5.4)

To reduce (5.1), let ∆P =0, then.

JR is called the reduced Jacobian matrix of the system. JR is the matrix which directly relates

the bus voltage magnitude and bus reactive power injection. Eliminating the real power and

angle part from the system steady state equations allows us to focus on the study of the

reactive demand and supply problem of the system as well as minimize computational effort.

The program developed also provides the option of performing eigen-analysis of the full

Jacobian matrix. If the full Jacobian is used, however, the results represent the relationship

between (∆, ∆V) and (∆P, ∆Q). Since ∆ is included in the formulation, it is difficult to

discern the relationship between ∆V and (∆P, ∆Q) which is of primary importance for

voltage stability analysis. Also modal analysis using the full Jacobian matrix is

computationally more expensive than using the reduced Jacobian. For these reasons, the

reduced Jacobian approach was considered.

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5.2 Modes of Voltage Instability

Let

RJ ξ η ……….(5.5)

where,

ξ = right eigenvector matrix of JR

= left eigenvector matrix of JR

= diagonal eigenvalue matrix of JR

From (5.3) and (5.5)

1V Q ………..(5.6)

And

1v q ………..(5.7)

where

v = ∆V = the vector of modal voltage variations

q= ∆Q = vector of modal reactive power variations

and

1 ………..(5.8)

Equation (5.7) represents uncoupled first order equations. Thus for the ith

mode:

i ii

1v qλ

………..(5.9)

The eigenvalue of the reduced Jacobian matrix identify different modes through which the

voltage of system could become unstable. The magnitude of the eigenvalues provides a

relative measure of the proximity to instability. If λi > 0, the ith

modal voltage and ith

modal

reactive power variation are along with the same direction, indicating that the system is

voltage stable. If λi < 0, the ith

modal voltage and ith

modal reactive power variation are along

with the opposite direction, indicating that the system is voltage unstable. In this sense, the

magnitude of λi determines the degree of stability of the ith

modal voltage. The smaller the

magnitude of positive λi , the closer ith

modal voltage is to being unstable.

[24]

Using modal analysis, the effect or participation of system buses in voltage instability and

critical modes near the point of collapse can be determined. Relative participation of kth

bus

to ith

mode is expressed by bus participation factor as follows:

ki ki ikP = ………..(5.10)

where ξki and ik are kth

element of the right and left eigenvectors corresponding to ith

eigenvalue of JR respectively.

Bus participation factors represent the area corresponding to each mode. The larger the

magnitude of Pki, the kth

bus is more effective corrective controls to improve voltage stability.

[25]

CHAPTER 6

PROPOSED METHOD FOR LOCATING FACTS DEVICES

Voltage collapse normally occurs when sources producing reactive power reach their limits

i.e. generators, SVCs or shunt capacitors, and there is not much reactive power supply to

support the load. Therefore, the reactive reserve margin is used as a voltage stability

indicator.

The most advanced solution to compensate reactive power is the use of a Voltage Source

Converter (VSC) incorporated as a variable source of reactive power. These systems offer

several advantages compared to standard reactive power compensation solutions. Reactive

power control generated by generators or capacitor banks alone normally is too slow for

sudden load changes and demanding applications, such as wind farms or arc furnaces.

Compared to other solutions a voltage source converter is able to provide continuous control,

very dynamic behavior due to fast response times and with single phase control also

compensation of unbalanced loads. The ultimate aim is to stabilize the grid voltage and

minimize any transient disturbances.

Voltage collapse is usually initiated by disturbances in a system vulnerable to voltage

instability. Voltage stability could be recognized by modal analysis of power system steady

state Jacobian matrix under contingency condition. If the smallest eigen values of reduced

Jacobian matrix are negative or very close to zero, the voltage instability is possible. Under

these conditions, it is necessary to increase the magnitude of critical modes until the system

security is ensured and voltage stability is achieved. This can be done via corrective

operations such as providing reactive power support with FACTS devices.

Voltage instability is due to the critical modes of reduced Jacobian matrix. Therefore, in the

given proposed method the objective is to determine system buses that have the most effect

on the critical modes. Critical modes are determined based on modal analysis of system

[26]

reduced Jacobian matrix under contingency conditions and the effectiveness of buses on

these critical modes is recognized by their participation factors.

In the proposed method, for each contingency a probabilistic index is defined which

evaluates the relative participation of each bus in voltage instability caused by all of the

critical eigenvalues corresponding to that contingency:

mij

i outage

jj=1

PPCM = P (k)

λ

………..(6.1)

where

PCMi = contribution of bus i to voltage instability caused by critical modes under kth

contingency state

Poutage = likelihood of kth

contingency occurring corresponding to outage of line k

m = number of critical eigenvalues in kth

contingency

Pij = participation factor of bus i to critical eigenvalue j

λj = critical eigenvalue j

Bus Participation Factors

The relative participation of bus k in mode i is given by the bus participation factor. It

determines the areas associated with each mode. The sum of all bus participations for each

mode is equal to unity because the right and left eigenvectors are normalized. The size of bus

participation in a given mode indicates the effectiveness of remedial actions applied at that

bus in stabilizing the mode.

There are generally two types of modes. The first type has very few buses with large

participations and all the other buses with close to zero participation, indicating that the mode

is much localized. The second type has many buses with small but similar degree of

participations, and the rest of the buses with close to zero participations; this indicates that

the mode is not localized. A typical localized mode occurs if a single load bus is connected to

a very strong network through a long transmission line. A typical non-localized mode occurs

when a region within a large system is loaded up and the main reactive support for this

region is exhausted.

[27]

The convention taken is that the term critical mode is used to identify all eigenvalue whose

magnitudes are smaller than a prescribed critical value (λcritical). The critical value is

determined based on the bus voltage magnitude profile in the system.

The probabilistic index defined above represents the relative contribution of each bus to

critical modes of kth contingency condition. Then, the total participation in all critical modes

(TPCM) for each bus was calculated considering all possible contingencies by following

equation:

L m

iji outage

jk=1 j=1

PTPCM = . P (k)

λ

………..(6.2)

where

TPCMi = the total participation of bus i in all critical modes under all possible contingencies

and L is the number of possible contingencies

For calculation of TPCM the outage of all lines is considered. If system has critical modes in

normal state (i.e. without any outage) due to special operating conditions, then this conditions

could be included in above relation with consideration of corresponding probability.

