Doctor of Philosophy · method, the on-line accurate modeling of the higher voltage (supply system) and the lower voltage level (aggregate load) based on the local measurements is

ON-LINE LOCAL LOAD MEASUREMENT BASED

VOLTAGE INSTABILITY PREDICTION

Momen Bahadornejad

B.Eng (Electrical Engineering)

M.Eng (Electrical Engineering)

A thesis submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

Centre for Built Environment and Engineering Research

School of Engineering Systems

Faculty of Built Environment and Engineering

Queensland University of Technology

2005

Statement of Original Authorship

The work contained in this thesis has not been previously submitted for a degree or diploma at any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another except where due reference is made.

Signed: _____________________________

Date: ______________________________

Acknowledgement

The work on this thesis has been a challenging, inspiring and interesting

experience. I started my study facing some problems causing serious difficulties

in my concentration on the research. Therefore, I have had the opportunity to

become indebted to many people. However, there are some people I am specially

obliged.

First and foremost, I would like to express my sincerest appreciation to my

supervisor, Prof. Gerard Ledwich for his time, patience, guidance,

encouragement, and financial support. Without his support and supervision this

thesis would never be a success. I would specially acknowledge his invaluable

skills in supervising non-English background students.

Many thanks to my associate supervisor Dr Bouchra Senadji for her beneficial

advices at different stages of my PhD. My special gratitude is due to Prof John

Bell, Research Assistant Dean to the Faculty of Built Environment and

Engineering for supporting me when I was facing financial problems. I would

also like to thank Dr Chuanli Zhang for providing real data used in this thesis.

Thanks also goes to the School of Engineering staff and REE postgraduate

students for being friendly and having contribution to a good working

environment. My particular thanks goes to Mr Dan Moradian for his technical

support.

I would like to acknowledge the last year scholarship provided by the Ministry of

Science, Research and Technology of Iran and express my appreciation to the

people in charge in the Power and Water Institute of Technology (PWIT) for

providing financial support for my PhD. My special thanks goes to my fellow

colleagues in PWIT who helped me over the years of my study. I am also very

grateful to Prof Javad Farhoudi, the former Iranian Scientific Counsellor in

Australia for his continuing support. God bless them all.

I am very happy and proud of making wonderful friends in Australia who made

me and my family feel Australia as the second home. In particular I wish to

thank my best friend Mr Ghavam Nourbakhsh who was like a brother to me

during my staying in Australia. Ghavam helped me to come to Australia and

supported me to pursue my PhD. I would also like to express my sincere

gratitude to Mrs and Mr Famouri, Mrs and Mr Thomas, and Mrs and Mr

O’Connor for their openness, friendship, support and hospitality. Many thanks to

the Ashgrove State School staff and all the students and their parents who helped

my sons and gave them self-confidence and taught us how to adapt ourselves to

this new environment.

I have to express my deepest thanks to our families for their love, sincere wishes,

and continuing supports. Particularly, I would like to appreciate my brothers-in-

law, Fareid and Fardin, for their numerous and invaluable helps over the years of

my study. Their generosity and help have been an inspiration to me.

Last but above all, I would like to express my most heartfelt gratitude to my

wife, Tahmineh, and my sons, Aidin and Arvin, for understanding, supporting

and encouraging me to finish this thesis. It was in Australia that I found how

great they are and I am really blessed having this family. Tahmineh took the

whole family responsibility and did what I never could. Without her generosity,

endeavours, and devotions, I may never have been able to survive and complete

my studies. Far from home and missing their relatives, Aidin and Arvin were so

patient and their endeavours to do their best made me feel stronger. Thanks to

God for having such a wonderful family. This thesis actually belongs to them.

Dedication

This work is dedicated to:

o my principal supervisor,

o my wife and my two sons,

o my mother and my mother-in-law.

Abstract

Voltage instability is a major concern in operation of power systems and it is

well known that voltage instability and collapse have led to blackout or

abnormally low voltages in a significant part of the power system.

Consequently, tracking the proximity of the power system to an insecure voltage

condition has become an important element of any protection and control

scheme.

The expected time until instability is a critical aspect. There are a few energy

management systems including voltage stability analysis function in the real-

time environment of control centres, these are based on assumptions (such as off-

line models of the system loads) that may lead the system to an insecure

operation and/or poor utilization of the resources.

Voltage instability is driven by the load dynamics, and investigations have

shown that load restoration due to the on-load tap changer (OLTC) action is the

main cause of the voltage instability. However, the aggregate loads seen from

bulk power delivery transformers are still the most uncertain power system

components, due to the uncertainty of the participation of individual loads and

shortcomings of the present approaches in the load modeling.

In order to develop and implement a true on-line voltage stability analysis

method, the on-line accurate modeling of the higher voltage (supply system) and

the lower voltage level (aggregate load) based on the local measurements is

required.

In this research, using the changes in the load bus measured voltage and current,

novel methods are developed to estimate the supply system equivalent and to

identify load parameters. Random changes in the load voltage and current are

processed to estimate the supply system Thevenin impedance and the composite

load components are identified in a peeling process using the load bus data

changes during a large disturbance in the system. The results are then used to

anticipate a possible long-term voltage instability caused by the on-load tap

changer operation following the disturbance. Work on the standard test system is

provided to validate the proposed methods.

II

The findings in this research are expected to provide a better understanding of

the load dynamics role in the voltage stability, and improve the reliability and

economy of the system operation by making it possible to decrease uncertainty

in security margins and determine accurately the transfer limits.

Key Words

Power system stability, Voltage stability, Long-term voltage instability, Voltage

collapse, Supply system modeling, Load restoration, Composite load, Induction

motor load, Constant impedance load, Constant power load, Load modeling,

Load peeling, Load parameters estimation, On-load tap changers, System

variations, Signal processing, On-line Identification

III

Table of Contents Abstract .................................................................................................................. I Key Words ............................................................................................................II Table of Contents ................................................................................................ III Table of Figures .................................................................................................. VI List of Tables....................................................................................................... XI Table of Symbols and Abbreviations............................................................... XIV Chapter 1 ............................................................................................................... 1 Introduction ........................................................................................................... 1

1.1 Motivation ................................................................................................. 3 1.2 Objectives and contributions..................................................................... 4 1.3 Thesis structure ......................................................................................... 6 1.4 Publications ............................................................................................... 8

Chapter 2 ............................................................................................................... 9 Literature Review of Local Data Based Voltage Stability Monitoring................. 9

2.1 Introduction ............................................................................................... 9 2.2 Definitions and Classification ................................................................. 11 2.3 Maximum Load Power: Nose curves...................................................... 13 2.4 Voltage Sensitivity of Loads................................................................... 17 2.5 Load Restoration and Voltage Stability .................................................. 19

2.5.1 Induction Motors................................................................................ 21 2.5.2 On-Load Tap Changers (OLTCs) ...................................................... 23 2.5.3 Thermostatic Loads............................................................................ 27

2.6 Literature Review on On-line Voltage Stability Analysis ....................... 27 2.7 Local Data Based Voltage Stability Monitoring...................................... 29 2.8 Literature Review on Aggregate Load Modeling .................................... 34

2.8.1 Component based and measurement based load modeling................ 36 2.9 Countermeasure to Long-term Voltage Instability .................................. 39 2.10 Summary ................................................................................................. 39

Chapter 3 ............................................................................................................. 41 Correlation Based System Thevenin Impedance Estimation .............................. 41

3.1 Introduction ............................................................................................. 41 3.2 Theory of the Correlation Based System Thevenin Impedance Estimation ………………………………………………………………………….42

3.2.1 Block diagram representation ........................................................... 43 3.2.2 Using system equivalent circuit ........................................................ 45 a) Voltage in phase and out-of-phase components with current ................ 46 b) Random changes in the load bus voltage and current............................ 47 Case 1: System with Constant Thevenin Voltage, 0EΔ = ...................... 48 Case 2: Random Changes in the Thevenin Source, 0EΔ ≠ .................... 49 Case 3: System with Dynamics................................................................... 49

3.3 Algorithm of the Correlation Based System Thevenin Impedance Estimation ....................................................................................................... 50 3.4 Simulation Results .................................................................................. 51

3.4.1 Case 1 System with Constant Thevenin, no dynamics ...................... 52 3.4.2 Case 2: Random Changes in the supply system, no dynamics .......... 55 3.4.3 Effect of the electrical distance between two loads in estimation ..... 56

IV

3.4.4 Case 3a: System with dynamics, no random changes in supply system 57 3.4.5 Case 3b: System with dynamics and random changes in supply system 62 3.4.6 Case 4: Simulation with Swing Reference Bus ................................. 63

3.5 Application of the Proposed Method to Real Data................................... 66 3.6 Summary................................................................................................... 69

Chapter 4.............................................................................................................. 71 On-line Load Characterization by Sequential Peeling ........................................ 71

4.1 Introduction .............................................................................................. 71 4.2 Theory of the On-line Load Characterization Using Load Bus Data ....... 73

4.2.1 Load active power peeling.................................................................. 74 4.2.2 Induction Motor Reactive Power Estimation ..................................... 75 4.2.3 fpk , fqk and ff pq kk Ratio Evaluation Using Load Bus Data .. 80

4.2.3 Removal of the Undesired Components from the Load Bus Data ..... 81 4.2.4 Estimation other Components of the Load Reactive Power............... 83

4.3 Algorithm of the Load Characterization by Sequential Peeling............... 84 4.4 Simulation................................................................................................. 87 4.5 Summary................................................................................................... 93

Chapter 5.............................................................................................................. 95 On-line Estimation of the Remaining Time to a Long-term Voltage Instability. 95

5.1 Introduction .............................................................................................. 95 5.2 Long-term Voltage Instability Prediction Considering Constant Impedance Load and OLTC ............................................................................ 96

5.2.1 System Description............................................................................. 96 5.2.2 Time to collapse estimation using impedance matching criteria........ 99 5.2.3 Time to collapse estimation using load power changes ................... 101 5.2.4 The Algorithms................................................................................. 102 A: Algorithm Based on the Impedance Matching Criteria........................ 102 B: Algorithm Based on the Load Power Changes..................................... 103 5.2.5 Simulation........................................................................................ 103

5.3 On-Line Voltage Collapse Prediction Considering Composite Load and ON-Load Tap Changer .................................................................................. 110

5.3.1 System Description.......................................................................... 110 5.3.2 Tap initial ratio estimation............................................................... 112 5.3.3 Taps to collapse estimation.............................................................. 114 5.3.4 The Algorithm ................................................................................. 115 5.3.5 Simulation........................................................................................ 116 5.3.6 Taps to collapse estimation with different disturbance sizes .......... 117 5.3.7 Effects of the load and measurement uncertainties on the estimation 119 5.3.8 Effect of the load composition on the voltage collapse................... 121

5.4 Summary................................................................................................. 122 Chapter 6............................................................................................................ 123 Case Study: BPA Test System........................................................................... 123

6.1 Introduction ............................................................................................ 123 6.2 Test System Description......................................................................... 124 6.3 Validation of the Proposed “System Thevenin Impedance Estimation” Method........................................................................................................... 128

V

6.3.1 Supply System without Random Changes ....................................... 133 6.3.2 Supply System with Random Changes ............................................ 134 6.3.3 Statistical Evaluation of the Estimated Thevenin Impedance.......... 137

6.4 Validation of the Proposed “Load Characterization by Sequential Peeling” Method .......................................................................................................... 138

6.4.1 Statistical Evaluation of the Estimated Power Components ........... 145 6.5 Validation of the Proposed “On-line Estimation of the Time to a Long-term Voltage Instability” Methods................................................................ 148

6.5.1 Maximizing OLTC Secondary Side Voltage Criterion................... 149 6.5.2 Impedance Matching Criterion ....................................................... 152 6.5.3 Simulation Results .......................................................................... 153

6.6 Summary ................................................................................................. 160 Chapter 7 ........................................................................................................... 161 Conclusions ....................................................................................................... 161

7.1 Summary of the Results ......................................................................... 162 7.2 Future Research...................................................................................... 166

References ......................................................................................................... 169 Appendix A: One-step-ahead Prediction for Removing Dynamic Component 179 Appendix B: Frequency Relation between Buses............................................. 180 Appendix C: BPA Test System Data ................................................................ 182

VI

Table of Figures Figure 1.1 The voltage collapse causing grid separation in the blackout in

southern Sweden and eastern Denmark, September 23, 2003 [5]. ................ 3

Figure 2.1 Power transfer limits with uncertainty margins [6]............................ 10

Figure 2.2 Classification of power system stability [8]....................................... 13

Figure 2.3 Figure 2.3 Two-bus system............................................................... 13

Figure 2.4 The so-called onion surface [6] as given by equation (2.4) . ............. 14

Figure 2.5 The onion surface projected onto the PV-plane. The vertex of each

curve is the maximum loadability point of the system................................ 16

Figure 2.6 Influence of the static load characteristic on P-V curves. The load

characteristic is shown for α equal to 2, 1, 0, -0.1, -0.4 and -0.5 [61] ........ 18

Figure 2.7 Per phase equivalent circuit of induction motor ................................ 19

Figure 2.8 Function of an OLTC control system [31]......................................... 23

Figure 2.9 Generator-line –LTC system [6] ........................................................ 24

Figure 2.10 P-V curves of generator-line-LTC system [6] ................................ 25

Figure 2.11 Two-bus system ............................................................................... 29

Figure 2.12 Maximal power transfer is reached (voltage instability) when the

apparent impedance of the load bus reaches the Thevenin circle [14] ........ 32

Figure 2.13 Load power response to a voltage drop............................................ 35

Figure 2.14 Block diagram of input-output transfer function model ................. 36

Figure 3.1. Time measurement of load voltage and current magnitudes ........... 42

Figure 3.2. Block diagram representation of proposed method for system

identification................................................................................................ 43

Figure 3.3 load connected to the equivalent circuit of the system ...................... 46

Figure 3.5 Four bus test system with two different variable loads...................... 51

Figure 3.6 Simulation of load admittance changes in case 1............................... 53

Figure 3.7 Magnitudes and phases of the load voltage and current in case 1 ..... 54

Figure 3.8 Difference of the real component of the load bus voltage and its auto-

correlation in case 1..................................................................................... 54

Figure 3.9 Simulation of load admittances changes in case 2 ............................. 56

Figure 3.10. Four bus test system with dynamic load ......................................... 58

Figure 3.11 Simulation of load #1 admittance changes in case 3a...................... 58

VII

Figure 3.12 Variation of generator angle in case 3a ........................................... 59

Figure 3.13 Variation of induction motor speed in case 3a ................................ 59

Figure 3.14 Magnitudes and angles of the load bus voltage and current ............ 60

Figure 3.15. Difference of the voltage imaginary component and its auto-

correlation in case 3a................................................................................... 61


correlation in case 3a after removing dynamic component ........................ 61

Figure 3.17. Simulation of load #1 and load #2 admittances in case 3b............ 62

Figure 3.18. Reference bus angle changes and its reflection in other buses ....... 63

Figure 3.19. Load bus voltage and angle after removing the effect of reference

bus rotation.................................................................................................. 64

Figure 3.20. Time measurement of load voltage and current magnitudes ......... 66

Figure 3.21. Time measurement of load voltage and current angles, angle of the

first data point in current is chosen as reference ......................................... 67

Figure 3.22. Real data load admittance magnitude and angle............................. 67

Figure 3.23. Real data voltage real component changes and its auto-correlation68

Figure 3.24. Residuals of the real data voltage real component changes and its

auto-correlation ........................................................................................... 69

Figure 4.1. One-line diagram a simple power system with composite load ....... 73

Figure 4.2 Equivalent circuit of a power system with composite load ............... 73

Figure 4.3 Active powers of different load components following a disturbance

in supply system.......................................................................................... 74

Figure 4.4 Per-phase equivalent circuit of induction motor............................... 76

Figure 4.5 Block diagram of a system………………………………………….81 Figure 4.6 Reactive power of different load components following a disturbance in supply system………………………………………………………………...83 Figure 4.7 Four bus test system………………………………………………...87 Figure 4.8 Simulation of load #2 admittance and system Thevenin impedance. 88

Figure 4.9 Simulation of load bus voltage and induction motor slip .................. 88

Figure 4.10 Simulation of load active and reactive powers ................................ 89

Figure 4.11 (a) and (b): changes in the load bus voltage and its autocorrelation

(c) and (d): load active power changes and its autocorrelation.................. 91

Figure 4.12 (a) and (b): voltage dependent load active power changes and its autocorrelation ……………………………………………………………… 91 Figure 4.13 (a) and (b): frequency dependent load active power changes and its

autocorrelation............................................................................................. 92

VIII

Figure 5.1. Simple system with OLTC and constant impedance load ................ 97

Figure 5.2. Changes in impedances, tap ratio, primary and secondary side

voltages and power ...................................................................................... 97

Figure 5.3. Changes in impedances, Load voltage and power ............................ 98

Figure 5.4. Simulation of load admittance ........................................................ 104

Figure 5.5 (a): System and load impedances, (b): Load real power, k=2.......... 105

Figure 5.6. (a): System and load impedances, (b): Load real power, k=2.5..... 105

Figure 5.7. (a): Changes in the system and load impedances,........................... 106

(b): changes in load real power, k=3.................................................................. 106

Figure 5.8. Comparison of estimated and actual values of power changes due to

tapping, k=2 ............................................................................................... 108


tapping, k=2.5 ............................................................................................ 108


tapping, k=3 ............................................................................................... 109

Figure 5.11 Simple power system with composite load and on load tap changer

................................................................................................................... 110

Figure 5.12 (a): tap position, primary and secondary voltages, (b): induction

motor slip, (c): load active powers, (d): load reactive powers .................. 111

Figure 5.13. Changes in the transformer secondary voltage and load power due

to taping ..................................................................................................... 112

Figure 5.14. Four bus test system with composite load..................................... 117

Figure 5.15. Simulation of the OLTC primary and secondary voltage changes,

(a): k=1.4, (b): k=2.5, (c): k=3 ............................................................... 118

Figure 5.16. Estimated OLTC secondary voltage for 10 successive tapings, k=3

................................................................................................................... 119

Figure 5.17. Time measurements of load voltage and current magnitudes in the

Brisbane load bus....................................................................................... 120

Figure 5.18. Simulation of load admittance with random changes ................... 120

Figure 5.19. Simulation of the load reactive powers, Induction motor: 60%,

Constant impedance: 35%, Constant power: 5%, k=2.9............................ 122

Figure 6.1. The BPA test system ....................................................................... 124

Figure 6.2. Overexcitation limiter characteristic ............................................... 125

Figure 6.3. Simulation of load admittance changes in bus 10........................... 127

IX

Figure 6.4. Tap ratio, magnitudes of bus 9 and bus 10 voltages....................... 127

Figure 6.5 Simulation of the magnitudes and angles of the bus 9 voltage and

current ....................................................................................................... 129

Figure 6.6 Magnitudes and angles in bus 9 voltage and current from disturbance

until start of OLTC operation................................................................... 129

Figure 6.7 Changes in the components of the bus 9 voltage and current from the

line trip disturbance until start of OLTC operation................................... 130

Figure 6.8 Auto correlations of the post-disturbance changes in the components

of the bus 9 voltage and current phasors................................................... 130

Figure 6.9 Post-disturbance changes in the load admittance magnitude and its

autocorrelation until start of OLTC operation .......................................... 131

Figure 6.10 Residuals of the post-disturbance changes in the components of the

load voltage and current until OLTC operation ........................................ 131

Figure 6.11 Autocorrelations of the residuals of the post-disturbance changes in

the load admittance magnitude until OLTC operation.............................. 132

Figure 6.12 Simulation of the magnitudes of the system Thevenin and load

impedances from the bus 9 view point...................................................... 133

Figure 6.13 Simulation of local and remote load admittances.......................... 134

Figure 6.14 Histograms of the components of the estimated Thevenin impedance

................................................................................................................... 138

Figure 6.15 Simulation of the load voltage and current magnitudes ................ 140

Figure 6.16 Simulation of the load total active and reactive powers ................ 140

Figure 6.17 Simulation of induction motor load power.................................... 141

Figure 6.18 Simulation of the constant impedance load power ........................ 141

Figure 6.19 Simulation of the constant power load .......................................... 142

Figure 6.20 Changes in the load post-disturbance active power....................... 143

Figure 6.21 Changes in the load post-disturbance reactive power.................... 143

Figure 6.22 Histograms of the estimated induction motor load active and reactive

power......................................................................................................... 146

Figure 6.23 Histograms of the estimated constant impedance load active and

reactive power ........................................................................................... 147

Figure 6.24 Histograms of the estimated constant power load active and reactive

power......................................................................................................... 147

X

Figure 6.25 Equivalent of the BPA test system with OLTC transformer and

composite load in bus 10. .......................................................................... 149

Figure 6.26 Simulation of the Thevenin voltage magnitude ............................. 151

Figure 6.27 Comparison of the estimated and actual values of the system

Thevenin impedance.................................................................................. 152

Figure 6.28. Simulation of the OLTC primary and secondary side voltages and

tap ratio in case1 ........................................................................................ 154


tap ratio in case2 ........................................................................................ 154


tap ratio in case3 ........................................................................................ 155

Figure 6.31. Estimated OLTC secondary side voltages, system Thevenin

impedance, and load impedance in case1.................................................. 157

Figure 6.32 Estimated OLTC secondary side voltages, system Thevenin

impedance, and load impedance in case2.................................................. 158

Figure 6.33 Comparison of the estimated OLTC secondary side voltages to the

actual values in case 3 ............................................................................... 159

Figure 6.34. Estimated system Thevenin impedance and load impedance and

their comparison to the actual values in case 3 ......................................... 159

Figure B.1 Three bus system ............................................................................. 180

XI

List of Tables Table 1-1 Voltage collapse incidents[4] ............................................................... 2

