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EFFECT OF SEMICONDUCTOR-CONTROLLED VOLTAGE INJECTION

BY UPFC AND W C ON

PO'WER SYSTEM S W n l r r r

bY

Ihlireza Alavian Mehr

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy,

Graduate Department of Electrical and Compter Engineering UniversiSr of Toronto

@ Copyright by Alireza Alavian Mehr 1998

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EFFECT OF SEMICONDUCTOR-CONTROLLED VOWAGE INJECTION

BY UPFC AND UETC ON

POWER SYSTEM STABILITY

Doctor of Philosophy Degree, 1998

AIireza Alavian Mehr Department of Electrical and Computer Engineering

University of Toronto

Commercial availability of various power semiconductor switches indicates

proliferation of power electronic based apparatus in utility power systems.

Furthemore, existing power system apparatus, e-g. mechanical phase shiRers

and mechanical tap changing transformers, will be retrofitted to utilize higher

switching speed of serniconductor switches. A group of these apparatus, i.e.,

unined power flow controuer W F C ) , static phase shifter (SPS), under-load

tap-changing (ULTC) transformer and static series capacitor (SSC), perform their

respective functions by means of injecting series controlled voltages in power

systems.

This thesis demonstrates that fast series voltage injection, for dynamic power

flow regdation, can result in voltage dynamics and even voltage instability. This

indicates that fast voltage injection by means of power electronic based apparatus

can couple voltage stability and angle stability phenomena. To investigate this

couphg phenomena, the voltage dependency of the load must be adequately

represented in the load model. The reported studies in this work is based on

representing the load by a combination of static and dynamic loads.

This thesis primarily investigates impacts of UPFC and semiconductor-

controlled ULTC on voltage stability and angle stability phenomena. An eigen

analysis approach is used for the studies. The eigen analysis results are validated

by digital the-domain simulations using a transient stability sofhare. Both the

eigen analysis and the transient stabdity software tools are tailored to account for

angle and voltage stability phenomena.

iii

Acknowledgments

1 wish to express m y sincere gratitude and appreciation to my supervisor,

Professor M.R. Iravani for his invaluable guidance, support and encouragement

throughout the preparation of this thesis.

1 extend m y appreciation to the members of my Ph.D. committee for their

carefid review of m y thesis and for many useful comments.

1 also thank graduate students and staff in the Power Group at the University

of Toronto for creating an atmosphere of friendship, mutual understanding and

cooperation. Special thanks are due to Dr. M. Parniani, Di. A. Nabavi-Niaki,

Dr. H. Karshenas, Mr. A. Mousavi and Mr. M. Tartibi for their helpful discussions

and suggestions.

Financial support f?om the Iranian Ministry of Culture and Higher Education,

and also from the University of Toronto are gratefully acknowiedged.

1 am grateM to my parents for their continued love, support and patience

throughout my studies.

The last but not the least, 1 offer m y heartfelt thanks to m y d e Fahimeh and

to my son Mshin for their invaluable encouragement and understanding.

Contents Abstract

Acknowledgments

Contents

CHApTExc1 1

Introduction ........................................................ 1 1.1 Overview ................................................................................................... . 1 1.2 Objectives ....................................................................................................... 3

1.3 Thesis organization ...................................................................................... 3

c-2 4

Methods of Analyses of Voltage Instability ............................. 4 Introduction ...................................................................................................... 4

Voltage Collapse ............................................................................................... 5

Voltage Instability Analysis ............................................................................ 6

Static Approach ................................................................................................ 7

........................*........ .. ...... 2 .4.1 V-Q SeILSitivity .. ... -8 ..................................................................... 2.4.2 Singdar Value Decomposition 8

......................................................................................... 2.4.3 Modal Analysis 1 0 ........................................................................... 2.4.4 Continuation Power Flow II

.................................................................. Small-Signal Dynamic Approach 12

.................................................................. Large-Signal Dynamic Approach 12

.................................................................................................... Conclusions 13

CHAPTER 3 15

System Representation for Simultaneous

................................... Voltage rrnd Angle Stability Study -15

.................................................................................................... 3.1 Introduction 15

........................................................................ 3.2 Synchronous Machine Model 16

3.3 Induction Machine ................................................................................... 18

3.4 Static Load Model ......................................................................................... 20

....................................................................... 3.5 Transmission Network Model 21

.................................................. 3.6 Mode1 of Conventional ULTC Transformer 22

................................................... ......................... 3.6.1 Principle of Operation ., 22 3.6.2 ULTC Mode1 ................................................................................................ 24

........................................................... 3.7 Model of Phase ShiRing Transformer 25

................................................................. 3.8 Static Phase ShiRer (SPS) Mode1 28

3.8.1 UPFC Mode1 ................................................................................................ 30 ................................................................... 3.8.2 Reduced-order Mode1 of UPFC 33

3.9 Mode1 of Two-terminal HVDC Systems ....................... .. ........................ 34

3.10 System Equations .......................................................................................... 37 .................................................................... 3.10.1 Eigen Analysis S o h a r e Tool 37

............................... 3 .10.2 Transient Stability Software Tao1 .................... ... -38

................................................................................................... 3.11 Conclusions 38

C-4 40

................ Impact of Fast Voltage Regdation on Voltage Stability -40

........................................................... ....................... 4.1 Introduction ...... .... 40

...................................... 4.2 Voltage Behaviour Under Steady-State Operation 41

......................................... 4.3 Voltage Behaviour Under Dynamic Conditions 4 4

4.3.1 Effect of Conventional and Fast Voltage Regulator .................................................................................. on Voltage Dynamics 46

.................................................................................... 4.3.2 Region of Attraction 48 ................. 4.3.3 Effect of Limits of a Voltage Regulator on Voltage Dynamics 51

...................... 4.3.4 Effect of UPFC overd time constant on Voltage Stability 53

4.4 Conclusions .................................................................................................... 53

c-5

Effect of Semiconductor Controiled ULTC

...................................... on Voltage and Angle Dynamics 54

.................................................................................................... 5.1 Introduction 54

5.2 Study System ................................................................................................. 54 .......................................................... 5.3 Effect of ULTC on Voltage Dynamics 55

................................... 5.3.1 Effect of Load Composition on Voltage Dynamics -58 .............................. 5.3.2 Effect of Transmission System on Voltage Dynamics 61

....................................................... 5.3.3 Effect of ULTC on System Dynamics -63 5.3.4 Combined Effect of ULTC and Transmission Network

on System D y n d c s ..................... .. ....................................................... 66

............................................................ 5.4 Validation of Eigen Analpis Resulta 68

.............................................................................................. 5.4.1 Software Tool 68 ............................................. 5.4.2 System Mode1 for Time-Domain Simulation 69

C-6 77

........................ EfEect of UPFC on Voltage and Angle Dynamics 77

................................................................. 6.1 Introduction .............................. .,., 77

6.2 Study System ................................................................................................. 78

6.3 Principles of Voltage Stability Enhancement by Means of a UPFC ........... 78 6.4 UPFC Control System ...................... .. .................................................. 80

........................................................ 6.4.1 Effect of UPFC on Voltage Dynamics 84 .............................. 6.4.2 Effect of System Configuration on Voltage D ynamics 85

..................... ..................... 6.4.3 Validation of Eigen Analysis Results ......... 87

................ 6.5 UPFC Control Strategy to PreventMitigate Voltage InstabiEty 90

.................................................................... 6.6 Voltage Stability Enhancement 94 ........................................... 6.6.1 Effect of UPFC on Voltage Dynamics ..... 94

.................................................................................................. 6.7 Conclusions 9 8

CEAP'mR7 100

.................. Conclusions and Recornmendations for Future Work 100

7.1 Conclusions .................................................................................................. 100

............................... 7.2 Suggestions for Future Work .. .................................. 102

...................................................................... A . 1 Synchronous Machine Mode1 107 ......................... .......................... A . 1.1 Stator Dynamics Not Included ...... 1 0 7

........................................................................ A.1.2 Stator Dynamics hcluded 113

....................................................................................... A.2 Induction Machine 117 ............................................... ..................................... k2.1 SimplXed Mode1 ... 118

.............................................................................................. A.3 HVDC System 122

.................................................... B . 1 Data for the Power System of Chapter 4 126

B.2 Parameters of the AC/DC Test System of Chapter 5 ................................. 127

B.3 Parameters of the Exciters of Chapter 5 .................................................... 128

v i i

Chapter

Introduction

1.1 Overview

Stability problems of interconnected power systems are traditionally

addressed under either (1) voltage stability problems or (2) angle stability

problems. Angle stability deals with electromechanical oscillations which are

closely associated with real-power flow Voltage stability deals with oscillatory or

monotonie variations of voltage, and primarily is due to insufficient reactive

power a t load centers.

Voltage instability, as compared with angle instability, is a relatively slow

process, such that the time span of a typical voltage collapse is in the range of a

few seconds up to tens of minutes. To the contrary, angle instability can occur

within a fraction of second to about a few seconds. Due to the major difference in

the time-intervals associated mith the phenomena of voltage and angle

instability they are treated and investigated as decoupled phenomena.

Conventionally, angle instability phenomenon is treated as a dynamic

phenomenon and voltage-instability is dealt with as a (quasi-) steady-state

phenomenon.

A widely accepted approach for enhancement of voltage stability is to inject

reactive power (VAR allocation) a t critical nodes of the system. There are

various techniques proposedladopted for enhancement of angle stability. These

indude utilization of semiconductor-controIled appmatus, e.g. static VAR Compensator

(SVC), thyristor-controlled series capacitor (TCSC), STATCOM, Unified Power Flow

Controller (UPFC), Static Series Capacitor (SSC), Static Phase-ShiRer (SPS), and

semiconductor-controlled Under-Load Tap Changing (ULTC) transformer. Selection of

either of these apparatus for angle stability enhancement depends on specific conditions

of the system under consideration, e.g. system configuration and the steady-state

function of the apparatus.

SVC and STATCOM inject reactive power in the system and enhance angle stability

by regulating the corresponding terminal voltage at a desired value. Either of SSC, SPS,

UPFC and ULTC injects a controlled series voltage in the system and enhances angle

stability by means of regulating phase-angle andlor magnifxde of the injected voltage.

For ease of reference, this group of apparatus which are capable of injecting controlled

series voltage in a system are referred to as dynamic voltage regulators (DVRs). TCSC

assists in angle stability enhancement through dynamic control of the series impedance

of the corresponding transmission Iine.

- The main objective and contribution of this work is to demonstrate that fast voltage

injection by means of DVRs, primarily for enhancement of angle stabiliQ can result in a

strong coupling between voltage stability and angle stabilitg phenornena. This indicates

that enhancement of angle stability by a fast acting DVR can lead to voltage instabiliw,

unless appropriate precautions are in place. This coupling phenomenon has not been

reported in the technical Eterature and the reasons are:

(1) Introduction of SSC, SPS, UPFC and thyristor-controlled ULTC in power systems is

a relatively new idea and no operational experience exists.

(2) This coupling phenomenon cannot be detectediinvestigated by existing

analytical (eigen analysis) and time-domain simulation tools. The reason is that

existing production grade bols are not capable of representing adequately the

load dynamics with respect to voltage phenomenon in the systern model.

The other contribution of this thesis is the enhancement of the existing eigen analysis .

and transient stability (TS) software tools to demonstrate the coupling phenomenon in a

large realistically sized power system. This thesis concentrates on the impact of fast voltage regulation, by means of UPFC, SPS and semiconductor-controlled ULTC, on

voltage and angle instabiliity phenornena.

1.2 Objectives

1.2 Objectives

The main objectives of the thesis are:

To explain the detrimentai effect of fast DVRs on the phenomenon of voltage

ins tability.

To demonstrate voltage instability phenomenon when either UPFC, SPS or

thyristor controlled ULTC is adopted for angle stabiliw enhancement in a

realistic sized and interconnected system.

To develop analytical (eigen analysis) and digital time-domain simulation tools

capable of demonstrating couphg between voltage and angle stabiliw

phenomena.

Chapter 2 provides an overview of existing methods for investigation of voltage

instabiliw and their limitations.

Chapter 3 introduces component models used for simultaneous investigation of

voltage and angle stability phenomenon. This chapter also briefiy introduces the eigen

analysis software tool and the transient stability software tool which have been

developed for investigation of coupling phenomenon between voltage and.angle

dynamics.

Chapter 4 presents the concept of voltage instability due to fast voltage control by

means of DVRs.

Chapter 5 demonstrates the effect of conventional (mechanical) under-load

tap-changing (ULTC) transformer and fast (semiconductor-controlled) ULTC on voltage

and angle instability phenomena. The studies are conducted o n a real size,

interconnected power system.

Chapter 6 investigates the effect of a UPFC on voltage and angle instability

phenomena. The studies are d s o conducted on a real sized power system.

Final conclusions are summdzed in Chapter 7.

Chapter

Methods of Analyses of Voltage Instability

2.1 Introduction

The phenornena of voltage instabilitg/collapse have been observed in power

systems and analyzed extensively during the past decade [1,2,3]. The

seriousneas of the consequences of these incidences presents strong motivations

for research in the causes of voltage instabilities. Normdy, a large number of

parameters and device controllers are involved in the performance and

dynamics of a power system with respect to voltage instability, and voltage

dynamic (instability) phenomenon can be observed and investigated in

realistically sized power transmission/distribution systems. The objective of this

chapter is to provide an oveMew of the existing techniques for investigation of

voltage dynamic phenomenon.

The primary function of an electrical power system is to generate power at a

satisfactory voltage and fiequency for the consumer. In the early development

stages of electriciQ the generators were situated close to the consumers and

very simple electrical networks were sufficient to Mfil this function. With the

increasing demand for electrical energy, considerable new developments

occurred on the electrical systems. The electricity supplied to consumers

increased in both magnitude and the geographicd area covered. The size of the

generators increased and higher voltage transmission networks were

superllnposed over the lower voltage distribution networks. Interconnection of

2.2 Voltage Collapse

electricie suppliers made it economically more attractive to locate generators at sources

of coal or water power at considerable distances from the consuners. Thus, instead of a

simple network between the generator and the consumer, a fa r more complex

transmission system is utilized.

The elements of a transmission system consume more reactive power as an attempt

is made to transfer more real power. Therefore, although the basic function of an

electrical system remains unaltered, namely, to deïiver power from the generator to the

consumer, in the interconnected power systems, another essential consideration is to

supply the reactive power to the intemediate elements of the system. As the reactive

losses of the elements are proportional to the square of the current components passing

through them, the reactive requirement of a power system can alter drastically with the

real power variations of the load.

As transmission line voltages are increased, greater conductor spacings are

necessary, thus creating higher line reactances which results in higher reactive losses

and higher line voltage drops. Thus, EHV transmission installations have increased the

reactive power requirements of a power system relative to the real power demand to such

a degree that the gross reactive power demand of a power systern is in the same order of

magnitude as the real power demand. EHV transmission systems are usually developed

for systems with major power stations remotely located from load centres. In such

developments, the significant reactive reserves carried by these generators is attenuated

by the relative high reactances between the generators and the loads. The tripping of a

heavily loaded E W line produces a major increase in the system reactive demand, and

may eventudy cause voltage collapse in the bulk power system.

2.2 Voltage Collapse

Stability is one of the most important issues in a power system operation and control.

Behavior of active power has been studied by the conventional stabilitp criteria, such as

steady state stability, dynamic stability, and transient stability. Voltage stability is

associated with the abiliw of a power system to maintain bus voltage magnitudes within

specifïed operating limits. hadequate voltage control has caused major blackouts in

some of the existing power grids which have been very costly to the systems [3,4].

Inability of the power systems to supply the required reactive power demands are

evident in al l the reported incidents.

Voltage collapse & a power system can occur following a sudden increase in reactive

2.3 Voltage I m a bility Analysis

power demand if there are inadequate effective reserves available. The scenarios of

voltage collapse situations in a power system vary depending on the nature of the system

and disturbances applied to the system. A conventional voltage conapse scenario has a

pattèrn as follows [l,2,3].

Assume that a power system is under medium to heavy load demand and certain key

transmission lines are more heavily loaded than their normal conditions. Voltage

instability, and eventually voltage collapse, can be triggered by loss of a heavily loaded

transmission line, loss of a generator and so on. Such disturbances impose a

substantially higher reactive power demand from the system by the extra loading of the

remaining lines, thus, resulting in voltage drops throughout the power system. The voltage levels will be quicldy restored at the power stations by the generator voltage

control systems, but will be below the required levels at the Ioad centres. Then, the MW

and MVAr of the system generators will increase. The transformer taps a t the load

centres will be raised to their maximum settings to restore the load voltages. Thus, active load demand inmeases, and results in further voltage &op dong the transmission lines.

Therefore, installed capacitor banks and the transmission iine capacitances supply less

reactive power. Consequently, some or dl the system generators will reach their full

excitation levels, and the generator terminal voltage will decrease. If the voltage level

drops below the critical value, protection systems trip the overloaded generators, which in turn will trigger a cascading drop of voltage and eventually the initiation of a

widespread voltage sagicollapse.

2.3 Voltage Instability Analysis

Voltage stability is the ability of a power system to maintain a steady acceptable

voltage at each bus under normal operating conditions, and aRer being subjected to a

disturbance. A system enters a state of voltage instability when a disturbance; e.g., an

increase in the load demand or a change in the system conditions, causes a progressive

and uncontrollable voltage decrease 151. The &EE working group on voltage stability has defined voltage stability as the ability to maintain voltage so that when load admittance

is increased, load power will increase, and both power and voltage are controllable [3].