TPCM demonstrates the relative contribution of each bus to system voltage instability under

all possible system states. According to the relation above, the larger the magnitude of bus

participation factor in critical modes, that bus is more effective in voltage instability. On the

other hand, the smaller the magnitude of positive λj, that mode is more critical. In addition,

bus contributions to voltage instability under contingencies are weighted by the likelihood of

contingencies occurring. Consequently, contingencies with higher probability will be more

important in locating FACTS devices.

TPCM values are calculated for every bus using above equation. Buses are then ranked by

their corresponding TPCM values. In general, the larger value a bus has the more effective it

will be. The bus with the largest TPCM is considered as the best location for one shunt

FACTS device, because according to definition of TPCM, that bus is more effective in more

[28]

probable contingencies (i.e. larger Poutage (k)) or is more effective in more critical modes (i.e.

smaller λj).

For a large- scale power system, more than one FACTS device may have to be installed in

order to achieve the desired performance. However, budgetary constraints force the utilities

to limit the number of FACTS devices to be placed in a given system. Given such a limit on

the total number of FACTS devices to be installed in a power system, the location of the next

controllers can be determined according to the ranking of buses in an iterative approach. At

each step, one FACTS device is installed at the bus with the largest TPCM value. Installation

of a controller in the determined location mitigates the critical modes caused by that bus and

other buses close to it. Therefore, the ranking of buses after the next iteration does not

necessarily match the previous one. The flowchart shown below demonstrates the proposed

strategy of FACTS devices locating.

[29]

6.1. Flowchart of Proposed Method

System data

σj < σcritical

K=1

Outage of line

k

Load flow

Modal analysis and determination of eigenvalues

Calculation of bus participation factor (Pij) for all critical modes

Ranking of buses based on associated TPCM values

Installation of FACTS at the top bus

End

All contingencies

are considered?

Need to install another

FACTS device?

TPCMi = ∑ i

k=k +1

No

No

Yes

No

[30]

CHAPTER 7

CASE STUDY

In this report standard IEEE 14 bus is considered. Given flowchart shown in figure shows the

proposed approach of placing FACTS in a power system.

Figure-7.1 : IEEE 14 bus System

[31]

We will analyze two cases:

Case 1: when load and generation of the system is scaled by the factor of 0.95.

Sub case 1: when voltage magnitude is maintained at 1.05pu





Performing load flow for the normal state of the system, the smallest eigenvalue of the

reduced Jacobian matrix is determined as λmin= 2.71. With the assumption of λcritical=1, the

calculated eigenvalue is not critical. Then, contingency analyses corresponding to line

outages are performed. Here it has been assumed that the failure probability of all lines is

assumed to be 0.02. The eigenvalue of reduced Jacobian matrix is calculated in each step. In

the table 2 three smallest eigenvalues of each state are shown and the corresponding critical

eigenvalues are specified by coloured cell from the table. It is clear that critical eigenvalues

exist only in two contingencies corresponding to the outage of line 1 and line 10. Using

modal analysis, bus participation factors associated with the critical eigenvalues are

calculated. The TPCM value of buses shown in table 3 is calculated using the formula given

above. From table 3 we can infer that bus 12 has the largest TPCM. So it is chosen as the best

location to place first FACTS device.

Table 2 the three smallest eigenvalues for different contingencies

Contingency

λmin1 λmin2 λmin3

1 line(1-2) 0.1202 2.5076 3.350

2 line(1-5) 2.6334 5.5253 7.6623

3 line(2-3) 2.5139 4.1422 5.5968

4 line(2-4) 2.6389 5.5318 7.6599

5 line(2-5) 2.6766 5.5468 7.6689

6 line(3-4) 2.6501 5.5453 7.6784

7 line(4-5) 2.6193 5.5263 7.6513

8 line(4-7) 2.4671 5.5083 7.6689

9 line(4-9) 2.4350 5.5267 7.6689

[32]

10 line(5-6) 0.4627 4.000 6.2199

11 line(6-11) 1.3764 4.5484 7.0003

12 line(6-12) 2.2747 3.4231 6.8293

13 line(6-13) 1.6374 3.7727 7.3455

14 line(7-8) 2.0172 6.5611 14.3417

15 line(7-9) 1.7331 5.4479 7.6090

16 line(9-10) 1.9465 3.0652 9.3284

17 line(9-14) 1.9693 3.0471 6.8017

18 line(10-11) 2.1765 5.3288 5.5991

19 line(12-13) 2.6711 4.1235 5.9822

20 line(13-14) 1.7820 5.4382 5.6578

Table 3- TPCM values of buses

Bus no. TPCM Bus No. TPCM

1 0 8 0.0119

2 0.0090 9 0.0166

3 0.0096 10 0.0183

4 0.0101 11 0.0207

5 0.0092 12 0.0242

6 0.0216 13 0.0235

7 0.0134 14 0.0212



After installing the shunt FACTS controller which may be a STATCOM at bus 12, it is now

changed into PV bus. The voltage of this bus is constant until the shunt FACTS device

reaches its reactive power limit. The voltage of the bus 12 is set at 1.05 pu. The sufficient

capacity to keep the voltage of bus 12 constant under all contingencies is 8 MVAr. After

installing STATCOM at bus 12, the contingency analysis was performed again. The result is

shown in table 4 and 5. According to table 4, it is clear that the smallest eigenvalue in each

contingency condition is increased considerably. However, the outage of line 1 still causes an

eigenvalue smaller than the critical value, because FACTS controller at a bus is installed

which is far from this line. When this line is out of circuit, injection of reactive power to bus

12 cannot influence considerably the reactive losses caused by the overload of line 2.