Table 3.1 Comparison of the estimated and actual values of the system Thevenin

impedance in case 1 .................................................................................... 55


impedance for a simulation time, T=100 sec and a time step ΔT = 0.02 sec

and different ratios of random changes in load #2 with respect to load #1. 56


impedance for a simulation time T=100 sec and a time step TΔ = 0.04 for

different values of 2Z ................................................................................. 56


impedance for a T=100 sec simulation time and a time step TΔ = 0.04 in

case 3, no changes in load #2 .................................................................... 62


impedance for a T=100 sec simulation time and a time step TΔ = 0.04 in

case 3b (random changes in load #2) .......................................................... 63


impedance for a 100 sec simulation time and 0.04 sec time steps for

different values of rotations in reference bus (bus 1) angle, without

random changes in supply system ........................................................... 65


impedance for a 100 sec simulation time and 0.04 sec time steps for

different values of rotations in reference bus (bus 1) angle, with random

changes in supply system .......................................................................... 65

Table 3.8. Estimated values of the system Thevenin impedance using 100 sec

successive time frames of Brisbane load centre measured voltage and

current phasors, starting at 9 am on 2002/06/06 ......................................... 69

Table 4.1 Comparison of the load estimated and actual powers for different

compositions of loads and a 10% change in the system impedance ........... 90

Table 4.2 Comparison of the load estimated and actual powers for different sizes

of the system disturbance (K) ..................................................................... 93

XII

Table 5.1. OLTC tap initial ratio, step size, lower limit, time delay and dead

band. .......................................................................................................... 104

Table 5.2. Actual and estimated system impedance, initial tap ratio and taps to

collapse for different values of k in (prefault)sk*Zt)(post faulsZ = ................... 106

Table 5.3. OLTC tap initial ratio, step size, lower limit, time delay, voltage

reference, and dead-band. .......................................................................... 117

Table 5.4. Estimated initial tap ratio and taps to collapse for different values of k

in %e)disturbanc(preZ/et)disturbanc(post Zk thth −−= ....................... 118

Table 5.5. Estimated initial tap ratio and taps to collapse for with different

disturbance sizes and random changes in the load .................................... 121

Table 5.6. Comparison of the taps to collapse for different load compositions 121


impedance, supply system without random changes, time step=0.04 sec. 134


impedance, supply system with random changes, Time step=0.04 sec..... 135


impedance, supply system with random changes, Time step=0.02 sec..... 135


impedance for different random change sizes for remote load (K), supply

system with random changes, Time step=0.02 sec.................................... 136

Table 6.5 Statistical parameters of the histograms of the components of the

estimated thevenin impedances ................................................................. 137

Table 6.6 Comparison of the load estimated and actual powers for different

compositions of loads, disturbance is the loss of one of the branches

between bus 5 and bus 6, P1 & Q1: Induction motor load, P2 & Q2:

Constant impedance load, P3&Q3: Constant power load ......................... 144

Table 6.7 Comparison of the load estimated and actual powers for different sizes

of the system disturbance. ......................................................................... 145

Table 6.8 Statistical parameters of the histograms of the components of the

estimated thevenin impedances ................................................................. 148

Table 6.9 mean values of the estimation error in the different components of the

load active and reactive power .................................................................. 148

XIII

Table 6.10 Comparison of the system post-disturbance estimated and actual

Thevenin impedance for different cases ................................................... 155

Table 6.11Comparison of the load post-disturbance and pre OLTC operation

estimated and actual power components. P1 & Q1: Induction motor load,

P2&Q2:Constant impedance load P3&Q3:Constant power

load……………………………................................................................ 156

XIV

Table of Symbols and Abbreviations OLTC On-load tap Changer

LTC Load tap changer

E Generator internal voltage

δ Voltage angle

ϕ Load impedance angle

f(.) An objective function f

s Induction motor slip

Δx Perturbation of variable x

x& The derivative of state variable

xf∂

∂ Derivative of the function f with respect to the variable x

thZ Thevenin impedance

thY Thevenin admittance

thE Thevenin voltage

thI Thevenin current

sZ System equivalent impedance

LZ Load impedance

LY Load admittance

pz An internal state variable

pT Active load recovery time constant

sα Steady-state load voltage dependence

tα Transient load voltage dependence

G(s) Transfer function

s Complex frequency variable in Laplace

w(t), d(t) white noise variables

E[] Expected value

W, D variances of w(t) and d(t) ∗

ΔI Complex conjugate of current changes

τ Time lag

)(Rxy τ Cross-correlation between processes x and y

XV

∑=

n

iix

1 Summation of the variable x from its ith to nth elements

TΔ Simulation time step

imag Imaginary component

sec Second

Auto Auto-correlation

Pu Per unit

T Total simulation time

deg Degree

11 QandP Induction motor load active and reactive power

22 QandP Constant impedance load active and reactive power

33 QandP Constant power load active and reactive power

pvK and fpK Voltage and frequency dependency coefficients of the load

active power

qvK and fqK Voltage and frequency dependency coefficients of the load

reactive power

of System nominal frequency

Absolute value

ω Frequency variable in the Fourier domain

h(t) Function in the time domain

)(H ω Function in the Fourier domain

Φ Fourier transform

EST Estimated value

ACT Actual value

ERR Estimation error

n Tap ratio, Tap position

sP VV and Primary and secondary side voltages of OLTC transformer

crn Tap position at voltage collapse point

nΔ Tap step

on Initial tap ratio

minn Tap lower limit

dT OLTC time delay

XVI

fT and mT OLTC intentional and mechanical time delays

mP Induction motor mechanical power

RV Reference voltage

FCL Full Load Current

OXL Over-excitation Limiter

diff Difference and approximate derivative

Xt OLTC transformer reactance

Chapter 1

Introduction For many decades power systems had been concerned of either angle stability or

thermal overload capabilities of lines. However, in the last three decades due to

the difficulties in the building of new transmission and generation facilities,

power systems have been loaded much more heavily than usual in the past. As a

consequence, many power systems around the world are experiencing voltage

problems leading to instabilities [1].

In normal operating conditions, voltage drops in the order of a few percent of the

nominal voltage are accepted between the generation and consumption points.

However, in some cases following a disturbance, voltages in some buses may

experience large and progressive falls. A possible outcome of voltage instability

is loss of load in an area, or tripping of transmission lines and other elements by

their protective systems leading to cascading outages[2, 3]. The term voltage

collapse is also often used. It is the process by which the sequence of events

accompanying voltage instability leads to a blackout or abnormally low voltages

in a significant part of the power system. Many voltage collapses have occurred

throughout the world as shown in Table 1.1. Investigations on the recent major

blackouts indicated that voltage instability is still one of the major causes of the

power system separation (Figure 1.1) [5]. In this event, voltage collapsed 100 sec

after the system fault and was recovered after power system separation.

Chapter 1. Introduction

2

TABLE 1-1 VOLTAGE COLLAPSE INCIDENTS[4]

Date Location Duration

13April1986

Winnipeg, Canada Nelson River HVDC link

1 Second

30Nov.1986 SE Brazil, Paraguay 2 seconds

17May1985 South Florida 4 seconds

22Aug.1987 Western Tennessee 10 seconds

27Dec.1983 Sweden 55 seconds

21May1983 Northern California 2 minutes

2Sep.1982 Florida 1-3 minutes

26Nov.1982 Florida 1-3 minutes

28Dec.1982 Florida 1-3 minutes

30Dec.1982 Florida 1-3 minutes

22Sep.1977 Jacksonville, Florida Few minutes

4Aug.1982 Belgium 4.5 minutes

20May1986 England 5 minutes

12Jan.1987 Western France 6-7 minutes

9Dec.1965 Brittany, France Unknown

10Nov.1976 Brittany, France Unknown

23July1987 Tokyo 20 minutes

19Dec.1978 France 26 minutes

22Aug.1970 Japan 30 minutes

22Sep.1970 New York State Several hours

20July1987 Illinois and Indiana Hours

11June1984 Northeast United States Hours

Voltage instability is driven by the load dynamics. In response to a disturbance,

power consumed by the loads tends to be restored by the action of the motor slip

adjustments (in seconds), tap-changing transformers (in minutes), and

thermostats (in a few hours). Load restoration causes further voltage reduction

on the high voltage network by increasing the reactive power consumption. A

run-down situation causing voltage instability occurs when the load dynamics


3

attempt to restore power consumption beyond the amount that can be provided

by the combined transmission and generation system.

Figure 1.1 The voltage collapse causing grid separation in the blackout in

southern Sweden and eastern Denmark, September 23, 2003 [5].

Although a system protection scheme may integrate and coordinate several types

of actions, action on load is the ultimate countermeasure. Shedding a proper

amount of load, at a proper place, within a proper time can be done indirectly

through a modified control of Load Tap Changers or directly as load shedding.

1.1 Motivation

This work is motivated by on-line identification of time to voltage instability,

that is a critical aspect, and the critical effect that the system and load

representation has on voltage stability studies.

Tracking the proximity of the power system to an insecure voltage condition is

an important element of any protection and control scheme. Many emergency

control measures are based on extensive off-line studies on voltage stability

modeling, computation of voltage collapse point and enhancement of power

system stability. A good overview of these areas is provided in [1, 6-9].

There are a few energy management systems including voltage stability analysis

function in the real-time environment of control centers [10]. In these methods,


4

in order to simplify the calculations, a number of approximations are used, which

introduce high or low uncertainty in the obtained transfer limits, according to the

used assumptions [1]. While, an optimistic approach may lead the system to

unacceptable values (security uncertainty), a pessimistic approach will avoid

risks by introducing larger security margins, but on the other hand it will lead to

a poor utilization of the resources (economic loss). These methods are also based

on off-line models of the system loads. The approaches to load modeling are

either component based or the measurement based, where one test is already

performed on the load [11].

The transfer limits across certain sections of the power system, depends on the

factors such as network topology, loading and generating conditions which vary

by time.

Load behavior is at the heart of voltage instability. The aggregate load seen from

bulk power delivery transformers are the most uncertain power system

components, due to the uncertainty of the participation of individual loads. In

order to develop and implement a true on-line voltage stability analysis method,

the real-time accurate modeling of the higher voltage (supply system) and lower

voltage level (aggregate load) at the load bus is required.

As far as loads restore to constant power due to the on-load tap changer (OLTC)

action, the long-term nature of phenomena together with a control on OLTC’s to

slow down the system degradation, might leave time to a computer to identify

the problem and trigger corrective actions.

1.2 Objectives and contributions

Based on the local measurements this project aimed to develop on-line methods

to predict a long term voltage instability caused by on-load tap changer. The


5

feasibility of the supply system equivalent estimation and load parameters

identification using the changes in the load bus measured voltage and current are

pointed out and then the results are used to identify a possible voltage collapse

resulting from a long-term voltage instability. The arising results may provide a

better understanding of the load dynamics role in the voltage stability, and

improve the reliability and economy of the system operation by making it

possible to decrease uncertainty in security margins and determine accurately the

transfer limits.

The main contributions of this thesis are:

• Correlation Based On-line Power System Thevenin Impedance

Estimation

A method is developed to estimate power system Thevenin impedance that is

based on signal processing on the measured data in the load bus. It is shown that

the cross-correlations of the changes in the load voltage and current with respect

to the changes in the load admittance can be used to estimate the system

Thevenin impedance. The method is validated by simulation. Any dynamic

components in the load voltage and current caused by system and/or load are

removed and the residuals are used for the estimation. Work on real data is also

provided to confirm the method.

• On-line Load Characterization by Sequential Peeling

A method is proposed that is based on the on-line measured load bus voltage and

current phasors during a disturbance. Load is considered as a combination of

induction motor, constant impedance, and constant power loads. The change in

the load active power due to the disturbance is used to identify the active power

of the load different components in a peeling process. Then the induction motor


6

reactive power is estimated using signal processing on the random changes in the

load total power. The other components of the load reactive power are then

estimated using the disturbance data.

• On-line anticipation of the Time to a Long-term Voltage Instability Caused

by the On-Load Tap Changer Operation

It is shown that the local load bus voltage and current can be used to anticipate

possible voltage instability and to estimate the taps to such instability in a system

consisting of constant impedance, induction motor, and constant power loads

behind a tap changer.

1.3 Thesis structure

Introduction (Chapter 1)

In chapter 1 an introduction to voltage stability and description of the facts that

have motivated the realization of the work held in this thesis are provided. The

novel contributions are briefly explained and the publications of the research

work are listed.

Literature Review of Local Data Based Voltage Stability Monitoring

(Chapter 2)

Chapter 2 outlines the fundamentals of voltage stability and some important

aspects such as voltage stability classification, voltage sensitivity of loads and

the role of on-load tap changers in voltage stability are highlighted. The literature

on the on-line voltage stability monitoring and load modeling is also reviewed

and discussed and the shortcomings are pointed out.


7

Correlation Based On-line System Thevenin Impedance Estimation

(Chapter 3)

Chapter 3 covers the achievements on the power system equivalent impedance

estimation from the load bus view point. It is clearly shown that the random

changes in the load bus voltage and current can be extracted and then to be used

to estimate the system Thevenin impedance.

On-line Load Characterization by Sequential Peeling (Chapter 4)

In chapter 4 it is shown that how the measured data in a load bus during a

disturbance can be used to estimate the parameters of the components of a

composite where load is considered as a combination of induction motor,

constant impedance, and constant power loads.

On-line Estimation of the Remaining Time to a Long-term Voltage

Instability (Chapter 5)

Chapter 5 is focused on the on-line identification of the expected time to voltage

instability. Using the local load bus voltage and current, the taps to a possible

voltage collapse in a system consisting of constant impedance, induction motor

and constant power loads behind a tap changer is anticipated. The proposed

method is confirmed by simulation.

Case Study: BPA Test System (Chapter 6)

In order to investigate and validate the proposed methods in this research, they

are applied to the BPA 10-bus test system. The loads are modified and complete

data of the system is provided.

Conclusions (Chapter 7)

Main conclusions and suggestions for future work are outlined.


8

1.4 Publications

Some results of this thesis have already been published in the publications

below. Chapter 3 is describing the contents of [1] and [2]. The results in [3] are

discussed in Chapter 4. The results in Chapter 5 are described in [4] and [5].

[1] M. Bahadornejad, G. Ledwich, “Studies in system Thevenin impedance

estimation from normal operational data”, proceedings of the 6th International

power Engineering Conference, IPEC 2003, 22-24 May 2003, Singapore.

[2] M. Bahadornejad, G. Ledwich, “System Thevenin impedance estimation

using signal processing on load bus data”, proceedings of the IEE Hong Kong

International Conference on Advances in Power System Control, Operation and

Management, APSCOM 2003, 11-14 Nov 2003, Hong Kong.

[3] Bahadornejad, M., and Ledwich, G., “On-line Load Characterization by

Sequential Peeling”, Presented in the 2004 International Conference on Power

System Technology, Powercon2004, 21-24 Nov 2004, Singapore.

[4] Bahadornejad, M., and Ledwich, G. , “Studies in the OLTC Effects on

Voltage Collapse Using Local Load Bus Data”, in Australasian Universities

Power Engineering Conference, AUPEC’2003, 28 Sept-1 Nov 2003,

Christchurch, New Zealand

[5] Bahadornejad, M., and Ledwich, G. , “On-line voltage Collapse Prediction

Considering Composite Load and On Load Tap Changer”, in Australasian

Universities Power Engineering Conference, AUPEC’2004, 26–29 September

2004, The University of Queensland, Brisbane, Australia

[6] Bahadornejad, M., and Ledwich, G., “Correlation Based Power System

Thevenin Impedance Estimation”, under preparation, to be submitted to the

journal “IEEE Transactions on Power Systems”

Chapter 2

Literature Review of Local Data Based Voltage Stability Monitoring

2.1 Introduction

Historically, (rotor) angle instability has been the dominant stability problem on

most power systems. As power systems have evolved, different forms of system

instability have emerged. Phenomena such as frequency stability, interarea

oscillations, and voltage stability have become great concerns to the power

system operation than in the past [8].The amount of power that can be

transferred between different parts of the power system is restricted by the

different stability limits. As an example, Figure 2.1 shows a problem concerning

stability limits [61]. These limits are generally difficult to determine with

sufficient accuracy and reliability due to the high uncertainty related to internal

and external factors, and therefore conservative criteria are often used for their

determination resulting in smaller secure operation areas. As it can be seen from

Figure 2.1 the voltage stability limit is strongly dependent on system loading.

Chapter 2. Literature Review of Local Data Based Voltage Stability Monitoring

10

Figure 2.1 Power transfer limits with uncertainty margins [6]

In the past two decades, in the prevailing open access environment, power

systems have been operated under much more stressed conditions than was usual

in the past. As a consequence, many power systems have experienced voltage

instability problems and voltage stability has become a factor leading to limit

power transfers [10]. Environmental pressures on transmission expansion,

increased electricity consumption in heavy load areas, new system loading

patterns due to the transmission open access and maximum profit environment

are some of the responsible factors for these conditions. The unstable behaviour

is characterized by slow (or sudden) voltage drops; sometimes escalating to the

form of a collapse [6].

An important element of any protection and control scheme is to track the

closeness of the system to a collapse in real time [12]. In order to achieve the

acceptability of the true system control/protection schemes, it is necessary to

provide them with “fall-back” position or a safety net based on local data, acting

Secure Operation

Voltage Stability Limit Thermal Limit

Angle Stability Limit

Load (MW)

Power Transfer Limit (MW)


11

if the centralized system fails for any reason, such as a possible failure of the

necessary hardware infrastructure [13] . Thus, despite the fact that voltage

instability is a system problem, there is still a need for devices that process only

local measurement [14, 15] .These controls are low cost and simple to build, and

hence, provide an attractive option for the utility industry. They can be used to

send alarms to control centre(s) when local monitoring indicates a locally weak

condition and/or to trip locally if the more severe conditions are encountered

[16] .

In order to create a basis for the next chapters, this chapter outlines some of the

fundamentals of voltage stability and gives a description of the phenomena that

contribute to voltage instability. The literature related to the voltage stability

monitoring and control based on the local measurements is also reviewed and the

existing methods are discussed.

2.2 Definitions and Classification

“Voltage stability refers to the ability of a power system to maintain steady

voltages at all buses in the system after being subjected to a disturbance from a

given initial operating condition” [16]. It depends on the system ability to restore

equilibrium between load demand and load supply from the power system.

The term voltage collapse is also often used. It is the process by which the

sequence of events accompanying voltage instability leads to a blackout or

abnormally low voltages in a significant part of the power system [7, 17, 18].

Stability is a condition of equilibrium between opposing forces. Depending on

the network topology, system operating condition and the form of disturbance,

different sets of opposing forces may experience sustained imbalance leading to

different forms of instability. A classification of the power system stability based


12

on time scale is shown in Figure 2.2. While rotor angle and frequency

instabilities are generator-driven phenomena, voltage instability is load-driven. It

should be noted that these terms do not exclude the affect of the other

components to the instability mechanism.

The time scale is divided into short and long-term time scales. When short-term

dynamics are stable they eventually die out some time after disturbance, and the

system enters a slower time frame.

The time frame of angle stability is that of the electromechanical dynamics of the

power system lasting in a few seconds.

During frequency excursions, the characteristic times of the processes and

devices that are activated will range from fraction of seconds, corresponding to

the response of devices such as underfrequency load shedding and generator

controls and protections, to several minutes, corresponding to the response of

devices such as prime mover energy supply systems and load voltage regulators.

Short-term voltage stability involves dynamics of fast acting load components

such as induction motors, electronically controlled loads, and HVDC converters.

The study period of interest is in the order of several seconds, and analysis

requires solution of appropriate system differential equations; this is similar to

analysis of rotor angle stability.

Long-term voltage stability involves slower acting equipments such as tap-

changing transformers, thermostatically controlled loads, and generator current

limiters. The study period of interest may extend to several or many minutes, and

long-term simulations are required for analysis of system dynamic

performance[10, 19, 20].


13

Power SystemStability

TransientStability

Small-Disturbance

Angle Stability

Short Term

Rotor AngleStability

Voltage Stability

Large-Disturbance

Voltage Stability

Small-DisturbanceVoltage Stability

Short Term Long Term

FrequencyStability

Long TermShort Term

Power SystemStability

TransientStability

Small-Disturbance

Angle Stability

Short Term


TransientStability

Small-Disturbance

Angle Stability

Short Term


Voltage Stability

Large-Disturbance

Voltage Stability



Voltage Stability

Large-Disturbance

Voltage Stability



FrequencyStability

Long TermShort Term

FrequencyStability

Long TermShort Term

Figure 2.2 Classification of power system stability [8]

2.3 Maximum Load Power: Nose curves One of the primary causes of power system instability is the transmission of

(large amounts of) power over long distances. In voltage stability, attention is

paid to power transfers between generation and load centers.

In the two bus simple system of Figure 2.3, a remote load is supplied by a strong

source (infinite bus) with constant voltage E through a transmission line

modeled as a series reactance that can be the Thevenin equivalent of a power

system seen from one bus.

Figure 2.3 Figure 2.3 Two-bus system

The receiving end voltage V and angle δ depend on the active and reactive

power transmitted through the line which under balanced three-phase, steady-

State sinusoidal conditions can be written as below:

0∠= EE jX δ∠=VV

jQP +

lll jXRZ +=

I


14

δsinX

EVP = (2.1)

XVδcos

XEVQ

2−= (2.2)

After eliminating δ using the trigonometric identity we get

02

222

=⎟⎠⎞

⎜⎝⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

XEVP

XVQ (2.3)

Solving equation (2.3) for V yields

QX

EPXEXXQEV

22

2

42

42−−±−= (2.4)

Figure 2.4 shows how the terminal voltage changes with the load powers

(dimensionless variables are used in the figure). In “normal” conditions, the

operating point lies on the upper part of the surface (corresponding to the

solution with the plus sign in equation (2.4)), with V close to E. Permanent

operation on the lower surface, characterized by a lower voltage and higher

current, is unacceptable.

Figure 2.4 The so-called onion surface as given by equation (2.4)

XV

2EQX

2EPX

020.

40.60.

80.

0

50.

1

20.

20.−

0

0

20.

40.