Based on the time frames of the reported voltage collapse incidences, voltage

instability can be categorized as transient and midterm voltage instability. The time

fiame involved in transient voltage instabiiity is nom fiactions of seconds to several

seconds. Induction motor dynamics aiid HVDC converters can be the major causes of

2.4 Static Appmach

instabilities in this t h e frame [3,6].

A voltage dip in the transmission system results i n more reactive demand by

induction motors, and in the stalling of induction motors, which worsens the voltage

instability condition. In this time fiame, generators and induction motors, due to their

dynamic interactions, should be represented as dynamic devices.

Midtem voltage instabilie has a time fiame of a few minutes. Load tap changers,

over-excitation of generators, load buildup by loads with constant energy controllers are

the main causes of voltage instabilities in this tirne name.

For the evaluation and analysis of voltage stability of power systems, three dinerent

approaches have been considered:

(1) Static approach

(2) Sm&-signal dynamic approach

(3) Large-signal dgnamic approach

A brief discussion of each approach folIows.

2.4 Static Approach

The static approach is based on dgebraic equations such as load flow equations. It

includes either a sensitivity analysis such as dQ/dV and dE/dV [7,8], a multiple load

flow solutions approach [9] or a feasibility of load flow [IO]. In a static approach, the

feasibiliw region, defked as the region with feasible load flow solutions, is studied. The

steady state region SS is described as [11,12]

SS(N) = { U ] ( U E FR) A ( y e S y ) )

where cl is the system driving function such as Vmf and omf. FR denotes the feasibility region given by

where x is a vector of state variables such as generator angle 6, anguiar frequency q generator interna1 voltage E, stator and rotor flux Y and system voltages Il In (2-2), h is a set of nonlinear fùnctions of the transmission system, and N is the network varying parameters.

The region in which all the operating limits are saeisfied, is given by S, such that:

where y is a vector of system variables, e.g. bus voltages, line fiows and reactive power

supplies, g(k,u) is a set of nonlinear functions of operating constraints.

To determine the voltage stability margin, the system is stressed until voltage

instability occurs. System conditions at different stress levels are obtained by solving the

load flow equations- In this approach properties such as sensitivities, existence and

multiple solutions of power flow equations are studied [13-161.

Various techniques based on the static approach have been developed for voltage

stability studies as described in the following sections.

V-Q sensitivi@ technique is based on sensitivity of bus voltage magnitude to reactive

. power injection at the same bus. A positive V-Q sensitivity means that the voltage at the

bus increases if reactive poweris injected into the bus. The system is voltage stable if all

V-Q sensitivities of the system buses are positive, and the system is unstable if V-Q sensitivity is negative for at least one bus.

V-Q sensitivity method does not identify unstable modes of the system. GeneraUy.

systern voltage instability does not occur due to voltage fluctuation of individual buses,

rather for individual modes to which each bus participates. Since the V-Q sensitivity

method does not give the unstable modes, it has limited application in the study ~f

voltage instability.

2.4.2 Singular Value Decomposition

The linearized steady state system equations are [19,20]

where, incremental change in bus real power injection,

incremental change in bus reactive power injection,

incrementd change in bus voltage angle,

AV = incremental change in bus voltage magnitude.

The jacobian matrix, 4 in the above equation contains the first derivatives of the

active power part, P. and reactive power part, Q, of the power flow equations with respect

to voltage magnitudes, V; and node angles, 8. If singular value decomposition is applied

to the power flow jacobian matrix, 4 then the matrix decomposition can be written as

i = t

- where n x n orthogonal matrices U and W are the right and left singular vector matrices

of J. The singular vectors ai and coi are the columns of the matrices U and W respectively,

and C is a diagonal matrizr with

where 0, 2 O for all i. The diagonal elements in the matrix Z are ordered so that

The minimum singular value, G, (J) , is a measure of how close toa singularity the power flow jacobian m a t h is. If the minimum singular value is equd to zero, then the

studied matrix is singular and no power flow solution can be obtained. The sin&&@ of

the jacobian matrix indicates that the inverse of the ma& does not exist. This can be

interpreted as an infinite sensitivity of the power flow solution to small perturbations in

the parameter values. At the point where on (J) = O several branches of equilibria corne together and the studied system will experience a qualitative change in the

structure of the solutions due to a small change in a parameter value. This point is called

a static bifurcation point of the power system [21-241.

The [AB ~ f j vector is then computed as

The inverse of the minimum singular value, a;', fkom a small disturbance point of

view, indicates the largest change in the state variables. Let

where un in the last colllmn of U. Then,

where w, is the last column of W. Voltage stability can be evaluated by on, un, and w,.

The smallest singular value, on, is an indicator of the distance to the steady-state

stability b i t . Also, un and w, corresponding to an indicate the most sensitive direction for changes of active and reactive power injections and sensitive voltages.

The modal analysis technique involves the computation of a small number of

eigenvalues and the associated eigenvectors of a reduced-order of the steady state

jacobian mat* of a power system [25]. The magnitudes of eigenvalues provide a relative

measure of proximity to instability. The eigenvectors, on the other hand, provide

information related to the mechanism of loss of voltage stabilitp.

Voltage stabiliw is afZected by both P and Q. Nevertheless, at each operating point we

can assume P t o be constant and evaluate voltage stability by considering the

incremental relationship between V and Q.

In (2-71, let

then

and,

where, .

J~ = JQV- J Q ~ J ; ~ JPV (2-13)

JR is cded the reduced jacobian matrix of the system, and it directly relates changes

in bus voltage magnitudes to increments in reactive power injection at the buses. Let,

where,

Therefore,

JR = R U

R = right eigenvector ma* of JR,

L = left eigenvector ma& of JR,

A = diagonal eigenvalue matrix of JR.

2.4 Static Approach

(2-14)

where Ri is the ia right eigenvector, and Li is the i* leR eigenvector of JR-

The magnitude of each eigenvalue determines the weakness of the corresponding

rnodai voltage. The srnaller the magnitude of X, the weaker the corresponding modal

voltage. If [Ail = 0, the i* modal voltage will collapse.

Singular value decomposition (SVD) technique and modal analysis technique

examine voltage stability of modes. Both techniques provide information regardhg the

voltage instability of the entire system.

2.4.4 Continuation Power Flow

The continuation method (or pathfinding method) is another class of methods for

soiving nonlinear steady-state equations which are used to determine voltage instability of power systems [26].

Multiple power flow solutions are likely to appear under heavy load conditions [9].

This is related to voltage instabüiQ especially when a pair of solutions are located close

to each other. However, when used to analyze voltage instability problems, most

conventional power flow tools break d m . The continuation power flow is a bo l capable

of dealing with the mathematical problems encountered during the analyses of voltage

problems.

Consider the solution of n nonlinear algebraic equations f ( x ) = O , where x is a

vector of n unknowns. Using the path-hding method, the problem is solved by analmg an easier version of the original problem, and then adding to the complexity until it becomes the original one.

2.5 Small-Signal Dynamic ApprocrcA

2.5 Small-Signal Dynamic Approach

In the analysis of voltage instabilities based on small-signal dynamic approaches,

dynamic devices are represented by differential equations, and then the linearized

differential equations dong with those representing the static part of the system are

solved to examine voltage stabiliw [1,17,18]. These linearized models are obtained nom a

nonlinear system model, which can be described as:

Machine and load dynamics: x' = f (x . u, t) Transmission System: h ( x , u,t) = O

Operating conditions: g ( x , u. t ) 5 0

From the dynamic equations of machines and loads, the general form of the

hearized dynamic model is:

hz' = L(X,U,N)AX+H(X.U,N)AU (2-19)

where hr, and Au are perturbations about an operating point. L(x,u,N) and H(xyuyN) are the linearized system and input matrices which depend on the states x, inputs u and the

network N. Since the system model is hem, the region of stable solutions of (2-19) in the

u space can be defïned by

where & is the ilh eigenvalue of L(x,,uJV).

2.6 Large-Signal Dynamic Approach

Most of the investigations associated with voltage instability involve the steady-state

response of the power system. This is as a resdt of the fact that the dynamics of a voltage

collapse process is complicated and has not been thoroughly understood. It is clear that

the voltage collapse dynamics cannot be descnbed solely by generator dynamics which is

traditionally believed to be responsible for transient angle instabiliw.

Various system components are involved in the dynamics of voltage collapse.

Unfortunately, the computation of the dynamics for large scale systems is an extremely

time consuming task and for some systems practicdly impossible. The key in this type of

analysis is to find the critical dynamic components. One approach to achieve this goal is

to eliminate the dynamic states which do not contribute to the a i t i ca i states.

2- 7 Conclusions

Conventionalls three mechanisms are assumed to play major roles in the dynamics of

a voltage collapse process. These are tap-changer dynamics, load dynamics particularly

those involving large percentage of induction motor loads, and generator field excitation

dgnamics [31. There are cases where the operating Iimits for a voltage collapse may not

be determined based on the system static models. This occurs when a static mode1 cannot

predict whether the system is stable or unstable at the bifurcation point.

Most multimachine dynamics can be described by a set of equations of

differential-algebraic form:

where r is a set of state variables, andy is a set of system variables and parameters. These differential-algebraic set of equations should be solved by digital time-domain techniques.

It is essential to employ appropriate models of the system components for evaluation and analysis of system voltage stability In this thesis, appropriate models for voltage stability analysis are introduced, and the voltage instability phenomenon caused by

interactions due to loads, ULTCs, UPFCs, AC/DC transmission systems and generators are analyzed using s m d signal and large signal dynamic methods described here.

2.7 Conclusions

A comprehensive review of the technical literature related to the phenornenon of

voltage stability in power systems reveals that voltage phenomenon is conventionally

viewed as a quasi-steady-state phenomenon. Therefore, extension of steady-state

andysis techniques are used for investigation/prediction of voltage collapse. Roliferation

of semiconductor-controlled power apparatus, e.g., HVDC converters, dynamic voltage

regulators, (DVR), static phase shifters, static under-load tap changers and unined power

flow controllers indicate that voltage dynamics cannot be viewed as a quasi-steady-state

phenomenon. This is as a result of rapid reactive power exchange among various

locations within a power system by means of semiconductor controlled devices. To the

best of our knowledge, as of now, no comprehensive investigation regarding the Wpact of

semiconductor-controlled devices on voltage dynamics has been published. The reasons

are: (1) analytical approaches and the corresponding existing software tools for

investigation of power system dynamics do not have adequate capabilities to represent

2-7 Conclusions

semiconductor-controlled devices and loads with respect to voltage dynamic

phenomenon, and (2) there are not enough operational experience and measurement

results to reveal the importance of the issue. It shodd be mentioned that a few reports

dealing with the phenomenon of voltage dynamics due to HVDC converter systems have

been recently published, which ernphasize the need for investigation of voltage stabiliw as a dynamic phenomenon.

Chapter

System Representation for Sirnultaneous Voltage and Angle Stability Study

rn 3.1 Introduction

The objective of this chapter is to introduce power system component models

and an overall system representation necessary for dynamic voltage stability

studies. The components are divided into two groups. The first group includes

conventional components, e.g., synchronous generators and transmission

networks. The second group consists of those components whose behaviors c a n

directly contribute and/or result in voltage instability. This group includes

induction machine loads, static loads with voltage dependency, under-load

tap-changers (ULTC), phase-shifting transformers, static phase shiRers (SPS), unifïed power flow controllers (UPFC), and HVDC converters.

This chapter introduces equivalent models of power system apparatus for

investigating voltage dynamics based on an small singnal stability approach

(linearized model) and a transient stabiIity simulation approach (nonlinear

model). When low-frequency dynamics, Le., f < 2 Hz, are of cancern, the

dynamics of transmission lines can be neglected and consequently the overall

system mode1 is substantially simplified. This chapter also provides the

component models when only low-Eequency dynamics are of concem.

3.2 Synchmnous Machine Model

Figure 3.1: Synchmnous machine d-axis and q-axis equivdent circuits

3.2 Synchronous Machine Model

Figure 3.1 shows the equivalent circuit representation of a synchronous machine in

the corresponding rotor d-q &ame [2q. The stator ia represented by one winding per axis.

The rotor is represented by (1) one equivalent damper circuit and one field winding on

the d-axis, and (2) two equivalent damper circuits on the q-axis. Based on the equivaent circuits of Figure 3.1, the dynamic behavior of the electrical system of the machine is given by (3-1) to (3-6).

3.2 Synchmnous Machine Mode1

where v, i, and R with d or q indices in (3-1) to (3-6) refer to the voltage, current, flux,

and resistance in d- or q- axes, respectively a> and CO, are rotor angdar frequency and the base angular fiequency of the synchronous machine. Also, indices f , 1, and 2 refer to the

field circuit and damper windings, and Ra is the armature resistance. Further details of (3-1) to (3-6) are given in Appendix A. The mechanical system of a synchronous machine which includes rotor body, rotating exciter and turbine sections is represented by a single

mass and its dynamics are given by (3-7) and (3-8).

where, H, Tm, kD and S are the inertia constant, mechanical torque, damping factor, and

the rotor angle of the synchronous machine, respectively The overall dynamic model of the synchronous machine, based on (3-1) to (3-8) is derived in Appendix A. Small-signal

dynamics of the machine is obtained by linearizing the &chronous machine differential

equations about an operating point. If the stator dynamics are neglected, the smd-signal model is given by

The pre-index i refers to the i-th synchronous machine variables, and As B, B, Cs, and Dus are given in Appendix A. AuDg and AiDQ are D-Q fkarne voltage and current vecturs

of the synchronous machine. QU includes control signals n o m governor and exciter se

systems. The vector of state variables is defined as

where 4 represents the state variables of the exciter system.

Small-signal dynamics of a synchronous machine, in the general case, when the

stator dynamics are included is given by

3.3 Induction Machine

where, the machine state variables are defïned as

Variables, input and output signals, and the associated matrices of (3-11) and (3-12)

are given in Appendix A.

3.3 Induction Machine

An induction machine as a nonlinear dynamic load has considerable impact on the

voltage dynamics of power systems. Consider an induction machine with a chosen

reference frame revolving at speed o. The machine equivalent circuits in the d and q axes

of the rotating reference fiame are shown in Figure 3.2 [28]. The machine dynamic mode1 is given by (3-13) to (3-19).

Figure 3.2: Equivalent circuits for an induction machine in a two axes rotating reference hune

3.3 Induction Machine

where or is the rotor angular fiequency, and a, is the base electncal angular fkequency. Indices s and r refer to the stator and rotor variables, respectively. m and TL are constants which depend on mechanical load. Te and Tm are the electrical and mechanical torques, and H is the inertial constant of the induction motor.

The small-signal dynamics of the induction machine j is developed from linearization

of (3- 13) to (3-19) about an operating point as given by (3-20) and (3-2 1); details are given in Appendix A.

Ai, = .c.(Az,) J J L~

where

,ATL is the mechanical load torque for the j-th induction machine. Matrices fii3 PiU pin and pi, are aiso given in Appendix A. If the stator dynamics are ignored, the

srnaIl-signal dynamic mode1 of the machine reduces to:

3.4 Static Load Madel

where,

/lin piun pin ,Ci,, and ,üiw are &O introduced in Appendix A.

3.4 Static Load Mode1

Voltage dependency of static loads is an important consideration in the phenornenon

of voltage dynamics. In general, a static load at bus Z including its voltage dependency

can be represented as [NI.

where, apb a~~ mi, and ni are constants which depend on the nature of the load. It should be noted that (3-24) and (3-25) are based on the assumption that network dynamics can

be neglected and the transmission network can be represented by algebraic equations which relate current and voltage phasors. If network dynamics are considered, then, static loads are mathematically modeled by ordinary differential equations and are represented as part of the network as will be described in Section 3.5.

Linearizing (3-24) and (3-25) provides a smd-signal mode1 of a static load:

where Ai, Aii, Au, and Aui are real and imaginary components of load current and voltage, and

3.5 Transmission Network Malel

3.5 Transmission Network Model

If network dynamics can be neglected, then algebrgc (phasor model) equations

representing relationships among network electrlcal quantities are used to represent the

electrical transmission network. In this case, the dynamics of machine stators are

neglected. Network dynamics can be neglected if only dynamics of interest are slow, e.g.

fkequencies about 2 Hz and less. This approach is used for network representation for

conventional angle stability studies.

If network dynamics are t o be taken into account, then each three-phase

transmission line is represented by its Ii model. Three-phase seriedparallel inductors,

capacitors and static loads are also represented by the corresponding lumped elements.

Thus the differential equations governing the dynamic behaviors of each set of

three-phase R, L and C components are developed. These equations are transformed into

the system globd d-q fiame as given by [28]:

u, = Ri,

where, (uJdq are the voltages and currents of the components in a dq frame. The overd network model is developed by combining the component equations and extracting a

state-space representation. It should be noted that the network state-space model is Iinear and consequently its small-signal and large-signal dynamics are given by one set of equations.

Ignoring the network dgnamics, inductor and capacitor equations will reduce to:

where XI and Y, are the reactance and susceptance of the network elements.