[33]

Table 4 the smallest eigenvalue associated with contingency after installation of STATCOM

at Bus 12

Contingency λmin Contingency λmin

Normal state 2.7971

1 line(1-2) 0.5987 11 line(6-11) 1.3906

2 line(1-5) 2.7109 12 line(6-12) 2.8100

3 line(2-3) 2.5755 13 line(6-13) 2.4062

4 line(2-4) 2.7171 14 line(7-8) 2.0627

5 line(2-5) 2.7574 15 line(7-9) 1.7804

6 line(3-4) 2.7289 16 line(9-10) 1.9465

7 line(4-5) 2.6969 17 line(9-14) 2.1620

8 line(4-7) 2.5353 18 line(10-11) 2.2323

9 line(4-9) 2.5043 19 line(12-13) 2.6711

10 line(5-6) 1.8798 20 line(13-14) 1.7820

Table 5 TPCM values of buses

Depending on the available budget, the placement of FACTS devices can proceed by

following the new ranked list of table 5, where bus 8 as a PV bus will be the second choice.

This means that reactive power generation capacity at this bus is need to be increased.

However, reactive power capacity of this bus can be kept constant and install the FACTS

device in next top bus which is bus 7. To keep the voltage of bus 7 and 12 constant under all

contingencies, FACTS devices of capacities 22MVAr and 11MVAr respectively need to be

installed at these buses. After installing the second FACTS device, all eigenvalues are

increased and the critical eigenvalues are disappeared. Table 6 represents the smallest

eigenvalue in each system state. Now, there is no critical eigenvalue and therefore, TPCM

value for all buses is zero.


1 0 8 0.0043

2 0.0026 9 0.0039

3 0.0029 10 0.0037

4 0.0027 11 0.0025

5 0.0022 12 0

6 0.0010 13 0.0008

7 0.0040 14 0.0028

[34]


at Bus 7


Normal state 3.8519

1 line(1-2) 2.0977 11 line(6-11) 1.9540

2 line(1-5) 2.7668 12 line(6-12) 3.8724

3 line(2-3) 3.7513 13 line(6-13) 3.1345

4 line(2-4) 3.8393 14 line(7-8) 2.0627

5 line(2-5) 3.8466 15 line(7-9) 1.7835

6 line(3-4) 3.8452 16 line(9-10) 1.9465

7 line(4-5) 3.8348 17 line(9-14) 2.1620

8 line(4-7) 3.8352 18 line(10-11) 3.5056

9 line(4-9) 3.5957 19 line(12-13) 3.6140

10 line(5-6) 2.5203 20 line(13-14) 2.4246


Sub case 2: when voltage magnitude is maintained at 1.0 pu

The sufficient capacity to keep the voltage of bus 12 constant under all contingencies is 39

MVAr. The result is shown in table 7 and 8. However, the outage of line 1 still causes an

eigenvalue smaller than the critical value, because FACTS controller at a bus is installed

which is far from this line. When this line is out of circuit, injection of reactive power to bus

12 cannot influence considerably the reactive losses caused by the overload of line 2.


at Bus 12

Contingency

λmin Contingency

λmin

Normal state 2.0085

1 line(1-2) 0.5646 11 line(6-11) 1.3340

2 line(1-5) 1.1580 12 line(6-12) 2.7948

3 line(2-3) 1.1261 13 line(6-13) 2.3643

4 line(2-4) 1.1921 14 line(7-8) 2.0174

5 line(2-5) 1.9253 15 line(7-9) 1.2001

6 line(3-4) 1.9251 16 line(9-10) 1.2815

7 line(4-5) 1.9453 17 line(9-14) 1.7183

8 line(4-7) 1.8600 18 line(10-11) 1.9767

9 line(4-9) 1.8111 19 line(12-13) 1.5518

10 line(5-6) 1.8361 20 line(13-14) 1.5732

[35]

Depending on the available budget, the placement of FACTS devices can proceed by

following the new ranked list of table 5, where bus 8 as a PV bus will be the second choice.

This means that reactive power generation capacity at this bus is need to be increased.

However, reactive power capacity of this bus can be kept constant and install the FACTS

device in next top bus which is bus 7. To keep the voltage of bus 7 and 12 constant under all

contingencies, FACTS devices of capacities 79MVAr and 39MVAr respectively need to be

installed at these buses. After installing the second FACTS device, all eigenvalues are

increased and the critical eigenvalues are disappeared. Table 9 represents the smallest

eigenvalue in each system state. Now, there is no critical eigenvalue and therefore, TPCM

value for all buses is zero.


.

Table 9 The smallest eigenvalue associated with contingency after installation of STATCOM

at Bus 7


Normal state 2.7020

1 line(1-2) 1.9799 11 line(6-11) 1.8328

2 line(1-5) 2.5187 12 line(6-12) 2.3486

3 line(2-3) 2.5317 13 line(6-13) 2.6823

4 line(2-4) 2.6570 14 line(7-8) 2.7020

5 line(2-5) 2.6749 15 line(7-9) 1.1823

6 line(3-4) 2.6599 16 line(9-10) 1.2832

7 line(4-5) 2.6689 17 line(9-14) 1.7677

8 line(4-7) 2.6532 18 line(10-11) 2.5957

9 line(4-9) 2.5642 19 line(12-13) 2.0387

10 line(5-6) 2.3899 20 line(13-14) 2.1618


1 0 8 0.0045

2 0.0028 9 0.0041

3 0.0031 10 0.0039

4 0.0029 11 0.0026

5 0.0024 12 0

6 0.0011 13 0.0008

7 0.0042 14 0.0030

[36]


In this case contingency of line (1-2) is not considered because line (1-2) is double circuit

line. In case of line overloaded by 30 % all lines are overstressed. Here it is assumed that if

one of the double circuit line is out then other will supply the power but in case of overloaded

line both the line will out simultaneously if contingency of line (1-2 ) is considered. So

outage of line (1-2) means that the system will collapse and iterative solution will not

converge. So in this case contingency of line (1-2) has been excluded. Here λmin= 0.4361

Table 10 the three smallest eigenvalues for different contingencies

Contingency

λmin1 λmin2 λmin3

1 line(1-2)