20.−

40.−


15

The condition to have at least one real solution for equation (2.4) is:

222

2

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛≤+

XE

XEQP (2.5)

The above inequality determines all combinations of active and reactive power

that the system can supply to the load. The equality in equation (2.5) corresponds

to the existence of a maximum load power, well-known from circuit theory [21].

More precisely, the Figure 2.4 shows a set of maximum load power points,

located on the “equator” of the surface. The projection of this limit curve onto

the (P,Q) plane is the parabola shown in Figure 2.4. In the (P,Q) load power

space, this parabola bounds the region where operation is feasible. All points

inside the parabola lead to two solutions for the terminal voltage V. For low

loading there are two equilibrium solutions; one with high voltage and the other

with low. The former is the stable equilibrium point(s.e.p.), and the latter one is

the unstable equilibrium point(u.e.p.) These equilibrium points approach each

other as the system is loaded slowly, up to the point where the two solutions in

(2.4) coalesce, i.e., the inner square root vanishes [22]. If the system is loaded

further, all system equilibrium disappears. The last equilibrium has been

identified as the steady-state voltage collapse point. At this bifurcation point, a

real eigenvalue of the load-flow Jacobian becomes zero, i.e., the Jacobian

becomes singular. The consequence of this loss of operating equilibrium is that

the system state changes dynamically. In particular, the dynamics can be such

that the system voltages fall in a voltage collapse [23].

In voltage stability analysis it is common to consider the curves which relate

voltage to (active or reactive) power. Such curves, referred to as P-V (or Q-V)

curves or nose curves are shown in Figure 2.5, for the simple system of Figure


16

2.3. These curves are indeed the projection of the onion surface projection onto

the PV-plane. The curves depend on how Q varies with P; in Fig. 2.5, a constant

power factor, i.e., ϕ= tanPQ , has been assumed for each curve.

The vertex of each curve determines the maximum power that can be

transmitted by the system and it is often called the point of maximum loadability

or point of collapse. Using the equality in equation (2.5) and substituting

ϕtanPQ = yields:

02

2222 =⎟

⎟⎠

⎞⎜⎜⎝

⎛−φ+

XEPtan

XEP (2.6)

Solving (2.6) for P will result in:

XE

φsinφcosPmax 21

2

+= (2.7)

Figure 2.5 The onion surface projected onto the PV-plane. The vertex of each

curve is the maximum loadability point of the system.

Substituting maxP from equation (2.7) into equation (2.4) one will get:

φsinEV Pmax +

=12

(2.8)

1EV

2EPX

20.−00.

40tan .φ −=

20.40.

00.00. 10. 20. 30. 40. 50. 60. 70. 80.

50.


17

Simply stated, voltage instability results from the attempt to operate the system

beyond maximum loadability point. This may result from a severe load increase

or, more realistically, from a large disturbance that increases X and/or decreases

E to the extent that the predisturbance load demand can no longer be satisfied.

When the point of collapse is reached the voltage starts decreasing quickly since

the reactive support of the system under these heavy loaded conditions is not

enough.

The set of the P-V curves in Figure 2.5 may also be considered as the different

load compensation cases, i.e.; by local reactive compensation it is possible to

increase the transfer capacity of the system, but at the same time the system

operates closer to the security margins, since the point of collapse is placed

closer to acceptable voltages.

2.4 Voltage Sensitivity of Loads

Loads are the driving force of voltage instability, and for this reason this

phenomenon has also been called load instability [1]. The term load refers to the

equivalent representation of the aggregate effect of many individual load devices

and the interconnecting distribution and subtransmission systems that are not

clearly represented in the system model.

The voltage dependency of loads is a critical aspect of voltage stability analysis

and the frequency dependence of loads is not of primary importance [1]. The

influence of the load characteristic in voltage stability is studied in this section.

Loads are a complex, time-varying mix of different devices. They were

historically modeled as constant admittances. More recently they have been

modeled as combination of constant impedance, constant current and constant


18

power (ZIP model), or in a voltage exponent form. Equation 2.9 describes the

voltage sensitivity, given by the parameter α, for general static models in

exponential form:

α

⎟⎟⎠

⎞⎜⎜⎝

⎛=

oo V

VPP (2.9)

In equation (2.9) oV is a reference voltage and oP is the active power consumed

under this voltage. Constant impedance, constant current and constant power

characteristics are obtained by using the typical values of α, 2, 1 and 0.

We assume that in Figure 2.3 at a certain time a large disturbance (such as

tripping of a transmission line) increases X, causing a significant change in the

system P-V characteristic. Figure 2.6 shows a P-V representation of the supply

Figure 2.6 Influence of the static load characteristic on P-V curves.

Typical values of α set at 2, 1, 0, -0.1, -0.4 and -0.5 [61]

(pu)V

40.−=α10.−=α0=α

2=α

1=α

50.−=α

Pre-disturbance

edisturbancPost −

(pu)P


19

system pre-disturbance and post-disturbance situation. Disturbance reduces the

maximum amount of power that can be transmitted by the system. The static

load characteristic for different values of α is also shown.

For a typical case where the load power is not affected by the voltage, i.e.,

constant power, the parameter is equal to zero. Those values, which are higher

than zero, express load voltage dependency that helps the stability of the system,

by providing some system relief. As exemplified in the figure, the larger this

parameter is, the further the new operating point is from the P-V nose. For a

static representation, this parameter expresses the instantaneous load-voltage

dependency.

Figure 2.6 shows an unstable situation when α is equal to –0.5. The negative

values of the parameter that are harmful to the stability are associated with a

combination of a dynamic restoration of the load and the discrete action of tap

changers that is explained in the next section. The admittances of such loads vary

with the supply voltage, either by their inherent design or by control loops

connected to the load devices, in order to consume constant power

2.5 Load Restoration and Voltage Stability

The static models ignore the dynamic behavior of loads that is important in

voltage stability studies. The power consumed by a static load is a function of

voltage only. However, the power consumed by a dynamic load is a function of

both voltage and time. At a voltage change, the dynamics of various load

components and control mechanisms tend to restore load power, at least to a

certain extent. This is referred as load restoration.

Consider that the power consumed by the load at any time depends upon the

instantaneous value of a load state variable, denoted as x:


20

( )x,V,zPP t= (2.10)

( )x,V,zQQ t= (2.11)

where, z is the load demand. tt Q,P are called the transient load characteristics.

If the load dynamics are described by the differential equation:

( )x,V,zfx =& (2.12)

Then the steady state of load dynamics is characterized by the following

algebraic equation:

( ) 0=x,V,zf (2.13)

Equation (2.12) can be used to obtain the state variable x as a function of z and

V:

( )V,zhx = (2.14)

Substituting (2.14) into (2.10) and (2.11) we obtain:

( )( ) ( )V,zPV,zh,V,zPP st == (2.15)

( )( ) ( )V,zQV,zh,V,zQQ st == (2.16)

where, ss Q,P , are the steady-state load characteristics.

The transition toward steady-state load characteristics is driven by the load

dynamics.

When loads are subjected to a step change in voltage, they will typically undergo

an initial (transient) step change in power. This will often be followed by a

period where the load recovers back to a new steady state value. This recovery

may be monotonic, or may involve some damped oscillatory behavior [23].

Three of the well-identified load power restoration mechanisms are:

• Induction motors

• Load behind load tap changer(OLTC)


21

• Thermostatic Loads

2.5.1 Induction Motors [1, 7, 24-26]

Induction motors are present in many industrial and commercial loads. The

electrical characteristics of single cage induction motors are often represented by

the equivalent circuit shown in Figure 2.7.

Figure 2.7 Per phase equivalent circuit of induction motor

Where, the parameters sR (stator resistance), sX (stator reactance), mX

(magnetizing reactance), rR (rotor resistance) and rX (rotor reactance) are

known. The rotor slip is indicated by s . The slip used in this model is the

frequency of the bus voltage minus the motor speed. Some programs incorrectly

use either average system frequency or 1.0 in place of the frequency [27].

When subject to a step drop in voltage, the motor active power first decreases as

the square of the voltage (constant impedance behavior), then recovers close to

its predisturbance value in the time frame of a second. The internal variable x of

this process is the rotor slip. For dynamic voltage stability studies a simplified

first order model with slip being the only state variable may be adequate. In fact,

a motor with constant mechanical torque and negligible stator losses restores to

constant mechanical power (demand variable z). Taking into account these losses

and more realistic torque behaviors, there is a small steady-state dependence of P

rXsXsR

sRR r=

mX

jQP +V

I


22

with respect to V. The steady-state dependence of the reactive power is a little

more complex. Q first decreases somewhat quadratically with V, reaches a

minimum, and then increases up to the point where the motor stalls due to low

voltage. In large three-phase industrial motors, the stalling voltage can be as low

as 0.7 pu while in smaller appliances (or heavily loaded motors) it is higher.

Based on the above equivalent circuit several levels of detail may be available,

as below [27]:

1. A dynamic model including the mechanical dynamics but not the flux

dynamics (first-order model),

2. addition of the rotor flux dynamics (third-order model) ,

3. Addition of the stator flux dynamics (fifth-order model).

Stator flux dynamics are normally ignored in stability analysis and for long-term

dynamic analysis the rotor flux dynamics may also be neglected, particularly.

Considering the above assumptions the induction motor state equation is as

below[28]:

ems PPdt

sdT −= (2.17)

In (2.17) mP is mechanical power and,

( )22

2

XsR

sRVPr

re

+= (2.18)

where rs XXX += is the total machine reactance.


23

2.5.2 On-Load Tap Changers (OLTCs) One of the key mechanisms in the load restoration is the voltage regulation

performed by the tap changing devices of main power transformers. The tap

changer controls the voltage of the distribution by changing the transformer

ratio. Normally the variable tap is on the high voltage side.

An early detailed description of a typical tap changer control system was given

in [29]. The proposed tap changer model was a complex nonlinear dynamic

model that encompassed some inherent time delays. More recent work presented

in [30] explored further the tap changer modeling issue, and presented simpler

yet still accurate models. Depending on the OLTC characteristic (type of time

delay), various discrete state dynamic models and corresponding continuous

approximations were derived.

Figure 2.8 Function of an OLTC control system [31]

The function of a typical OLTC control system is shown in Figure 2.8. The

system remains in the state wait as long as the voltage deviation ( rvv − ) is less

than the function voltage ( functionv ). When the limit is exceeded, a transition to

the state count occurs. Upon entering count, a timer is started and is kept

running until either it reaches the delay time dT , causing a transition to the state

action; or the voltage deviation becomes less than the reset voltage ( resetv ),

causing a transition to the state wait and reset of the timer. When entering the

ready

count

wait

action

functionr uvv >− dcount TT >

resetr uvv <−


24

state action, a control pulse to operate the tap changer is given. After the

mechanical delay time ( mT ), the tap operation is completed and the control

system receives a ready signal from the tap changer. The control system then

returns to state wait [31, 32].

The time delay is tuned by the time delay parameter Td0. The actual time delay

can then be either fixed ( dod TT = ), or inversely proportional to the voltage

deviation ( rdod vvTT −∝ ). The variable tap ratio has a limited regulation

range and each step is usually in the range of 0.5%-1.5%.

Load Restoration through LTC

The load restoration by LTC is indirect. In Figure 2.9, when LTC restores 2V to

its reference value the load power, which depends on voltage, is also restored. In

the analysis of LTC dynamics the fast dynamics of the generators and induction

motors can be replaced by their steady-state equations [6].

Figure 2.9 Generator-line –LTC system [6]

Figure 2.10 shows three different load transient characteristics for different

values of n. Following a disturbance due to the initial drop in the primary

voltage operating point will move from point O to point A. Then due to the LTC

halla

This figure is not available online. Please consult the hardcopy thesis available from the QUT Library


25

Figure 2.10 P-V curves of generator-line-LTC system [6]

action the secondary side voltage will change and the operating point will move

to point B, causing more drops in primary side voltage. The vertical dashed line

in Figure 2.10 shows the load characteristics when its voltage has restored to its

reference value. When the voltage near the loads (i.e., on the regulated side of

the device) is restored to its pre-disturbance level, the voltage dependence as

viewed from the bulk system is eliminated. While the system voltage may have

changed significantly, the voltage seen by the load has returned to its

predisturbance state. Therefore the power consumed, regardless of the load

voltage dependence, is relatively constant [8, 33]. This is the steady-state load

characteristic seen from the primary side.

LTC Reverse Control Action

In references [34-42] reverse control action of LTC in association with voltage

collapse is investigated. It is shown theoretically that behind a critical condition

of tap position, the secondary voltage drops if the tap position is raised aiming at

raising the secondary voltage. The above critical condition for occurrence of

no

n1

n2

OA

B

P1

V1

Post- disturbance


26

reverse action coincides with the power matching condition under which the

power consumed by load is maximized [21]. All practical feeders serve

components of voltage sensitive loads. By changing the tap the effective load

admittance (seen from bulk power side) decreases, so that eventually it can

become smaller than that of the equivalent network feeding the load bus. After

this point the process of load restoration becomes unstable, as successive tap

changes decrease distribution voltage further and further away from its setpoint

[43]. The proposed voltage instability predictor in [12] is based on comparison of

system and load impedance, too.

In other words, the power matching condition is nothing but the voltage collapse

condition. This fact suggests that the reverse action of tap changer is closely

related to the voltage collapse. Hence, tap locking action could be one of the

countermeasures to avoid voltage collapse. There are, however, exceptions such

as a feeder serving almost exclusively heavily loaded induction motors [43]. The

reactive consumption of induction motor is dependent on supply voltage, and

demonstrates a negative slope of reactive power to voltage in heavily loaded,

compensated cases. In such a case increase of distribution side voltage will

decrease reactive consumption and thus will result in increasing the transmission

side voltage as well.

Yorino and Galina in [37] have investigated the sensitivity of load voltage to tap

position with both load exact characteristics and constant power load and have

concluded that it can be used to assess the occurrence of reverse control action .

In [35] Hong and Wang have presented an approach for estimating voltage

stability region concluding the stability region of LTCs.


27

2.5.3 Thermostatic Loads [1, 7, 44, 45]

An aspect of load behavior that contributes significantly to the voltage stability

problem is the effect of thermostatic controls. The voltage dependence of loads

in a system, particularly loads such as resistive space and water heating, can give

considerable load power relief following a voltage depression induced by a

system disturbance. However, this reduction in power does not remove the need

to deliver energy, e.g., to maintain constant temperature. Eventually, the reduced

power consumption of the individual loads results in thermostats leaving loads

connected longer. The aggregate effect is to push the nominal load power up

towards a level that will produce the pre-disturbance actual power at the

depressed voltage. The state variable (x) and demand variable (z) for this process

are connected equipment and energy requirement, respectively. The time

constant associated with this resetting action is open to investigation, but values

between 10 and 30 minutes have been suggested [33]. However, for a large

enough voltage drop, the aggregate load power does not recover to its

predisturbance value, owing to the fact that the heaters stay on permanently, thus

giving a mere impedance load characteristic in the steady state. It is not always

necessary to take into account “thermostat” characteristics, unless significant

numbers of on-load tap-changers will reach regulation limits following a

disturbance.

2.6 Literature Review on On-line Voltage Stability Analysis

Very few online methods have been proposed in the past to predict voltage

collapse of a power system and to take corrective actions [9, 46-49]. A few such

methods are discussed here which utilities use predominantly to study voltage

collapses.


28

Voltage stability evaluation using voltage at bus alone will not give a correct

indication of an impending collapse [50]. Sometimes, voltage levels may fall

below limits due to heavy loading, during peak hours. This is only a low voltage

condition and may not lead to voltage collapse. Consequently just monitoring

voltage levels should not be used as a voltage collapse warning index. Voltage

levels have to be studied as function of some other key system parameter such as

real power or reactive power [49].

Voltage stability analysis is still widely studied in industries by computing the P-

V and Q-V curves at selected load buses [1]. Generally, such curves are

generated, by executing large number of load flows, using conventional methods

and models. While such procedures can be automated, they are time-consuming

and do not readily provide information, useful in gaining insight into the causes

of stability problems. In addition, these procedures focus on individual buses;

that is, the stability characteristics are established by stressing each bus

independently. This may unrealistically distort the stability condition of the

system. Also, the buses selected for Q-V and P-V analysis must be chosen very

carefully, and a large number of such curves may be required to obtain complete

information.

The sensitivity analysis using Jacobian matrix evaluates the dV/dQ factor for

voltage stability analysis [51, 52]. When the system is heavily loaded, generators

may reach their reactive limits. Since sensitivity analysis is based on load flows,

a P-V bus in this case will be replaced as P-Q bus and there will be drastic

changes in the sensitivities.

Moreover, the magnitudes of the sensitivities for different system conditions do

not provide a direct measure of the relative degree of stability.


29

Modal analysis method is an indirect calculation of the dV/dQ sensitivities [53].

When the system is heavily stressed both dV/dQ and dV/dP sensitivities play an

important role in voltage collapse prediction. The implicit assumption is that the

active power dynamics don’t play any role in the voltage collapse, which is not

valid when the system is heavily loaded. A small positive eigenvalue is itself an

indication of danger. This method predicts voltage collapse only when one of the

calculated eigenvalues becomes negative [49].

2.7 Local Data Based Voltage Stability Monitoring

Voltage stability is threatened when a disturbance increases the reactive power

demand beyond the sustainable capacity of the available reactive power

resources [8]. Since the reactive power can not be transferred to areas far away,

voltage instability events usually appear due to localized shortage of reactive

power or voltage control [54]. This makes it possible for decentralized voltage

stability on-line monitoring just with the local bus information, while networks

with long distance to the area being monitored can be equalized.

The local voltage stability monitoring and control is based on a two-bus

equivalent system as shown in the Figure 2.11. The supply system is represented

by its Thevenin equivalent seen from the terminals of the load bus of interest. In

this section the literature related to the system Thevenin impedance identification

is reviewed.

Figure 2.11 Two-bus system

P+jQthhth jXtRZ +=VthE

I

ljXlRlZ +=


30

From electric circuit theory it is known that load power is maximized when the

load impedance is the complex conjugate of the transmission impedance. This

case however is not suited for power system applications. The first problem is

that in a transmission system the resistance can be negligible compared to the

reactance, and thus the maximum power goes to infinity. The other problem is

that, a highly capacitive load would be required to match the dominantly

inductive nature of the system impedance. Also, half of the energy must be

consumed in transmission system [6].

A modified derivation, closer to power system applications is made by assuming

that the power factor of the load is specified. Specifying the load power factor

φ cos is equivalent to having a load impedance of the form (Figure 2.11):

φtanjRRjXRZ lllll +=+= (2.19)

Under this assumption, the load active power is given by:

22

2

φ)tanR(X)R(RERP

lthlth

l

+++= (2.20)

Maximizing P over the variable lR , the necessary extremum condition is:

0=∂∂

lRP

(2.21)

The solution of (2.21) provides the following criterion:

( ) ( ) 01 2222 =+−+ φtanRXR lthth (2.22)

which is equivalent to:

thl ZZ = (2.23)

The second derivative is given by:

( )φtanRR

Pl

l

22

212 +−=

∂∂

(2.24)


31

which is always negative, indicating that the solution is a maximum. Thus in this

case the load power is maximized when the load impedance becomes equal in

magnitude to the transmission or Thevenin impedance;

Maximal power transfer thl ZZ =⇔

For a lossless transmission the load maximum active power and receiving end

voltage will be, respectively:

thmax X

EφsinφcosP

21

2

+= (2.25)

φsin12 +=

EV Pmax (2.26)

which are the same power limit and voltage magnitude for a lossless line derived

from load flow in equations (2.7) and (2.8). This means that the impedance

matching point is nothing but the voltage collapse point.

Based on this fact, Vu et. al. have proposed a Voltage Instability Predictor(VIP)

[15]. In their work the ratio between the voltage and current phasors measured at

the load bus is used to compute the apparent impedance of the load. Local bus

and the rest of the system are treated as Thevenin equivalent circuit.

The impedance plane is separated into two regions, the Thevenin impedance

circle and the rest of the plane (Figure 2.12). As load varies, its apparent

impedance traces a path in the plane, and voltage instability occurs when the

load impedance crosses the Thevenin circle Indeed, VIP can be viewed as a

voltage Relay with an adaptive set-point [14].

In that work [14], using only local phasor measurements of voltage (V) and

current (I), the Thevenin equivalent is estimated, using Ohm’s law in equation

(2.27).


32

IZVE th+= (2.27)

The Thevenin equivalent parameters are estimated by the measurements taken at

two or more different times [15]. It is assumed that source voltage and Thevenin

impedance are constant.

The reports on the feasibility study of the VIP for a realistic 7000 bus system is

provided in[12], where the proximity to voltage collapse is expressed in terms of

the distance between two voltage curves or two impedance curves.

Figure 2.12 Maximal power transfer is reached (voltage instability) when the

apparent impedance of the load bus reaches the Thevenin circle [14]

In [16], a new measure; power margin, is proposed to describe the proximity to

collapse in terms of power.

The proposed VIP can precisely predict voltage collapse if the system Thevenin

impedance be estimated accurately. Using VIP the proximity to voltage collapse

can be described in terms of power and can be looked upon as the power

available to be pushed through the VIP location before the network collapses.

The device has several potential applications. VIP is a voltage relay with an

adaptive set-point. It can be used to impose a limit on the loading at each bus,

and sheds load when the limit is exceeded. It can also be used to enhance

existing voltage controllers such as SVCs. Coordinated control can be obtained if

Circle of radius

thZ

lZ

r

x


33

communication links are available; however, in case of emergency, the device

carries out its own decision.

Using the same method for Thevenin equivalent estimation, a method has been

proposed by Haque in [48] to estimate on-line the maximum permissible loading

and voltage stability margin at different buses of a system.

Looking at the voltage and current at the connection point will provide the

Thevenin impedance of the system if there were no variation in the system.

The major difficulty is that there are system variations; such as system dynamics

and dynamic loads, changing the Thevenin equivalent of the supply at the

connection point. It can be shown that in this case the apparent estimated

impedance will be equal to the load impedance, interpreting that the system

voltage is collapsing, even though this is not the case.