3.6 Model of Conventional ULTC Transformer

3.6 Mode1 of Conventional UlLTC Transfomer

3.6.1 Principle of Operation

F'dlowing a disturbance in a power system, normally, voltage a t the load side either

sags or rises; however, tap changing equipment starts to restore the load-side voltage to a

controlled level, and thus the load. Under ioad tap changing (ULTC) transformers and

distribution voltage regdators act similarly in regulating load-side voltage. A voltage

relay monitors load-side voltage. If voltage drifts, or jumps, outside a dead-band, a time

delay is energized. If the relay times out, the tap changing mechanism will be energized

and tapping takes place until the voltage is located in-band, or until the maximum or

minimum tap is reached. Once in-band, the voltage relay and timer mechanism are reset.

The fûnctional block diagram of a ULTC is shown in Figure 3.3, which consists of the

following elements [27] :

Tap changing mechanisrn driven by a motor unit,

Voltage regulator which consists of a measuring element and a time-delay

element

Figure 3.4 shows a block diagram of a ULTC ccntrol system. The measuTing element

Measured Voltages I I Tap Position

Reference Voltages

Figure 3.3: h c t i o n a l block diagram of ULTC

of the voltage regulator consists of an adjustable dead band relay with hysteresis. The input to the regulator is a voltage error, namely V,, .

Measuring Motor Drive

Unit

Tirne Delay - Element

Tap-Changer Element

3.6 Modd of Conventional ULTC Transfomer

The output of the measuring element is e, which takes values of O, 1, or - 1 , depending

on the input V, With a reguiator dead band of D and a hysteresis band of E, the output is:

O for -D 5 V,,$ D

O for D < V e , s D + e ; K, increasing

O for -D -E S Ver$-D ; Km decreasing

1 for V, ,>D+E

1 for DCV, , ,SD+E ; V, decreasing

-1 for Vorr<-D-&

-1 for -D-ESV',S-D ; Lincreasing

An adjustable time delay relay is used to reduce the effect of short-time voltage

Transformer

figure 3.4: Block diagram of ULTC control system

variations and to avoid unnecessary tap-position changes. The output of the time-delay

element is

O for t S Td; e arbitrary

1 for t > T d ; e = l

-1 for t > T d ; e = - 1

where Td is the adjustable time delay of the regulator. Td it is constant ( Td = Th ) for a regulator with independent tune delay. For a regulator with variable time-delay characteristics, Td is a function of initial tirne delay settïng Th, voltage error V, and dead-band D:

D Td = ,TT

err do

The motor drive unit and the tap-changer mechanism may be represented by a .

simple time delay TM inherent to the equipment. The output signal Ani represents an

incremental change in tap position, and is equal to O, 1, or -1. Incremental change of tap

. position hi in the i-th operation is:

O for tsTM p b arbitrary

1 for t > T M 3 b = l

-1 for t>TM 9 b = - 1

The per unit tums ratio after the i-th operation is

where Ani represents the per unit turns ratio step corresponding to a change in tap position by one step.

Figure 3.5 shows the tap changing logic for the tap movement. AV is the dead-band

and An is the step size corresponding to the tap movement logic of Figure 3.5. Each tap

change is a discrete t h e event. When the load-side voltage of a ULTC stays over V,+AV (or below V,-AV' for a certain time period, known as time-delay, the turns-ratio n is

decreased (increased) by hn. The continuous mode1 of the tapchanger can be represented

by the fist-order differential equation (3-35) which is the first-order approximation to

discrete t h e event tap changing.

3.7 Model of Phase Shifting Transformer

Figure 3.5: bgic'for tap movement of a ULTC

where T is the ULTC the-constant, and it is in the order of several seconds to minutes.

Having VI and V' representing the primary and secondas. side voltages of the

tap-changer, linearizing (3-35) and expressing it in a d q fkame, we deduce:

where - -

The primary and secondary voltages are related by:

Simiiar expressions relating currents at both sides of the tap-changer can be derived

as:

3.7 Model of Phase Shifting Transformer

A phase shifter injects a voltage in series with the transmission line voltage. The

injected voltage may be in quadrature angle, in-phase, or at any angle with respect to the

line voltage. The phase shifter's ability to control magnitude and phase angle of the

3.7 Model of Phase Shifing Transfomer

injected voltage is used to improve transient stabilitg, to eenhance damping and to control

steady-state power flow (29321.

A conventional phase-shifter can be approximated by a transformer having a complex

turns ratio, = nL0. Therefore, the voltage equations of a conventional phase shifter

in a d q fiame are:

Linearizing (3-39) about an operating point,

where indices 1 and 2 refer to the primary and secondary sides of the phase shif'ter, respectively 8 is the phase shift £kom the primary side to the secondary side of phase shifter; it is positive when Y1 leads V2, and

Similady, the current equation can be found to be:

Linearizing (3-41) leads to:

where

3-7 Model of Phase Shifting Transformer

It should be noted that the ULTC model is a special case of the phase shifting

transformer model given by (3-40) and (3-43) where phase shiR angle 0 is set equal to a

fuced value; i.e-, zero in this case.

In most practical system configurations, a phase shiRing transformer (or a ULTC) is

located at the texminal bus of the load for which voltage regdation must be carried out.

For such a condition, the load model and the phase shifting transformer model can be

combined and represented by a single mode1 as follows.

Consider a dynamic load with the atate space equations given by (3-22), and a phase

shifter which is represented by (3-36). Combining the load dynamic equations with the

ULTC or the phase shifber dynamic equation, and based on (3-40) and (3-42):

where, index 1 refers to the primary side of the phase shifter, and

The combination of load and phase shifter models is summarized in the block diagram of Figure 3.6.

3.8 Static Phase Shifier (SPS) Model

Figure 3.6: Block diagram of combination of dynamic load and phase shifter

3.8 Static Phase Shifter (SPS) Model

Figure 3.7 shows a schematic diagram of a phase shifter. The input to the phase

shifier is the three-phase voltage provided by an excitation transformer (ET). The output

of the phase shifter is a three-phase voltage (Vp) injected in the system by the series

boosting transformer (BT). The converter controls the magnitude andor phase-angle of

the injected voltage.

Converter l -

Figure 3.7: Schematic diagram of a static phase shiRsr

A conventional phase-shifter utilizes mechanical switches to control the injected

voltage Vp. A static phase shifter (SPS) utüizes semiconductor controlled converters ta

regulate the injected voltage. The speed of operation of a SPS is sigaincantly higher than

that of a conventional phase shifter. The impact of high speed voltage regulation, by

means of either a SPS or a thyristor controlled UETC, on voltage dynamics is the prime

concem of this fhesis work. There exists various power eIectronic converter topologies for

implementation in a SPS [29,35,40]. Principles of operations, limitations, and technical

merits of various SPS configurations have been reported in the technical fiterature [35]

and are not forther explained here. Among various circuit configurations proposed for a

SPS, the configuration shown in Figure 3.8 has attracted significant attention and is

referred to as Unined Power Flow Controller (UPFC). A UPFC uses two Voltage Source

Converters (VSCs) and a dc link as the interface between ET and BT. Each VSC has

independent control over phase-angle, magnitude and fkequency of its ac side voltage.

Depending upon the ccntrol strategy adopted, a UPFC can independently perforn the

following control fiuictions within a transmission system:

0 Line power flow regulation

Line reactive power flow regulation - Voltage regdation at bus B - Voltage regulation at bus E

Line , E

ET Inverter Inverter

figure 3.8: Schematic diagram of a UPFC

A UPFC is capable of performing the above functions during steady-state,

small-signal dynamics and large signal dynamics of power systems. Various control

strategies, applications and modelling requirements of a UPFC are addressed in E341. As mentioned previously, investigation of the impact of rapid voltage control at bus B of Figure 3.8, on voltage stabilitg, is one of the objectives of this thesis.

3.8 Static Phase S h i , , r (SPS) Mo&

figure 3.9: UPFC equivalent circuit at niadamental fkequency

3.8.3 UPFC Mode1 -

Based upon the principles of operation of a VSC, the UPFC of Figure 3.8 can be

represented by the equivalent circuit of Figure 3.9 1341. The circuit diagram of Figure 3.9

is a UPF'C representation only at the fundamental fiequency of operation. Fundamental

fkequency is defined as the nominal fiecpency of the power system. In Figure 3.9, each

VSC is represented by a three-phase voltage source and a unidirectional curent source

at the ac side and dc side respectively. Resistances and inductances identifg leakage

eiements of boosting and excitation transformers.

The three-phase voltages representing inverter 1 are expressed as [34]:

u,, = V,,cos (ot + 6,) Ueb = Vem Cos ( ~ t + 6e - 120') u,, = V,, COS ( ~ t + 6, - 240°)

1 where, V,, = ~ r n , v ~ , ; me and 6, are amplitude and phase angle of PWM control signal of the excitation inverter, and v h is the dc link voltage. Similar expressions are valid for

the boosting inverter with rnb and &, as the amplitude and phase angle of its control signal. With reference to Figure 3.9, the dynamic mode1 of the ac side can be written as:

" e a + reiea = us,, - V,,cos (ot + 6,)

3.8 Static Phase Shi'fter (SPS) ModeL

(3-47)

The dc iink dynamics are expressed by:

Equations (3-47) to (3-49) describe the overdl dynamic model of the UPFC. The UPFC mathematical model is expressed in a d-q frame, using matrix Pu as the transformation matrix.

3 Pu = diag {P, P, 2)

where P is the Park's transformation ma&

A state space small-signal model for the UPFC is obtained aRer transformation of

(3-47) to (3-49) to a dq name and linearization about an operating point:

3.8 Static Phase Shifter (SPS) Model

where,

A, = COI

1 1 1 1 B , = d i a g l y . p r g - 1

e e b b

3.8 Static Phase Shifter (SPS) Model

The state variables and control signals are:

3.8.4 Reduced-order Model of UPFC

Similar to the reduced-order equations of synchronous and induction machine, by

ignoring the network dynamics a reduced-order mode1 can be obtained for the UPFC. Denning,

equation (3-52) can be written as:

Let A i i , = O, then ikom (3-531,

3.9 M&l of ?ho-teminal NVDC Systems

A, = --5#-'9 I l 1

Bu, = -3#,'B,

Bu, = A, - S#r:A,

In order to combine UPFC model with the nefmork model, we d e h e a set of currents

pointing out f?om UPFC busses from both excitation and boosting sides as:

such that:

Moreover, (3-54) can be partitioned to separate indices related to the excitation and

boosting sides of UPFC such that

Ce, Cbv..., Dbm are sub-matrices obtained according ta the partitioning of (3-54). Substituting (3 -59) into (3-57) gives the 6nd output UPFC. currents:

Ai, = C,,Axvu + D,Avdq + D,AU,, (3-60)

where,

3.9 Mode1 of Two-terminal HVDC Systems

High voltage direct current transmission systems are integral parts of interconnected

power systems. HVDC connections are either in the form of a link, where the converter

stations are interfaced by a dc transmission line, or in the form of a back-to-back system,

where the converters are physically a t the same location. Multi-terminal HVDC systems

are not considered in this thesis. Operation of an HVDC converter either as a rectifier or

as an inverter is accompanied by consumption of reactive power. This reactive power is to

be provided by the system (transmitted to the converter station) or locally supplied by

capacitor banks, fdters and static (rotating) VAR generators. hadequate reactive power

at an inverter station can result in commutation failure and collapse of the EKûC Iink. Reactive power demaad of a HVDC converter under those operating conditions where

reactive power supply to the system is not optimal, can result in voltage instability and

even voltage collapse.

Neglecting harmonies and representing curent and voltages of a HVDC converter

only a t the fundamental fkequency, the equations that govem the relationship of current

and voltages of rectifier and inverter stations in a d q ficame are [26,27,36] :

- 3& 3 . "' - 'q Di i COS yi-EX,iti

where i,, i i , u, and vi are dc currents and voltages at rectifier and inverter sides. T, and Ti are tap changers of rectifier and inverter bridges:~, and vii are ac side voltages at

rectifier and inverter bridges. COS@, and cos(+ are the power factors given at the ac sides of the rectifier and inverter terminais. a, and yi are the rectifier ignition angle and inverter extinction angle, respectivelp. Also, voltages and currents with indices of d and q

refer to those in d-q h e .

The rectifier current controller can be modelled as:

Figure 3.10: DC transmission

cos a, = x, + k, [ iref - i,]

where x, is an intemal state variable, and imfis the seference current for i, . K i and kp are coefficients used in the control circuit of the rectifier. Simiiarly, the inverter extinction angle control at the inverter side can be represented as:

Using the Tkection representation of DC transmission line of Figure 3.10

where, p is a denvative operator. Equations (3-61) to (3-75) provide a state-space model of a two-terminal HVDC system. Linearizing (3-61) to (3-75) gives a smdl-signal dynamic

model of the HVDC system:

where

3.10 System Equations

Matrices As, B,, B,,, B,, Cs, D,, Qu. and Dm are given in Appendix A.

Rearranging (3-76) yields:

where

A d = l ~ i ~ ~ Aiqr Aidi A$J

3.10 System Equations

The mathematical models of the power system components deduced in Section 3.2 to

3.9, including mathematical models of controls, are used to develop the overall

mathematical model of a power system for

eigenvalue analysis, and

transient stability studies.

3.10.1 Eigen halysis Software Tool

The developed eigenvalue analysis package utilizes the linearized mathematical

models of the system components and (1) constructs a state-spacs mathematical model of

the system, and (2) uses a Q-R eigen analysis approach to calculate al1 the system

eigenvalues. The developed state-space formulation also provides eigenvectors,

3.11 Conclusions

participation factor and the transfer fiuiction with respect to a defhed set of single-input

single-outputs (SISO).

The approach adopted to systematically construct the system matrix is described in

[44,46]. Contribution of this thesis t o the development of the eigen-analysis package is:

r Extemion of the approach provided in [44,461 for systematic inclusion of models of

induction motor load, static voltage dependent load, ULTC, UPFC, static

phase-shifter, two-terminal INDC system and their correspondhg control in the

overall system matris;.

The eigen analysis package utilizes MATLAB routines for numerical eigen-value

calculations. The package provides the option either to include dynamics of transmission

lines in the overall system equations or to represent Iines as static components based on

their phasor models. Steady-state values, about which the linearization process is carried

out, are calculated fkom a three-phase power flow program [43].

3.10.2 Transient Stability Software Tool

The non-linear mathematical models of the power system components descnbed in

Section 3.2 to 3.9 and mathematical models of control system were formulated as a

system of differential-dgebraic equations [42]. The Werentid equations describe the

dpnamics of the system and the algebraic equations represent the network phasor model.

Systematic development of the overd system equations fkom the mathematical models

of the components are deseribed in 1431. A multi-step integration algorithm [47,48] is

adopted to numerically integrate the equations. The main feature of multi-step

integration techniques over other techniques, e.g., Runge-Kutta methods is that they can

handle stiE equations very efficiently. Thus, the sof'tware package can be used as a

computationally efficient tool t o investigate both short-term and mid-term transient

stability. The steady-state initial values for multi-step numerical integration were

calcdated fiom a three-phase power-flow program [431. The contribution of this thesis to

the development of the s o h a r e package for transient stability studies is systematic

inclusion of the dynamic models of induction machine load, voltage dependent static

loads, UWC, UPFC and static phase shifker in the overall system differential-algebraic

equations [48].

3.11 Conclusions

The main objective of this chapter is to develop power system study tools applicable

3.11 Conclusions

for simultaneous voltage and angle stability studies, in particular when the system

under investigation includes semiconductor-controlled apparatus, e-g. stat ic

phase-shifters. To meet the objective, state-space models of induction motor load,

under-load tap-changer m T C ) , H M C two terminal system, static phase-shifter and

their controls are developed. The developed models are combined with the state space

models of conventional power system components, i.e., synchronous machines,

transmission lines and static loads.

The overall system model is tdored for (1) eigen-analysis and (2) transient stabiliw

studies. Eigen analysis is based on the Iinearized system mathematical mode1 about an

operating point. The developed eigen-analysis s o h a r e bol permits both inclusion and

exclusion of network dynamics in the overall system model. The Iinearized model can

represent system dpamics within the fkequency range of a fiaction of a Hz (e.g., 0.001

Hz) up t o about 60 Hz. Transient stability studies are performed based on step-by-step

numerical integration of the system (nonlinear) equations.

The eigen-analysis and t rans ien t stabil i ty software packages provide a

comprehensive set of tools to investigate voltage and angle dynamics of power systems

under small-signal perturbations and large signal transients. Application of the software

tools are reported in Chapter 5 and 6 of this thesis.

The salient advantage of the above software packages over the exissting production

grade eigen-analysis and transient stability programs is that they can simultaneously

represent both angle and voltage dynamics of a power system. Thus, they can be used for

comprehensive stability analysis of power systems. The existing software tools are

fomulated with respect to angle stability phenomenon and do not accurately predict

voltage dynamics. This shortcoming is due to lack andlor inadequacy of cornponent

models with respect to voltage dynamic phenomenon.

Chapter

4.1 Introduction

Realistic scenarios corresponding to voltage dynamics and instability only

can be observed and investigated in large, interco~ected power systems, i.e.,

systems which include tens of buses and several generating and load centers.

Such studies are conducted based upon the time-domain simulation approach,

and plots of voltage versus time are used to identify voltage behaviour. The

studies provide accurate results but do not give much insight into the nature of

voltage dynamics. Eigen-analysis is an approximate method which provides

more information on the nature of voltage dynamicslinstability, e.g. level of

participation of each component in voltage behaviour. However, this method also

does not provide a clear picture with respect to the nature of voltage dynamics,

particularly in large systems.