2 line(1-5) 0.2126 2.4178 4.2210

3 line(2-3) 0.0420 1.6399 2.5058

4 line(2-4) 0.3435 2.6426 5.0805

5 line(2-5) 0.3726 2.7064 5.1902

6 line(3-4) 0.4214 2.7715 3.5489

7 line(4-5) 0.3877 2.3298 5.1267

8 line(4-7) 0.2835 2.1950 4.6918

9 line(4-9) 0.3714 2.7260 4.8339

10 line(5-6) 0.0540 1.6501 4.1749

11 line(6-11) 0.4131 1.6571 3.4896

12 line(6-12) 0.4171 2.3385 4.6398

13 line(6-13) 0.3831 2.3968 4.2838

14 line(7-8) 0.4191 3.3105 5.2955

15 line(7-9) 0.2852 1.6746 3.3423

16 line(9-10) 0.3853 1.4649 3.0724

17 line(9-14) 0.3825 1.4869 3.5869

18 line(10-11) 0.4351 1.6412 4.5321

19 line(12-13) 0.4358 2.5315 4.2794

20 line(13-14) 0.4296 1.7648 3.5948



1 0 8 0.1122

2 0.0093 9 0.1661

3 0.0942 10 0.1892

4 0.0573 11 0.1937

5 0.0429 12 0.2034

6 0.1696 13 0.2045

7 0.1226 14 0.2205

[37]

Sub case 1: when voltage magnitude is maintained at 1.0 pu

Table 12 The smallest eigenvalue associated with contingency after installation of

STATCOM at Bus 14


Normal state 1.1964

1 line(1-2) 11 line(6-11) 1.0729

2 line(1-5) 0.9250 12 line(6-12) 1.1743

3 line(2-3) 0.8694 13 line(6-13) 1.1761

4 line(2-4) 1.0839 14 line(7-8) 1.2691

5 line(2-5) 1.1200 15 line(7-9) 1.3600

6 line(3-4) 1.1868 16 line(9-10) 0.7479

7 line(4-5) 1.1325 17 line(9-14) 0.7792

8 line(4-7) 1.0196 18 line(10-11) 1.1384

9 line(4-9) 1.1622 19 line(12-13) 1.1396

10 line(5-6) 0.9054 20 line(13-14) 0.7914

Table 13- TPCM values of buses

Table 14 the smallest eigenvalue associated with contingency after installation of

STATCOM at Bus 10


Normal state 2.0868

1 line(1-2) 11 line(6-11) 1.6811

2 line(1-5) 1.8001 12 line(6-12) 1.7980

3 line(2-3) 1.6487 13 line(6-13) 2.0172

4 line(2-4) 2.0038 14 line(7-8) 2.1164

5 line(2-5) 2.0076 15 line(7-9) 2.1414

6 line(3-4) 2.0828 16 line(9-10) 1.7695

7 line(4-5) 2.0640 17 line(9-14) 2.0557

8 line(4-7) 2.0702 18 line(10-11) 1.2281

9 line(4-9) 2.0624 19 line(12-13) 1.8468

10 line(5-6) 1.4844 20 line(13-14) 1.3191


1 0 8 0.0117

2 0.0010 9 0.0098

3 0.0060 10 0.0232

4 0.0048 11 0.0227

5 0.0041 12 0.0196

6 0.0165 13 0.0151

7 0.0098 14 0

[38]

At bus 14 = (30 MVAr) should be installed

At bus 10 = (30 MVAr) should be installed


After installing the shunt FACTS controller at bus 14, it is now changed into PV bus. The

voltage of this bus is constant until the shunt FACTS device reaches its reactive power limit.

The sufficient capacity to keep the voltage of bus 14 constant under all contingencies is 42

MVAr. After installing STATCOM at bus 14, the contingency analysis was performed again.

The result is shown in table 15 and 16. According to table 15, it is clear that the smallest

eigenvalue in each contingency condition is increased considerably.

Table 15 the smallest eigenvalue associated with contingency after installation of

STATCOM at Bus 14


Normal state 3.5642

1 line(1-2) 11 line(6-11) 1.1705

2 line(1-5) 0.9562 12 line(6-12) 3.0202

3 line(2-3) 0.9219 13 line(6-13) 3.3657

4 line(2-4) 1.1151 14 line(7-8) 1.3059

5 line(2-5) 1.1511 15 line(7-9) 2.9256

6 line(3-4) 3.2910 16 line(9-10) 0.7881

7 line(4-5) 1.6518 17 line(9-14) 2.6717

8 line(4-7) 3.4971 18 line(10-11) 3.0428

9 line(4-9) 3.4678 19 line(12-13) 3.5642

10 line(5-6) 0.9459 20 line(13-14) 3.5627



1 0 8 0.0063

2 0.0006 9 0.0045

3 0.0045 10 0.0167

4 0.0030 11 0.0155

5 0.0026 12 0.0117

6 0.0104 13 0.0082

7 0.0051 14 0

[39]

Depending on the budget, the placement of FACTS devices can proceed by following the

new ranked list of table 16, where bus 10 as a PV bus will be the second choice. This means

that reactive power generation capacity at this bus is need to be increased. To keep the

voltage of bus 14 and 10 constant under all contingencies, FACTS devices of capacities

25MVAr need to be installed at these buses. After installing the second FACTS device, all

eigenvalues are increased and the critical eigenvalues are disappeared. Table 17 shows the

smallest eigenvalue in each system state. Now, there is no critical eigenvalue and therefore,

TPCM value for all buses is zero.

Table 17 The smallest eigenvalue associated with contingency after installation of

STATCOM at Bus 10


Normal state 5.1801

1 line(1-2) 11 line(6-11) 5.1731

2 line(1-5) 1.8728 12 line(6-12) 3.0202

3 line(2-3) 1.0007 13 line(6-13) 2.0669

4 line(2-4) 2.0770 14 line(7-8) 4.5473

5 line(2-5) 2.1614 15 line(7-9) 4.9207

6 line(3-4) 3.4841 16 line(9-10) 4.4147

7 line(4-5) 4.4192 17 line(9-14) 5.0900

8 line(4-7) 4.6784 18 line(10-11) 5.1765

9 line(4-9) 4.9641 19 line(12-13) 4.1006

10 line(5-6) 1.0972 20 line(13-14) 5.1756

[40]

CHAPTER 8

MATLAB PROGRAMS:

8.1 Mybusout

% This program prints the power flow solution in a tabulated form

% on the screen.

%

% Copyright (C) 1998 by H. Saadat.

