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?

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

the voltage-weak points, and what areas arc involved?

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.

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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 also 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

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.

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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.

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.

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.

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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 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.

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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 controllability and

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

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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

VoltageControl

Electric generatorsSynchronous CondensersConventional Transformer tap-changerConventional Shunt Capacitor/ReactorConventional 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 andReactive PowerFlow Control

Generator schedulesTransmission line switchingPhase Angle Regulator (PAR)Series Capacitor (switched or fixed)High Voltage Direct CurrentTransmission (HVDC)

Interphase Power Controller (IPC)Thyristor controlled Series Capacitor(TCSC)Thyristor Controlled Series Reactor(TCSR)Thyristor Controlled Phase ShiftingTransformer (TCPST)UPFCStatic Synchronous Series Compensator(SSSC)Interline Power Flow Controller (IPFC)

TransientStability

Braking ResistorExcitation EnhancementSpecial Protection SystemsIndependent Pole TrippingFast Relay SchemesFast ValvingLine SectioningHVDC

Thyristor Controlled Braking Resistor(TCBR)SVC, STATCOM, TCSC, TCPST, UPFCBESS, SMES, SSSC, CSC, IPFC

DynamicStability

Power System StabilizerHVDC

TCSC, SVC, STATCOM, UPFC, SSSC,TCPST, BESS, SMES, SSSC,CSC, IPFC

Short Circuit Switched series reactors Thyristor switched series reactor,

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CurrentLimiting

Open circuit breaker arrangements

TCSC,IPC, SSSC, UPFC; These are secondaryfunctions of these controllers and theireffectiveness 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:

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

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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 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.

Figure 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 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

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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.

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.

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Figure 2: STATCOM circuit diagram

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). 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 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.

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Figure 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 4) This configuration

serves as a series compensator whose output voltage is in quadrature with,

and controlled, independently of the transmission line current.

Figure 4: Circuit diagram for a Static Synchronous Series Compensator

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

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a dc capacitor link. Figure 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 5: Circuit Diagram of a Unified Power Flow Controller

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-

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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.

Over the last decades, there has been a growing interest in algorithms

inspired by observing natural phenomenon. It has been shown that these

algorithms are good alternatives as tools

in solving complex computational problems. Various heuristic approaches

have been adopted in research, including genetic algorithm, tabu search,

simulated annealing etc. Some used a genetic algorithm to determine the

best location of a given set of FACTS devices in a deregulated electricity

market. The optimal locations of FACTS devices are obtained for Var

planning. A methodology is carried out using a genetic algorithm to find the

optimal number and location of thyristor-controlled phase shifters in a power

system. In this paper, power system stability is used as an index for optimal

allocation of SVCs. For this, several SVCs are placed in a power system based

on their primary function, which is the voltage stability. To locate SVCs based

on the voltage stability, two methods are used: modal analysis and genetic

algorithm.

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4. 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 eigen values 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

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quantities (voltage magnitude, voltage angle, injected active power and

injected reactive power) can be

expressed by power flow equations as follows:

4.1 REDUCED JACOBIAN MATRIX

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

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.

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

in (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.

To reduce (1), let ∆P =0, then.

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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.

4.2 MODES OF VOLTAGE INSTABILITY

Let

where,

ξ = right eigenvector matrix of JR

= left eigenvector matrix of JR

= diagonal eigenvalue matrix of JR

From (3) and (5)

And

Where

V = ∆V = the vector of modal voltage variations

q= ∆Q = vector of modal reactive power variations

and

Equation (7) represents uncoupled first order equations. Thus for the i th

mode:

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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 σi > 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

σi < 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 σi determines the degree of stability of the ith

modal voltage. The smaller the magnitude of positive σ i , the closer ith modal

voltage is to being unstable.

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:

(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.

5. 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.

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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 reduced Jacobian matrix

under contingency conditions and the effectiveness of buses on these critical

modes is recognized by their participation factors.

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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:

(11)

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

σj = critical eigenvalue j

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

eigenvalue whose magnitudes are smaller than a prescribed critical value

(σcritical). 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:

(12)

where

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

contingencies and L is the number of possible contingencies

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

(i.e. smaller σj).

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.