The objective of this chapter is t o investigate the phenomenon of voltage

dynamics in a simple, srnall system and provide a qualitative explanation of how

fast voltage regdation may have an adverse impact on voltage stability. Instead

of using tirne-domain simulation and eigen analysis software tools described in

Chapter 3, basic equations which govern voltage behaviour of the system are

developed in a step-by-step fashion to provide a better explanation of the

phenomenon.

4.2 Voltage Behaviour Under Steady-State Operation

D m Y Load

figure 4.1: A power system with voltage regdator

4.2 Voltage Behaviour Under Steady-State Operation

To demonstrate the effects of dynamic loads and fast voltage regdation by either a

ULTC or a UPFC on voltage stability ofpower systems, the radial power systern shown in

Figure 4.1 is considered. The network shown in Figure 4.1 coaaects a generating area to

a load area through a voltage regulating device, and the system data are given in

Appendix B.

In voltage stabiüty studies, not only the effects of voltage dependency of load should be considered, but also the dependency of the load to voltage deviations dV/d t should

be taken into account. Different models for power system loads which include voltage

dependency a d o r voltage deviation ( d V / d t ) dependency, have been reported in the

literature [37-391. A general yet simple mode1 of load for voltage stability studies can be

expressed by

The load mode1 of (4.1) consists of two parts; i.e., a voltage dependent static part, and

a dynamic part which is dependent on the voltage rate of change. The system nequency

is assumed t o remain unchanged as it is the case in most voltage collapse incidents. The

static parts of the load are described by polynomial models, and the dpamic parts by the

e s t order dinerential equations:

4 2 Vottage Behaviour Under Steady-State Operation

k, and kp are positive constant parameters which represent the dynamic nature of the load. JP and iq are constant power loads. b1 and al represent constant current loads, and

b2 and a2 refer tO constant impedance loads.

Equation (4.4) states that a reduction in reactive power is an indication of voltage

reduction and vise versa. Under a steady state condition the dynarnic term vanishes, and

the static equations describe the load behaviour.

O .5 1 Load Voitage [PU]

Figure 4.2: Q-V plots of the study system (V, = 1. I pu, Xl = 0.5 pu).

Neglecting the system losses, the load active power and reactive power, in terms of

the parameters of the system of Figure 4.1, are given by:

where q represents the overall reactance of the transmission network interfacing the generating area and the load centez V, and V are the sendîng and receiving end voltages,

4.2 Voltuge Bekuiour Un&r Steady-State Operation

Figure 4.3: Leaf diagrams of the study system for various values of the load level J!Q

respectively. n is the voltage ratio of 'the second- side with respect to the primary side of the voltage regulator. In a ULTC this ratio is referred to as the tap ratio. Figure 4.2 shows the Q-V characteristics of the system. Figure 4.2 identifies P and Q that can be

delivered to the load centre under steady-state conditions.

Figure 4.3 depicts n-V plots of the voltage regulator for various levels of constant

power load. Equations (4.2) and (4.3) are used to represent the static part of the load. In

a conventional ULTC, n varies within 20% to 30% of the nominal tap ratio. However,

steady-state voltage regdation of a UPFC can be within S O % of the nominal voltage

value. Each plot in Figure 4.3 is referred to as a Ieaf diagram. The area of each leaf

diagram is a measure of relative proximity to voltage instability corresponding to the

steady-state conditions [3]. This is due to the fact that a leaf diagram with a large area

4.3 Voltage Behavwur Under Dynamic Condifions

also has a large region of attraction, while a leaf diagram with small area has small region of attraction and it is more prone to voltage instabiliw.

When load increases, the area of the corresponding leaf diagram shrinks until it

vanishes a t the bifurcation point. The shrinkage is a result of load dependency on

voltage. The smaller the area of the diagram, the more prone is the system to voltage

instability. Variables Q, V; and n in Figures 4.2 and 4.3 also assume some numerical

values which camot be permitted in a real system. Therefore, the reader should be

cautioned that only a limited portion of Q-V and P n region of Figures 4.2 and 4.3 are

practically acceptable. However, the whole Q-Vand P n region are illustrated to provide a

better description of the systern behaviour.

4.3 Voltage Behaviour Under Dynamic Conditions

Voltage dynamics of the power system in Figure 4.1 is described by (4.1) to (4.6) and

can be simplified as:

where,

such that

1 vsv PSzn v f l = , [ q ' o d " ' " V " V ) - ~ - ~ s ] Q

P, and Qs are the static parts of the loads. The equilibrium points associated with the systern correspond to li = O. With V, and n, as the voltage and tums-ratio of the voltage regulator at the equilibrium point, and Pso and Qs, as the load power values when evaluated at Vo, the equilibrium points for V, are obtrcined firom:

4.3 Vol fage Behav w ur Under ISynamic Conditions

Equation (4.8) identifies a leafdiagram which depends upon load level ho (P,,, Q,) . Figure 4.4 depicts a schematic graph of (4.8) for an arbitrary load level to ident* the

structure of equilibrium points. The nature of equilibrie points, obtained by solving

(4.8) at V,, are identified by an eigenvalue analysis of (4.9) which is obtained from

linearization of (4.7).

where

1 An, = -T A,, = O

The equilibrium points of the linearized system in Figure 4.4 are shown by points A

and B, where point A is a spiral point or a stable node, and point B is a saddle point. The

left part of the leaf diagram of Figure 4.4, corresponding to the spiral points as well as

stable nodes, contains stable equilibrium points. To the contrary, the right part of the leaf

diagram, corresponding to the saddle points, contains unstable equilibrium conditions.

Figure 4.4 also shows that the trajectories near to a spiral point (point A) eventudly rest

at that point, and the system is stable. Conversely, the trajectories close to a saddle point

(point B) detract and indicate unstable conditions.

Consequently, fkom a small-signal analysis point of view, equilibrium points located

on the leR part of the leaf diagram are stable. Therefore, any incoming trajectories to the equilibrium points are attracted by these points. On the other hand, the right part of the

leaf diagram identines unstable equilibrium points where no trajecto~es can rest. This is

4.3 Voltage Behauiour Under Dynarnic Conditions

Figure 4.4: Equilibrium points and the corresponding trajedories of the study system.

an important characteristic of the leaf diagram which can show the behaviour of the

system under both small-signal and large-signal dynamics. A description of large-signal

stability in close relation with the leaf diagram will be given in the next sections. The following sections also explore the efZects of time constant of voltage control devices such

as under load tap changer and UPFC on voltage stability of a power system, under

different operating conditions.

4.3.1 Effect of Conventional and Fast Voltage Regulator on Voltage Dynamics

Figure 4.5 shows the system dynamic response to a disturbance. Figure 4.5 is

obtained by solving the se t of nonlinear dynamic equations given by (4.7). The

disturbance is introduced as a result of a sudden change in the Ioad characteristic fiom

load level LI to load level L2 as given in Figure 4.5. The pre- and post-disturbance load

parameters are also given in Figure 4.5.

Point O in Figure 4.5 corresponds to the pre-disturbance operating condition. Point A

is the desired post-disturbance steady-state operating condition which is also a stable

spiral point. Point A is determined &om the system steady-state equations based on the

post-disturbance load parameters. Although the steady-state equations indicate stable

operating points (O and A) corresponding to pre- and post-disturbance load parameters,

due tu the system nonlinear characteristic, the deparhire fiom O may not conclude a t A

and the system c a n become unstable. Figure 4.5 shows that slow behaviour of voltage

regulator CT =300.0s and T=25.0s) results in stable departure of the operating point from

4.3 Voltage Behuiour Under Dynarnic Conditions

O to A. To the contrary, due to the fast response of voltage regulator (T=20.0s and

T=l.Os), the departure is not successful and voltage instability is experienced. The voltage coilapse phenornenon in the latter two cases can be described as follows.

When T is large, the Q-V trajectory follows the system conditions, as shown in

Figure 4.5 (a), and directly approaches the equilibrium point without becoming close to

the saddle node. Thus, the system remains voltage stable. A better insight is obtained

from doser examination of Figures 4.4 and 4.6. Figure 4.6 shows plots of V(n) for various

values of 2: which are obtained by solving the set of nonlinear equations of (4.6). Figure

4.6 also shows t h e leaf diagram corresponding t o the desired post disturbance

stëady-state condition. The starting point for trajectories is point O. The desired end

point for trajectories is the spiral point A- When T is relatively large, the trajectory

approaches to an equilibrium point (either A or B) via a spiral path - which is also a

stable path - shown in Figure 4.4, and finally is attracted by the stable operating point A.

In contrast, when T is relatively small (c25s), the trajectory approaches to an

equilibrium point towards the saddle node shown in Figure 4.4. Thus, the trajectory is

Figure 4.5: Trajedories of Q-V to identify departure fkom the pre-disturbance point O to the desired post-disturbance point A for various values of voltage regulator time-constant. Load parameters at O are aOz0.28, al=-0.28,a2=0.2, bO=bl=b2=0, Load parameters at A are a0=0.68, al=-0.28, a2=0.2, bO=bI=b2=0.

4.3 Voltage Behavwur Under Dynarnic Conditions

figure 4.6: Plots of V(n) for various T values and the post-disturbance leaf diagram.

placed among the system eigenvectors such that state n increases and state V decreases.

In this case, voltage sags monotonicdy, and voltage collapse occurs.

4.3.2 Region of Attraction

Voltage behaviour also can be determined by examination of steady-state and

dynamic characteristics of the system given by (4.101, (4.11), and Figure 4.7.

Equations (4.8) and (4.10) give all the equilibrium points, $ for which = O . This is shown by the leaf diagram corresponding to the post-disturbance load level in Figure

4.6. The a x a bounded by the leaf diagram 3 corresponds to O> 0, and the area outside

that to < 0. 3 consists of two çubsets: %! and ZR. is loci of all stable points, while 36! is that of unstable points. These regions are marked in Figure 4.7. Similarly, equation

(4.81, shown by a dashed line in Figure 4.6, corresponds to li = O . For any point on the

system trajectory above that lins, is negative. On the other hand, points on the system

trajectory which are below that, are characterized by >i > 0 .

4.3 Voltage Behavwur Under Qmarnic Conditions

Figure 4.7: Dynamic characteriatics of the system trajectories.

The region of attraction is defined by the set of alI trajectories. gr$, such that following

a disturbance, they start from a stable operating point and rest a t another stable

operating point. The pertinent trajectory is a stable trajectory, by definition. The region

of instability is complernentary to the region of attraction, and the trajectories are

entitled by unstable set of f i

Based on the above defînitions, (4.7). (4.10), and (4.12). the region of instability can be

characterized by

4.3 Voltage Behvwur Under Dynarnic Conditions

where,

If a disturbance, e.g. reduction in load level, forces the system trajectory to move

above V,, the origin of trajectory will be inside the target leaf diagram. Therefore,

V > 0, and voltage rises. This resdts in a negative n , and the fxajectory would be a

stable one.

However, oRen the voltage drops &ter most disturbances, due to an increase in load

level, o r a generator/bansmission outage. In this case, the pattern of a trajectory can

reveal if the trajectory is stable or unstable. The trajectorg, stable or unstable, starts with

a direction characterized by lh O and < 0. The reason can be described using

Figure 4.7. Following a disturbance, the initial condition is located outside the target leaf

diagram which results in falling the voltage below 5.

Consequently, following a voltage reduction, a trajectory can have three patterns:

(1) It terminates at the stable operating point, A, without passing it. The trajectory is an

element of rh with very large T.

(ii) It passes the stable operating point, A, inside the leaf diagram and crosses line

V = V, , n = O . The trajectory is an element of 34 with moderate III

(üi) It passes the stable operating point, A, and crosses the subset 3R of the leaf

diagram. The trajectory is an element of with smail T.

The trajectories in the former two patterns are in the region of attraction, and thoae

in the latter pattern are in the region of instabiliw. AU trajectories enter the leaf diagram

during which voltage recovers. h pattern 2, there is an instance in which n = O ; fiom then on, the trajectory returns to the stable operating point from outside the leaf

diagram. The reason is that, inside the leaf diagram, V > O and voltage keeps rising

until it hits the leaf diagram and passes it. Then, becomes negative and voltage

stabilizes, as f i É O .

In contrast, in pattern 3, the trajectories cross region 3& with = O , and enter the

4-3 Voltage Behviour Under Dynamic Conditions

region with v< O . Voltage will then falI monotonicallg, for 6 > O . The plane shown in Figure 4.8 illustrates the voltage instability region. The region of instability is a set of

hyperbolic functions with respect to voltage regulator's time constants. As it is shown,

the smder the II: the more prone the system to instabili@

Figure 4.8: Region of instability

4.3.3 Effect of Limits of a Voltage Regulator on Voltage Dynamics

For a voltage control limited to 20%, Figure 4.9 shows the system response t o the

slow and the fast voltage regulator operations. Figure 4.9 indicates that voltage

regulator action stabilizes the load voltage at 0.78 per unit. Figure 4.9 concludes that

imposing the voltage control lixniters in a DVR can result in stabilizing the voltage

dynamics at a reduced voltage.

4.3 Voltage Behaviaur Under Dynarnic Conditions

Figure 4.9: Changes in the load voltage and reactive power as a r e d t of the disturbance when the tap setting of the voltage regdator is limited.

Figure 4.10: Dynamic response of the system of to a sudden inerease in the load reactive power demand for UPFC

4.4 Conclusions

4.3.4 Effect of UPFC overd time constant on Voltage Stabiüty

Figure 4.10 shows the response of the system of Figure 4.1 to the same disturbance

and operating conditions as described in section 4.3.1, if the voltage regulator consists of

a UPFC. The UPFC confzols the magnitude of the voltage, and does not injectlabsorb

reactive power. Figure 4.10 indicates that if the UPFC's time-constant is relatively large

(about 3091, a new steady-state operating point is achieved and the system is voltage

stable. However, if UPFC rapidly responds to the disturbance (T=ls), voltage collapse

occurs.

4.4 Conclusions

The impact of a fast voltage regulator on the phenomenon of voltage instability is

investigated and compared with that of a conventional (slow) voltage regulator. The

studies conclude that a fast dynamic voltage regdator accelerates and causes voltage

collapse phenomenon in a power system as compared to a slow voltage regulator. The

concept and fundarnentals of this phenomenon is demoykated in a radial power system.

The inability of a power system to supply the required reactive power is the main cause

of the voltage instability. A power system with a fast dynamic voltage regulator becomes

voltage unstable as it cannot supply this requirement. On the other hand, the dynamic

trajectory of a voltage regulator with a 'slow speed of response is attracted by the region

of attraction, and the system is relatively more stable with respect to voltage *

phenomenon.

Chapter

Effect of Semiconductor Controlled ULTC on Voltage and Angle Dynamics

5.1 Introduction

Throughout the world there have been several reported incidents of voltage

instability caused by unexpected changes i n load levels, sometimes in

combination with unusual conditions in the systems, or by network disturbances

such as the loss of an important transmission line, a transformer o r a generator.

In addition to the present severe operating conditions of power systems there

are newly introduced system components, such as semiconductor controlled

ULTC and WFC, which exhibit adverse effects on voltage stability and these

effects have not been studied,

This chapter examines the phenomenon of voltage dynamics in contrast with

angle dynamics, due to rapid voltage control in a relatively large (28-busbar),

inter-connected HVDC/AC power transmission system with potential problems

in t e m s of angle stability. Effects of load model and ULTC time-constant on

voltage dynamics are discussed. The studies are conducted based on an eigen

analpsis appmach. The eigen study results are validated by digital the-domain

solution of the nonlinear system model, using a transient stabiliw program.

5.2 Study System

Figure 5.1 shows schematic diagram of the power system selected for the

5.3 Efect of UZTC on Voltage Dynamics

studies. Figure 5.1 represents a skeleton of the high voltage transmission system at the

western part of the United States. The system includes eight equivalent generating

stations with a total capacity of 20050 MVA. The system supplies nine major load

centers. The AC transmission lines are at 500 kV and the DC tie is composed of a bipole

DC Link rated at k450 kV, 2000 MW. The system parameters are given in Appendix B.

Under a Ml-load condition, Load#3 requires 950 MVA at 500 kV G5 is rated at 150

MVA. Therefore, under fiIl-load conditions, Load#3 is dominantly supplied by remote

generators through the transmission grid, primarily through the comdor composed of

lines LI0 to L13. The system is to provide this load without experiencing voltage o r angle

instability when lines LI1 t o L16, L25, L30 and L31 are in service. If line L25 is

disconnected, Load#3 and Load#4 are to be supplied through the transmission comdor

which is composed of lines L10 to L13. If Load#3 and load#4 are at their rated values,

then voltage sag at buses A, B, C and D is experienced. This voltage sag at buses C and D is beyond the acceptable limits. Thus, partial load shedding at bus D has to be performed

to maintain the rest of the load in service. The approaches to overcome this problem are:

(1) to install multiple static VAr compensators at buses A, C and D, (2) to install

controlledseries capacitors in lines LI0 to L13, or (3) to install a power regulator to

maintain the required power flow within the acceptable voltage limits. The aim in this

thesis is not to provide guidelines to choose among alternative DVRs, but to investigate

impact of a selected DM( on voltage/angle dynamics. Figure 5.1 shows that a power flow

regulator between buses A and B can be Lsed to maintain the required power flow within

acceptable voltage lirnits. Therequired voltage regdation to ensure power flow can be

carried out by either a ULTC or a UPFC.