%clc

disp(tech)

fprintf(' Maximum Power Mismatch = %g \n', maxerror)

fprintf(' No. of Iterations = %g \n\n', iter)

head =[' Bus Voltage Angle ------Load------ ---Generation--- Injected'

' No. Mag. Degree MW Mvar MW Mvar Mvar '

' '];

disp(head)

for n=1:bus_no

fprintf(' %5g', n), fprintf(' %7.3f', Vm(n)),

fprintf(' %8.3f', deltad(n)), fprintf(' %9.3f', Pd(n)),

fprintf(' %9.3f', Qd(n)), fprintf(' %9.3f', Pg(n)),

fprintf(' %9.3f ', Qg(n)), fprintf(' %8.3f\n', Qsh(n))

end

8.2 Mylfybus

% This program obtains th Bus Admittance Matrix for power flow solution

% Copyright (c) 1998 by H. Saadat

[41]

j=sqrt(-1); i = sqrt(-1);

nl = linedata(:,1); nr = linedata(:,2);

R = linedata(:,3);X = linedata(:,4);

Bc = j*linedata(:,5); a = linedata(:, 6);

nbr=length(linedata(:,1));

nbus = max(max(nl), max(nr));

Z = R + j*X; y= ones(nbr,1)./Z; %branch admittance

for n = 1:nbr

if a(n) <= 0 a(n) = 1; else end

Ybus=zeros(nbus,nbus); % initialize Ybus to zero

% formation of the off diagonal elements

for k=1:nbr;

Ybus(nl(k),nr(k))=Ybus(nl(k),nr(k))-y(k)/a(k);

Ybus(nr(k),nl(k))=Ybus(nl(k),nr(k));

end

end

% formation of the diagonal elements

for n=1:nbus

for k=1:nbr

if nl(k)==n

Ybus(n,n) = Ybus(n,n)+y(k)/(a(k)^2) + Bc(k);

elseif nr(k)==n

Ybus(n,n) = Ybus(n,n)+y(k) +Bc(k);

else, end

end

end

if condition==14

Ybus(8,8)=Ybus(8,8)+j*0.19;

else

Ybus(9,9)=Ybus(9,9)+j*0.19;

[42]

end

clear Pgg

8.3 Mylfnewton

% Power flow solution by Newton-Raphson method

clc;

bus_no=length(busdata(:,1));

bus_type=ones(bus_no,2);

Pd=zeros(bus_no,1);Qd=zeros(bus_no,1);

Pg=zeros(bus_no,1);Qg=zeros(bus_no,1);

bus_type(:,1)=bus_type(:,1).*busdata(:,2);

bus_type(:,2)=bus_type(:,1);

Vm=busdata(:,3);

delta=busdata(:,4);

Pd=busdata(:,5);Qd=busdata(:,6);

Pg=busdata(:,7);Qg=busdata(:,8);

Qmin=busdata(:,9); Qmax=busdata(:,10);

Qsh=busdata(:,11);

Ym=abs(Ybus); t=angle(Ybus); %calculates the magnitude and angle of Ybus elements.

maxerror=1 ; converge=1;

iter = 0;

%=========================

% Start of iterations

%=========================

clear J JPd JPv JQd JQv dPQ ddeltaV

while maxerror >= accuracy

while maxerror >= accuracy & iter <= maxiter % Test for max. power mismatch

iter = iter+1;

[43]

Psch=(Pg(2:bus_no)-Pd(2:bus_no))/basemva;

Qschtotal=(Qg(2:bus_no)-Qd(2:bus_no))/basemva;

PV_no=sum(bus_type(2:bus_no,2))/2; %calculates the number of PV buses.

Pcal=zeros(bus_no-1,1);

Qcal=zeros(bus_no-1-PV_no,1);

Qsch=Qcal;

j=0;

for i=2:bus_no

if bus_type(i,2)~=2

Qsch(i-1-j,1)=Qschtotal(i-1);

else

j=j+1;

end

end

JPd=zeros(bus_no-1,bus_no-1);

JPv=zeros(bus_no-1,bus_no-1-PV_no);

JQd=zeros(bus_no-1-PV_no,bus_no-1);

JQv=zeros(bus_no-1-PV_no,bus_no-1-PV_no);

for i=2:bus_no

for j=2:bus_no

if i==j

for n=1:bus_no

if n~=i

JPd(i-1,i-1)=JPd(i-1,i-1)-Vm(i)*Vm(n)*Ym(i,n)*sin(delta(i)-delta(n)-t(i,n));%

diagonal elements of JPd

end

end

else

[44]

JPd(i-1,j-1)=Vm(i)*Vm(j)*Ym(i,j)*sin(delta(i)-delta(j)-t(i,j)); % off diagonal

elements of JPd

end

end

end

for i=2:bus_no

PV_counter=0;

for j=2:bus_no

if bus_type(j,2)==2

PV_counter=PV_counter+1;

else

if i==j

for n=1:bus_no

JPv(i-1,i-1-PV_counter)=JPv(i-1,i-1-PV_counter)+Vm(n)*Ym(i,n)*cos(delta(i)-

delta(n)-t(i,n));

end

JPv(i-1,i-1-PV_counter)=JPv(i-1,j-1-PV_counter)+Vm(i)*Ym(i,i)*cos(t(i,i)); %

diagonal elements of JPv

else

JPv(i-1,j-1-PV_counter)=Vm(i)*Ym(i,j)*cos(delta(i)-delta(j)-t(i,j)); % off diagonal

elements of JPd

end

end

end

end

PV_counter=0;

for i=2:bus_no

if bus_type(i,2)==2

PV_counter=PV_counter+1;

[45]

else

for j=2:bus_no

if i==j

for n=1:bus_no

if n~=i

JQd(i-1-PV_counter,i-1)=JQd(i-1-PV_counter,i-

1)+Vm(i)*Vm(n)*Ym(i,n)*cos(delta(i)-delta(n)-t(i,n));% diagonal elements of JQd

end

end

else

JQd(i-1-PV_counter,j-1)=-Vm(i)*Vm(j)*Ym(i,j)*cos(delta(i)-delta(j)-t(i,j)); % off

diagonal elements of JQd

end

end

end

end

PV_counterQ=0;

for i=2:bus_no

if bus_type(i,2)==2

PV_counterQ=PV_counterQ+1;

else

PV_counterV=0;

for j=2:bus_no

if bus_type(j,2)==2

PV_counterV=PV_counterV+1;

else

if i==j

for n=1:bus_no

JQv(i-1-PV_counterQ,i-1-PV_counterV)=JQv(i-1-PV_counterQ,i-1-

PV_counterV)+Vm(n)* Ym(i,n)*sin(delta(i)-delta(n)-t(i,n));