Taking the measurements at two different times and using equation (2.27) one

gets:

( )121212 IIZVVEE th −+−=− (2.28)

Assuming constant Thevenin source we will get:

IZV thΔ+Δ=0 (2.29)

Thus:

IVZth Δ

Δ−= (2.30)

It is also obvious that:

( )1212 IIZVV l −=− (2.31)

Thus:

thl ZIVZ −=ΔΔ

= (2.32)


34

Warland and Holen [55] tried to overcome the problems arising when the

Thevenin equivalent is not constant(the usual case). They have introduced the

VIP++ that estimates the Thevenin equivalent parameters by adding the

knowledge from the surrounding area, using phasor measurement units on two

buses. The proposed method loses the locality feature of data in the estimation

process.

Beguvic and Novosel in [13] and Soliman et.al, in [56] using the local bus past

data together with the recent measurements have proposed a recursive least

square method to predict Thevenin impedance. However they do not discuss how

the method will work if a disturbance changes the system impedance and/or

there are system dynamics.

The method introduced by Palethorpe and et.al. in [57] employs a power

electronic circuit to inject a small current disturbance onto the energized power

system, and the measurements of the disturbance current and resultant voltage

transients are used to identify the system impedance. In that work the main

problem is that they have not taken into account the system and load changes.

2.8 Literature Review on Aggregate Load Modeling

While data for most transmission and generation elements is well established or

can be readily determined from measurement, good load data and models yet

remain difficult to reliably ascertain [58].

The load seen by a bulk power delivery transformer is an aggregate of many

individual loads, fed through distribution lines and MV/LV transformers,

compensated by switched capacitors, etc. While typical data can be obtained for

individual equipments [59], the real problem is to determine the composition of

the load that varies not only from one bus to another but also with the season, the


35

time of the day, etc. Thus the aggregate loads are the most uncertain power

system components.

A typical load response to a step change in voltage is illustrated in Figure 2.13.

Figure 2.13 Load power response to a voltage drop

The load recovery shown in Fig. 2.13 is characterized by three parameters,

steady state load-voltage dependence, transient load-voltage dependence and a

load-recovery time constant [60].

pu

pus

o

oss

pu

put

o

ott

maxpp

minp

oOp

oOs

pp

V)P(

V/VP/Pand

V)P(

V/VP/P

zzzwith

VVPz

VVPP)V(P

dtdz

Tts

Δ

Δ=

ΔΔ

≅αΔ

Δ=

ΔΔ

≅α

⟨⟨

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=−=

αα

(2.33)

where pz is an internal state variable(load recovery). oV and oP are the voltage

and power consumption before a voltage change. pT is the active load recovery

time constant and can be obtained from a least-square fit of the time response.

Similar relationships hold for the reactive power [6]. sα and tα are the steady-

state and transient load-voltage dependence, respectively. In [61-63] using the

normal operation data the load parameters time-varying characteristic and their

dependency with weather and season of the year have been studied and have

P

t tPΔ

sPΔ

V VΔ


36

been concluded that the active and reactive time constants and transient time

load-voltage dependence exhibit a strong time dependency, even during the day.

It has been proven that load representation has a critical effect on voltage

stability analysis results when loads are subjected to large variations in system

voltages [1]. Consequently, more and more studies have been done to better

understand the nature of load dynamics [11, 27, 28, 60, 64-73].

Load modeling has typically been conducted using two basic approaches:

Component based modeling and measurement based modeling.

2.8.1 Component based and measurement based load modeling In this approach a survey of the individual devices which make up the aggregate

load is made and based on known characteristics of the individual devices a

composite load is synthesized [58]. Some Shortcomings of the method are as

below:

• Large number of individual load devices

• Devices are not connected simultaneously

• Device characteristic varies by the age and manufacturer

Measurement approach to identify the load model and its parameters comes from

real load and may give better modeling results over the component based

approach [11].

The most convenient load model from the point of view of parameter

identification is the input-output transfer function load model (Fig. 2.14).

Transfer function G(s) is of the general form:

Figure 2.14 Block diagram of input-output transfer function model

G(s) VΔ PΔ


37

G(s) = K01

201

2

asasbsbs

++

++ (2.34)

where K is the gain constant.

Xu et al. in [69] exploring the determination of load characteristic from field

tests performed on the B.C. Hydro system, have proposed a load test procedure.

It is shown that load response due to disturbances caused by transformer tap

changing are sufficient to capture the dynamic and static load characteristics. It is

also shown that the amount of voltage drop is not very critical for determination

of load parameters and a 5% voltage drop will be sufficient. Their tests indicated

also that the load tap changers and feeder voltage regulators are the main cause

of load recovery dynamics. The influence of voltage variations on the estimation

of the load model parameters has been investigated by Dai et al [74]. It is shown

that first order models are more robust and input-output transfer function model

is not influenced by the delay of the transducers.

In[11] generic algorithms and evolutionary programming based identification is

used by Zhu et al., to identify the power system load model based on data from

field measurement. Beguvic and Mills in [75] proposed a simple recursive

algorithm for real-time identification of the basic components of the composite

load models and, proposing a monitoring system, investigated their effect on the

voltage stability boundary.

Disadvantages of the measurement based models are:

• They are valid only for the particular time and location of measurement

• They are valid only for the small voltage changes

• They do not account the rotational non-linearities of induction motors


38

Load models should be able to capture, with acceptable accuracy, the load

behavior. The models should also be physically based, derived from “easy to

obtain” data and suitable for use in both static and dynamic analysis. In a real-

time control, the control effect should be improved if load parameters can be

real-time identified.

2.9 Countermeasure to Long-term Voltage Instability

Although a system protection scheme may integrate and coordinate several types

of actions, action on load is the ultimate countermeasure. Shedding a proper

amount of load, at a proper place, within a proper time can be done indirectly

through a modified control of OLTCs or directly as load shedding.

Emergency control of OLTC can be achieved by OLTC blocking, by bringing

back the taps to predetermined positions, or by reducing OLTC set-points [43].

In the latter case, the sensitivity of load to voltage is eliminated. From the

viewpoint of customer voltage quality, although the distribution voltage remains

low, it is on the average less sensitive to transmission transients [10].

The role of automatic and non-automatic OLTCs in emergency voltage control is

investigated in [43]. It is shown that how tap blocking of bulk power delivery

transformers can prevent an approaching voltage collapse. The problems and

limitations of this countermeasure are also discussed. It is also shown that

OLTCs at higher voltage levels (including those of generator step-up

transformers) can help maximize the loadability margin either by automatic

control, or by selection of tap adjustments using off-line studies.


39

2.10 Summary

The fundamental materials that are needed in the following chapters are

reviewed in this chapter. The related works to the on-line voltage stability

monitoring and load modeling were also reviewed and discussed.

In chapter 3 a method is developed to estimate power system Thevenin

impedance that is based on signal processing on the measured data in the load

bus.

Chapter 3

Correlation Based System Thevenin Impedance Estimation

3.1 Introduction As it was stated in chapter 2 the local voltage stability monitoring and control is

based on a two-bus equivalent system where the supply system is represented by

its Thevenin equivalent seen from the terminals of the load bus of interest. In this

chapter a method is developed to estimate power system Thevenin impedance

that is based on signal processing on the random changes of the measured

voltage and current phasors in the load bus. It is shown that the cross-

correlations of the changes in the load voltage and current with respect to the

changes in the load admittance can be used to estimate the system Thevenin

impedance. It is assumed that the power system is linear and the customer load

acts as a random walk process [76]. The required steps are clearly shown.

Chapter 3. Correlation Based System Thevenin Impedance Estimation

41

Dynamic components in the load voltage and current caused by system and/or

load are removed and the residuals are used for the estimation. The method is

validated by simulation and work on real data is also provided to reconfirm the

method.

3.2 Theory of the Correlation Based System Thevenin

Impedance Estimation

There is evidence that the changes in composite customer load in periods up to

10 seconds are unpredictable. The idea that the costumer load changes are

uncorrelated white noise, and hence, the composite load would be the integral of

white noise [77] is confirmed in [76]. Changes in the customer load cause

changes in the load bus voltage and magnitude (Figure 3.1).

Figure 3.1. Time measurement of load voltage and current magnitudes


42

In the next sections the theory of the system Thevenin impedance estimation

using the changes in the local load bus voltage and current is provided. Two

different methods are used that the first method is based on the block diagram

representation of power system and the other approach uses the system

equivalent circuit to develop the theory.

3.2.1 Block diagram representation The block diagram of the interaction between a power system and a local load is

shown in Figure 3.2. VΔ and IΔ are the changes of the local load bus voltage

and current phasors, respectively. SZ and LY are the system equivalent

impedance and the load admittance, respectively. The system mode is partly

exited by the changes in the local load and partly from changes in the other

loads, here treated as w(t) and d(t) as white noise uncorrelated, respectively.

Figure 3.2. Block diagram representation of proposed method for system

identification

In Figure 3.2, one can write:

( )SL

sLZY

tdZYtwI++

=Δ1)( (3.1)

))()((1

twtdZY

ZVSL

S −+

=Δ (3.2)

Local Load

Power System

w(t)

d(t)

∑∑ sZ+

IΔ

LY

+

_ +

VΔ Local Load

Changes

Other Loads Changes


43

There is no clear relation between VΔ and IΔ , but, multiplying VΔ by IΔ and

IΔ by itself will result in:

( )21][

SL

sLS

ZY)t(w)t(d)t(d)t(dZY)t(w)t(w)t(d)t(w

ZIV+

−+−=ΔΔ (3.3)

( )222

12

SL

SLSL

ZY)t(d)t(dZY)t(w)t(dZY)t(w)t(w

II+

++=ΔΔ (3.4)

Applying the expectation function (mean value) to equations (3.3) and (3.4) will

yield:

[ ] [ ] [ ] [ ] [ ]{ }( )21 SL

sLS

ZY)t(w)t(dE)t(d)t(dEZY)t(w)t(wE)t(d)t(wE

ZIVE+

−+−=ΔΔ

(3.5)

[ ] [ ] [ ] [ ]( )2

22

12

SL

SLSL

ZY)t(d)t(dEZY)t(w)t(dEZY)t(w)t(wE

IIE+

++=ΔΔ (3.6)

Where, E denotes expected or mean value. Now considering that w(t) and d(t)

are uncorrelated white noises, i.e.; [ ] 0=)t(d)t(wE , equations (3.5) and (3.6)

become:

[ ]( )21 SL

sLS

ZYDZYW

ZIVE+

+−=ΔΔ (3.7)

[ ]( )2

22

1 SL

SL

ZYDZYW

IIE+

+=ΔΔ (3.8)

Where, D and W are variances of d and w, respectively. The ratio of equation

(3.7) with respect to equation (3.8) will have two components as below:

[ ][ ] DZYW

WZDZYW

DYZIIEIVE

SL

s

SL

Ls2222

2

+−

+=

ΔΔΔΔ

(3.9)

In the case when the majority of the modal disturbance is originated in the local

load section being examined, i.e.; 0≈D , equation (3.9) can approximate:


44

[ ][ ] SZ

IIEIVE

−≈ΔΔΔΔ

(3.10)

The negative sign in equation (3.10) is because current is leaving the supply

system. In the case 0≈W , (3.9) can be used to identify the load. If both W and D

are present equation (3.10) can not be used to estimate the system impedance. In

this case ΔV and ΔI should be correlated with a quantity that is correlated with

w but uncorrelated with d. This quantity for example can be the changes in the

admittance (or impedance) of the local load. Let this quantity be LYΔ , then

considering [ ] 0=Δ )t(dYE L it can easily be shown that:

[ ] [ ]SL

LSL ZY

w(t)Δ(EZΔVΔYE

+−=

1 (3.11)

and,

[ ] [ ]SL

LL ZY

w(t)Δ(EΔIΔYE+

=1

(3.12)

Hence:

[ ][ ] S

L

L ZΔIΔYEΔVΔYE

−= (3.13)

It should be notified that the terms in equations (3.10) and (3.13) are

instantaneous changes in quantities. This means that any possible dynamic

component caused by load and/or system needs to be removed from voltage and

current differences before using the equations.

3.2.2 Using system equivalent circuit

In Figure (3.3) power system is represented by its Thevenin equivalent. It is

assumed that:

1. The system is linear.


45

2. Load admittance has a fixed real and imaginary components plus

uncorrelated and unpredictable changes in each component.

3. There are random changes in the supply system causing random

changes in the Thevenin equivalent.

The changes in the supply system and local load will give rise to the

changes in the load current and bus voltage.

Figure 3.3 load connected to the equivalent circuit of the system

It was shown in section 3.2 that the changes in the load bus voltage and current

can be used to estimate the system Thevenin impedance. At the first glance it is

expected that the changes in the voltage in phase and out-of-phase components

with respect to current can be respectively used to estimate the real and

imaginary components of the system Thevenin impedance, respectively. This

hypothesis is examined in the next section.

a) Voltage in phase and out-of-phase components with current In Figure 3.3 one can write:

VIZE =− (3.14)

Let δ= jEeE α= jVeV and β= jIeI . Multiplying both sides of equation (3.14)

by β− je1 yields:

)()( α-βjδ-βj eVIZeE =− (3.15)

System equivalent

Ijxrz +=

V

_

+EY


46

In another form,

ϕ+ϕ=+−−+− sincos)()sin()cos( jVVIjxrβδjEβδE (3.16)

Where RVV =ϕcos and XVV =ϕsin are the voltage in phase and out of phase

components with respect to current, respectively, as shown in Figure 3.4.

Figure 3.4 Voltage components with respect to current

The difference of equation (3.16) at two different times becomes the two

following equations:

12)()cos()cos( 12111222 RR VVIIrβδEβδE −=−−−−− (3.17)

12)()sin()sin( 12111222 XX VVIIxβδEβδE −=−−−−− (3.18)

The two first terms in each of the above equations are dependent on the

magnitude and angle of the system Thevenin voltage and the current angle, and

hence, these terms can not be removed by correlation methods. These equations

can not be used to estimate the system Thevenin impedance components because

there is no information available about the system Thevenin voltage.

b) Random changes in the load bus voltage and current [78] Considering changes to E , V and I equation (3.14) can be rewritten as

following:

V

I

βαϕ −=

ϕcosVVR =

ϕsinVVX =

Im

Re


47

( ) ( ) VΔVIΔIZEΔE +=+−+ (3.19)

Subtracting equation (3.14)) from equation (3.19) gives:

VΔIZΔEΔ =− (3.20)

Multiplying both sides of equation (3.20) by *IΔ 1, the complex conjugate of IΔ ,

and getting the expectation value yields:

⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡−⎥⎦

⎤⎢⎣⎡ **

S* IΔVΔΕIΔIΔΕZIΔEΔΕ (3.21)

Where E means expectation or mean value and [ ]ZmeanZS = . Solving equation

(3.21) for SZ gives:

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡−⎥⎦

⎤⎢⎣⎡

=*

**

SIΔIΔΕ

IΔVΔΕIΔEΔΕZ (3.22)

In equation (3.22) the possible small random changes in the system Thevenin

impedance have been cancelled by averaging the impedance over the sampling

time frame. The rest of this section is devoted to the application of equation

(3.22) in different cases in power system.

Case 1: System with Constant Thevenin Voltage, 0EΔ = In this case it can easily be shown that equation (3.22) will reduce to the

following equation:

⎥⎦⎤

⎢⎣⎡ ΔΔ

⎥⎦⎤

⎢⎣⎡ ΔΔ

−=*

*

SIIΕ

IVΕZ (3.23)

1 For the complex processes (t)iZ and (t)jZ cross-correlation functions is defined by

)](t*(t)[)( τ+=τ jiZZ ZZER [peebles].


48

Being available the data of load voltage and current at different times, in this

case equation (3.23) will be applicable, even though it is not a realistic case.

Case 2: Random Changes in the Thevenin Source, 0EΔ ≠ The random changes in the system, e.g., random changes in the other loads in

the system, will cause random changes in the system Thevenin voltage.

In this case 0≠EΔ , and because E and I are correlated variables, equation

(3.23) can not be used to estimate the system impedance. Correlating the

variables in equation (3.22) with YΔ , the changes in the local load admittance,

can be used to eliminate EΔ from equation because E and Y are not correlated,

i.e.; [ ] 0=YE ΔΔΕ . This means that in this case equation (3.20) will estimate the

system Thevenin impedance if the quantities in the equation are correlated

with YΔ , hence:

[ ][ ]YIΕ

YVΕZS ΔΔΔΔ

−= (3.24)

Case 3: System with Dynamics System variations; such as system dynamics and dynamic loads, will impose

dynamic variations to the Thevenin equivalent of the supply at the connection

point. Consequently, load voltage and load currents will also contain dynamic

components. In equations (3.23) and (3.24) the effect of the dynamic component

in EΔ can not be cancelled by getting its correlation with respect to the changes

in the load current, in equation (3.23), and/or changes in the load admittance, in

equation (3.24), and consequently estimation fails.

Subtracting dynamic components from load voltage and load current differences

can be a useful method for the elimination of the dynamics in equations (3.23)

and (3.24). This can be achieved by fitting a curve to the real and imaginary


49

components of the measured load voltage and current differences, individually,

and then subtracting the predicted curves from the real ones. The residuals then

may be used for the estimation of the system impedance using equation (3.23)

and/or equation (3.24).

3.3 Algorithm of the Correlation Based System Thevenin

Impedance Estimation

Considering all above cases the system Thevenin impedance can be estimated

using the following steps:

Step1: Providing the initial local load bus data

Compute load current (simulation) and/or load admittance (real data) using

VYI = equation in each time step.

Step2: Refining the local load bus data

Compute the changes in the local load bus voltage, current and admittance using

the following equations:

( )∑−

=+ −=

1

11

n

kkk VVVΔ (3.25)

( )∑−

=+ −−=

1

11

n

kkk IIIΔ (3.26)

( )∑−

=+ −=

1

11

n

kkk YYΔY (3.27)

Where n is the number of the data points. The sign of the current difference is

changed to represent IΔ a current feeding the system.

This differentiation is needed because the estimation method uses the random

changes in the load voltage, current and admittance.


50

Step3: Removing the dynamic components

Remove any dynamic components, caused by system and/or load,

from IΔVΔ and . As it was already stated in this chapter the method uses only

the instantaneous changes in quantities. In this thesis one-step prediction method

can be used to remove the dynamic components (Ref. to Appendix A).

Step4: Thevenin impedance estimation

Estimate system Thevenin impedance using equations (3.23) and/or (3.24).

Equation (3.23) can only be used when there are no changes in the supply

system. It is noticeable that the estimated thZ is the average of the system

equivalent impedance over the data time frame and the small possible random

changes in system Thevenin impedance are ignored. However, if a disturbance in

the system causes a sudden change in the supply system the estimation process

must be renewed to identify the system new equivalent impedance. This situation

can be recognized by a sudden change in the bus voltage while the load

impedance remains almost constant.

3.4 Simulation Results In this section, the algorithm of the system Thevenin impedance estimation is

confirmed by simulating a test system. The proposed method is applied to the

four bus system shown in Figure 3.5.

Figure 3.5 Four bus test system with two different variable loads

I

2Z1

Load #1 Load #2

23 41Z 3Z


51

Bus 1 is modelled as an infinite bus. The changes in load admittances are

modelled as unpredictable random processes. These changes give rise to the

changes in the local load bus (bus 3) voltage and current. Also, the changes in

load #2 cause changes in the system Thevenin voltage, seen from the bus 3 view

point.

Dynamic components are also created in the load voltage and current by

choosing an initial value for the angle of the synchronous machine in bus 2

different from its steady state value, and/or, adding dynamic loads; such as

induction motors, to the system. These dynamic components of the load voltage

and current are removed using a “one-step prediction” algorithm.

The changes in the voltage and current of load #1 are then used to estimate the

system Thevenin impedance (seen from the bus 3).

3.4.1 Case 1 System with Constant Thevenin, no dynamics In this case passive load #2 is modelled as a constant admittance. The passive

load #1 admittance has a fixed real and imaginary component (constant power

factor) plus a small separate stochastic change in each component.

The small load changes in each step are considered to be normally distributed

random variables. These changes in the load will give rise to variations in the

voltage and load current in bus 3.

The magnitude of the simulated load admittance for a 0.02 sec sampling time

and 100 sec total simulation time is shown in Figure 3.6. Figure 3.7 indicates the

magnitudes and the angles of the load current and load bus voltage.


52

Figure 3.6 Simulation of load admittance changes in case 1

Figure 3.7 Magnitudes and phases of the load voltage and current in case 1


53

Bus 3 voltage and current and the load admittance are differentiated using

equations (3.25) - (3.27). The difference of the real component of the load bus

voltage and its auto-correlation are shown in Figure 3.8.

Figure 3.8 Difference of the real component of the load bus voltage and its auto-

correlation in case 1

Equations (3.23) and (3.24) are used to estimate the system Thevenin impedance.

The results of the simulations with different values for Thevenin impedance thZ

are shown in Table 3.1.

Referring to table 3.1, for both equations the estimated values are very close to

the actual values. The small differences are due to the applying one-step

prediction program.


54

TABLE 3.1 COMPARISON OF THE ESTIMATED AND ACTUAL VALUES OF THE SYSTEM

THEVENIN IMPEDANCE IN CASE 1

thZ (actual) Estimated value using (3.23)

Estimated value using (3.24)

0.0499 + 0.1065i

0.0983 + 0.1915i

0.0842 + 0.3553i

0.0497 + 0.1064i

0.0977 + 0.1913i

0.0837 + 0.3553

0.0502 + 0.1064i

0.0980 + 0.1927i

0.0843 + 0.3546i

3.4.2 Case 2: Random Changes in the supply system, no dynamics In this case the changes in the components of both load admittances are taken as

uncorrelated random changes (Figure 3.9). The voltage phasor and the current

phasor to the load in bus 3 are measured and used for estimation of the system

Thevenin impedance.

Figure 3.9 Simulation of load admittances changes in case 2


55

Equations (3.25) - (3.27) were used to find the differences of the bus 3 voltage

and current and the load admittance. System Thevenin impedance was estimated

using (3.23) and (3.24). The estimation results for different values of the ratios of

random changes in two loads are shown in Table 3.2. As it is expected the

achieved results by equation (3.23) are not close to the actual values and even in

the last case (ratio=5) the components of the estimated impedance are negative.