This chapter examines the impact of a thyristor-controlled ULTC on voltage and

angle dynamics of the system.

5.3 Effect of UETC on Voltage Dynamics

The system component for eigen analysis are represented according to the models

described in Chapter 3. Excitation system models of generating units G1 to G5 are

represented by the block diagram of Figure 5.2 [41]. Excitation systems of generating

units G6 and G8 are represented by the block diagram of Figure 5.3. ~ara.m&rs of the

excitation systems are given in Appendix B. Dynamics of the governor systems are

neglected and input mechanical power to each unit is assumed to remain cbnstant during system dynamics.

5.3 Eficf of ULTC on Voltage Dynarnics

Figure 5.1: Schematic diagram of study system

5.3 Efict of ULTC on Voltage Dynamics

Generator 1 Terminal Voltage

Figure 5.2: Ercitation system Tgpe ST3 used for generating units 01 to 05

Generatxu Terminal Voltage

U~rnin Figure 5.3: Excitation system Type BBCI used for generating Utlits G7 and G8

The ULTC is installed between buses A and B of Figure 5.1. The ULTC injects a

quadrature phase-angle voltage, with respect to the voltage phasor a t bus A. The

magnitude of the injected voltage can be up to 46% of its rated terminal voltage. The

ULTC is rated at 800 MVA, 550 kV. The leakage impedance of the ULTC is assumed to

remain constant at 0.001+j0.06 per unit.

Figure 5.4 shows the block diagram of the ULTC control system. Measured power Pm is compared with the reference value (Pref) and the power deviation (AP) is used as the

error signal to adjust changes in the magnitude (AV) of voltage at busbar B of Figure 5.1.

the power deviation signal is conditioned by means of a low-pass filter. The fûnction of .

the low-pass filter is to minimize the effect of power oscillatory signals due to system

resonance modes and mechanical torsional modes which are usually within the

fkequency range of 10 to 50 Hz. The conditioned &P is fed to a PI controller. Transfer

fùnction k3/ (1 + sTd) represents the time-constant associated with the operation of

5.3 E ' c t of ULTC on Voltage Dynarnics

Auxifiary Signal 1

Figure 5.4: Block diagram of ULTC control

the ULTC. Time constant Td varies in a wide range depending upon the type of ULTC. Impact of Q on the stability of the overall system is investigated in the following

sections. In a semiconductor-controlled ULTC, auxiliary s i p a l s can be added to either

junctions (1) or (2) to enhance system damping for higher fkequency modes, e.g. torsional

modes. Such stability signal has not been considered in the reported studies. Typical

parameters of the control system of Figure 5.4 is given in Appendia B.

5.3.1 Effect of Load Composition on Voltage Dynamics

Figure 5.5 shows loci of the eigenvalues associated with the two least stable voltage

modes of the system at different load levels corresponding to Load#3. The ULTC control

is disabled, and the magnitude of the injected voltage is adjusted according to the load

flow study results. Load#3 is cornposed of 20% induction motor load a t 0.87 lagging

power factor and 80% k e d impedance load at 0.91 lagging power factor. Contribution

factors indicate that one mode is primarily innuenced by Load#3 and one mode by the

voltage regulator of the HVDC inverter. Both modes are well damped and changes in damping factors and frequencies of both modes, due to changes in Load#S, are

insignifïcant.

Figure 5.6 shows loci of the voltage mode eigenvalues associated with Load#3 and the

HVDC converter when Load#3 is composed of 60% induction motor at 0.87 lagging power

factor and 40% fixed impedance at 0.91 lagging power factor. Cornparison of Figure 5.5

5.3 Efict of ULTC on Voltage Dynamics

-25 -20 -15 -10 Real WS)

Figure 5.5: Least stable voltage mode eigenvalues of the test system. ULTC control is disabled. Load#3 is composed of 20% induction motor and 80% h e d - impedance.

I I I 0.15

-25 -20 -15 -10 Real W s ) ]Figure 5.6: Least stable voltage mode eigenvaiues of the test system.ULn: contml is

disabled. Load#3 is composed of 60% induction motor and 40% fked impedance.

5.3 Efict of ULTC on Voltage Dynamics

and Figure 5.6 shows that for the given condition, the voltage mode eigenvalues

associated with Load#3 change as a function of load#3 composition, but these changes

are practically insignismt. Comparison of Figure 5.5 and Figure 5.6 also indicates that

the impact of Load#3 composition on the voltage mode associated with the HVDC is

Figure 5.7: Voltage mode eigenvalues of the test system associated with Load#3.ULTC control is disabled. Load#3 is composed of 100% indudion motor.

Figure 5.7 shows loci of the eigenvalue of the voltage mode associated with Load#3

when it is composed of 100% induction motor operating at about 0.87 lagging power

factor. Comparison of Figure 5.7 with Figure 5.5 and Figure 5.6 shows that:

0 Increase in the percentage of induction motur load for a given operating condition

( k e d load MVA) reduces the damping of the voltage mode.

0 For the given system configuration and the operating condition (ULTC control

disabled), the voltage mode eigenvalue associated with Load#3 is well damped

and is not of any practical concem.

The voltage mode eigenvalue associated with the KVDC inverter control is not

iduenced by various conditions considered for Load#3, and is not discussed any M e r .

5.3 Effect of ULTC on Voltage Dynarnics

5.3.2 Effect of Transmission System on Voltage Dynamics

Figures 5.8 to 5.11 illustrate eigenvalue loci of the voltage mode associated with

Load#3 and the inertial mode of G5 when either of lines LI0 or L25 are out of service and

the ULTC control is disabled. Figure 5.8 shows the impact of magnitude and composition

of Load#3 on the eigenvalue of voltage mode when line LI0 is out of s e ~ c e . Figure 5.8

shows that the damping of the voltage mode is signincantly affected by the composition

of the load, i.e. percentage of induction rnotor load. Figure 5.8 also indicates that the

instability of the voltage mode can be of practical concern when the load is

predominantly induction motor. Figure 5.9 shows the effect of magnitude and - composition of Load#3 on the loci of the eigenvalue associated with the inertial mode of

G5 when line LI0 is out of service. Figure 5.9 indicates that when Load#3 is adjusted a t

about 100% of its rated value, generator G5 becomes unstable (locus C) or very close to

instability point (loci A and BI. Thus, for the system to be operational when line LI0 is

out of service, either Load#3 must be adequately reduced or other counter-measures, e.g.

ULTCILTPFC, must be adopted to maintain angle s tabw.

Figure . 5.10 . shows loci of the eigenvalue associated with the voltage mode when line

L25 is out of service and load#4 is adjusted to 765 MVA and a power factor of 0.87

lagging. Figure 5.10 indicates that voltage instability can occur when Load#3 becomes

1 Im (rad/s)

-20 -15 -10 -5 Real (Lfs) fime 5.8: Voltage mode eigenvalue associated with Load#3 when L10 is out of

service (ULTC control disabled). A.- Load#3 is composed of 20% induction motor and 80% fixed impedance. B: Load#3 is composed of 60% induction motor and 40% e e d impedance. C: Load#3 is composed of 100% induction motor.

5.3 Efict of ULTC on Voltage Llynamics

O Real 0e2

Figure 5.9: InertiaI mode eigenvalue of generator G5 when LI0 is out of service (ULTC control is disabled). A: Load#3 is composed of 20% induction motor and 80% fked impedance. B: Load#3 is composed of 60% induction motor and 40% fixed impedance. C: Load#3 is composed of 100% induction motor.

more than 40%. Figure 5.11 shows loci of the eigenvalue of the inertial mode of generator

G5 under the same operating conditions. Figure 5.11 indicates that angle instabiïity i s

also experienced when Load#3 is larger than 40% of the rated value. Figures 5.5 to 5.11

demonstrate that:

+ When the transmission system is fully in service (Lines LI0 and L25 are in

service), the osciliatory voltage mode associated with Load#3 is stable and its

damping is adequately large so that the mode is not of any practical concern with

respect to voltage instability (Figures 5.5 to 5.7).

When line LI0 is out of service, damping of the voltage mode is noticeably

reduced, however, it remains stable (Figure 5.8). When LI0 is out of service and Load#3 is composed of only induction motor load, Go experiences angle instability

(Figure 5.8).

5.3 Effect of ULTC on Voltage Dynarnics

Real (Us)

Figure 5.10: Voltage mode eigenvalue associated with Load#3 when L25 is out of seMce m T C control disabled). A. Luad#3 is composed of 20% induction motor and 80% k e d impedance. B: Load#3 is composed of 60% induction motor and 40% fixed impedance. C: Load#3 is composed of 100% induction motor.

When line L25 is out of service both the inertial mode of G5 and the voltage mode

of Load#3 exhibit instability (Figures 5.10 and 5.11). Since system disruption due

to angle instability occurs much faster than that of voltage instabiliw, the angle

instabilitp becomes of more iqunediate concern and remedial measures for

enhancement of angle stability must be activated fi&.

5.3.3 Effect of ULTC on System Dynamics

Figures 5.13 and 5.12 show the effect of the speed of response of the ULTC, i.e.

5.3 Efict of ULTC on Voltage Dynamics

-0.8 -0.6 -0.4 -0.2 O 0.2 Red (Us)

Figure 5.11: Inertial mode eigenvalues of generator G5 when L25 is out of senke (ULTC control disabled). A: Load#3 is composed of 20% induction motor and 80% fked impedance. B: Load#3 is composed of 60% induction motor .and 40% fked irnpedance. C: Load#3 is composed of 100% induction motor.

time-constant Td, on the inertial mode of G5 and the osdatory voltage mode of Load#3.

The initial steady-state operating conditions for the study case reported in Figures 5.13

and 5.12 are: (1) Load#3 is adjusted at 100% induction motur load, (2) lines LI0 and L25

are in service, and (3) G5 delivers 142 M'VA at 0.96 lagging power factor. Steady state

power flow studies show that under the given operating conditions, the ULTC should

adjust the steady-state magnitude ofinjectedvoltage at 0.16 per unit so that the system

can meet the load requirements.

Figure 5.13 shows the locus of the eigenvaIue of the inertial mode of G5 when

time-constant Td varies nom 0.50 seconds to 0.015 seconds. Figure 5.13 indicates that

the inertial mode can be made stable by decreasing the speed of response of the ULTC. This pattern of change of the inertial eigenvalue is predictable, since smaller

time-constant (G) indicates that power transfer is adjusted more rapidly by the ULTC.

5.3 Efict of ULTC on Voltage Dynamics

-20 -15 -10 Real (Us)

Figure 5.12: EEect of the speed of response of UETC qn the voltage mode of Load#3

I A) ULTC control disabled B) Td = 0.50 s

C) Td = 0.10 s

Real (Us)

Figure 5.13: Impact of the speed ofresponse of ULTC on the inertial mode of 05.

5.3 Efic tof ULTCon Voltage Dynarnics

Therefore, the power-angle of G5 does not have to be readjusted to meet the load

requirements. Figure 5.12 shows the locus of the eigenvalue of the voltage mode

corresponding to the same time-constant and other control parameters obtained for the

study cases of Figure 5.13. Figure 5.12 shows that the damping of the voltage oscillatory

mode is reduced as the speed of response of the ULTC is increased (Td decreased).

However, for the given operating conditions, the voltage mode is highly stable and

reduction of its damping is not of practical concern. Figure 5.12 also illustrates that

on-load tap-changing for real power flow regulation (points B, C and D) has adverse

effect on the stability of voltage mode as compared with the case where no on-load

tap-changing (point A) is introduced.

Cornparison of Figure 5.13 and Figure 5.12 indicates that changes in the speed of

response of a ULTC, which is controlled for real power flow regulation, has the opposite

effect on the direction of movement of the eigenvalues associated with the inertial mode

(e.g. for G5) and voltage mode (e-g. foreLoad#3). This cornparison also shows that the

effect of ULTC time-constant on the bestial mode is more signincant as compared with

that of voltage mode, i.e. the damping of the inertial mode is changed fmm about 0.2 to

0.5 (60%), while the damping of the voltage mode is changed firom about 14 to 19 (26%). A

series of eigenvalue studies, similar to those reported in Figures 5.13 and 5.12, are

conducted to examine the impact of various ULTC control parameters (Figure 5.4) on the

inertial and voltage modes of the system. The studies conclude that:

Damping of the voltage modes is only sensitive to Td (Figure 5.4) and other

control parameters do not noticeably influence the voltage mode.

Frequency of the voltage mode is insensitive (fairly constant) to al l parameters

of the control system as long as the voltage mode is highly s table

( 1 real part 1 > 5 US 1. The fkequency of the voltage mode is also affected by Td when damping of the mode is small (e.g., point (1) of curve C in Figure 5.8).

5.3.4 Combined Effect of ULTC and Transmission Network on System Dynamics

Figures 5.14 and 5.15 show the combined effect of the speed of response of ULTC (Td) and the transmission system on the voltage mode of load#3 and the inertial mode of G5. .

Initially Iine LI0 is out of service, Load#3 is adjusted a t 100% induction motor load and

G5 delivers 142 MW at 0.96 to the system.

Figure 5.14 clearly shows that to maintain angle stability of G5, the ULTC must have

5.3 Efict of VLTC on Voltage Dynumics

A) ULTC control disabled B) Td = 0.50 s

C) Td = 0.10 s

D) = 0.015 s

-

3 0.2 Real W s )

. . Figure 5.14: Impact of the speed of response of uL'I!C on the inertial

mode of G5 (Line L10 is out of service).

I A) ITLTC control disabled B) Td = 0.50 s

c) Td = 0.10 S

Figure 5.15: Impact of the speed of reapome of ULTC on the voltage mo'de of Load#3 (line LI0 is out of service).

5-4 VaCiihtion of Eigen Analysis Results

a rapid response (Td 10.01 s ) to be able to meet real power flow transfer to the load

area. Only a semiconductor-controlled ULTC can have the required speed of response

and a conventiond ULTC cannot be used under the given condition when LI0 is out of

service. ln contrast to the locus of the eigenvalue of the inertial mode, Figure 5-15 shows

that if a fast response of ULTC is adopted to maintain angle stabilitp, voltage mode of

Load#3 can become unstable (point D of Figure 5.15). It is interesting to note that if

Load#3 is represented by a fixed impedance instead of an induction machine, no

oscillatory instability corresponding to the voltage mode can be detected in the analysis.

Cornparison of the study results reported in Section 5.3.3 and 5.3.4 show that:

kicreasing the speed of response of the ULTC has opposite effects on the

dampings of the inertial and voltage modes of the system.

Due to the selected initial operating conditions and the system configuration, the

results of Section 5.3.3 do not show system instabiliw while the results of Section

5.3.4 indicate either voltage or angle instability

5.4 Validation of Eigen Analysis Results

The study results reported in Section 5.3 are based on the Iùiearized mathematical

model of the study system of Figure 5.1. The eigen andysis results do not include the

effect of nonlinearities, e.g., magnetic saturations of generators, transfomers and control

limits. To validate the eigen analysis results, the case studies reported in Section 5.3 are

also conducted on a detailed nonlinear mathematical mode1 of the study system. This r eqùes (a) development of the set of nonlinear equations which govern the system

dynamic behaviour and (b) numencal integration of the equations to obtain time-domain

responses associated with the test cases. Quantitative agreement between the

corresponding eigen analysis and the-domain response is used as the measure for

validity of modehg approach, assumptions, numencal results and conclusions.

5.4.1 Software Tool

The software tu01 used for time-domain simulation of the study system is a Transient Stability Program (TSP) developed at the Universitg of Toronto [42]. The TSP version

used for the studies reported in this thesis is an enhanced version of the original

program. The major program enhancements which were canied for the purpose of these

studies are (1) the addition of an induction machine model, (2) the addition of HVDC

5.4 V d i d a h of Eigen Analysis ~ e s u i t s

converter and DC line models, (3) the addition of UPFC and ULTC models and (4) the

addition of control models for HVDC, ULTC, UPFC and a synchronous machine

excitation system.

5.4.2 System Mode1 for Time-Domain Simulation

The mathematical model of each apparatus of the studied system of Figure 5.1 is

briefly described in the following paragraphs.

The electrical system models of the synchronous machines G1 to G8 are represented

in the corresponding d-q reference fkame as described by (3-1) to (3-8). Magnetic

saturation characteristic of each machine is also included in the model [283.

Excitation system of each synchronous machine is also included in the model

according to the block diagrams of Figures 5.2 and 5.3. The nodinear limits associated

with each excitation system are also included in the model. The dynamics of the

governors are not included in the TSP model and the mech8ILical input of each machine

is assumed to remain constant.

Therotating shaR system of each turbine-generator unit is represented by a single,

rigid rotating mass. Damping of each mass is also represented by a constant damping

coefficient. The dynamics of each rotating mass is represented by a second-order

differential equation with constant coefficients.

Except Load#3, dl loads are represented by fixed RL elements which is referred to as

fixed impedance load. Load #3 is represented by a shunt combination of an equivalent

induction motor and a fixed impedance load.

Electrical systems of the equivalent induction machine is represented by (3-13) to

(3-19) of Chapter 3. The magnetic saturation characteristic of the machine is also

included in the model [28]. the mechanical system of the induction machine is

represented by a single, rotating mass. The mathematical models described in Chapter 3

are used to represent the ULTC for tirne-domain studies.