[46]

end

JQv(i-1-PV_counterQ,i-1-PV_counterV)=JQv(i-1-PV_counterQ,i-1-

PV_counterV)-Vm(i)* Ym(i,i)*sin(t(i,i)); % diagonal elements of JPv

else

JQv(i-1-PV_counterQ,j-1-PV_counterV)=Vm(i)*Ym(i,j)*sin(delta(i)-delta(j)-

t(i,j)); % off diagonal elements of JPd

end

end

end

end

end

for k=2:bus_no

for n=1:bus_no

Pcal(k-1,1)=Pcal(k-1,1)+Vm(k)*Vm(n)*Ym(k,n)*cos(delta(k)-delta(n)-t(k,n));

end

end

j=0;

for k=2:bus_no

if bus_type(k,2)~=2

for n=1:bus_no

Qcal(k-1-j,1)=Qcal(k-1-j,1)+Vm(k)*Vm(n)*Ym(k,n)*sin(delta(k)-delta(n)-t(k,n));

end

else

j=j+1;

end

end

dP=Psch-Pcal; dQ=Qsch-Qcal;

dPQ=[dP;dQ];

J=[JPd,JPv;JQd,JQv];

[47]

ddeltaV=J\dPQ;

u=length(ddeltaV);

ddelta=ddeltaV(1:bus_no-1);

dVm=ddeltaV(bus_no:u);

delta(2:bus_no)=delta(2:bus_no)+ddelta;

maxerror=max(abs(dPQ));

j=0;

for i=2:bus_no

if bus_type(i,2)~=2

Vm(i)=Vm(i)+dVm(i-j-1);

else

j=j+1;

end

end

if iter == maxiter & maxerror > accuracy

fprintf('\nWARNING: Iterative solution did not converged after ')

fprintf('%g', iter), fprintf(' iterations.\n\n')

fprintf('Press Enter to terminate the iterations and print the results \n')

converge = 0; pause, else, end

end

for i=2:bus_no

if (bus_type(i,1)==2)&(bus_type(i,2)==2) %generator reaches its limits

Qcal(i)=0;

for n=1:bus_no

Qcal(i)=Qcal(i)+Vm(i)*Vm(n)*Ym(i,n)*sin(delta(i)-delta(n)-t(i,n));

end

Qgen(i)=Qcal(i)+Qd(i)/basemva;

if Qgen(i)>Qmax(i)/basemva

bus_type(i,2)=0;

[48]

Qg(i)=Qmax(i);

maxerror=1;

elseif Qgen(i)<Qmin(i)/basemva

bus_type(i,2)=0;

Qg(i)=Qmin(i);

maxerror=1;

end

end

end

if iter >= maxiter converge = 0; maxerror=0; end

end

if converge ~= 1

tech= (' ITERATIVE SOLUTION DID NOT CONVERGE'); else,

tech=(' Power Flow Solution by Newton-Raphson Method');

end

deltad=180/pi*delta;

Vm; delta;

Pcal=zeros(bus_no,1);Qcal=zeros(bus_no,1);

for i=1:bus_no

for n=1:bus_no

Pcal(i)=Pcal(i)+Vm(i)*Vm(n)*Ym(i,n)*cos(delta(i)-delta(n)-t(i,n));

Qcal(i)=Qcal(i)+Vm(i)*Vm(n)*Ym(i,n)*sin(delta(i)-delta(n)-t(i,n));

end

end

Pcal=Pcal*basemva;

Qcal=Qcal*basemva;

for i=1:bus_no

Pg(i)=Pcal(i)+Pd(i);

Qg(i)=Qcal(i)+Qd(i);

[49]

end

PQ_buses=zeros(bus_no-1-PV_no,1);

j=0;

for i=1:bus_no

if bus_type(i,2)==0

j=j+1;

PQ_buses(j)=i;

end

end

8.4 Main

clc;

clear all;

basemva=100; accuracy=.001;

maxiter=50; critical_eigenvalue=1.0;

% BUS BUS Voltage Angle --Load-- --------Generator--------- Injected

% No code Mag. Degree MW Mvar MW Mvar Qmin Qmax Mvar

busdata=[ 1 1 1.060 0 0.0 0.0 0.0 0.0 0 0 0;

2 2 1.045 0 21.7 12.7 40.0 0.0 -40.0 50.0 0;

3 2 1.010 0 94.2 19.0 0.0 0.0 0.0 40.0 0;

4 0 1 0 47.8 -3.9 0.0 0.0 0.0 0.0 0;

5 0 1 0 7.6 1.6 0.0 0.0 0.0 0.0 0;

6 2 1.070 0 11.2 7.5 0.0 0.0 -6.0 24.0 0;

7 0 1 0 0.0 0.0 0.0 0.0 0.0 0.0 0;

8 2 1.090 0 0.0 0.0 0.0 0.0 -6.0 24.0 0;

9 0 1 0 29.5 16.6 0.0 0.0 0.0 0.0 0;

10 0 1 0 9.0 5.8 0.0 0.0 0.0 0.0 0;

11 0 1 0 3.5 1.8 0.0 0.0 0.0 0.0 0;

12 0 1 0 6.1 1.6 0.0 0.0 0.0 0.0 0;

13 0 1 0 13.5 5.8 0.0 0.0 0.0 0.0 0;

14 0 1 0 14.9 5.0 0.0 0.0 0.0 0.0 0];

[50]

load_scale=0.95;

busdata(:,5)=busdata(:,5)*load_scale; busdata(:,6)=busdata(:,6)*load_scale;

busdata(:,7)=busdata(:,7)*load_scale;

%Line Data

%

% Bus Bus R X B tap setting

% nl nr pu pu pu value

linedata=[ 1 2 0.01938 0.05917 0.0528 1 ;