THEVENIN IMPEDANCE FOR A SIMULATION TIME, T=100 SEC AND A TIME STEP ΔT =

0.02 SEC AND DIFFERENT RATIOS OF RANDOM CHANGES IN LOAD #2 WITH RESPECT TO

LOAD #1.

Ratio of changes



0.5 1 2 5

0.0499 + 0.1065i 0.0499 + 0.1064i 0.0501 + 0.1064i 0.0505 + 0.1060i

0.0474 + 0.1044i 0.0416 + 0.0991i 0.0181 + 0.0800i -0.1155 - 0.0275i

0.0498 + 0.1057i 0.0497 + 0.1081i 0.0470 + 0.1068i 0.0495 + 0.1118i

3.4.3 Effect of the electrical distance between two loads in estimation In this part, the effect of the electrical distance between changing loads in the

estimation results is investigated by choosing different values for 2Z in Figure

3.6. Results of estimation are shown in Table 3.3. In this simulation the ratio of

the changes in load #2 is chosen to be twice the changes in load #1. In the latter

case in Table 3.3 two loads are connected to the same bus.


THEVENIN IMPEDANCE FOR A SIMULATION TIME T=100 SEC AND A TIME STEP TΔ = 0.04

FOR DIFFERENT VALUES OF 2Z

2Z



0.0+0.2i 0.0+0.1i 0.0+0.0i

0.0542 + 0.1324i 0.0508 + 0.1166i 0.0510 + 0.1168i

0.0330 + 0.1143i 0.0232 + 0.0927i 0.0147 + 0.0617i

0.0517 + 0.1313i 0.0507 + 0.1139i 0.0493 + 0.0905i


56

In this case, equation (3.23) fails to estimate the system Thevenin impedance,

due to the changes in the supply system caused by the changes in load #2.

Equation (3.24) has reasonable accuracy except for the latter case that both loads

are connected to bus 3. Indeed in this case, load #2 is a part of the local load, and

to have a good estimation result, its current also should be included in the load

current.

3.4.4 Case 3a: System with dynamics, no random changes in supply system In this case a combination of passive and induction motor loads are connected to

bus 3 (Figure 3.10).

Figure 3.10. Four bus test system with dynamic load

Bus 1 is modelled as infinite bus. The synchronous generator in bus 2 is

modelled by its electromechanical equation. The changes in the load admittances

are taken as random thus the connected load operates as a random walk process.

The motor size and time constants are taken as random over a range. The passive

and active load admittances are first modelled as a set of random impedances;

they are then scaled by the random walk process to model the random switching

ON and OFF of load elements.

2Z1 23 41Z 3Z

Mz

Load #2

z

I

Load #1


57

This changing of the connected load give rise to variations in the angle of the

synchronous generator connected to bus 2. The changes in the angle and local

frequency of generator cause changes in voltage and frequency at the load bus.

The bus 3 frequency deviation is calculated as a proportion to the generator bus

frequency deviation (ref. to Appendix A). These changes create a dynamic

response in the induction motor model. This dynamic in turn will cause changes

in the composite load power, voltage and current.

The total load #1 admittance, generator angle and induction motor slip for a 0.04

sec sampling time and 10 sec of simulation time are shown in Figures 3.11, 3.12

and 3.13, respectively. Figure 3.14 shows the magnitudes and angles of the load

#1 voltage and current. The presence of the dynamic oscillation in load voltage

and current can easily be seen in Figure 3.14.

0 1 2 3 4 5 6 7 8 9 101

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045abs(yload # 1)

Time(sec)

Adm

ittan

ce(p

u)

Figure 3.11 Simulation of load #1 admittance changes in case 3a


58

0 1 2 3 4 5 6 7 8 9 10-7.8

-7.6

-7.4

-7.2

-7

-6.8

-6.6

-6.4

-6.2del2

Time(sec)

Ang

le(d

eg)

Figure 3.12 Variation of generator angle in case 3a

0 1 2 3 4 5 6 7 8 9 100.0256

0.0258

0.026

0.0262

0.0264

0.0266

0.0268Motor slip

Time(sec)

Slip

(pu)

Figure 3.13 Variation of induction motor speed in case 3a


59

0 2 4 6 8 100.932

0.933

0.934

0.935

0.936load bus voltage magnitude

Vol

tage

(pu)

0 2 4 6 8 100.93

0.94

0.95

0.96

0.97

0.98load current magnitude

Cur

rent

(pu)

0 2 4 6 8 10-8

-7.5

-7

-6.5load bus voltage angle

Time(sec)

Ang

le(d

eg)

0 2 4 6 8 10144.2

144.4

144.6

144.8

145

145.2load current angle

Time(sec)

Ang

le(d

eg)

Figure 3.14 Magnitudes and angles of the load bus voltage and current

Equations (3.25) and (3.26) are applied to the load voltage and current

components. The difference of the voltage imaginary component and its auto-

correlation are shown in Figure 3.15. One- step prediction algorithm is applied

to the voltage difference and the predicted curve is subtracted from the actual

curve. The residual and its autocorrelation are shown in Figure 3.16. Equations

(3.23) and (3.24) were applied to the residuals. The estimated and actual

Thevenin impedances are compared in Table 3.4. The estimated values of the

system impedance are very close to the actual values because in this case there is

no random variation in the supply system. This agrees with the proposed

estimation theory.


60

0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4x 10-4 Voltage imaginary component

Time(sec)

Vol

tage

imag

(pu)

-8 -6 -4 -2 0 2 4 6 8 10-1

-0.5

0

0.5

1x 10-4 Auto(voltage imag)

lag(sec)

Cor

rela

tion


correlation in case 3a

0 1 2 3 4 5 6-1

-0.5

0

0.5

1x 10

-4 voltage imag residual

Time(sec)

Res

idua

l(pu)

-8 -6 -4 -2 0 2 4 6 8 10-5

0

5

10

15

20x 10

-8 Auto(voltage imag residual)

lag(sec)

Cor

rela

tion


correlation in case 3a after removing dynamic component


61


THEVENIN IMPEDANCE FOR A T=100 SEC SIMULATION TIME AND A TIME STEP TΔ = 0.04

IN CASE 3, NO CHANGES IN LOAD #2



0.0334 + 0.0710i

0.0667 + 0.1333i

0.0 + 0.1333i

0.0333 + 0.0707i

0.0665 + 0.1331i

-0.0000 + 0.1332i

0.0337 + 0.0706i

0.0668 + 0.1331i

0.0002 + 0.1330i

3.4.5 Case 3b: System with dynamics and random changes in supply system This case is similar to case 3a but random changes are also added to load #2 in

Figure 3.10. Simulation of load #1 and load #2 admittances is shown in Figure

3.17.

0 1 2 3 4 5 6 7 8 9 101

1.05

1.1

1.15

1.2abs(yload # 1)

Adm

ittan

ce(p

u)

0 1 2 3 4 5 6 7 8 9 100.34

0.36

0.38

0.4

0.42

0.44

0.46abs(yload # 2)

Time(sec)

adm

ittan

ce(p

u)

Figure 3.17. Simulation of load #1 and load #2 admittances in case 3b

The same procedure as case 3a is applied to the load bus voltage and current

components and the residuals, after removing the dynamics, are used to estimate


62

system Thevenin impedance. The estimated results are shown in Table 3.5.

Equation (3.23) fails but, the estimated values by (3.24) agree with the actual

values of the system impedance. The results confirm the developed theory for

estimation.


THEVENIN IMPEDANCE FOR A T=100 SEC SIMULATION TIME AND A TIME STEP TΔ = 0.04

IN CASE 3B (RANDOM CHANGES IN LOAD #2)



0.0334 + 0.0710i

0.0667 + 0.1333i

0.0000 + 0.1333i

0.0277 + 0.0660i

0.0571 + 0.1240i

-0.0057 + 0.1277i

0.0348 + 0.0690i

0.0689 + 0.1300i

0.0016 + 0.1329i

3.4.6 Case 4: Simulation with Swing Reference Bus In this case the angle of voltage in bus1 (reference bus) in Figure 3.5 rotates with

a constant velocity2. Dynamic component is also imposed to the system by

choosing the initial value of the bus 2 voltage angle different from its steady-

state value. A simulation of different phasors angles is shown in Figure 3.18.

0 20 40 60-150

-100

-50

0Reference bus angle

Del

ta1(

pu)

0 20 40 60-150

-100

-50

0

50Generator bus angle

Del

ta2(

pu)

0 20 40 60-150

-100

-50

0Load voltage angle

Del

ta3(

pu)

0 20 40 60-200

-150

-100

-50

0 Current angle

Cur

rent

ang

le(p

u)

Time(sec) Time(sec)

Figure 3.18. Reference bus angle changes and its reflection in other buses

2 This may happen if the measurement unit is not completely synchronized with the system.


63

As it can be seen in Figure 3.18 the reference bus angle changes are reflected on

the angles of other buses in system. These changes were removed from load bus

voltage and current and the result is shown in Figure 3.19.

0 20 40 60-10

-5

0

5 Voltage angle without rotationD

elta

3(pu

)

Time(sec)0 20 40 60

-42

-40

-38

-36

-34Current angle without rotation

Cur

rent

ang

le(p

u)

Time(sec)

Figure 3.19. Load bus voltage and angle after removing the effect of reference

bus rotation

The real and imaginary components of the load bus voltage and current, with

reference bus rotation effects and after removing these effects, were

differentiated and after removing the dynamic components the residuals were

used to estimate the system Thevenin impedance. The results for different

rotation sizes in the reference bus angle and two different cases, with and

without random changes in supply system, are shown in Tables 3.6 and 3.7,

respectively.

When there are no random changes in the supply system, Table 3.6 illustrates

that, equation (3.23) can be used to estimate the system Thevenin impedance in

either case, with reference bus rotation effects on the measured data and/or

removing those effects from data. However, equation (3.24) can only be used to

estimate the system impedance after removing the reference bus effects from

data.

In the case that there are random changes in the supply system, equation (3.23)

fails to estimate the Thevenin impedance, but, after removing the reference bus


64

effect, equation (3.24) can be used for most cases. The above simulation results

concede with the proposed theory.


THEVENIN IMPEDANCE FOR A 100 SEC SIMULATION TIME AND 0.04 SEC TIME STEPS FOR

DIFFERENT VALUES OF ROTATIONS IN REFERENCE BUS (BUS 1) ANGLE, WITHOUT

RANDOM CHANGES IN SUPPLY SYSTEM

)secdeg(1

dtdδ



-3.60

-36

-72

-180

0.0509 + 0.1167i

0.0509 + 0.1167i

0.0509 + 0.1167i

0.0509 + 0.1167i

*0.0483 + 0.1160i **0.0488 + 0.1151i

0.0507 + 0.1169i 0.0491 + 0.1129i

0.0518 + 0.1158i 0.0517 + 0.1139i

0.0551 + 0.1095i 0.0486 + 0.1052i

0.0025 + 0.1269i 0.0507 + 0.1173i

0.0230 + 0.0481i 0.0503 + 0.1159i

0.1128 + 0.2440i 0.0487 + 0.1149i

0.0178 + 0.0851i 0.0363 + 0.1233i

* Estimated value with reference bus rotation ** Estimated value after removing the effect of reference bus rotation


THEVENIN IMPEDANCE FOR A 100 SEC SIMULATION TIME AND 0.04 SEC TIME STEPS FOR

DIFFERENT VALUES OF ROTATIONS IN REFERENCE BUS (BUS 1) ANGLE, WITH RANDOM

CHANGES IN SUPPLY SYSTEM

)secdeg(1

dtdδ



-3.60

-36

-72

-180

0.0509 + 0.1168i

0.0509 + 0.1167i

0.0509 + 0.1167i

0.0509 + 0.1169i

*0.0399 + 0.1099i **0.0403 + 0.1083i

0.0413 + 0.1095i 0.0438 + 0.1062i

0.0452 + 0.1065i 0.0442 + 0.1058i

0.0487 + 0.1020i 0.0427 + 0.0960i

0.0391 + 0.0959i 0.0542 + 0.1142i

0.1863 + 0.1388i 0.0572 + 0.1149i

0.0527 + 0.1717i 0.0566 + 0.1130i

-0.1703 + 0.0186i 0.0688 + 0.1098i

* Estimated value with reference bus rotation ** Estimated value after removing the effect of reference bus rotation


65

3.5 Application of the Proposed Method to Real Data3

The proposed method in this project was applied to the measured real data

obtained from a load bus in Brisbane. The magnitude and phase of the measured

load voltage and current for a sample 100 sec time frame are shown in Figures

3.20 and 3.21, respectively. The angle of the first data point in current is chosen

as reference. The measured current is related to one feeder of six different

feeders connected to the load bus. The load admittance magnitude and angle

connected to this feeder is shown in Figure 3.22.

In this study five other load feeders are considered as a part of the supply system

for which the Thevenin impedance is estimated.

0 10 20 30 40 50 60 70 80 90 1008165

8170

8175

8180

8185

8190Voltage magnitude

Vol

tage

(V)

0 10 20 30 40 50 60 70 80 90 1002490

2495

2500

2505

2510

2515current magnitude

Cur

rent

(A)

Time(sec)

Figure 3.20. Time measurement of load voltage and current magnitudes in a

residential feeder in South Pine substation, Brisbane

3 real data obtained from a phasor measurement unit in the Australian electric power system installed at major load centres by QUT


66

0 10 20 30 40 50 60 70 80 90 10010

11

12

13

14

15Voltage angle

Vol

tage

ang

le(D

eg)

0 10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1Current angle

Cur

rent

ang

le(D

eg)

Time(sec)

Figure 3.21. Time measurement of load voltage and current angles, angle of the

first data point in current is chosen as reference

0 10 20 30 40 50 60 70 80 90 1000.304

0.306

0.308

Load admittance magnitude

Adm

ittan

ce(m

ho)

0 10 20 30 40 50 60 70 80 90 100-15

-14

-13

-12Load admittance angle

Time(sec)

Ang

le(D

eg)

Figure 3.22. Real data load admittance magnitude and angle

The changes in the voltage real component and its auto-correlation are shown in

Figure 3.23. Voltage imaginary and current components also have similar


67

changes. One step prediction method was used to remove dynamic components

of voltage and current changes. The residuals of the changes in the voltage real

component and its auto-correlation are shown in Figure 3.24.

Equations (3.23) and (3.24) were applied to the residuals to estimate the system

Thevenin impedance. Sample results for 100 sec successive time frames of data

are shown in Table 3.8. As it can be seen in Table 3.8, equation (3.23) fails to

estimate the system Thevenin impedance. The real components of the estimated

values by this equation are all negative and the imaginary components are

inconsistence. The estimated results by equation (3.24) are close together and the

differences in the results are mostly due to the random changes in the other loads

connected to the load bus. The results obtained from real data agree with the

theory and simulation results.

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4Voltage real changes

Vr(V

)

Time(sec)

-4 -3 -2 -1 0 1 2 3 4-1000

0

1000

2000

3000 Auto(voltage real changes)

Aut

o of

Vr

Lag(sec)

Figure 3.22. Real data voltage real component changes and its auto-correlation


68

0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4Voltage real residuals

Vr-r

esid

uals

(V)

Time(sec)

-4 -3 -2 -1 0 1 2 3 4-1000

0

1000

2000

3000Auto(voltage real residuals)

Aut

o of

Vr-r

esid

uals

Lag(sec)

Figure 3.23. Residuals of the real data voltage real component changes and its

auto-correlation

TABLE 3.8. ESTIMATED VALUES OF THE SYSTEM THEVENIN IMPEDANCE USING 100 SEC

SUCCESSIVE TIME FRAMES OF BRISBANE LOAD CENTRE MEASURED VOLTAGE AND

CURRENT PHASORS, STARTING AT 9 AM ON 2002/06/06

3.6 Summary

In this chapter, based on signal processing of the measured data in the local load

bus, a method was introduced to estimate power system Thevenin impedance.

Estimated value

using Eq. (3.23) (ohms)

Estimated value

using Eq. (3.24) (ohms)

-0.0452 + 0.0368i

-0.0465 + 0.0411i

-0.0345 + 0.0263i

-0.0389 + 0.0261i

0.3324 + 0.4993i

0.3389 + 0.4845i

0.3396 + 0.5083i

0.3433 + 0.5245i


69

Block diagram representation of the system was firstly used to develop the

theory. Then, different cases were discussed in detail by applying circuit theory

to the Thevenin equivalent circuit of the supply system. It was shown that the

ratio of the correlations of the changes in the load voltage and load admittance

with respect to the correlation of the changes in the load current and load

admittance can be used to estimate the system Thevenin impedance. The

required steps of the estimation algorithm were clearly shown.

The method was confirmed by simulation on a four bus test system. It was

shown that the dynamic components in the load voltage and current can be

removed and the residuals can then be used to estimate the system Thevenin

impedance.

The proposed method was also applied to the real data obtained from a load bus

in Brisbane load centre. The results agreed with the theory and simulation

results.

Chapter 4

On-line Load Characterization by Sequential Peeling

4.1 Introduction Load characteristics play an important role to the power system dynamic

stability. This is particularly true for voltage stability studies in which loads may

be subjected to large variations in system voltages and in which the short term

and long term load characteristics may come into play. Following a disturbance

in system, the dynamics of various load components and control mechanisms

tend to restore load power. The restored consumption may be beyond the supply

system capability causing voltage instability problem to the system [1].

Modeling of generators and transmission system has been well studied and many

models are now provided by manufacturers but the load behaviour is still one of

the biggest uncertainties in the prediction of voltage instability, and in spite of

Chapter 4. On-line Load Characterization by Sequential Peeling

71

significant attention given to load modelling by researchers, it remains a

challenge to properly develop load models which can be used with confidence in

studies.

A load model which is to be used in on-line voltage stability studies has the

following requirements [58]:

• It should be able to capture load behaviour when subjected to practical

variations in system voltages.

• Load model should be able to capture the effects of rotating load

dynamics such as motor stalling which may be the limiting factor in some

cases.

• It must be possible to derive the model from local load bus measured

data.

• In order to reduce the computational burden in the voltage stability

studies, the model should be simple enough.

• The model should be physically based.

In this chapter, considering the above requirements and based on the local load

bus data, a method is developed to characterize on-line the load behaviour. It is

shown that the measured load bus voltage and current during a disturbance in the

supply system can be used to identify parameters of a composite load consisting

induction motor, constant power and constant impedance load. The changes in

the load active power due to the disturbance are used to identify the active power

of the different load components in a peeling process. Then, the induction motor

reactive power is estimated using the random changes in the load active and

reactive power. The other components of the load reactive power are also

estimated using the disturbance data.


72

The theory is provided and the peeling algorithm is clearly shown. The proposed

method is validated by simulation on a test system.

4.2 Theory of the On-line Load Characterization Using Load Bus

Data

This section is devoted to the theory of the identification of a composite load

consisting of constant impedance, constant power and induction motor loads

(Figure 4.1). The induction motor can be considered as an equivalent to an

aggregate of induction motors.

Figure 4.1. One-line diagram a simple power system with composite load

The equivalent circuit of the system of Figure 4.1 is shown in Figure 4.2. Supply

system is shown by its Thevenin equivalent. X is the sum of the motor stator and

rotor inductances and the magnetizing circuit is ignored for simplification. rR is

the rotor resistance and s denotes to motor slip. The objective is to identify the

active and reactive power of the load components, using load bus measured

voltage and current [79].

Figure 4.2 Equivalent circuit of a power system with composite load

V

LR

LXconstant33 =+ jQP

X

22 jQP +

11 jQP +

thZ

jQP+

I

sRR r=

M Induction Motor

V

I Constant Power

Transmission

P+jQ Constant Impedance

G


73

4.2.1 Load active power peeling

It is assumed that the induction motor is a load with constant mechanical torque

with negligible stator losses. Following a disturbance, motor active power

restores to its predisturbance value[10]. In Figure 4.2, following a sudden

reduction in the magnitude of load bus voltage V ,due to a system disturbance,

the motor slip s increases, resulting reduction in R, until 1P matches its

predisturbance value (motor mechanical power) at time 2t (Figure 4.3). The

constant power load remains unchanged.

Figure 4.3 Active powers of different load components following a disturbance

in supply system

In Figure 4.3, PΔ , the difference of the load total power at times 2t and

−1t (predisturbance), is the final change in constant impedance load power due to

disturbance. We can write:

221

Ptt

Ptt

PP Δ==

−=

=Δ − (4.1)


74

Considering that the power of constant impedance load is proportional to the

squared voltage, one can write:

⎟⎟⎠

⎞⎜⎜⎝

⎛

=−

==Δ − 2

2

1

22 2 tt

Vtt

VKP pv (4.2)

Where, 2pvK is the voltage dependency coefficient of the constant impedance

load. Having measured voltage at different times, equations (4.1) and (4.2) can

be used to compute2pvK . Using this coefficient and the bus voltage magnitude,

then, 2P can be computed at any instant of time.

Peeling 2P from the total active power the residual power, resP , will be the sum

of 31 and PP . At the instant of disturbance 1t , the induction motor firstly acts as a

constant impedance load. Thus:

111

Ptt

Ptt

PP resresres Δ==

−=

=Δ +− (4.3)

Where:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=−

==Δ +−

1

2

1

21 1 tt

Vtt

VKP pv (4.4)

Using equations (4.3) and (4.4) one can compute 1pvK coefficient and this

factor in turn can be used to calculate the motor active power at −= 1tt that is pre-

disturbance 1P . Peeling 1P from resP the reminder will be 3P ; the constant power

load active power.

4.2.2 Induction Motor Reactive Power Estimation The electrical characteristics of single cage induction motors are often

represented by the equivalent circuit shown in Figure 4.3, where sR (stator


75

resistance), sX (stator reactance), mX (magnetizing reactance), rR (rotor

resistance) and rX (rotor reactance) are motor different parameters. The rotor

slip is indicated by s . It is important to note the “slip” used in this model is the

frequency of the bus voltage minus the motor speed. Some programs incorrectly

use either average system frequency or 1.0 in place of the bus frequency.