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5.1. FLOW CHART OF PROPOSED METHOD

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System

σj <

K=1

Outage of

Load flow

Modal analysis and determination of eigenvalues

Calculation of bus participation factor (Pij) for all

Ranking of buses based on associated TPCM values

Installation of FACTS at the top bus

All contingencies

Need to install

TPCMi = ∑k =1

L

PCM i

k=k +1

No

No

Yes

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6. CASE STUDY

A simple case has been taken in which IEEE 14 bus is considered. Given flowchart shown in

figure shows the proposed approach of placing FACTS in a power system. The load and

generation of the system is scaled by the factor of 0.95

Performing load flow for the normal state of the system, the smallest eigen value of the reduced

Jacobian matrix is determined as σmin= 2.71. With the assumption of σcritical=1, the calculated

eigen value 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 show and the corresponding critical eigen values are

specified by coloured cell from the table it is clear that critical eigen values 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- smallest eigen values for The three different contingenciesContingency σmin1 σmin2 σmin3

1 line(1-2) 0.1202 2.5076 3.3502 line(1-5) 2.6334 5.5253 7.66233 line(2-3) 2.5139 4.1422 5.59684 line(2-4) 2.6389 5.5318 7.65995 line(2-5) 2.6766 5.5468 7.66896 line(3-4) 2.6501 5.5453 7.67847 line(4-5) 2.6193 5.5263 7.65138 line(4-7) 2.4671 5.5083 7.66899 line(4-9) 2.4350 5.5267 7.668910 line(5-6) 0.4627 4.000 6.219911 line(6-11) 1.3764 4.5484 7.000312 line(6-12) 2.2747 3.4231 6.829313 line(6-13) 1.6374 3.7727 7.345514 line(7-8) 2.0172 6.5611 14.341715 line(7-9) 1.7331 5.4479 7.609016 line(9-10) 1.9465 3.0652 9.328417 line(9-14) 1.9693 3.0471 6.801718 line(10-11) 2.1765 5.3288 5.599119 line(12-13) 2.6711 4.1235 5.9822

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20 line(13-14) 1.7820 5.4382 5.6578 Table 3- TPCM values of buses

Bus no. TPCM Bus No. TPCM1 0 8 0.01192 0.0090 9 0.01663 0.0096 10 0.01834 0.0101 11 0.02075 0.0092 12 0.02426 0.0216 13 0.02357 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 eigen value in

each contingency condition is increased considerably. However, the outage

of line 1 still causes an eigen value 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.

Table 4- The smallest eigen value associated with contingency after installation of STATCOM at Bus 12Contingency min Contingency min

Normal state 2.79711 line(1-2) 0.5987 11 line(6-11) 1.39062 line(1-5) 2.7109 12 line(6-12) 2.81013 line(2-3) 2.5755 13 line(6-13) 2.40624 line(2-4) 2.7171 14 line(7-8) 2.06265 line(2-5) 2.7576 15 line(7-9) 1.78066 line(3-4) 2.7290 16 line(9-10) 1.94657 line(4-5) 2.6970 17 line(9-14) 2.16238 line(4-7) 2.5354 18 line(10-11) 2.23249 line(4-9) 2.5044 19 line(12-13) 2.671210 line(5-6) 1.8798 20 line(13-14) 1.7820

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Table 5- TPCM values of busesBus no. TPCM Bus No. TPCM1 0 8 0.00432 0.0026 9 0.00393 0.0029 10 0.00374 0.0027 11 0.00255 0.0022 12 06 0.0010 13 0.00087 0.0040 14 0.0028

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 200MVAr 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.

Table 6- The smallest eigen value associated with contingency after installation of STATCOM at Bus 7Contingency min Contingency min

Normal state 3.85191 line(1-2) 2.0977 11 line(6-11) 1.95402 line(1-5) 2.7668 12 line(6-12) 3.87243 line(2-3) 3.7513 13 line(6-13) 3.13454 line(2-4) 3.8393 14 line(7-8) 3.85205 line(2-5) 3.8466 15 line(7-9) 1.78356 line(3-4) 3.8452 16 line(9-10) 1.94657 line(4-5) 3.8348 17 line(9-14) 2.16208 line(4-7) 3.8352 18 line(10-11) 3.5056

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9 line(4-9) 3.5957 19 line(12-13) 3.614010 line(5-6) 2.5203 20 line(13-14) 2.4246

Table 7- Bus participation factors in critical contingency conditions

ContingencyBus

1 line (1-2) 10 line(5-6)

2 0.0544 03 0.0577 04 0.0603 0.00175 0.0550 0.00086 0.0816 0.18677 0.0773 0.01338 0.0716 09 0.0874 0.048610 0.0905 0.075711 0.0890 0.137312 0.0893 0.217213 0.0900 0.197114 0.0961 0.1217

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7. 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 0.2 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 in this paper, 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.

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. In this paper,

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 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. Results obtained

from simulations show that the proposed method could allocate FACTS devices with a simple

and straightforward approach in order to improve system voltage stability with consideration of

contingency conditions.

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REFERENCES

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

Publication on Transmission System Application Requirements for FACTS Controllers”

2. B. Gao, Student Member IEEE G.K. Morison P. Kundur. Fellow IEEE “VOLTAGE

STABILITY EVALUATION USING MODAL ANALYSIS”

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

4. N.G. Hingorani, L. Gyugyi, Understanding FACTS, IEEE Press

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