1 5 0.05403 0.22304 0.0492 1 ;

2 3 0.04699 0.19797 0.0438 1 ;

2 4 0.05811 0.17632 0.0340 1 ;

2 5 0.05695 0.17388 0.0346 1 ;

3 4 0.06701 0.17103 0.0128 1 ;

4 5 0.01335 0.04211 0.0 1 ;

4 7 0.0 0.20912 0.0 0.978 ;

4 9 0.0 0.55618 0.0 0.969 ;

5 6 0.0 0.25202 0.0 0.932 ;

6 11 0.09498 0.19890 0.0 1 ;

6 12 0.12291 0.25581 0.0 1 ;

6 13 0.06615 0.13207 0.0 1 ;

7 8 0.0 0.17615 0.0 1 ;

7 9 0.0 0.11001 0.0 1 ;

9 10 0.03181 0.08450 0.0 1 ;

9 14 0.12711 0.27038 0.0 1 ;

10 11 0.08205 0.19207 0.0 1 ;

12 13 0.22092 0.19988 0.0 1 ;

13 14 0.17093 0.34802 0.0 1 ];

linedata(:,5)=linedata(:,5)/2;

line_no=length(linedata(:,1));

[51]

linedata_outage_percent=[2;2;2;2;2;2;2;2;2;2;2;2;2;2;2;2;2;2;2;2]/100;

condition=0;

mylfybus % forms the bus admittance matrix

myLFNEWTON % power flow solution by Newton-Raphson method

myBUSOUT % prints the power flow solution on the screen

%================================================

% calculation of eigenvalues

%================================================

JR=JQv-JQd*(inv(JPd)*JPv);

[right_eigenvectors,lambda]=eig(JR)

left_eigenvectors=inv(right_eigenvectors')

lambda_row=eig(JR)',min_lambda=min(lambda_row), min_vec(1)=min_lambda;

eigenvalue_no=length(lambda_row);

%================================================

% bus participation factors

%================================================

bus_participation_factors=right_eigenvectors.*left_eigenvectors;

first_row=[1:eigenvalue_no];

bus_participation_factors_result=[0,first_row;PQ_buses,bus_participation_factors]

Pnormal=1;

for k=1:line_no

Pnormal=Pnormal*(1-linedata_outage_percent(k));

end

NormalTPCM=zeros(bus_no,1)';

for k=1:eigenvalue_no

if lambda_row(k)<=critical_eigenvalue

[52]

PQ_no=length(PQ_buses);

for j=1:PQ_no

NormalTPCM(PQ_buses(j))=NormalTPCM(PQ_buses(j))+Pnormal*bus_participation_facto

rs(j,k)/lambda_row(k);

end

end

end

input('press "Enter" key ...');

%================================================

% Contingency Analysis

%================================================

TPCM=zeros(bus_no,1)';

initial_line_no=line_no;

linedata_saved=linedata;

mainbusdata=busdata;

for condition=[1:13,15:20]

busdata=mainbusdata;

if linedata_outage_percent(condition)~=0

new_linedata=zeros(initial_line_no-1,6);

if condition==1

new_linedata(1:initial_line_no-1,:)=linedata(2:initial_line_no,:);

elseif condition==initial_line_no

new_linedata(1:initial_line_no-1,:)=linedata(1:initial_line_no-1,:);

else

new_linedata(1:condition-1,:)=linedata(1:condition-1,:);

new_linedata(condition:initial_line_no-1,:)=linedata(condition+1:initial_line_no,:);

end

linedata=new_linedata;



[53]


%====================================================

%calculation of eigenvalues in contingency conditions

%====================================================


[right_eigenvectors,lambda]=eig(JR)

left_eigenvectors=inv(right_eigenvectors')

lambda_row=eig(JR)',

min_lambda=min(lambda_row),min_vec(condition+1)=min_lambda;


%===================================================

%bus participation factors in contingency conditions

%===================================================







for j=1:PQ_no

TPCM(PQ_buses(j))=TPCM(PQ_buses(j))+linedata_outage_percent(condition)*bus_particip

ation_factors(j,k)/lambda_row(k);

end

end

end

end

condition, input('Press "Enter" key...');

linedata=linedata_saved;

end

[54]

%================================================

% Contingency Analysis for line 14

%================================================

condition=14;

% IEEE 14-BUS TEST SYSTEM

% BUS BUS Voltage Angle --Load-- --------Generator--------- Injected

% No code Mag. Degree MW Mvar MW Mvar Qmin Qmax Mvar

busdata=[ 1 1 1.060 0 0.0 0.0 0.0 0.0 0 0 0;

2 2 1.045 0 21.7 12.7 40.0 0.0 -40.0 50.0 0;

3 2 1.010 0 94.2 19.0 0.0 0.0 0.0 40.0 0;

4 0 1 0 47.8 -3.9 0.0 0.0 0.0 0.0 0;

5 0 1 0 7.6 1.6 0.0 0.0 0.0 0.0 0;

6 2 1.070 0 11.2 7.5 0.0 0.0 -6.0 24.0 0;

7 0 1.05 0 0.0 0.0 0.0 0.0 0.0 0.0 0;

8 0 1 0 29.5 16.6 0.0 0.0 0.0 0.0 0;

9 0 1 0 9.0 5.8 0.0 0.0 0.0 0.0 0;

10 0 1 0 3.5 1.8 0.0 0.0 0.0 0.0 0;

11 0 1.05 0 6.1 1.6 0.0 0.0 0.0 0.0 0;

12 0 1 0 13.5 5.8 0.0 0.0 0.0 0.0 0;

13 0 1 0 14.9 5.0 0.0 0.0 0.0 0.0 0];

busdata(:,5)=busdata(:,5)*load_scale; busdata(:,6)=busdata(:,6)*load_scale;

busdata(:,7)=busdata(:,7)*load_scale;

%Line Data

%

[55]

% Bus Bus R X B tap setting

% nl nr pu pu pu value

linedata=[ 1 2 0.01938 0.05917 0.0528 1 ;

1 5 0.05403 0.22304 0.0492 1 ;

2 3 0.04699 0.19797 0.0438 1 ;