Figure 4.4 Per-phase equivalent circuit of induction motor

In Figure 4.4, neglecting magnetizing current and stator resistance the induction

motor current and its complex power are given respectively by:

( )222

2

22 XsR

VXsjRs

Xs

R

VjXs

R

jXs

RVI

r

r

r

r

r +

−=

+⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

=+

= (4.5)

222

22

222

2

XsR

VXsj

XsR

VRsjQPVIS

rr

r

++

+=+== ∗ (4.6)

where rs XXX += , and V is the voltage magnitude. Around the normal

operational region of low slip, i.e., 22 XsRr >> , P and Q in equation (4.6) can

be reduced to:

rRVs

P2

≈ (4.7)

2

22

rR

VXsQ ≈ (4.8)

jQP+V

I

mXsRr

SX rXSR


76

Having P already estimated, equations (4.7) and (4.8) could be used to estimate

Q if X was known, but this is not the case.

It was shown in section 3.2.1 that the changes in the local load bus data due to

the random changes in the supply system can be used to identify the load, and

vice versa. In this section it will be shown how the frequency dependent changes

in the local load bus measured data can be used to estimate the induction motor

reactive power.

Induction motor is the only frequency dependent component in the composite

load in Figure 4.4. Changes in the system frequency, f, will cause changes in the

induction motor real power (P) and reactive power (Q). The changes in P and Q

with respect to frequency are defined by:

ff qp kfQ,k

fP

=∂∂

=∂∂ (4.9)

Now using the standard relationship between motor slip and frequency,

fff

s r−= and around the normal operational region of low slip (i.e. ff r ≈ ),

( )

o

rrff

ff

ffffs 1

22≈=

−−=

∂∂ (4.10)

where 0f is the system nominal frequency. From equations (4.9) and (4.10) we

can write:

sP

ffs

sP

fPk

op f ∂

∂≈

∂∂

∂∂

=∂∂

=1 (4.11)

And,

sQ

ffs

sQ

fQk

oq f ∂

∂≈

∂∂

∂∂

=∂∂

=1 (4.12)

Substituting P from equation (4.7) in (4.11) yields:


77

21 VfR

kor

p f≈ (4.13)

Using equations (4.8) and (4.12) one can get:

22

2 VfR

Xskor

q f≈ (4.14)

An expression also can be easily found for the ratio of fpk and

fqk as below:

rp

q

RXs

k

k

f

f 2= (4.15)

Now equations (4.13)-(4.15) and the assumption of small slip can be used to

deduce expressions for P and Q from equations (4.7) and (4.8). These estimates

are specified by:

( )r

oqor

orrr R/X

fk

XfR

fRRXVs

RVs

P f

22222

=××

×=≈ (4.16)

( )rp

oqor

rorr R/Xk

fk

XfR

RXs

fR

VXs

R

VXsQ

f

f

4422

2

2

2

2

22

=××=≈ (4.17)

Equations (4.16) and (4.17) can be used to estimate the total induction motor

active and reactive power in a large system if rRX ratio is already known and

fpk and fqk can be estimated. However, having P already estimated, equation

(4.16) may be used to estimate rRX ratio, and then, to estimate Q using (4.17).

Alternatively, the following equation can be used to save the computation time,

and also, to lessen the estimation error. It can easily be shown from equations

(4.16) and (4.17) that:

f

f

p

q

k

kPQ2

= (4.18)

Having the estimated value of P, equation (4.18) can be used to estimate the

induction motor reactive power Q if the ff pq kk ratio is estimated.


78

4.2.3 fpk , fqk and ff pq kk Ratio Evaluation Using Load Bus Data

It has been confirmed in [8] that the costumer load changes are uncorrelated

white noise. It was shown in chapter 2 that the changes in the local bus voltage

and current, due to the changes in system, can be used to identify the local load,

and vice versa. Using this idea in chapter 3 the random changes in the local load

bus were used to estimate the system Thevenin impedance. Now it is shown how

the load active and reactive power changes, due to the random changes in the

supply system, can be used to evaluatefpk ,

fqk and the ff pq kk ratio.

In Figure 4.4, the constant impedance and constant power components of load

are not frequency dependent loads. Hence, fpk and

fqk coefficients defined by

(4.9) can respectively be considered as the load total active and reactive power

dependency coefficients with respect to frequency. Considering that the total

load power is a function of voltage and frequency one can write:

vf PPvvPf

fPP Δ+Δ=Δ

∂∂

+Δ∂∂

=Δ (4.19)

vf QQvvQf

fQQ Δ+Δ=Δ

∂∂

+Δ∂∂

=Δ (4.20)

In equations (4.19) and (4.20) the subscripts f and v denote the frequency

dependent and the voltage dependent component, respectively. Equations (4.9),

(4.19) and (4.20) can be used to show:

fP

ff

fP

fPk f

p f Δ

Δ=

Δ×⎟⎟⎠

⎞⎜⎜⎝

⎛Δ×

∂∂

=∂∂

=1 (4.21)

fQ

ff

fQ

fQk f

q f Δ

Δ=

Δ×⎟⎟⎠

⎞⎜⎜⎝

⎛Δ×

∂∂

=∂∂

=1 (4.22)


79

f

f

p

q

PQ

ffP

ffQ

fPfQ

k

k

f

f

Δ

Δ=

Δ∂∂

Δ∂∂

=

∂∂∂∂

= (4.23)

Using the random changes in the local load measured data the following

equations can be used to evaluatefpk ,

fqk and the ff pq kk ratio:

][][

ffEfPE

k fp f ΔΔ

ΔΔ= (4.24)

][][

ffEfQE

k fq f ΔΔ

ΔΔ= (4.25)

][][

ff

ff

p

q

PPEPQE

k

k

f

f

ΔΔ

ΔΔ= (4.26)

where E is the expected or mean value. Equations (4.24)-(4.26) can be used to

estimate the coefficients and their ratio if the frequency dependent components

are extracted from PΔ and QΔ , the changes in the load total active and reactive

power.

Correlating PΔ and QΔ with fΔ in equations (4.19) and (4.20) will eliminate the

effect of the voltage dependent components vPΔ and vQΔ if these components

are not correlated with the frequency changes. The problem is that there are

some correlation due to the correlation between fΔ and VΔ . The frequency

changes in the system cause the changes in the induction motor slip that in turn

cause changes in the load bus voltage. The changes in the load bus voltage also

influence the bus frequency through their effect on the bus voltage angle.

Therefore, removal of the voltage dependent components is the only way to use

equations (4.24)-(4.26) in the estimation process.


80

One of the methods that can be used to remove the undesired components from

the load bus power and frequency changes is described in the next section.

4.2.3 Removal of the Undesired Components from the Load Bus Data

Consider the system shown in Figure 4.5. A perturbation signal i(t) is applied to

a linear system with impulse response h(t) and the response of the system is y(t).

Figure 4.5 Block diagram of a system

Before y(t) can be measured, it is corrupted with noise n(t) which is due to the

other sources so that the observable signal is

( ) ( ) ( )tntytz += (4.27)

The time domain input-output relation for the linear system of Figure 4.5 is as

below [80]:

( ) ( ) ( ) ( )τ+−τ=τ ∫∞

iniiiz RdttR.thR)

110

1 (4.28)

where R denotes the correlation function. Assuming that i(t) and n(t) are not

correlated and/or their cross-correlation is ignorable comparing to the auto-

correlation of i(t), equation (4.28) reduces to:

( ) ( ) ( ) 110

1 dttR.thtR iiiz −τ= ∫∞

(4.29)

The frequency domain relationship for the system of Figure 4.5 is obtained by

Fourier transformation of equation (4.29):

( ) ( ) ( )ωΦω=ωΦ iiiz .H (4.30)

∑Input i(t)

Noise n(t)

y(t) z(t) + + h(t)


81

where H and Φ are the transfer function and the Fourier transform, respectively.

Equation (4.30) may be rewritten as

( ) ( )( )ωΦωΦ

=ωii

izH (4.31)

Equation (4.31) can be used to calculate and remove the undesired components

from the load bus power and frequency changes.

Now, considering vΔ as the input signal, vPΔ and vQΔ as two different outputs,

and PΔ and QΔ as the observable outputs, the following procedure can be used

to calculate and remove the voltage dependent components of the changes in the

load total active and reactive power:

1. Find the transfer functions from V to P and Q using the following

equations :

)()(

)(Hvv

vp

ωΦ

ωΦ=ω1 (4.32)

)()(

)(Hvv

vq

ωΦ

ωΦ=ω2 (4.33)

Where )(vp ωΦ and )(vq ωΦ are the Fourier transforms of cross-correlations of

V to P and Q, respectively and )(vv ωΦ is the Fourier transform of

autocorrelation of V.

2. Using the estimated )(H ω1 and )(H ω2 find vPΔ and vQΔ the load power

changes components associate with vΔ , in the time domain.

3. Subtract vPΔ and vQΔ from PΔ and QΔ , respectively.

vres PPP Δ−Δ=Δ (4.34)

vres QQQ Δ−Δ=Δ (4.35)


82

The residuals in equations (4.34) and (4.35) are fPΔ and fQΔ if there are no

changes associate with the local load. Otherwise, these changes should be

removed from the residuals.

4.2.4 Estimation other Components of the Load Reactive Power

A simulation of the load reactive power changes due to a disturbance is shown in

Figure 4.6. The post-disturbance induction motor reactive power 1Q is identified

using the above described method. At the instant of disturbance, 1t , the induction

motor also acts as a constant impedance load. We can write:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=−

==

=−

==Δ +−+−

1

2

1

2

11 ttV

ttVK

ttQ

ttQQ qv (4.36)

Figure 4.6 Reactive power of different load components following a disturbance

in supply system

Equation (4.36) is used to evaluate the reactive power voltage dependency


83

coefficient qvK which can then be used in (4.37) to calculate the reactive power

of the constant power load:

−− =−

==

1

2

13

ttVK

ttQQ qv (4.37)

The estimated 1Q and 3Q are then peeled off from the load total reactive power at

the new steady-state time in Figure 4.6 to estimate the constant impedance load

reactive power :2Q

32

122

2 Qtt

Qtt

Qtt

Q −=

−=

==

(4.38)

The estimated active and reactive powers can then be used to obtain the load

parameters in Figure 4.2.

4.3 Algorithm of the Load Characterization by Sequential

Peeling

The procedure of the load characterization using the sequential peeling method is

explained in this section. Load is considered as a combination of induction

motor, constant impedance, and constant power loads.

Step1: Providing the initial local load bus data

Compute load bus active and reactive power and frequency and current

(simulation) and/or load admittance (real data) using VYI = equation in each

time step.

Step2: Peeling load active power

2.1. Use equations (4.1) and (4.2) to find 2pvK the voltage dependency

coefficient of the constant impedance load and then using (4.39) calculate its


84

predisturbance and post-disturbance steady-state active power; i.e., −= 12

ttP and

22 tt

P=

in Figure 4.3, respectively.

22 2

VKP pv= (4.39)

2.2. After peeling −= 12

ttP from the load total power, use equations (4.3) and

(4.4) to find 1pvK coefficient. Then, calculate the induction motor active power

using equation (4.40).

−− ==

==

1

2

11 1 tt

VKtt

PP pvm (4.40)

2.3. Use equation (4.41) to find the active power of the constant power load.

( ) −=−−=

1123

ttPPPP (4.41)

Step3: Induction motor reactive power estimation

3.1. Refining the local load bus data

Compute the changes in the load bus voltage, active and reactive power,

frequency and load admittance using the following equations:

( )∑−

=+ −=

1

11

n

kkk VVΔV (4.42)

( )∑−

=+ −=

1

11

n

kkk PPΔP (4.43)

( )∑−

=+ −=

1

11

n

kkk QQΔQ (4.44)


85

( )∑−

=+ −=

1

11

n

kkk ffΔf (4.45)

Where n is the number of the data points. This differentiation is required because

the estimation method uses the random changes in the load bus data.

3.2. Removing the dynamic components

The estimation method uses only the instantaneous changes in quantities. In this

thesis one-step prediction method is used to remove the dynamic components

from the data computed in the sub-step 3.1.

3.3. Removing the components associate with the voltage changes ΔV

Estimation method uses only the frequency dependent components of the

changes in the load bus power. Using equations (4.32) and (4.33), estimate the

transfer functions from load bus voltage to the active and reactive power. Then,

use the transfer functions to estimate the load power changes associate with the

voltage changes, in the time domain and subtract them from the load power total

changes.

3.4. Induction motor reactive power estimation

Estimate induction motor reactive power using equation (4.46):

( )( )ff

ff

PPEPQEPQ

ΔΔ

ΔΔ=

2 (4.46)

Where, E is the expectation value.

Step4: Estimation the other components of the load reactive power

Estimate reactive power of the constant power load using equations (4.36) and

(4.37). Then, use equation (4.38) to estimate the reactive power of the constant

impedance load.


86

4.4 Simulation

The proposed method is applied to the test system shown in Figure 4.7. Bus 1 is

assumed to be an infinite bus. The synchronous generator dynamics in bus 2 is

simulated by its electromechanical swing equation. Random changes in the

passive load #2 cause random changes in the voltage and current of bus 3.

Figure 4.7 Four bus test system

To simulate a disturbance, at a certain time the system impedances are changed

from one level to another level causing sudden changes in the quantities of load

#1. Figure 4.8 shows a simulation of the load#2 admittance and the supply

system Thevenin impedance from the bus 3 view point.

Load #1 is simulated as a combination of induction motor, constant impedance,

and constant power loads. The induction motor load is simulated as a constant

demand load; i.e., the motor dynamics restores the motor active power

consumption to its per-disturbance value. The bus 3 frequency deviation is

calculated as a proportion to the generator bus frequency deviation (Ref. to

Appendix B). A simulation of the bus 3 voltage and the induction motor slip is

shown in Figure 4.9. Figure 4.10 shows the simulated load total active and

reactive power and their components.

I

Z Load #2

1V 3V 4V 2V1Z 2Z 3Z

Z M Load #1

Infinite


87

Equations (4.39)-(4.41) are used to estimate the load active power components.

The comparisons of the estimated and actual active powers for the different

compositions of the loads and a 10% change in the system Thevenin impedance

due to the disturbance are shown in Table 4.1.

Figure 4.8 Simulation of load #2 admittance and system Thevenin impedance

Figure 4.9 Simulation of load bus voltage and induction motor slip


88

Figure 4.10 Simulation of load active and reactive powers

A one-step prediction algorithm was applied to the post-disturbance active and

reactive powers of load #1 and the bus 3 voltage and the predicted curves were

subtracted from the actual ones. The residuals of the load bus voltage and active

power and their autocorrelations are shown in Fig. 4.11. The method described in

the section 4.2.3 was applied to the residuals to extract the frequency dependent

components of the random changes in the load active and reactive powers.

Figure 4.12 shows the voltage dependent changes in the load active power and

its autocorrelation. The frequency dependent changes in the load active power

and its autocorrelation are shown in Figure 4.13. Using the extracted frequency

dependent components in the load power (4.) was used to estimate the induction

motor reactive power and the load other reactive power components were

anticipated on the similar way to the identification of the load active power

components. The estimated results and the actual values for different load


89

compositions are also compared in Table 4.1.

As it can be seen from Table 4.1, the estimated powers are very close to their

actual values.

TABLE 4.1

COMPARISON OF THE LOAD ESTIMATED AND ACTUAL POWERS FOR DIFFERENT

COMPOSITIONS OF LOADS AND A 10% CHANGE IN THE SYSTEM IMPEDANCE

1P 2P 3P 1Q 2Q 3Q

EST1.(PU)

ACT2.(PU)

ERR3 (%)

0.3007 0.3775 0.0498

0.3000 0.3780 0.0500

0.2254 -0.1336 -0.3428

0.0872 0.1889 0.0199

0.0870 0.1890 0.0200

0.2081 -0.0395 -0.5321

EST.(PU)

ACT.(PU)

ERR (%)

0.3999 0.2781 0.0498

0.4000 0.2778 0.0500

-0.0292 0.0990 -0.3175

0.1756 0.1826 0.0198

0.1728 0.1852 0.0200

1.6159 -1.4201 -0.8137

EST.(PU)

ACT.(PU)

ERR (%)

0.3013 0.2853 0.0996

0.3000 0.2862 0.1000

0.4226 -0.3069 -0.3900

0.0855 0.0962 0.0998

0.0861 0.0954 0.1000

-0.6846 0.8684 -0.2392

EST.(PU)

ACT.(PU)

ERR (%)

0.3004 0.3816 0.0002

0.3000 0.3822 0

0.1187 -0.1524 -

0.0855 0.1914 0.0001

0.0859 0.1911 0

-0.4589 0.1336 -

EST.(PU)

ACT.(PU)

ERR (%)

0.3007 0.2020 0.0496

0.3000 0.2023 0.0500

0.2437 -0.1415 -0.8898

0.0798 0.0008 0.0197

0.0804 0 0.0200

-0.7248 - -1.2701

EST.(PU)

ACT.(PU)

ERR (%)

0.2006 0.0025 0.0020

0.2000 0 0

0.2830 - -

0.0329 -0.0010 -0.0006

0.0325 0 0

1.1613 - -

1 EST: Estimated value 2 ACT: Actual value 3 ERR: Estimation error


90

Figure 4.11 (a) and (b): changes in the load bus voltage and its autocorrelation (c) and (d): load active power changes and its autocorrelation

Figure 4.12 (a) and (b): voltage dependent load active power changes and its

autocorrelation


91

Figure 4.13 (a) and (b): frequency dependent load active power changes and its

autocorrelation

The results of the estimation for a fixed initial load combination and different

system disturbance sizes are compared to the actual values in Table 4.2. As it is

seen in the Table 4.2, those values estimated based on the disturbance data are

estimated more accurate for the larger disturbances. The induction motor load

reactive power estimation, which is based on the signal processing in the load

bus data random changes, is also of reasonable accuracy.

Knowing load voltage, the estimated load power components can be used to

identify the load different parameters in Figure 4.2.


92

TABLE 4.2

COMPARISON OF THE LOAD ESTIMATED AND ACTUAL POWERS FOR DIFFERENT SIZES OF

THE SYSTEM DISTURBANCE (K)4

K (%) 1P 2P 3P 1Q 2Q 3Q

1

EST5

ACT

ERR

0.3047 0.2907 0.0492

0.3000 0.2946 0.0500

1.5685 -1.3223 -1.6226

0.0842 0.0977 0.0495

0.0832 0.0982 0.0500

1.1235 -0.4564 -0.9736

5

EST

ACT

ERR

0.3003 0.2920 0.0509

0.3000 0.2931 0.0500

0.0987 -0.3951 1.7245

0.0830 0.0978 0.0505

0.0837 0.0977 0.0500

-0.7673 0.1277 1.0348

10

EST

ACT

ERR

0.3005 0.2917 0.0492

0.3000 0.2914 0.0500

0.1719 0.1048 -1.6421

0.0840 0.0979 0.0495

0.0843 0.0971 0.0500

-0.3301 0.7937 -0.9854

20

EST

ACT

ERR

0.3002 0.2875 0.0501

0.3000 0.2878 0.0500

0.0561 -0.0918 0.1909

0.0848 0.0966 0.0501

0.0855 0.0959 0.0500

-0.7991 0.6526 0.1146

50

EST

ACT

ERR

0.3002 0.2769 0.0499

0.3000 0.2770 0.0500

0.0591 -0.0445 -0.1078

0.0896 0.0922 0.0500

0.0895 0.0923 0.0500

0.1611 -0.1211 -0.0647

100

EST

ACT

ERR

0.3000 0.2587 0.0499

0.3000 0.2586 0.0500

0.0006 0.0276 -0.1434

0.0981 0.0854 0.0500

0.0973 0.0862 0.0500

0.9002 -0.9658 -0.0860

4.5 Summary

Based on the measured load bus voltage and current phasors during a large

disturbance in the power system a method was introduced to identify on-line the

load parameters. It was shown that the measured disturbance data and the

random changes in the local load bus quantities can be used to estimate the real

and reactive powers of the different components of the load, individually. The

change in the load active power due to the disturbance was used to identify the

active power of the load different components in a peeling process. Then the 4 K is the percentage of increase in the power system Thevenin impedance due to the disturbance. 5 The abbreviations in the second column are the same as Table 4.1.


93

induction motor reactive power was estimated using the random changes in the

load measured power. The other components of the load reactive power was also

estimated using the disturbance data. The proposed method was confirmed by

simulation.

Chapter 5

On-line Estimation of the Remaining Time to a Long-

term Voltage Instability

5.1 Introduction Load power restoration in constant demand loads, such as TV’s and computers,

induction motors, on-load tap changer (OLTC) controlled loads and constant

energy heating loads, is the main cause of voltage collapse [33]. To stop the

evolution of an unstable scenario before its conclusion to a voltage collapse, the

time to identify the instability is a critical aspect. Many emergency control

measures to deal with voltage collapse are based on extensive off-line studies.

The feasibility of the supply system Thevenin impedance estimation and load

parameter identification using the changes in the load measured voltage and

current were pointed out in chapters 2 and 3, respectively. It was shown in

chapter 2 that the ratio of the cross-correlation of the changes in the load voltage

and current with respect to the changes in the load admittance can be used to

estimate the system Thevenin impedance. Based on the measured load bus

Chapter5. On-line Estimation of the Remaining Time to a Long-term Voltage Instability

95

voltage and current phasors during a disturbance in the power system a method is

introduced in chapter 3 to identify on-line the load parameters. In that method

the change in the load active power due to the disturbance is used to identify the

active power of the load different components in a peeling process. Then the

induction motor reactive power is estimated using the random changes in the

load measured power. The other components of the load reactive power are also

estimated using the disturbance data. The estimated power components can be

used to obtain the load parameters.