The steady-state initial conditions for each case study (eigen analysis and TSP simulation) are obtained f?om a power flow program. The power flow program generates

steady-state initial values in formats which are directly used by the eigen analysis and

the TSP s o m m e tools. Detailed procedures for systematic development of the overall

system model âom individual component models are described in [43].

5-4 VaCidation of Eigen Analysks Results

The following approach is adopted to obtain a quantitative comparison between the

eigen analysis and the corresponding TSP results. Modal fiequenues and dampings are

calculated Çom the TSP results and compared with the corresponding values obtained

Born eigen analysis [34]. Damping and frequency of the oscillatory modes are determined

by (a) performing the simulation for several (up to 10) cycles of the oscillatory mode of

concern, and (b) n u m e r i d y fitting

to the oscilIations. a and o correspond to the damping and fkequency of the oscillatory mode. Modal frequency o is obtained by performing an FFT on the time response obtained from the TSP. If the time response includes more than one (dominant) oscillatory mode, then: (a) fïrst the frequencies of the modes outside the a~ceptable~range

are determined based on an FFT analysis, and (2) then no& filters are used to eIiminate

the oscillations from the time response [34].

Figure 5.16: The-domain response (O < t < 90 s) of rotor angle of 0 5 to a smd-signal disturbance (80% motor load, LI0 out of s e ~ c e , ULTC controUer disabled)

Figure 5.16 shows deviations in the generator angle (6) of unit G5 as a result of a

small-signal disturbance. The small-signal disturbance is introduced by energizing a

three-phase constant-impedance load at busbar D at time t = O S . The load is rated at

45 MVA and 0.80 lagging power factor. Prior to the disturbance, Load#3 is adjusted at

80% induction motor load and iine L10 is out of service. The ULTC controller is disabled

5.4 Validation of Eigen Analysis Results

Figure 5.17: Time-domain response (90 < t c 120 s) ofrotor angle of G5 to a smd-signal disturbance (80% induction motor load, LI0 out of service, ULTC controner disabled)

and its tap position is adjusted for steady-state power transfer corresponding to 80%

induction motor load at busbar D- Figure 5.16 indicates that the disturbance excites rotor

oscillations a t the frequency of o = 7.932 rad/s (1.26 Hz) and the damping of

u = +0.052 (Us). Eigen analyses studies predict the same unstable behaviour at o> = 8.069

radis and damping coefficient of a = 0.061(1/s). Both eigen analysis and TSP results

closely agree with respect to the rotor behaviour during smd-signal dynamics.

Figure 5.17 shows the rotor dynamic for the time period of 90 s to 120 S. Figure 5.17

indicates that approximately after t = 95 s, the rotor oscillations become nonlinear and

f indy at about t = 117 s, G5 losses synchronization. In practice, power system stabilizer

(PSS) of G5 must mitigate the rotor oscillations before angle instability occurs.

Figures 5.18 compares eigen values corresponding to the voltage mode of Load#3

obtained from eigen value analysis- (linear model) and time-domain simulation -

(nonlinear mode). Figure 5.18 shows close agreement between the eigen analysis and

time-domain simulation results for the operating conditions B, C and D. Under operating

condition A, i.e. when the ULTC control is disabled, unit G5 experiences angle instabiütg

approximately 22 seconds after the disturbance. Within the time interval of 22 seconds,

5.4 Validation of E@en Analysis Results

A) ULTC control disabled B) Td = 0.50 s

C) T, = 0.10 s D) Td = 0.015 s

Eigen Analysis -

i

1

TSP

-0.6 -0.4 -0.2 O 0.2 Real (Ys)

figure 5.18: Cornparison of eigenvalues of voltage mode obtained fkom eigen andysis and time-domain simulation Une LI0 is out of service, Load#3 is 100% induction motor load)

Voltage

(pu)

Figure 5.19: Voltage dynamic at bus D due to a small-signal disturbance (Line LI0 is out of service, Load#3 is 100% induction motor load), time constant of L E C is Td = 0.015 S.

5.4 VdicEafion of Eigen Analysis Resdts

the voltage mode does not complete even one swing and consequently the damping and

nequency of the voltage mode cannot be determined from the simulation results.

Figure 5.19 shows time-domain voltage response of the system (at busbar D) due to

the small-signal disturbance obtained fiom TSP. Figure 5.19 shows that the disturbance

results in an aperiodic voltage collapae. This highly nonlinear behaviour cannot be

predicted based on the eigenvalue studies. However, the eigenvalue results indicate that

the operatiag point is not stable and even a small disturbance will lead to voltage

instabZty

-Figure 5.20 compares the eigenvalue analysis results and the simulation results

corresponding to the inertial o s d a t o r y mode of unit G5 under the same initial operating

- conditions and small-signal disturbance described for Figure 5.18. Time-domain

response of the rotor angle of unit G5 corresponding to the operating conditions B, C and

D of Figure 5.20 are illustrateci in Figure 5.21. Both tirne-domain simulation results and

the eigen analysis results for the corresponding conditions predict the same pattern of

behaviour.

I A) ULTC control disabled B) Td = 0.50 s

C) Td = 0.10 s

Im (radfs) Eigen Analysis

,' #'

/'

/ TSP

Figure 5.20: Cornparison of eigenvalues of the inertial mode of 65 obtained fîom eigen analysis and the-domain simulation (Line L10 is out of service, Load#3 is 100% induction motor load)

5.4 VCrticEation of Eigen Analysis Results

-401 I 1 I t I 1 1 * ? J

O 2 4 6 8 10 12 14 16 18 20 Time (Seconds)

(BI 1 O

A8 5

(degree) O

-5 O 2 4 6 8 10 12 14 16 18 20

Tirne (Seconds)

Time (Seconds)

(Dl 4 I 1 1 1 1 , I ,

4 1 1 l I 1 I t I * O 2 4 6 8 t O 12 14 16 18 20

Tirne (Seconds)

Figure 5.21: Power-angle dynamies ofunit 6 5 due to a smd-signal disturbance (Line LI0 is out of senrice, had#3 is 100% induction motor load) A: ULTC controller disabled B: Time-constant of ULTC Td = 0.50 s C: Time-constant of ULTC Td = 0.10 s D: Time-constant of ULTC Td = 0.015 s

5.5 Conclusions

Cornparison of Figure 5.18 and Figure 5.20 indicates that:

Power flow regulation by means of the ULTC has opposite efTects on the

stability of the angle and the voltage modes.

Angle mode instability occurs when power flow regulation of the ULTC is slow

and thus the generator has to supply power to the load during the system

dynamics. This instability is prevented if the ULTC power flow regulation is

carried out façt enough to prevent rotor oscillations.

Voltage mode instability is due to the systems inability t o provide reactive

power to the loads.

5.5 Conclusions

This chapter examines the effects of a semiconductor-controlled under-load

tap-changing (ULTC) transformer on smd-signal dynarnics of an interco~ected power

system. Both voltage modes and electromechanical modes are sirnultaneously considered

in the stabïlity evaluation. The studies are conducted based on an eigenvalue analysis

approach. The eigen-analysis results are validated based on a digital time-domain

simulation of the study system, using a transient stability program. The studies conclude

that:

Under heavy-load condi t ions (s t ressed system), a fas t - response

(semiconductor-controlled) ULTC is required to dynamically regulate real power

flow and to maintain small-signal angle stability of the system. The UXTC must

be fast enough to dynamically modulate real power flow to counteract the

inertial mode of synchronous machines. The studies show the conventional

ULTCs (slow-response) can not provide the required damping for angle stab-

Although real-power modulation by means of a fast-response ULTC can enhance

angle stability, it can reduce the damping of voltage mode(s). Therefore,

enhancement of small-signal angle stability can result in voltage instabilitp.

Therefore, in contrast to the conventional practice, the phenomena of voltage

and angle stability, in the presence of semiconductor-controlled ULTC, &mnot be

independently investigated.

Under light-load conditions, the stability of the oscillatorg voltage mode is not

sensitive t o the speed of response of the ULTC. Therefore, it is a fairly

5.5 Conclusions

reasonable assumption to decouple voltage and angle small-signal instability

under such conditions. Such an assumption si@cantly simplines the system

model which is of importance .in the stability evaluation of large, intercomected

systems.

To detect smd-signal voltage instabilitg, the load must be properly modelled in

the overd system formulation, Le., dynamic voltage dependency of load has to

be incorporated in the model. If the load is represented by fixed RLC components, then the oscillatory voltage dynamics and consequently voltage

instability cannot be identified in neither the eigen analysis nor in the

time-domain simulation approaches.

Chapter

Effect of UPFC on Voltage and Angle Dvnamics

6.1 Introduction

Among the proposed/installed semiconductor-controlled power equipment,

the Unifïed Power Flow Controller (UPFC) as emerged as the most technically

attractive apparatus for stability enhancement and optimal steady-state power

flow regulation of large power systems. The studies reported in this chapter

demonstrate that if a UPFC is adopted to perform the function of either a

phase-shifter or a ULTC, it can contribute to and even cause voltage instability

However, a UPFC has the capability to regulate real-power flow and inject

reactive-power in t h e system. If these two capabilities of a UPFC are

simultaneously adopted, both voltage and angle stability c m be enhanced by a

UPFC.

This chapter proposes and examines a UPFC control strategy for

simultaneously enhancing of voltage and angle stability. The studies are

conducted on the test system presented in Chapter 5 using the eigenvalue

analysis approach described in Chapter 3. Eigen analysis results are validated

by digital computer tirne-domain simulations, based upon the use of the

Transient Stability Rogram (TSP) described in Chapter 3.

6.2 Study System

6.2 Study System

Figure 6.1 shows a single line diagram of the study system which is used to

investigate the effect of a UPFC on voltage and angle dynamics. The systern parameters

are the same as those used for the study cases of Chapter 5, Figure 5.1. The UPFC is installed between buses A and B of the study system. The shunt converter of the UPFC is connected to bus A The UPFC injects a controlled voltage in the system and dynamically

regulates the voltage at bus B.

6.3 Principles of Voltage Stability Enhancement by Means of a UPFC

Figure 6.2 shows the schematic diagram of a radiai transmission system which is equipped witb a UPFC. The UPFC is p h d y installed to regdate real power flow fkom

sending end (source) to the receiving end (load). Real power flow is dynamically

regulated by controlling phase-angle ( S B ) and magnitude (V') of the injected voltage.

Increase of real power flow fiom the source to the load is accompanied by reactive power

demand fkom both ends of the transmission line 1343. The amount and direction of flow of

reactive power from each end is a function of XI, Xz, XB, XE, VB, VE, 6*, ÔB and 6 [34],

where,

In practice, the sending-end and the receiving-end have limitations on the amount of

reactive power they can supply to the line to meet the requirement for the desired real

power hansfer. This limit is usually more stringent at the load end. This reactive power

demand is a major contributor to the phenornenon of voltage instability when real power

fiow nom sending- to the receiving-end is to be increased by means of a UPFC. Chapter 5

shows that rapid real power flow regulation is another factor which can initiate

oscillatory voltage dynamics and even voltage instabiliw.

When either a ULTC or a phase shifter is used ta control real power flow of a

transmission corridor, demand for reactive power fkom both ends of the comdor changes

as real power flow is varied. In practical configurations, the corridor ends are not usually

connected to stiff voltage sources; thus a change in reactive power is accompanied by

changes in voltage magnitude. An increase in reactive power demand results in voltage

drop. Voltage drop is followed by reduction in real power transfer. Thus, the ULTC (phase-shiRer) further increases the phase-shifk angle to meet the demand for real power

6.3 Principles of Voltage Stability Enhancement by Means of a UPFC

Figure 6.1: Schematic diagram of the studied system

6.4 UPFC Control Systern

1 - UPFC 1

Schematic diagram of a radial line equivalent with a UPFC for power fiow control.

t r d e r . hcrease in the phase-shift angle ca. be accoxnpanied by further reactive power

demand. This process can result in monotonic voltage decline or voltage swings

depending on the overall system parameters, nature of the load and operating conditions.

Voltage instabilitg may result if no couritermeasure is provided. Conceptua&, this type

of voltage instabilitp can be avoided if reactive power demand of the line is limited while

real power flow is increased (regulated at the desired value).

Power system apparatus which provide direct control over phase-angle, i.e. ULTCs, conventional mechanical phase-shifters and thyristor-controlled phase-shifters can only

regulate real power flow, and the required reactive power for the transfer of the real

power must be provided by the system. Under such circumstances, the phenomenon of

voltage instability can be avoided either by limiting the amount of real power flow or

installing reactive power sources, e.g. SVCs and fked capacitors, preferably at multiple

locations within the system. This approach is not cost effective. Furthemore dynamic

coordination of controllers among reactive power sources and a real power fiow controller

is not a trivial task.

The above voltage instability phenomenon can be prevented based on the use of a

UPFC to (1) control real power flow through the line and (2) inject reactive power in the

system t o meet voltage stability criteria. In this respect, the UPFC is utilized as a

combination of a phase-shifter (or ULTC) and a static VAR compensator. The following

sections present two different UPFC control strategies and their effects on voltage

dynamics of the test system of Figure 6.1.

6.4 UPFC Control System

The objective of this section is to study the impact of a UPFC on voltage dynamics

6-4 UPFC Control Systern

associated with Load#3 when the UPFC is instded between buses A and B of the study

system of Figure 6.1. The primary function of the UPFC is to adjust power flow to meet

real power requirements of Load#3. The UPFC carries out this function by controlling

magnitude and/or phase-angle of voltage at bus B. The real power requirement of Load#3

is primarily provided through the transmission comdor which is composed of line LI0 to

L13, Figure 6.1. Reactive power requirement of Load#3 is also p d a l l y provided through

the same corridor by means of the UPFC. The sending-end of the corridor is not

connected t o a stifholtage source and it varies between the upper and lower limits on

the voltage of the sending-end, e.g., 1.11 per unit and 0.84 per unit respectively. These

limits are identified based on optimal steady-state operation of the overall system. Thus, during steady-state condition, there is a limit to the amount of reactive power that can be

supplied through the comdor to Load#3.

During dynamic conditions, voltage limits are temporarily violated. A dpnamic

reduction in voltage results in a smaller amount of reactive power transfer to Load#3. In response, the UPFC may change the magnituddphase-angle of the injected voltage to

compensate for the reduction in reactive power flow. Without proper provisions, the

UPFC operation can result in M e r voltage &op and consequently voltage instability.

collapse. This type of instability can be prevented, if the UPFC can maintain the voltage

at bus A within a Iimit by injecting reactive power in the system. This is achieved if the

shunt converter of the UPFC is operated in the capacitive mode. It should be noted that

thyristor-controlled phase-shifters and ULTCs are not capable of injecting reactive power

in the system and therefore cannot assist in enhancing voltage stability under the

described scenario.

Figure 6.3 shows the UPFC control system for power fïow regdation [34]. The UPFC

controls power flow to Load#3 by regdating phase-angle of voltage at bus B. Figure 6.4 shows the details of the UPFC control system. Figure 6.4(a) shows a block diagram of

the feedback loop for controlling Measured three-phase power Pm is compared with

the reference power signal, and the error signal is fed to a PI controller. The control block

with the time constant TpZ represents the signal generating system. It should be noted

that in the studied system of Figure 6.1 torsional oscillations are not included in the

formulation. Thus, signal hP does not include any high eequency oseillatory components.

In practical applications where high fkequency power components are present, adequate

filtering must be provided to avoid sustained (unstable) high frequency oscillations of

generating units due to the cont~ol hop. Similady, if the power signal is derived from a

transmission comdor where inter-area oscillations are experienced, provisions must be

I I V&nf 1 I A d a r y Al.udïaxy Signais

Signals Figure 6.3: The UPFC control system

made to eliminate the effect of tie oscillations on the control. In the system of Figure 6.1,

due to the relatively small size of G5 with respect to the other generating units, no

oscillations associated with inter-area modes are present in the measured power signal.

However, disturbances at Iines L14, LIS, L30, and Load#3 area wilI excite the inertial mode of unit G5. Consequently, frequency deviation signal A q measured at bus B,

includes the effect of rotor oscillations of unit G5. Aco is used as the auxiliary signal of the

control system of &, Figure 6.4(a), to mitigate the inertial mode of G5. The compensator

block is used to provide phase and gain conditioning for Acri The block with time constant

Td is used to Vary the speed of response of the auxiliarg loop. .

Figure 6.4(b) shows that the control system regulates the rms magnitude of the

injected voltage. During steady-state conditions, modulation index Mg corresponds to

0.16 per unit voltage injection. This is to ensure that changes in SB by means of the

controller of Figure 6.4(a) can effectively regulate real power transfer to Load#3 under al1 operating conditions. This control strategy for 6B and Mg gives the priority to real

power flow control of Load#3 area and determines the reactive power flow according to

the system operating conditions. This indicates that the control loops of Figure 6.4(a) and

(b) reach their limits due to power fiow requkments and not due to reactive power flow

requirements.