2 4 0.05811 0.17632 0.0340 1 ;

2 5 0.05695 0.17388 0.0346 1 ;

3 4 0.06701 0.17103 0.0128 1 ;

4 5 0.01335 0.04211 0.0 1 ;

4 7 0.0 0.20912 0.0 0.978 ;

4 8 0.0 0.55618 0.0 0.969 ;

5 6 0.0 0.25202 0.0 0.932 ;

6 10 0.09498 0.19890 0.0 1 ;

6 11 0.12291 0.25581 0.0 1 ;

6 12 0.06615 0.13207 0.0 1 ;

7 8 0.0 0.11001 0.0 1 ;

8 9 0.03181 0.08450 0.0 1 ;

8 13 0.12711 0.27038 0.0 1 ;

9 10 0.08205 0.19207 0.0 1 ;

11 12 0.22092 0.19988 0.0 1 ;

12 13 0.17093 0.34802 0.0 1 ];

linedata(:,5)=linedata(:,5)/2;




%====================================================

%calculation of eigenvalues in contingency conditions

%====================================================


[right_eigenvectors,lambda]=eig(JR);

left_eigenvectors=inv(right_eigenvectors');

[56]

lambda_row=eig(JR)',min_lambda=min(lambda_row),min_vec(condition+1)=min_lambda;


%===================================================

%bus participation factors in contingency condition

%===================================================



for j=1:length(PQ_buses)

if PQ_buses(j)>7

PQ_buses(j)=PQ_buses(j)+1;

end

end





for j=1:PQ_no

TPCM(PQ_buses(j))=TPCM(PQ_buses(j))+linedata_outage_percent(condition)*bus_particip

ation_factors(j,k)/lambda_row(k);

end

end

end

condition, input('Press "Enter" key ...');

disp('=============================================================');

disp(' Results ');

disp('=============================================================');

TPCM_values=zeros(2,bus_no+1);

TPCM_values(1,:)=[1:1:14];

TPCM_values(2,:)=TPCM

[57]

CHAPTER 9

CONCLUSION

In earlier methods number of SVC installed is more in number which is considerably

reduced. Also there is improvement in system voltage in contingency conditions as well as

normal state. In addition, the proposed method has less time consuming calculations. When

SVC was used, the optimal allocations are 0.19, 0.25 and 0.25 pu at buses 10, 13 and 14

respectively and these reactive power are fully used for the outage of line 1. On the other

hand, the optimal FACTS devices allocations obtained by the proposed method are 0.22 and

0.11 pu at buses 7 and 12 respectively. Therefore, the number of STATCOMs to be installed

is decreased as well as their reactive power capacity. The reason is that the allocated FACTS

devices proposed earlier are applied only in one area of the network (i.e. at three close

buses). This causes a non-uniform reactive power supply in the network. However, the

method proposed here, allocates FACTS devices in two separated areas of the network that

leads to a more uniform reactive power supply in the system. Consequently, it will be

effective in more contingency conditions correspond to the outage of lines.

Other important point to notice is that when the load and generation is scaled by factor of

0.95 then the weakest bus is 12 followed by bus 7 but when line is overloaded and load and

generation is scaled by factor of 1.3 then the weakest bus is 14 followed by bus 10. So

voltage instability could be at different bus depending on the loading of the line. Also, when

voltage magnitude of line set at 1.0 pu then the rating of FACTS devices to be installed is

less than that when voltage magnitude is set at 1.05 pu.

Application of FACTS devices can improve considerably the system voltage stability and

prevent voltage collapse. Nevertheless, location of FACTS devices strongly influences their

damping effect. Therefore, optimal location of FACTS is a very important issue. Here, the

application of FACTS devices to extend voltage stability margin in contingency conditions is

investigated. A probabilistic index based on modal analysis and calculation of bus

participation factors was defined which can be used to rank of system buses based on their

[58]

effect on system voltage stability enhancement under all possible contingencies. The

proposed method selects the most effective bus to voltage instability as the best place for

installing FACTS.

[59]

CHAPTER 10

REFERENCES

1. Hadi Saadat, Power System Analysis, TATA McGRAW HILL, 2006.

2. P. Kundur, Power System Stability and Control, McGraw-Hill

3. D P Kothari & I J Nagrath, Power system engineering, TATA McGRAW HILL, June

2008.

4. Narain G. Hingorani, Laszlo Gyugyi Understanding FACTS: Concepts and

Technology of Flexible AC Transmission Systems, Wiley-IEEE Press, December

2001.

5. Enrique Acha, Claudio R. Fuerte- Esquivel, Hugo Ambriz-Perez and Cesar Angeles-

Camacho, FACTS: Modelling and Simulation in Power Networks, John Wiley &

Sons, Ltd

6. B. Gao, Student Member IEEE G.K. Morison P. Kundur. Fellow IEEE, “Voltage

stability evaluation using Modal Analysis”, System Planning Division, Ontario

Hydro,Ontario. Canada, transactions on power Systems, Vol. 7, No. 4. November

1992

7. H.J.C.P. Pinto N. Martins X. Vieira F° A. Bianco P. Gomes M. G. dos Santos,

“Modal Analysis for voltage stability: Application at Base Case and Point of

Collapse”, Bulk Power System Voltage Phenomena - III Voltage Stability, Security

& Control, Davos, Switzerland, 22-26 August 1994

8. Wenjuan Zhang, Fangxing Li, Leon M. Tolbert, “Optimal Allocation of Shunt

Dynamic Var Source SVC and STATCOM: A Survey”, IEEE paper,2006

9. J. E. Candelo, N. G. Caicedo, F. Castro-Aranda, “Proposal for the Solution of

Voltage Stability Using Coordination of Facts Devices”, IEEE PES Transmission

and Distribution Conference and Exposition Latin America, Venezuela, 2006

10. Nimit Boonpirom, Kitti Paitoonwattanakij, “Static Voltage Stability Enhancement

using FACTS”, 2005

[60]

11. D. G. Ramey, Fellow, IEEE, M. Henderson, Sr. Member, IEEE “Overview of a

Special Publication on Transmission System Application Requirements for

FACTS Controllers”

88481823 improving voltage stability in power systems using modal analysis

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