In this chapter, methods are developed to anticipate voltage instability for a

system consisting of load behind an OLTC. The analysis theory is provided for

the constant impedance load case, and also, a composite load containing constant

power, constant impedance and induction motor loads. It is shown that the local

load bus measured voltage and current phasors can be used to identify a possible

voltage collapse resulting from a long-term voltage instability caused by the on-

load tap changer, and to estimate the time to such a collapse. It is assumed that

the supply system Thevenin impedance and the load parameters are identified,

prior to the OLTC operation. The proposed methods are confirmed by

simulation.

5.2 Long-term Voltage Instability Prediction Considering

Constant Impedance Load and OLTC

5.2.1 System Description Consider a simple model system shown in Figure 5.1, where the constant

impedance load is supplied through an OLTC. The Tap ratio is shown by n. E

and sZ are the supply system Thevenin equivalent voltage and impedance,


96

respectively. The OLTC transformer impedance is ignored. At time T, a major

disturbance causes a significant sudden increase in sZ (Figure 5.2a). This change

in the system impedance, in turn, will cause significant sudden drops

in PV,V andsp , the load primary and secondary voltages and real power,

respectively.

Figure 5.1. Simple system with OLTC and constant impedance load

OLTC attempts to restore the load side voltage to its setpoint value by decreasing

the tap ratio, n. These tap changes decrease the load impedance, seen from the

primary side (Figure 5.2a). Tap changes will stop in two cases: sV reaches its set

point value, and/or, Tap reaches its limit.

0 50 100

0.2

0.4

0.6

0.8

1abs(Zload & Zs)

(a) Time(sec)

Per

uni

t

ZloadZs

0 50 1000.75

0.8

0.85

0.9

0.95Real power

(b) Time(sec)

Per

uni

t

0 50 1000.85

0.9

0.95

1Vs, Vp, Tap position

(c) Time(sec)

Per

uni

t

TapVsVp

Figure 5.2. Changes in impedances, tap ratio, primary and secondary side

voltages and power

n:1VsE

IpVp

P+jQ

Zs ZL


97

However, it is possible that the load impedance becomes equal in magnitude to

the impedance of the equivalent system feeding the bus (Figure 5.3a). In this

case any further tap changes beyond this point will have reverse effect causing

decreases in the load side voltage and consequently load power (Figures 3b

&3c). In the critical point of impedance matching, the load voltage and

power, PVs and , will be on their maximum values [81].

Figure 5.3. Changes in impedances, Load voltage and power

Also, the changes in the real power due to tapping in the two sides of the critical

point will have different signs. Thus, each of the following approaches may be

used to identify the remaining time to a possible collapse:

5.2.2 Time to collapse estimation using impedance matching criteria

This approach is based on the comparison of magnitudes of the load primary side

view impedance and system Thevenin impedance.


98

For a constant impedance load and two successive tap ratios in Figure 5.1, one

can write:

2221

2

21 11

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛== +++

ii

i

i

i

Li

Li

P

P

nn

nnn

nn

Zn

ZnZ

Z

i

i (5.1)

where in is the tap ratio after ith tapping and nΔ is the size of each tap step and:

iP

PLiP I

VZnZ i

i== 2 (5.2)

In equation (5.2) PP IV and are the OLTC primary side measured voltage and

current. Putting i=0 in equation (5.1) and using equation (5.2), the initial tap ratio

can be computed.

Also it can be shown that after the ith tapping the value of pZ will be:

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=

o

ipp n

nZZ

oi (5.3)

where op nZo

and are the initial values of the primary side view load impedance

and tap ratio, respectively. At the voltage collapse point the load impedance

becomes equal in magnitude to the supply system Thevenin impedance. Putting

SZZi p = in (5.3) we can compute cri nn = , the required tap ratio that

makes load impedance equal in magnitude to the system impedance:

op

Socr

Z

Znn = (5.4)

Voltage will not collapse if mincr nn ≥ where minn is the OLTC lower limit.

The number of the tap changes to collapse cri can be computed using (5.5):

nnn

i crocr Δ

−= (5.5)


99

The calculated cri should be rounded to the lower integer if it is between two

integers. The tap changing logic at time instant it is as following:

⎪⎪⎩

⎪⎪⎨

⎧

>−<Δ−

<+>Δ+

=+

otherwiseandif

andif

1

i

minio

ssi

maxio

ssi

in

nndVVnn

nndVVnn

n (5.6)

Where d and osV are half of the OLTC dead band and load voltage reference

value, respectively. There are two modes of OLTC operation. In sequential mode

tap changes starts after an initial fixed time delay and continues at constant time

intervals until the secondary side voltage error is brought back inside the OLTC

deadband, or until the tap limit is reached. The initial time delay is in the range

of 30-60 sec and the subsequent taping time intervals are usually around 10 sec.

The voltage error dead band is usually in the range of ± 1%-2%.

In non-sequential mode of operation there is no distinction between first and

subsequent taps and all time delays are given by the same formula [3].

Now it should be checked to see whether crn will exceed the tap limit or not. If

yes, the collapse will not happen. But if the answer is no, then other quantities

should be checked.

Using cri and on the magnitude of sV for crnn = may be computed and

checked to see whether it exceeds the reference value or not. If yes, collapse will

not happen, otherwise, system will experience a voltage collapse. It can easily be

shown that:

( ) ( )i

o

o

i

Ps

Ps

P

P

i

oosis ZZ

ZZ*

Z

Z*

nn

*VV+

+= (5.7)


100

Where i is the number of tap changes and iPZ is defined by (5.2). ( )osV is the

post-disturbance secondary side voltage, before the OLTC operation. Putting

crii = in (5.7) the secondary voltage at critical point can be computed.

In an alternative method one may compute the amount of the load power

(apparent, active or reactive) and compare it to its predisturbance value. For a

constant impedance load:

( )( )

( )( )

2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

os

is

o

iV

V

PP

(5.8)

In equation (5.8) ( )oP is the post-disturbance load real power and before the

OLTC operation.

In the following subsection another method will be introduced that is based on

the changes in the load power due to the OLTC operation.

5.2.3 Time to collapse estimation using load power changes

At the voltage collapse point the load power is in its peak value. Therefore, the

changes in the power prior to this point are positive and after that point the

changes are negative. Hence, the tap ratio related to the sign changing point of

the load power changes will identify the time of voltage collapse.

Using a least-square based method the value of the third change can be estimated

using the values of the first and second changes in load power. The subsequent

changes are also estimated using each pair of the changes until the point that the

change in load power becomes negative.

It can be shown that for the algebraic equation

θ= xy (5.9)

an estimation for θ can readily be as following:


101

( ) YXXXˆ T1T −=θ (5.10)

[ ]Tβα=θ can be estimated Putting:

( ) [ ]T341 PPY ΔΔ= and ( )

⎥⎦

⎤⎢⎣

⎡ΔΔΔΔ

=12

231X PPPP

in (5.10). ( ) [ ]T452 PPY ΔΔ= can then

be computed putting ( )⎥⎦

⎤⎢⎣

⎡ΔΔΔΔ

=23

342XPPPP

in (5.9). iPΔ is the change in load active

power due to the ith tapping. This one-step ahead computation uses the estimated

power changes to estimate a new point and it continues until the estimation of

the jth point that jPΔ becomes negative. If at the (j-1)th point of tapping the tap

ratio exceeds the OLTC limit, collapse will not happen, otherwise, using the

estimated power changes the amount of the load power at this point should be

computed and be compared to the load pre-disturbance power. If the estimated

power exceeds the pre-disturbance power, collapse will not happen, because this

means that secondary voltage will reach its reference value prior to this point of

tapping and tapping process will stop.

5.2.4 The Algorithms Based on the theory developed in section 5.2, the procedures of the voltage

instability prediction for a system with a constant power load behind an OLTC

are explained in this section. The first method is based on the impedance

matching criteria and the second method uses the changes in the load power due

to the OLTC operation to predict possible long-term voltage instability.

A: Algorithm Based on the Impedance Matching Criteria

Step1: Estimate the supply system Thevenin impedance using the proposed

method in chapter 3.


102

Step 2: Putting i=0 in (5.1) and using (5.2) estimate the tap initial ratio.

Step 3: Estimate crn the required tap ratio and cri related number of taping that

make load impedance equal in magnitude to the system impedance using (5.4)

and (5.5), respectively and rounding the result to the lower integer, if it is

between two integers. If mincr nn ≤ load bus voltage is stable, otherwise go to

step 4.

Step 4: Using (5.7) and (5.8) compute the load power (apparent, active or

reactive) for the critical point crii = and compare it to its predisturbance value:

B: Algorithm Based on the Load Power Changes

Step 1: After the OLTC third operation, estimate the subsequent changes in load

power due to the successive OLTC operation in a one-step ahead method using

equations (5.9) and (5.10) as described in section 5.2.3. The estimation

continues until the estimated power change becomes negative. At the point of the

last positive estimated value if the tap ratio exceeds the OLTC limit, collapse

will not happen, otherwise go to the next step.

Step 2: Using the estimated power changes in step 1, compute the amount of the

load power at the point of the last positive estimated value and compare it to the

load pre-disturbance power. If the estimated power exceeds the pre-disturbance

power, collapse will not happen; otherwise voltage collapse will start by the next

OLTC operation if it is not avoided.

5.2.5 Simulation The proposed methods are applied to the system shown in Figure 5.1. The load

happenwillcollapseotherwise

happennotwillcollapsePnnP edisturbancprerc

→

⇒≥= − if


103

admittance is modelled as a constant impedance load plus a small randomly

changing component (Figure 5.4). OLTC quantities are set to the values shown

in Table 5.1. The time interval for all tap changes is assumed to be 10 sec.

Considering the values in Table 1; the tap limit will be reached by ten successive

tap changes. At the time instance T= 10 sec there is a significant increase in the

system impedance, and this in turn, causes sudden drops in the secondary voltage

and load power. OLTC starts to restore load voltage and power by successively

changing the tap ratio. Now let: .The system and load impedances and load real

power for three different values of k are shown in Figures 5.5, 5.6 and 5.7.

Figure 5.4. Simulation of load admittance

TABLE 5.1. OLTC TAP INITIAL RATIO, STEP SIZE, LOWER LIMIT, TIME DELAY AND DEAD

BAND.

on nΔ % minn TΔ (sec) d% 1 1.5 0.85 10 2±


104

Figure 5.5 (a): System and load impedances, (b): Load real power, k=2

Figure 5.6. (a): System and load impedances, (b): Load real power, k=2.5


105

Figure 5.7. (a): Changes in the system and load impedances,

(b): changes in load real power, k=3

The method described in section 3.3 was used to estimate post-fault system

impedance. Tap initial ratio and the number of taps to collapse were estimated

using equations (5.2) and (5.4), respectively. The results are shown in Table 5.2.

The estimated numbers of taps have been rounded to their lower integer

numbers.

TABLE 5.2. ACTUAL AND ESTIMATED SYSTEM IMPEDANCE, INITIAL TAP RATIO AND TAPS

TO COLLAPSE FOR DIFFERENT VALUES OF K IN (prefault)sk*Zt)(post faulsZ =

sZ (pu) on (pu)

k Actual

Estimated

A

E

Taps to collapse

2

0.1+0.4i

0.1+0.4i

1

0.9999

15

2.5

0.125+0.5i

0.125+0.5i

1

1.0005

10

3

0.15+0.6i

0.15+0.6i

1

0.9994

5


106

As it can be seen from Table 5.2, estimated values for the system impedance are

the same as the actual values and the estimated initial tap ratios are very close to

the actual value. In the first case (k=2) collapse will not happen because as it

was stated before OLTC limit will be reached by 10 tap changes. In the second

case (k=2.5) also collapse will not happen because in this case the critical tap

ratio is equal to the tap limit and OLTC operation will be stopped.

In the third case (k=3) the estimated taps to collapse is less than 10. Now it

should be checked to see whether OLTC operation will stop before the critical

point or not. Equation (5.8) is used to compute the load real power in the critical

point. The result is as following:

onncrnnP..P

===<= 0815420

Thus tap operation will continue and voltage will collapse.

The above method of voltage collapse identification is based on the estimated tap

initial position. The size of a tap step is usually in the range of 0.5%-1.5%. So,

any error in estimation of the tap initial ratio may result in an incorrect value

of on , and therefore, invalid identification of voltage collapse.

Now we use equations (5.9) and (5.10) to identify the voltage collapse. The sizes

of the actual and estimated changes in power for different values of k are shown

in Figures 5.8, 5.9 and 5.10. As it can be seen the estimated and actual values are

close together.

Estimated power changes show that the power changes due to tapping for k=2

and k=2.5 are still positive at tenth tapping ( minnn = ). This means that in these

cases collapse will not happen. But for k=3 the sign of the power change

becomes negative at the sixth tap change. Thus, in this case the fifth tap change


107

is the critical point.


tapping, k=2


tapping, k=2.5


108


tapping, k=3

The values of the power changes at the first five tap changes are added to the

post-fault (before OLTC operation) power value to compute the load power at

the critical tap ratio. The result is as following:

onncrnnP..P

===<= 0815420

Thus tap operation will continue and voltage will collapse. In this method, for

identification of a voltage collapse one should wait until the time of the fourth

change in power. Therefore, it will be useful just in the cases that there is left a

lot of time to collapse.

Simulation results shown in Figures 5.5, 5.6 and 5.7 confirm the above achieved

results by the both proposed methods in this section.


109

5.3 On-Line Voltage Collapse Prediction Considering Composite

Load and ON-Load Tap Changer

5.3.1 System Description A simple power system is shown in Figure 5.11, where a combination of the

induction motor load, constant impedance load, and constant power load is

supplied through an OLTC. The supply system is shown by its equivalent

Thevenin circuit. The impedance of the OLTC transformer is ignored. The

induction motor load is modelled as a constant demand load where following any

changes in the load bus voltage the motor slip s is changed and hence the

variable resistance R is adjusted to a new value. X is the sum of the stator and

rotor reactances and the rotor resistance is shown by rR . For simplicity the stator

resistance and the magnetizing circuit are ignored.

Figure 5.11 Simple power system with composite load and on load tap changer

The Tap ratio is shown by n. At time T, a major disturbance causes a significant

sudden increase in the supply system impedance. This change in the system

impedance, in turn, causes significant sudden drops in the load primary and

secondary voltages and real power, respectively (Figure 5.12).

It is assumed that following any changes in the load voltage the constant power

load impedance is immediately adjusted according to equation (5.11).

X

LX

3V

22 jQP +LRI

2V1V

QjP+

1:n

thZ

33

23

3 jQP

VZ

−=

33 jQP +

Rs

Rr=

11 jQP +


110

33

23

3 jQPV

Z−

= (5.11)

The motor slip, s , increases, resulting reduction in R, until matches its

predisturbance value (motor demand) (Figure 5.12).

OLTC attempts to restore the load side voltage to its setpoint value by decreasing

the tap ratio. Each tap change causes a sudden change in the load voltage and

hence activates motor dynamics causing changes in the motor slip until motor

active power is recovered at a new steady state condition (Figure 5.12).

Figure 5.12 (a): tap position, primary and secondary voltages, (b): induction

motor slip, (c): load active powers, (d): load reactive powers

The taping process continues until the voltage at bus 3 is maximized. At this

point, while the active powers of the induction motor and constant power loads

are recovered, the active power of the constant impedance load, and hence, the


111

load total active power, are also maximized (Figure 5.13). After this point the

OLTC reverse action starts and the process of the load voltage restoration

becomes unstable. Tap changes will stop in two cases: the OLTC secondary side

voltage reaches its set point value, and/or, Tap reaches its limit.

Figure 5.13. Changes in the transformer secondary voltage and load power due

to taping

5.3.2 Tap initial ratio estimation Following any sudden change in the load voltage the induction motor acts

primarily as a constant impedance load, and therefore, right after each taping the

impedances of the induction motor and constant impedance components of load

remain constant but the impedance of the constant power component of load is

adjusted in real time according to (5.11). Assuming that the active and reactive

powers of the constant power load are identified, equation (5.12) can be used to

estimate the pre- and post-taping admittances of this component from the bus 2


112

view point [82]:

22

333

V

jQPY P

−= (5.12)

It can easily be shown that the tap initial ratio can be estimated using equation

(5.13):

+=

−=

−

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−

1

1

3

32

Tt

Tt

PTP

PTP

o

o

YY

YY

nnn

(5.13)

Where, 2VIYTP = is the total load admittance from the bus 2 view point and

1T is the time instant of the first taping. The size of each tap step nΔ is assumed

to be known, otherwise, using equation (5.13) for two successive taping, on and

nΔ can both be determined.

The tap changing logic is according to equation (5.6). In this paper the tap

operating mode is considered to be in sequence. This mode of operation consists

of a sequence of tap changes starting after an initial fixed or constant time delay

and continuing at constant time intervals. The first tap time delay, if not constant,

can be determined by the following formula:

mfdOLTC TTVdTT ++Δ

= (5.14)

Where, VΔ is the difference between the controlled and reference voltages, d is

half of the OLTC dead-band, dT is the maximum time delay of the inverse-time

characteristic, fT is the intentional time delay, and mT is the mechanical time

delay [6].


113

5.3.3 Taps to collapse estimation

Once the tap initial ratio is estimated, the following method can be used to

estimate the time to collapse.

Let thY,Y,Y,Y and321 to be defined in Figure 5.11 as the induction motor load,

constant impedance load, constant power load, and the supply system Thevenin

admittances, respectively, where:

23

12

3

11

V

PRXj

V

PY −= (5.15)

LL jRX

jBGY+

=−=1

222 (5.16)

23

32

3

33

V

Qj

V

PY −= (5.17)

ththth

th jBGZ

Y +==1 (5.18)

It can easily be shown in Figure 5.11 that:

22

222

3BG

InV th

+= (5.19)

Where n is the tap ratio and the other terms are as below:

2VYII thth += (5.20)

23

3122

V

PPGnGG th

+++= (5.21)

2

3

32

3

122

V

Q

V

PRXBnBB th +++= (5.22)

It can also be shown that:

22

23

1XR

VRP

+= (5.23)


114

Solving equation (5.23) for R will result in:

1

221

43

23

24

PXPVV

R−±

= (5.24)

It was found by investigation that the above solution is only valid with the plus

sign. Finally, replacing R in equation (5.22) from equation (5.24), and

considering that following each taping the induction motor active power 1P is

recovered in steady state to the motor constant demand mP , equation (5.19) can

be rewritten as follows:

223BG

InV th

+= (5.25)

Where:

23

322

V

PPGnGG m

th+

++= (5.26)

23

324

32

3

23

22

24 V

Q

XPVV

VXPBnBB

m

mth +

−+++= (5.27)

...,,,,ininn o 3210=Δ−= (5.28)

Considering different values for i in equation (5.28), equation (5.25) can be

solved for 3V until this variable is maximized at crii = . At this point if either of

the constraints of OLTC in equation (5.6) is not yet met, collapse will happen.

5.3.4 The Algorithm Assuming that following a disturbance, the system Thevenin impedance and the

load parameters have already been identified by the proposed methods in [6] and

[7], respectively; the following algorithm can be used to anticipate a possible

voltage collapse and to estimate time to such a collapse:


115

1. Estimate the tap initial position using equation (3).

2. with different values of n from equation (5.28) solve equation (5.25) for

the voltage magnitude 3V until it is maximized.

3. Check the OLTC constraints in equation (4) for the point which 3V is

maximized. Voltage will start to collapse in the next taping if neither the voltage

is recovered to its set point nor tap has exceeded its limit.

5.3.5 Simulation The proposed method was applied to a four bus test system (Figure 4). Bus 1 is

assumed to be an infinite bus. The synchronous generator dynamics in bus 2 is

simulated by its electromechanical swing equation. The bus 2 frequency

deviation is calculated as a proportion to the generator bus frequency deviation

[7]. The load in bus 3 is a composition of induction motor, constant impedance,

and constant power loads. The induction motor load is simulated as a constant

demand load; i.e., following any changes in the load bus voltage the motor

dynamics restores the motor active power consumption to its previous value in a

few seconds.

Following a disturbance, OLTC; that is simulated by its discrete-time logic

described in equation (5.6), restores load voltage and load power by changing the

tap ratio. Voltage collapse happens when the OLTC reverse control action starts,

i.e., when secondary voltage drops if the tap position n is decreased aiming at

raising the secondary voltage. It is assumed that the voltage and current at the

primary side of OLTC are measured and the load post-disturbance parameters

have been identified before the OLTC starts to operate.


116

Figure 5.14. Four bus test system with composite load

TABLE 5.3. OLTC TAP INITIAL RATIO, STEP SIZE, LOWER LIMIT, TIME DELAY, VOLTAGE

REFERENCE, AND DEAD-BAND.

on nΔ % minn OLTCT (sec) (pu)RV d% 1 1.5 0.85 20 0.98 2±

5.3.6 Taps to collapse estimation with different disturbance sizes To simulate a disturbance, at a certain time, 21 and ZZ are changed from one

level to another level causing sudden changes in the quantities of load. Now

defining k as the ratio of the supply system Thevenin impedance to its pre-

disturbance value; the magnitude of the voltage in bus 3 is shown in Figure 5.15

for three different values of k1 and the same pre-disturbance load composition.

In the first case (Figure 5.15a) the tap operation stops because the regulated

voltage reaches its set-point value. In the second case (Figure 5.15b) tap reaches

its lower limit minn and tap operation stops. In the third case (Figure 5.15c)

voltage will collapse if it is not avoided.

Three step algorithm of section 5.3.4 was used to estimate tap initial ratio and the

number of taps to collapse for the above cases. Gauss-Seidel method was used to

1 %e)disturbanc(preZ/et)disturbanc(post Zk thth −−=

Infinite Bus

1n

4V

I

M Z

1V1Z 2Z

3V

2V


117

solve equation (5.25). The results are shown in Table 5.4.

Figure 5.15. Simulation of the OLTC primary and secondary voltage changes,

(a): k=1.4, (b): k=2.5, (c): k=3

TABLE 5.4. ESTIMATED INITIAL TAP RATIO AND TAPS TO COLLAPSE FOR DIFFERENT

VALUES OF K IN %e)disturbanc(preZ/et)disturbanc(post Zk thth −−=

K Estimated

on (pu) Taps to

maxV Tap change

limit Voltage

Collapse?