Figure 6.4(c) shows the voltage control system of the dc-1Uik capacitor of the UPFC. Measured capacitor voltage is compared with a fixed reference value and the error is

used to regulate power exchange between the excitation converter and the system to

maintain capacitor voltage. Voltage a t bus B of the system (Figure 6.1) during

steady-state condition, depending upon the magnitude of Load#3, varies &om 0.73 per

6.4 UPFC Cuntrol System

HPF Am L

Compensaior 1 +sTd

Figure 6.4: Block diagram of UPFC control system

(a) Control loop of SB (c) DC link voltage control loop

Qmf ~urjliary signal

(b) Voltage control loop of bus B (d) Control loop of mE

6-4 UPFC Control System

unit up to 0.94 per unit. This voltage is not regulated by the UPFC. The excitation

converter of the UPFC is controlled not to exchange any reactive power with the system.

Figure 6.4(d) shows the control block diagram of r n ~ which performs this function.

Theréfore, the UPFC neither absorbs nor injects reactive power in the system.

A) UPFC conttol disabled B) Td = 0.20 s

C) Td = 0.10 s

- i 0.20

D) Td = 0.015 s

Figure 6.5: Effect of UPFC time-constant on the voltage mode associated with Load#3.

1 i

6.4.1 Effect of UPFC on Voltage Dynamics

0.15

Figure 6.5 shows the effect of UPFC time-constant (Td of Figure 6.4(a)) on the

eigenvalue associated with the voltage dynamics of Load#3. Load#3 is initially adjusted

at 100% induction motor load. Lines L10 and L25 are in s e ~ c e , and G5 delivers 142

MVA at 0.96 lagging power factor. Eigenvalue A on Figure 6.5 corresponds to the voltage

mode of Load#3 when the UPFC controls a& disabled, and according to the power flow

conditions, the injected voltage is adjusted so that the required power of Load#3 is

provided by G5 and the system. Figure 6.5 indicates that under this operating condition

the voltage mode of Load#3 is highly stable and the phenornenon of voltage dynamics is

not a concern. Points B, C, and D on Figure 6.5 show the eigenvalue of the voltage mode

under the same steady-state operating condition when the UPFC control is in s e ~ c e .

Figure 6.5 indicates that reducing Td can effectively reduce darnping of the voltage mode

associated with Load#3. However, for all the conditions, the voltage mode is well damped

-20 -15 -10 Real (Us)

6.4 UPFC Control System

Figure 6.6: Effect of UPFC tirnecorntant Td on the hertial mode of G5.

and voltage instability is not a concern for the given operating conditions and the UPFC control parameters.

Figure 6.6 shows the locus of the eigenvalue associated with the inertial mode of G5 when the UPFC time-constant Td is varied fkom 0.015 s to 0.20 S. Figure 6.6 indicates

that slower response of the UPFC results in smaller damping of the inertial mode.

However, for d l the given operating conditions, the inertial mode of G5 is positively

damped for the whole range of variation Td. Faster response (smaller time-constant) of

the UPFC results in lesser dependance of Load#3 on G5 and more on power hansfer 6om

the system. Thus, the inertial mode of G5 exhibits higher damping (Point D of Figure 6.6)

when the UPFC the-constant has its smallest value. Cornparison of Figure 6.5 and

Figure 6.6 indicates that reducing the time constant of power flow controller of the UPFC has the opposite effects on the damping of the voltage mode of Load#3 and the inertial

mode of G5. Increasing the time constant (1) reduces the damping of the voltage mode and (21.increases the damping of the inertial mode.

6.4.2 Effect of System Codguration on Voltage Dynamics

The impact of the UPFC on the voltage mode, when line LI0 of Figure 6.1 is out of

service, is illustrated in Figure 6.7. It is assumed that Load#3 is 100% induction motor

- A) UPFC control disabled B) Td = 0.20 8

c) Td = 0.10 s D) Td = 0.015 s

4

-0.6 -0.4 . -0.2 O 0.2 R e d (Ys)

Figure 6.7: EfFect of UPFC tirneanstant on the eigenvalue of voltage mode associated with Load#3 when line LI0 is out of service.

l A) UPFC conhl disabled

B) Td = 0.20 s

C) Td=O.lOs

D) Td = 0.015 s

Figure 6.8:

0.2 Real (Us)

Effect of UPFC tirneamtant rd on the eigenvalue of the inertial mode of G5 when Iine LI0 ia out of service.

6.4 UPFC Control System

load. Figure 6.7 shows the locus of the eigenvalue associated with the voltage mode when

Td is varied Born 0.015 s to 0.20 S. When the UPFC control is disabled and the initial

injected voltage is adjusted to meet the load requirement, the voltage mode is stable

(Point A of Figure 6.7). If the UPFC control is activated to adjust the power flow

according to the requirsments of Load#3, the stabiliw of the voltage mode depends on the

value of Td. Figure 6.7 shows that when T , is either 0.20 s or 0.10 s, the voltage mode is

stable. However, when Td is 0.015 s, voltage instability is experienced.

In contra& to the detrimental eEect of UPF'C on the voltage mode eigenvalue (Figure

6.7) when Tdis s m d , faster response of the UPFC enhances stabiliw of the inertial mode + of generator G5. Figure 6.8 shows that when the time constant of the UPF'C power flow

regulator is large, the inertid mode of G5 is unstable, i.e., points A and B. However, the

inertial mode is stable when the tirne-constant is small (Point D of Figure 6.8), and the

damping is adequately large to provide reliable operation of the system.

6.4.3 Validation of Eigen Analysis Results

The study results reported in Sections 6.4.1 and 6.4.2 are based on the linearized

mathematical model of the study system of Fïgure 6.1. The eigen analysis resdts do not

include the effect of nonlinearities, e.g. magnetic saturations of generators and

transformer and control limits. To validate the results of the linearized model, test cases

of Sections 6-4.1 and 6.4.2 are also conducted on a detailed nonlinear mathematical

model of the study system. The digital time-domain transient stability program and the

component modeIs of the system are the same as those described in Chapters 3 and 5.

However, the ULTC model is replaced by the UPFC model. Details of the UPFC-mode1

are also given in Chapter 3.

Figures 6.9 and 6.10 compare corresponding eigenvalues obtained from the

eigen analysis (linear model) approach and time-domain simulation (nonlinear

model) approach. The methodology to obtain eigenvalues of the oscillatory modes

fkom time-domain simulation results are descfibed in Section 5.4.2 of Chapter 5.

Figures 6.9 and 6.10 show quantitative agreement between the time-domain

results and the fkequency-domain eigen analysis results.

Figure 6.11 shows the tirne-domain simulation resdt comesponding to the dynamics

of voltage at bus D of the study system when the UPFC is in service to regulate power

flow (Td = 0.015 s) to Load#3 area. The oscillations in the voltage are excited as a result

6.4 UPFC Contrul System

Tirne-Domain Simulation

Eigen Analysis

1 0.2 Real (Us)

Figure 6.9: Compmison of eigenvalues of voltage mode obtained h m eigen analysis and tirne-domain simulation (Line LI0 is out of service, Load#3 is 100% induction motor load)

I A) UPFC controI disabled B) Td = 0.20 s

Cl Td= 0.10 s

TSP Simulation

Method

Eigenvalue Analysis 1

O 0.2 Red (Us)

Figure 6.10: Cornparison of eigenvalues of the inertid &ode of 0 5 obtained h m eigen analysis and time-domain simulation (Line LI0 is out of service, Load#3 is 100% indudion motor load)

of a sudden energization of a fixed impedance Ioad at bus D. The load is rated at 63 MVA

and 0.83 lagging power factor. Prior t o the disturbance, Load#S is h e d at 100% induction

motor Ioad and h e LI0 is disconnected. Figure 6.11 indicates that the voltage oscillatory

mode is unstable and its amplitude increases with time. This is consistent with the

eigenvalue results (Eigenvalue D of Figure 6.9).

For the given operating condition and parameters, the disturbance also excites the

inertial mode of G5, but the UPFC control can effectively mitigate the inertial mode, and

therefore, angle instability is not experienced. Figure 6.11 also shows that about 45

seconds after the disturbance, the system experiences nonlkiear voltage behaviour and

finally voltage collapse. The instant of voltage collapse cannot be predicted based upon

eigen analysis results.

0L 1 I I I I J

O 1 O 20 3 0 4 O 5 0 6 O Tim e (Seconds)

Figure 6.11: Voltage dynamics at bus D to a smaii-signal disturbance (Line LI0 is out of service, Load#3 is 100% induction motor load, UPFC time-constant is 0.015 s)

Figure 6.12 shows deviation in the rotor position of unit G5 as a result of the above

disturbance when the UPFC is in service to regulate power flow to Load#3 (Td = 0.20 s).

Figure 6.12 shows that the rotor oscillation cannot be mitigated by the UPFC and the

amplitude of oscillation grows. This is consistent with the eigen analysis results of

Figure 6.10, point B. Figure 6.12 shows that approximateb 35 seconds after the

disturbance, the rotor oscillations become nonlinear and the system experiences angle

6.5 UPFC Control Sttategy to Prevent/Mitigate Voltage Instability

Cornparison of Figure 6.11 and Figure 6.12 indicates that the post disturbance time

intemals for which either angle or voltage collapse is expenenced are comparable.

-50 I I 1 I I I , 1 1 1 O 5 10 15 20 2 5 30 35 40 45 50

Tirne (Seconds)

Figure 6.12: Inertial mode dynamics of unit G5 due ta a small signal disturbance (Line LI0 is out of senrice, Load#3 is 100% induction motor load, UPFC time-constant is 0.20 SI

6.5 UPFC Control Strategy to Prevent/Mitigate Voltage Instability

Figure 6.13 shows a control strategy adopted for the UPFC to prevent instability of

the voltage oscillatory mode associated with Load#3, under heavy real power transfer to

the load area. Further details of each control loop are illustrated in Figure 6.14. The main difference between the control strategies described in this section and that of

Section 6.4 is as foIlows. The control strategy of Section 6.4 does not permit any reactive

power exchange between the shunt converter of the UPFC and the system. The control

strategy of this section utilizes the shunt converter of UPFC as a static VAR system to

regulate the voltage at bus B (Figure 6.1) at a pre-specified reference value. Under heavy

load conditions of Load#3 area, the shunt converter is operated in the capacitive mode

and injects reactive power in the systern.

Figure 6.14(a) shows the details of the control loop associated with power flow

6.5 UPFC Contml Stnrtegy to Prevent/M&gate Voltage Insta6iZii-y

regdation. Measured power Pm is passed thmugh a low-pass filter (LPF,) to eliminate

high fiequency modes. The fïltered signal is then compared with the reference signal

(Pd. and the error signal is provided to a PI controller to determine the angle of the

injected voltage t jB. To enhance angle stability for unit G5, an auxiliary controller is

added to the controi loop to provide power flow modulation by means of the UPFC, Figure 6.14(a). A frequency deviation signal (ho) is measured and passed through a high-pass

(EIPF,). The function of this filter is to eliminate the effect of inter-area modes on the

control loop. The fIltered signal is then passed through a Iead-lag compensator for phase

and gain adjustment. The transfer function with the time constant Td is used to

investigate the impact of the au.x%ary loop time-constant on the voltage mode. Instead of

junction (Z ) , the a d a r y signal (Ma) can also be added to junction (1).

Vdcmf A d a r y Auxiliary Signais

Signals

VE: Figure 6.13: (a): UPFC control system to prevent voltage ïnstability

(b): Voltage phasors of the UPFC system

Figure 6.14(b) shows the block diagram of the dc link voltage control syst.em. The

capacitor voltage (V&) is compared with the desired value (Vk+) and the error signal is

provided to a PI controller t o control $. 6E controls power exchange betweenthe dc

capacitor and the transmission system by means of VSC-E.

6.5 UPFC ControL Stnztegy to Pmvent lMitigate Voltage Ikstability

AuxiIiary Signal

Atilcillary Signai Awciliary Signal

Figure 6.14: Alternative UPFC control scheme.

(a) A8B controller, (b) AQontroller, (cl AmfiontrolIer

6-5 UPFC Control Strategy to Preuent/Mitigate Voltage Instubility

Figure 6.14(b) shows the control system used to regulate voltage a t the terminal of

the shunt converter (V'). In the study system of Figure 6.1, the shunt converter of the

UPFC is connected to bus A Depending upon the line loading condition, the magnitude

of VEt for the study system of Figure 6.1 c a n Vary fkom 0.73 per unit to 0.94 per unit.

Lower values of Vn correspond to higher levels of real power flow. To minimize reactive

power demand from the system during heavy loading conditions of the line, the

magnitude of Va is dynamically adjusted to 1.02 per unit. This requires the operation of

VSC-E (Figure 6.13) as a VAR compensator under ail operating conditions of the UPFC

system. This is accomplished by comparing the magnitude of V' with the set value of Vset =1.02 per unit and using the error signal to adjust the interna1 voltage of VSC-E by

means of modulation index m ~ . The amount of injected reactive power in the system is a

function of real power transfer to the load area. The injected reactive power reduces

reactive power burden f?om the sending ends with respect to Load#3.

Injected reactive power by VSC-E ensures that the required reactive power demand

of Load#3, under stressed conditions, is available. Thus, voltage collapse and instability

at Load#3 can be prevented regardless of the amount of real power exchange between the

Load#3 iiiea and the rest of the system.

Figure 6.13(b) shows that for a given value of VEt, dependhg upon the magnitude and

phase angle of injected voltage VB, the magnitude of VBt can Vary over a relatively wide range. Existing insrdation coordination- practice of HVAC transmission systems permits

steady-state operation up to 1.5 per unit voltage at a busbar when there exists no direct

load comection to the bus. This indicates ihat the magnitude of the injected voltage by

the UPFC can be up to 0.48 per unit.

I V ~ d m o + = I V ~ t l m m + IV~Imax = 1 .O2 + 0.48 = 1 -5 per unit (6-2)

Magnitude of the injected voltage can be dynamically controlled by the UPFC

through modulation index mg- Dunng time intervals where maximum real power

transfer to Load #3 is required, it is advantageous to inject maximum voltage in the

system by means of the UPFC. The reason can be explained using the phasor diagram of

Figure 6.13b). The dominant variable for real power fiow contml is 6. The desired value

of 6 can be achieved by means of simultaneous adjustment of 6B and 1 VB 1 . For a given

vdue of 6, a larger 1 VB 1 results in a s m d e r vdue of 6 ~ . Thus, to ensure that control

system of SB, Figure 6.14(a), can provide power flow control under extreme load

conditions, the injected voltage 1 V' ( must have its largest value. In the studies reported

in this chapter, the magnitude of the injected voltage is kept constant at 0.48 per unit.

6-6 Voltage Stabilie Enhancement

6.6 Voltage Stability Enhancement

This section demonstrates that the control strategy of Section 6.5 enables the UPFC to provide rapid dynamic power flow regulation to Load#3 and simultaneously prevent

voltage instabilie of the system.

6.6.1 Effect of UPFC on Voltage Dynamics

Initidy, line LI0 is out of semice and Load#3 is set at 950 MVA (100%) of the rated

value) which is composed of 100% equivalent induction motor load. The UPFC which is installed between buses A and B, Figure 6.1, adjusts the power fiow to the requirement of

Load#3 under the given system condition. Figure 6.15 shows the eigenvalue of the

voltage mode associated with Load#3 when time constant Td of the UPFC (Figure

6.14(a)) is varied fkom 0.015 s to 0.2 S. Figure 6.15 shows that for the whole range of

variation of the time constant, the oscillatory voltage mode is stable. Figure 6.15 also

illustrates the voltage mode eigenvalues when the control strategy of Section 6.4 is

adopted for the UPFC. Cornparison of the corresponding eigenvalues demonstrate that

when the UPFC is controlled to inject reactive power in the spstem a t bus A, the

phenomenon of voltage instability is prevented regardless of the speed of voltage

regulation.

A) UPFC mntrol disabled B) Td = 0.20 s

C) Td = 0.10 s 0.204 D) Td = 0.015 a

0.202

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 O 0.2 Real (l/s)

Figure 6.15: Effect of UPFC time-codstant Td on the eigenvdue of voltage mode associated with Load#3 when line L10 is out of service.

Figure 6.16 shows the effect of the UPFC power flow control on the inertial mode of

G5. Figure 6.16 indicates that slow response of the voltage regulator (Td = 0.20 s) does

6.6 Voltage Stability Enhancement

I a UPFC eontrol of Section 6.5 r UPFC control of Section 6.4

-0.6 -0.4 -0.2 O 0.2 Real (Us)

Fi- 6.16: Ened ofUPFC time-constant Td on the eigenvaiue ofthe inertial mode ofG5 when line LI0 is out of service.

not maintain the stability of the inertial mode. However, faster power regulation, e.g.

Td = 0.10 s or Td = 0.015 s, can ensve that the inertial mode remains stable. Figure 6.16

also provides a quantitative cornparison between the corresponding eigenvalues for the

cases that control strategies of Section 4 and Section 5 are adopted for the UPFC. The cornparison demonstrates that: (1) slow voltage regulation (Td = 0.20 s) cannot maintain

angle stability for both control strategies, and (2) control strategy of Section 5 increases

the damping of the inertial mode.

Evaluation of Figures 6.15 and 6.16 indicates that simultaneous stability of voltage

and angle can be maintained if (1) rapid voltage regulation is camed out (Td L 0.10 s 1 and (2) adequate reactive power is injected in the system at bus A (control strategy of

Section 6.5).

Figure 6.17 compares eigenvalues of the voltage mode obtained from (1) the

linearized model of the system based on the eigenvalue approach and the nonlinear

model of the system based on the digital tirne-domain simulation method. The approach

described in Section 5.4 is used to calculate eigenvalues of the voltage mode from the

time-domain responses. Figure 6.17 depicts close quantitative agreement between the

corresponding eigenvalues obtained fkom the two methods. Figure 6.18 compares the

tirne-domain simulation and eigen analysis results corresponding to the inertial mode of

6.6 Voltage StabiZity Enhancement

B) Td = 0.20 s

CI Td =O.Io s 0.208 D) Td = 0.015 s

0.206

Tim~ime-Domain Simulation Methoci 0.204

0.202

1 I 1 I 1 1 1 I I 1 1 1 1 I

-1.2 -1.0 -0.8 -0-6 -0.4 -0.2 O 0.2 Real (l/d

Figure 6.17: Cornparison of eigenvalues of voltage mode obtained h m eigen andysis and he-domain simulation when line L10 is out of service.

0 Eigen Annipis The-Domain Simulatin0

-0.6 -0.4 -0.2 O 0.2 Real (Us)

Figaw 6.10: Cornpariaon of eigenvalues of the inertial mode of G!5 obtained h m eigen analysis and tirnedomain simulation when line L10 is out of service. (contml strategy of Figure 6.13 is used for the UPFC).

6.6 Voltage Stubility Enhancement

unit G5. Agreement between the tirne-domain simulation results and the eigen andysis

results in the indication of the vdidity of the resuIts and conclusions.

Figure 6.19 shows variations of voltage at bus D corresponding to a small-signal

disturbance in the system. The system is initidy supplying 950 MVA at a power factor of

0.91 to Load#3 and G5 provides 148 MVA at 0.98 lagging power factor. The UPFC operates based on the control strategy of Figure 6.13. Td is adjusted at 0.015 S. A

smd-signal disturbance is imposed on the system by energizing a passive load at bus D. The disturbance load is rated at 63 MVA and 0.82 lagging power factor. Figure 6.19

-121 I I I I l . , I

O 10 20 30 40 50 60 70 80 9 0 Tirne (Seconds)

Figure 6.19: Voltage mode dynamics at bus D due to a srnail-si al disturbance (line LI0 is out ofaervïce, control strate of(contro~ strategy o f R w e 6.13 is us& for the UPFC, and Td = o.oI~%.

shows tirne-domain variations of voltage at the bus after the disturbance. Figure 6.19

illustrates that the disturbance results in voltage oscillations at a fkequency of about

0.03 Hz. The voltage oscillations are positively damped and the system remallis voltage

stable. This is consistent with the corresponding eigen analysis results which is depicted

on Figure 6.17, data point D. To the contrast, Figure 6.11 shows that for the same

operating conditions and disturbance scenario, voltage becomes unstable and finally

voltage collapse occurs if the control strategy of Section 6.4 is adopted for the UPFC.

Figure 6.20 shows dynamics of the inertial mode of unit G5 for the same operating

condition, when the UPFC is operated based on the control strategy of Figure 6.13 and

6.7 Conclusions

1 1 1 1 I 1

O 10 20 30 40 50 60 Tirne (Seconds)

Figure 6.20: Inertial mode dynamics of u+t G5 due to a am&-signal distubance (2ine LO is out of service, control strategy of Figure 6.13 ii used for the UPFC, and Td = 0.20 S)

Td is adjusted at 0.20 S. Figure 6.20 indicates that slow power flow control by means of

the UPFC cannot maintain angle stability of G5. This is also in agreement with the

eigenvdue results (data point B of Figure 6.16).

6.7 Conclusions

This chapter examines the effect of voltage control, by means of a Unified Power Flow

Controller (UPFC), on the damping of the system eigenvdues associated with the

inertial modeb) and voltage mode(s) of a power system. The UPFC is used primady to

regulate power flow t o a load area through dynamic control of magnitude and

phase-angle of the injected voltage. Two control strategies for the UPFC are examined:

First Strategy: The UPFC regulates a senes injected voltage (phase-angle and

magnitude) and does not exchange reactive power with the system. Under this

strategy, the UPFC operates similar to a phase-shifter or an on-load

tap-changing transformer.

6.7 Conclusions

Second Strategy: The UPFC regulates the series injected voltage (both

magnitude and phase angle), and its shun t converter maintains i t s

corresponding terminal voltage at a pre-specified value. Thus, the shunt

converter of the UPFC is controlled to perform as a forced-commutated static

VAR compensator (STATCOMI.

Eigen andysis and digital time-domain simulation results show that:

When a UPFC adopts the first control strategy, to regulate real power flow of a

voltage dependent load, damping(s) of voltage oscillatory mode(s) can be noticeably

affected by the UPFC control system. OsciIlatoiy voltage instability is observed

when rapid voltage regulation is carried out. Slow voltage regulation may not result

in voltage instability; however, it may cause angle instability. The conceptual reason

behind the voltage instability is inadequate capacity of the system to provide the

required reactive power for the load. The results indicate that based on the rirst

control strategy full speed of response of a UPFC cannot be exploited.

When the UPFC adopts the second control strategy, stability of both voltage and

angle oscillatory modes are enhanced. The reason is that the second control

strategy guarantees that the UPFC provides control over real power flow and

reduces (minimizes) the system dependance on the sending-end reactive power.

In contrast to the conventional practice, voltage instability and angle instability

cannot be considered as decoupled dynamic phenomena when a power system is

equi p ped with semiconductor-controlled apparatus.

The phenornenon of voltage dynamics c m be evaluated only if the voltage

dependency of the load is incorporated in the load model. Therefore, based on

conventional practice, where load is represented by constant power loads or

constant impedance loads, the interdependency of angle and voltage dynamics

cannot be studied.

Chapter

Conclusions and Recommendations for Future Work

7.1 Conclusions

This thesis examines effect of dynamic voltage regulators (DVRs) on the

angle stability and voltage stability phenomena of power systems. DVRs are

power electronic based apparatus which are capable of injecting a series

controlled voltage in a power system. Two specific DVR circuit topologies which

are considered in this thesis are: (1) semiconductor-controlled under-load

tap-changing (ULTC) transformer and (2) unined power flow controller (UPFC).

The studies are conducted on a realistically-sized, interconnected power

system which include eight generating stations, 31 transmission lines, 20 high voltage buses and one HVDC link. The studies reported in this thesis carried out

using an eigen analysis technique. Eigen analysis results are validated by

digital time-domain simulation of the system, using a transient stability

software tool. Both the eigen analysis and the transient stability tools are

developed and tailored for the studies reported in the thesis.

The thesis concludes that:

Fast voltage injection by means of a DVR, for enhancement of angle

stability, c m result in voltage instabiliQ. This indicates that in contrast

to the conventional approach, the phenomena of angle stability and

7.1 Conclusions

voltage stability cannot be treated as decoupled phenomena. This is in spite of

the fact that the t h e &ames associated with angle instability and voltage

instability are significantly different. The reason for this type of voltage

instability is incapability of the system to supply required reactive power

agsociated with the real power transfer imposed by series voltage injection. In

mathematical terms, fast voltage injection may result in voltage instability by

imposing unstable dynamic trajectones which are retarded e01n the region of

attraction.

The couphg phenomenon between voltage and angle dynamics can be detected

from the analysis, if voltage dependency of the load is adequately represented in

the load model. In these studies, the load is represented by equivalent induction

motors which dynamically incorporates voltage dependency in its model.

Under light load conditions, fast voltage injection, for enhancement of angle

stabiliw, does not affect voltage dynamics, i.e. no couphg between voltage and

angle dynamics is detectable.

4 C o u p h g between voltage and angle dynamics is reduced if series voltage

injection is camied out slower, i.e. voltage instability due to fast voltage injection

can be prevented if the overall time constant associated with the voltage control

loop of DVR is increased to a pre-specifïed value. This value depends upon the

system configuration, parameters and load characteristics. However, slow

voltage injection may not be useful in enhancing angle s t a b w .

Conventional phase-shifters which use mechanical switches (instead of

electronic switches) do not cause voltage instability since the time constants of

their responses are much larger than the limit value which causes voltage

ins tability.

A semiconductor-controlled ULTC which is equipped with an a d a r y control

for enhancing angle stability, can cause voltage instability

A UPFC, which is used to enhance angle stability through series voltage

injection, can also lead to voltage instability. However, shunt converter of a

UPFC can be independently controlled to act as a VAR generator. Thus the

reactive power demand of the load can be fully or partially met by the shunt

converter, and as a result voltage instability due to fast voltage injection be

prevented. This is another salient advantage of a UPFC as compared with other

7.2 Suggestkons for Future Work

power electronic apparatus which have been adopted/proposed for power system

applications.

Investigation of the coupling phenomenon of voltage and angle dynamics

requires representation of voltage dependency of the load in the study tools, i.e.

eigen analysis and transient stabilitp software programs. The lack of a proper

load mode1 obscures the voltage instability phenomenon, however, it does not

affect the results with respect to angle instabifiw.

7.2 Suggestions for Future Work

The following suggestions are for further studies in continuation of this work

The ULTC and the UPFC control systems provided in Chapters 5 and 6 have not

been optimized to achieve the best performance fkom the system. Therefore,

there is a need to establish criteria for achieving the optimum response of the

system based on adjustment of the control parameters. This may be carried by

considering the system as a multiple input - multiple output (MIMO) rather

than the single input - single output (SIS01 which has been used in Chapters 5

and 6.

0 The IEEE Working Group on Load M o d e h g has suggested various load models

for voltage stabilitg studies. This work only considers induction motor load and

does not investigate the coupling phenomenon of voltage and angle dynamics

with different load models. It is to be mentioned that none of the above load

models have been verinedlvalidated. Thus, it is recommended to investigate the

degree of coupling between voltage and angle dynamics for diffeient load

models, to establish accuracy and sensitivity of such models.

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[48] M.R. kavani, "A Sofbvare Tool for Coordination of Controllers in Power Systems," IEEE Dans., Vol. PWRS-5, NU. 1, pp. 119-125, 1990.

Appendîx

Al Synchronous Machine Model

Per unit stator and rotor flux linkage equations are [27,28]:

kl.1 Stator Dynamics Not Imicliided

Mer eliminating the stator dynamics and linearizing (3-1) to (3-81, and (A-1)

and (A-2), we get:

A1 Synchronous Machine Model

AS Synchronous M d i n e Model

ad. -Zn a d . ~ " a q . ~ " a q . 1

The reference &ame of machine's d-q axes is transferred to the common D-Q axes using the transformation shown in Figure k2. This transformation can be f o d a t e d as:

where

A I Synchronous Machine Mo&Z

Therefore, the state space equations of each synchronous machine in the common

reference f k n e are:

where the pre-index i refers to the i-th synchronous machine. .AuDg and AiDQ are the z z

voltage terminal and the stator D-Q axes currents of the i-th synchronous machine in the common reference frame, and

Figure k 1: Coordinate transformation.

IL0

AS Synchronous Machine Model

1-2 BaiMoid,, + B,Z;,!$frC6 A,, + BaiZ;A C,

Figure A.2 shows the representation of the exciter block diagram associated with the

synchronous machine. State space equations of the exciter system can be written in a general form as [27]:

adu

A I Synchronous Machine M&l

Figure A.2: Block diagram representation of syachronous machine with exciter

where Xdu is the unsaturated value of Xd, and h+Ez is the vector of exciter state space variables. Overall state space equations of synchronous machine is obtained from (A-9) to

(A-13) as:

where

k l . 2 Stator Dynamics Included

In a similar manner described in the preceding sub-section, the exact mode1 of

synchronous machine including stator dynamics can be derived £kom (3-1) to (3-8). Let

L (:,:2) = The sub-mah consisting of the fist two columns of L-' .

A I Synchronous Machine Model

L ( 3 = The sub-matrix consisüng of the third column of L-l .

Exact mode1 of synchronous machine which indudes stator dynamics is

A I Synchronous Machine Model

.B = L vsy

.D L vsny

Including the exciter model nom (A-11) to (A-13) into (A-16) and (A-17) yields:

where

Psn =

L

r

A1 Spchmnous Machine Mode2

A2 Induction Machine

A2 Induction Machine

Expressing flux linkages of induction machine relating to currents, we get:

where,

Linearizing (3-13) to (3-19) and (A-201, and setting the rotor voltage to zero, yields the induction machine state equation:

where,

A2 Induction Mcrchine

T h e induction machine state equations may be written in a standard form as:

where,

p p p p p p p p p p p p - - - - - - - - - - - - - - - - -

- - - - - -

E 1 ( : 1 : denotes a mat* consisting the first two columns of E-1, and E-1 (3) consists of the f3kh col- of E-l . The pre-index j refers to the? induction machine in the power system.

With the dynamics of the stator equations of induction machine neglected, voltage

equations written in a synchronously rotating reference Game, from (3-13) to (3-16)

becomes:

A 2 Induction Machine

where p is a derivative operator, and

Including torque equation in (A-24), and linearizing the equations give:

Partitioning the matrix in (A-25) yields:

where 111 l, II i2, IIz1, I Ip , and IIu are defined based on (A-25), and

A2 Induction Machine

Further, the stator and rotor cments can be calculated f2om state variables by

linearizing (A-20).

where,

Eliminating Aydqs in (A-25) and (A-28) and rearranging the outcome with respect to

the machine inputs and outputs yields:

where,

A2 Induction Mahine

A3 HVDC System

Makices A , B,, Bs,, B, Cm D,, Qu, and Dm in (3-76) are as the following:

- a b - * if (i, JI = (32) "dc

- ; if (i, JI = (5,3)

O ; Otherwise

- 1 O -kiuii sin bi

O -kivii sin yi 1 - l u d i o -v ~ E o

O O

1 kicos ai ki cos ai

- Cui = -'di0 -'qio

k p s $i ki cos di

Appendix

Test Systems

. - _ .- ., .,". . /- .

B.l Data for the Power System of Chapter 4

Transfomers and the equivalent transmission line data in Figure 4.1 are as

Table B-1: Transformer and transmission line data

Transmission Zr 0.2150 UEh

Transformer X, 0.0006 pu

Per unit system is based on 100 MVA and 500 KV as the base power and base

voltage, respectively The per unit voltage at the sending end terminal is set at

1.1 pu.

B.2 Parameters of the AC/DC Test Systern of Chcrpter 5

B.2 Parameters of the ACIDC Test System of Chapter 5

'liab1e B-2: Generator data

Generator -ber

B.3 Putameîers of the Excifers of Chapter 5

B.3 Parameters of the Ekciters of Chapter 5

Table B-3: Excitation system data for nnits G1 to 6 5

Table B-4: Excitation system data for units G6 and GS

B.3 Parameters of the Exciters of Chapter 5

Table B-5: AC line parameters

Line No.

LI

L2

L3

L4

L5

L6

Length(de1

108

210

210

185

185

160

Xi (CUmiIe)

O. 1043

0.3618

0.3618

0.3618

0.3618

0.3618

R(Wmile)

0.0019

0.0028

0.0028

0,0028

0.0028

0.0028

X& (Wmile)

0.4616

0.4516

0.4516

0.4516

O .45 16

0.4516

k* (%)

O

O

O

O

O

O

B.3 Parameters of the Erciters of Chapter 5

* k is the seriea compemation level

130

RWmile)

0.0118

0,0019

0.0023

0.0064

0.0064

0.0064

0.0064

0,0064

0.0064

0.0116

0.0084

0.0084

0.0084

0.0140

0.184

O. 181

X, «Unde)

0.5227

0.4618

0.3714

0.4762

0.4762

0.4762

0.4762

0.4762

0.4762

0.3962

0.3742

0.3742

0.1023

O. 1642

0.3216

0.2123

<

Line No.

L16

LI7

L18

LI9

L20

L21

L22

L23

L24

L25

L26

L27

L28

L29

L30

L31

Length(mi1e)

115

210

164

145

145

145

123

123

123

136

187

187

69 +

73

50

78

& (Nmile)

0.2373

0.4611

0.4322

0.3914

0.3914

0.3914

0.3914

0.3914

0.3914

0.3412

0.4118

0.4118

0.4102

0.3714

O. 1684

0.4311

- k* (%)

35

40

O

O

O

O -

O

O

O

40

35

35

O

O

O

O

B.3 Parameters of the Exfiters of Chapter 5

Table B-6: Load data

Load# 1 MW 1 Power Factor 1 1

2

1470

3

4

0.79 Lagging 1

0.85 Lagging

1572

5

0.93 Lagging

Induction Motor h a d +

Passive Load

1720

1200

7

0.79 Lagging

0.85 Lagging

8

2100

9

0.84 Lagging

2700 0.93 Lagging

1650 0.95 Lagging

B.3 Parameters of the Exciters of Chapter 5

Table B-?= Transformer Data - --

Transformer #

Tl

kV

25/525

X%

10%

R%

1%

B.3 Parameters of the Exciters of Chapter 5

!t'able B4k Generator data - -

G1 G2 G3 G4

# of paraIlel uiits 2 4 3 4

Rated voltage (kV) 25 26 25 20

# of poIes 2 2 2 2

Rated fkequency (Hz) 60 60 60 60

~1 (PU) 0.223 0.205 0.241 0.246

H (s) 2.74 3.06 2.60 3.43

G7 is an infinite bus at 25 kV

B.3 Parameters of the Exciters of Chapter 5

Tàble B-9= Induction machine data

TSble B-10: Data for the UEI'C control

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