1.40 1 29 10 No 2.50 1 12 10 No 3.00 1 4 10 Yes

Estimated values for the tap initial ratio in Table 5.4 are the same as its actual

value 1=on . In the first and second cases collapse will not happen because as it

Time (Sec)

(a)

(b)

(c)


118

was stated before OLTC limit will be reached by 10 tap changes. In the third

case (k=3) the estimated taps to maxV is less than 10. Now it should be checked

to see whether OLTC operation will stop before the critical point or not. The

Estimated secondary (bus 3) voltage for different tap numbers is shown in Figure

5.16. As it can be seen in Figure 5.16 the estimated maxV is not in the OLTC

dead band zone (Table 5.3), and hence, tap operation will continue and voltage

will collapse. Simulation results shown in Figure 5.15 confirm all above

estimated results.

Figure 5.16. Estimated OLTC secondary voltage for 10 successive tapings, k=3

5.3.7 Effects of the load and measurement uncertainties on the estimation In periods up to 10 seconds the changes in the customer composite load are

unpredictable (Figure 5.17). These small random changes and also, measurement

uncertainties may affect the results of the proposed estimation method in this

section. To investigate this matter random changes are added to the load


119

admittance (Figure 5.18).

Figure 5.17. Time measurements of load voltage and current magnitudes in the

Brisbane load bus.

Figure 5.18. Simulation of load admittance with random changes

The estimation results for the same load composition and the same disturbance

sizes in Table 5.4 are shown in Table 5.5. As it can be seen the estimated taps to

collapse are the same as Table 5.4 and the estimated tap initial ratios are very

close to the actual value 1=on .


120

TABLE 5.5. ESTIMATED INITIAL TAP RATIO AND TAPS TO COLLAPSE FOR DIFFERENT

DISTURBANCE SIZES AND RANDOM CHANGES IN THE LOAD

k Estimated on (pu) Taps to maxV Collapse

1.40 0.9983 29 No 2.50 0.9991 12 No 3.00 0.9998 4 Yes

5.3.8 Effect of the load composition on the voltage collapse Four different compositions for the total load were chosen. The constant power

load is the same for all compositions. The estimated taps to collapse following a

large disturbance (k=3.6) are compared in Table 5.6. It can be seen in Table 5.6

that, as it is expected, larger induction motor load in the composition makes

safer the system voltage in the long-term, but increases the risk of the short-

term voltage instability. This result can be explained using Figure 5.19.

Following a disturbance, the induction motor active and reactive power

restoration, due to the slip increase, may bring the load total consumption

beyond the supply system capacity, resulting in short-term voltage instability

(case 4). However, if the load voltage is stable in the short-term, then the

reductions in the induction motor reactive power, due to the successive tapings,

improves the voltage stability in the long-term, through cancelling the increase in

the constant impedance load consumption.

TABLE 5.6. COMPARISON OF THE TAPS TO COLLAPSE FOR DIFFERENT LOAD

COMPOSITIONS

Case

Percentage in total load (%)

Taps to Instability

Voltage Collapse

1 P1=15, P2=80, P3=5 3 Yes 2 P1=30, P2=65, P3=5 4 Yes 3 P1=45, P2=50, P3=5 5 Yes 4 P1=60, P2=35, P3=5 0 Yes(Short- term)


121

Figure 5.19. Simulation of the load reactive powers, Induction motor: 60%,

Constant impedance: 35%, Constant power: 5%, k=2.9

5.4 Summary

In this chapter, based on the measured data in the local load bus, methods were

introduced to anticipate voltage instability for a system consisting of load behind

an OLTC. Theories were provided for the constant impedance load case, and

also, a composition of the constant power, constant impedance and induction

motor loads. It was shown that the local load bus measured voltage and current

phasors can be used to identify a possible voltage collapse resulting from a long-

term voltage instability caused by the on-load tap changer, and to estimate the

time to such a collapse. Two methods were developed for the first case based on

the impedance matching and the sudden changes in the load power due to the tap

operation, respectively. The load side voltage maximizing criterion was used to

anticipate the voltage instability in the system feeding a composite load behind

an OLTC. It was assumed that the supply system Thevenin impedance and the

load parameters are identified, prior to the OLTC operation. The proposed

methods were confirmed by simulation.

Induction motor load

Constant power load Constant impedance load

Chapter 6

Case Study: BPA Test System

6.1 Introduction Operating of power systems nearer to their physical limits has made the long

term dynamics a significant concern for many utilities in the world. As a

consequence, power system engineers have had increasing interest in advanced

simulations [6, 33, 83, 84]. CIGRE Task Force 38-02-08 tackled this problem

and compared methods and tools on different study cases proposed by the

members.

One the purposes of the task force was to provide the researchers or practitioners

with benchmarks or test systems to be used for the assessment of existing tools

or of new developments in the field of long term dynamics. The Task Force

gathered five test systems, three of them displaying voltage stability problems.

The so called "BPA1 test system" was presented by Canada. This system has

been used by researchers to illustrate mechanisms of voltage instability in time

domain simulation [1, 19, 31, 32, 85, 86] .

1 Bonneville Power Administration (http://www.bpa.gov)

Chapter 6. Case Study: BPA Test System

123

In order to investigate and validate the proposed methods in this research, they

are applied to BPA test system.

6.2 Test System Description Figure 6.1 shows a single line diagram of the BPA test system. It is based on the

system described in [87], with loads and generators outputs modified. A local

area is fed by a remote generation area through five transmission lines. Generator

G3 in the local area, in addition to the supplying a part of the active power,

regulates the voltage at bus 6, to maximize the transfer capacity of the five

transmission lines.

Figure 6.1. The BPA test system [87]

In this study, generator G1 is modeled as an infinite bus, and neglecting the

stator transformer voltages and rotor subtransient dynamics, generators G2 and

G3 are represented by third order models. Considering the time interval of

interest in long term voltage stability, this simplification is acceptable. Excitation

systems and integral type overexcitation limiters are also added to the generation

system. The generators parameters are the same as in [19]. The limiter

characteristic is shown in Figure 6.2. If the field current exceeds fdmax1I the

1

2

34 5 6

7

8 9 10

Local Area

OLTC

G1

G2

G3

Subtransmission Equivalent

Generation Area


124

current is ramped down to its continuous limit with time delay dependent on the

level of field current. The limiter block diagram and parameters are provided

from [7].

Figure 6.2. Overexcitation limiter characteristic

Load at bus 10 is modeled as a composite of induction motor, constant power

and constant impedance loads. Ignoring the flux dynamics, induction motor

dynamics are modeled with the rotor motion equation [6]. Following any

changes in the load bus voltage, the motor dynamics restore the motor active

power. The other system load at bus 7 is modeled as constant impedance load.

Small random changes are also added to both loads.

The on-load tap changer (OLTC) supplying the composite load is modeled in

detail. OLTC is assumed to be EU1 sequential mode type with 60 seconds time

delay for the first tap movement [6]. In order to reduce the simulation time the

time delay for the subsequent tap movements is set to be 10 seconds. The

deadband is assumed to be %1± p.u. of the controlled bus voltage. Tap range is

15± steps, and the step size is 1%. Complete data of the system is provided in

Appendix C.

fdI

FCL1.05Ifdmax1 ×=

FCL1.6Ifdmax2 ×=

Time (sec) 30 0

FCL=Full Load


125

The disturbance considered is the sudden change, without a fault, in the

impedance of the transmission corridor between bus 5 and bus 6, for example,

loss of one of the branches.

A simulation of the magnitude of the composite load admittance from the bus 9

view point, including the compensation admittance, for a 0.04 sec sampling time,

a disturbance at 50 sec time (5% increase in the transmission corridor

impedance), 1% tap step size, and 115 sec total simulation time is shown in

Figure 6.3. Figure 6.4 shows the tap ratio and magnitudes of bus 9 and bus 10

voltages.

At time 34 sec in the simulation the overexcitation limiter (OXL) limits the field

current of the generator G3 and voltage support of bus 6, the load area end of

transmission corridor, provided by this generator, is lost. As a consequent,

voltages at OLTC primary and secondary sides are slightly decreased, causing an

increase in the load admittance through increase of the induction motor slip and

constant power load admittance.

At time 50 sec in the simulation, a disturbance in the system causes sudden

decrease in the OLTC primary and secondary side voltages. The load admittance

has increased in response to the voltage reduction. The small increase in the load

voltages after disturbance is due to the voltage control applied by generator G2.

At time 110 sec, the OLTC has operated to recover the secondary side voltage

causing further decrease in the primary side voltage and an increase in the load

admittance, seen from the OLTC primary side. Any further operation of the

OLTC will reduce load voltage stability margin by decreasing the load

impedance magnitude and bringing it closer to the value of the system

impedance after tripping of the line.


126

20 30 40 50 60 70 80 90 100 110

2.93

2.94

2.95

2.96

2.97

2.98

2.99

abs(Load addmittance)-bus 9

addm

ittan

ce(p

u)

Time (sec)

OLTC operation

Disturbance

overexcitationlimiter of G3

Figure 6.3. Simulation of load admittance changes in bus 10.

20 30 40 50 60 70 80 90 100 110

0.975

0.98

0.985

0.99

0.995

1

tap ratio,primary and secondary voltages

Time(sec)

tap ratioabs(V10)abs(V9)

OLTC operation

Disturbance

OXL(G3)

Figure 6.4. Tap ratio, magnitudes of bus 9 and bus 10 voltages


127

6.3 Validation of the Proposed “System Thevenin Impedance

Estimation” Method

In this section, the proposed method for the system Thevenin impedance

estimation in this research work is applied to the BPA test system. The supply

system is viewed from bus 9, i.e., the primary side of OLTC.

A simulation of the magnitudes and angles of the bus 9 (OLTC primary side)

voltage and current is shown in Figure 6.5. The disturbance considered at time

50 sec is an increase, without fault, in the transmission corridor impedance

between bus 5 and bus 6. At time 110 sec OLTC operates to restore the load side

voltage. The dynamics and random changes in the magnitudes and angles of the

voltage and current are caused by supply system and load. These signals, for the

time span between disturbance and tap operation, are highlighted in Figure 6.6.

For the same time period, the real and imaginary components of the changes in

the bus 9 voltage and current phasors and their autocorrelations are shown in

Figures 6.7 and 6.8, respectively. The autocorrelations confirm the presence of

electromechanical dynamics in the signals. A simulation of the changes in the

load admittance and its autocorrelation is also indicated in Figure 6.9. The DC

offset shown by the autocorrelation in Figure 6.9 is due to the induction motor

load slip change in response to the voltage step caused by the disturbance in the

system.

A one-step ahead prediction algorithm was applied to the post-disturbance

changes in the real and imaginary components of bus 9 voltage and current

phasors and the estimated signals were subtracted from the actual ones. The

residuals are shown in Figure 6.10. The autocorrelations of residuals in Figure

6.11 confirm that the residuals are free from dynamic components.


128

40 60 80 1000.98

0.985

0.99

0.995

1abs(voltage)-bus 9

Per

uni

t

40 60 80 1002.88

2.9

2.92

2.94

2.96abs(current)-bus 9

Per

uni

t

40 60 80 100-32.4

-32.3

-32.2

-32.1

-32

-31.9angle(voltage)-bus 9

Deg

Time (sec)40 60 80 100

-41.6

-41.4

-41.2

-41

-40.8angle(current)-bus 9

Deg

Time (sec)

Figure 6.5 Simulation of the magnitudes and angles of the bus 9 voltage and

current

60 80 100

0.985

0.9852

0.9854

0.9856

0.9858

Per

uni

t

abs(voltage)

60 70 80 90 1002.886

2.888

2.89

2.892

2.894

2.896

60 80 100-32.08

-32.07

-32.06

-32.05

-32.04

Deg

60 80 100

-41.02

-41

-40.98

-40.96

Time (sec)

abs(current)

angle(voltage) angle(current)

Time (sec)

Figure 6.6 Magnitudes and angles in bus 9 voltage and current from disturbance

until start of OLTC operation.


129

20 40 60-4

-2

0

2

x 10-5imag(diff(V9))

20 40 60

-2

-1

0

1

2

x 10-4

20 40 60

-2

-1

0

1

2x 10-4

20 40 60

-4

-2

0

2

4

x 10-5 real(diff(I9))

imag(diff(I9))

Time (sec)Time (sec)

real(diff(V9))

Figure 6.7 Changes in the components of the bus 9 voltage and current from the

line trip disturbance until start of OLTC operation

-10 0 10-1

0

1

2

3

4x 10-7 Auto(voltage real)

lag(sec)-10 0 10-5

0

5

10

15x 10-8 Auto(current real)

lag(sec)

-10 0 10-2

0

2

4

6x 10-6 Auto(voltage imag)

lag(sec)-10 0 10-1

0

1

2

3x 10-5 Auto(current imag)

lag(sec)

Figure 6.8 Auto correlations of the post-disturbance changes in the components

of the bus 9 voltage and current phasors


130

10 20 30 40 50 60

-1

0

1

2

3x 10-4

Time(sec)

diff(abs(Yload))

-8 -6 -4 -2 0 2 4 6 81

2

3

4x 10-5 Auto(Yload changes)

Lag(sec)

Figure 6.9 Post-disturbance changes in the load admittance magnitude and its

autocorrelation until start of OLTC operation

0 20 40 60-4

-2

0

2

4x 10-5 Residuals(voltage real)

0 20 40 60-4

-2

0

2

4x 10-4

0 20 40 60-4

-2

0

2

4x 10-5

Time(sec) 0 20 40 60-4

-2

0

2

4x 10-4 Residuals(current imagl)

Residuals(current real)

Residuals(voltage imag)

Time(sec)

Figure 6.10 Residuals of the post-disturbance changes in the components of the

load voltage and current until OLTC operation


131

-10 0 10-5

0

5

10

15x 10-8

Auto(voltage real residu)

-10 0 10-2

0

2

4

6

8x 10-8 Auto(current real residu)

-10 0 10-1

0

1

2

3

4x 10-6 Auto(voltage imag residu)

lag(sec)-10 0 10-2

0

2

4

6x 10-6 Auto(current imag residu)

lag(sec)

Figure 6.11 Autocorrelations of the residuals of the post-disturbance changes in

the load admittance magnitude until OLTC operation

The above method of the residual extraction will be applied to the signals in

different simulations through this section.

The system Thevenin impedance estimation method is validated by choosing

different values for the system Thevenin impedance. This impedance may be

changed by:

• Applying a disturbance, for example losing one or two of the lines in the

transmission corridor between bus 5 and bus 6. Other sizes of disturbance

are also applicable.

• Changing the ratio of different tap changers, except the tap changer

between bus 9 and bus 10.

• Changing the size of the reactive compensations in buses 6, 7, and 8.

• Changing the load connected to bus 7.


132

6.3.1 Supply System without Random Changes

In this case the load in bus 7 is simulated as a constant impedance load without

random changes. A simulation of the system Thevenin impedance and the

composite load impedance, connected to bus 10, both from the bus 9 view point,

is shown in Figure 6.12. The proposed algorithm is section 3.3 is applied to the

60 sec post-disturbance bus 9 voltage and current phasors to estimate the system

impedance. The results of the simulations for a 0.04 sec time step and different

values of Thevenin impedance are shown in Table 6.1. as it can be seen, in this

case the estimated system Thevenin impedances are very close to the actual

values.

20 40 60 80 100 120 140

0.15

0.2

0.25

0.3

0.35

Time(sec)

Per

uni

t

Magnitudes of system Thevenin and load impedances

Thevenin ImpedanceLoad Impedance

Figure 6.12 Simulation of the magnitudes of the system Thevenin and load

impedances from the bus 9 view point


133


THEVENIN IMPEDANCE, SUPPLY SYSTEM WITHOUT RANDOM CHANGES, TIME STEP=0.04

SEC

thZ (actual) Estimated value using equation (3.23)

Estimated value using equation (3.24)

0.0338 + 0.1513i

0.0355 + 0.1543i

0.0384 + 0.1656i

0.0271 + 0.1651i

0.0328 + 0.1508i

0.0315 + 0.1540i

0.0348 + 0.1651i

0.0244 + 0.1640i

0.0345 + 0.1515i

0.0349 + 0.1549i

0.0375 + 0.1674i

0.0285 + 0.1645i

6.3.2 Supply System with Random Changes

In this case random changes are added to the load in bus 7. Consequently, the

random changes in the bus 9 voltage and current are partly due to these random

changes in supply system, and partly due to the random changes in the composite

load connected to bus 10. A simulation of the admittances of both loads is

shown in Figure 6.13. The results of the simulations for a 0.04 sec time step and

the same values of Thevenin impedance in Table 1 are shown in Table 6.2.

20 40 60 80 100

2.92

2.94

2.96

2.98

3

3.02

3.04

3.06

3.08

Time(sec)

Per

uni

t

Magnitudes of local and remote load addmittances

abs(Yload)-Bus7

abs(Yload)-Bus10

Disturbance

OLTC operation

Figure 6.13 Simulation of local and remote load admittances


134


THEVENIN IMPEDANCE, SUPPLY SYSTEM WITH RANDOM CHANGES, TIME STEP=0.04 SEC



0.0338 + 0.1513i

0.0355 + 0.1543i

0.0384 + 0.1656i

0.0271 + 0.1650i

0.0257 + 0.1444i

0.0229 + 0.1487i

0.0276 + 0.1594i

0.0150 + 0.1588i

0.0338 + 0.1491i

0.0332 + 0.1563i

0.0367 + 0.1682i

0.0263 + 0.1660i

As it can be seen from Table 6.2, the estimated result using equation (3.24) are

still of reasonable accuracy. However, the obtained results from equation (3.23)

are not quantify to the actual values, as it was expected.

The number of data points was doubled by reducing the sampling time to half.

The estimation results for system Thevenin impedance are compared to the

actual values in Table 6.3. As it was expected, the results are improved,

comparing to the estimated results in Table 6.2.


THEVENIN IMPEDANCE, SUPPLY SYSTEM WITH RANDOM CHANGES, TIME STEP=0.02 SEC



0.0338 + 0.1513i

0.0355 + 0.1543i

0.0384 + 0.1656i

0.0271 + 0.1651i

0.0258 + 0.1464i

0.0265 + 0.1487i

0.0288 + 0.1603i

0.0162 + 0.1571i

0.0344 + 0.1505i

0.0353 + 0.1566i

0.0376 + 0.1661i

0.0255 + 0.1659i

For the final simulation in this section, by choosing different values of the ratios

of random changes in two loads, the effect of the size of the random changes in

the supply system in estimation result is investigated. The results are shown in

Table 6.4.


135


THEVENIN IMPEDANCE FOR DIFFERENT RANDOM CHANGE SIZES FOR REMOTE LOAD (K),

SUPPLY SYSTEM WITH RANDOM CHANGES, TIME STEP=0.02 SEC

K

thZ (actual)

Estimated value using equation

(3.23)

Estimated value using equation

(3.24) 0.1

0.2

0.5

1

1.5

2

0.0355 + 0.1543i

0.0355 + 0.1543i

0.0355 + 0.1543i

0.0355 + 0.1544i

0.0355 + 0.1544i

0.0355 + 0.1544i

0.0326 + 0.1539i

0.0324 + 0.1532i

0.0255 + 0.1487i

0.0019 + 0.1335i

-0.0287 + 0.1133i

-0.0608 + 0.0900i

0.0343 + 0.1549i

0.0350 + 0.1535i

0.0358 + 0.1545i

0.0324 + 0.1553i

0.0372 + 0.1518i

0.0353 + 0.1485i K is the change in the remote load as a fraction of the change in the local load.

Table 6.4 indicates that the estimated results are affected by the size of the

supply system random changes. For large values of K, as it was expected,

equation 3.23 completely fails to estimate the Thevenin impedance and even for

some cases the components of the estimated values are negative. In fact,

increasing the size of the random changes in the remote load will increase their

contribution to the changes in the bus 9 voltage and current phasors.

The estimation error for equation 3.24, in the last case in Table 6.4 is less than

4% that is still of reasonable accuracy.

It is noticeable that the electrical distance between the remote load and the local

bus is also an important factor in the contribution of the supply system random

changes to the local load bus changes, i.e., the smaller the electrical distance the

bigger the contribution. In the BPA test system, the load connected to bus 7 is

separated of bus 9 by only two transformers and the subtransmission line

between bus 8 and bus 9, i.e., the remote load is electrically close to the local

load bus. In such cases, the estimation result can be improved by using more data

points by increasing the length of the data time frame, and/or, decreasing the


136

sampling time. In this research work, the time frame is restricted to the time

difference between disturbance occurring time and the OLTC operation.

However, in the cases that the local load random changes are highly dominated

by the remote load random changes, the changes in the bus 9 voltage and current

caused by the OLTC operation may be used as an alternative to estimate the

system Thevenin impedance from the bus 9 view point.

6.3.3 Statistical Evaluation of the Estimated Thevenin Impedance

The proposed method in this thesis uses the normal random changes in the load

voltage and current to estimate the system Thevenin impedance. Hence, the

estimation result, imitating the data, should be normally distributed around the

actual value. To evaluate the integrity of the method, in this section different

results for one case are statistically evaluated. The histograms of the real and

imaginary components of the 200 estimated Thevenin impedances, for a 0.04 sec

simulation time step, are shown in Figure 6.14. Different parameters of the

histograms are shown in Table 6.5.

TABLE 6.5 STATISTICAL PARAMETERS OF THE COMPONENTS OF THE ESTIMATED

THEVENIN IMPEDANCES

Estimated component of Thevenin impedance

Mean value (pu)

Median (pu)

Standard deviation (pu)

Coefficient of variation (%)

Real component 0.0519 0.0517 0.0039 7.56 Imaginary component 0.1630 0.1631 0.0053 3.22

The coefficient of variation is defined to be: %100mean value

deviationstandardCV ×= .

As it is seen in Table 6.5, the mean values are very close to the medians, i.e., the

estimated values are almost normally distributed. The standard deviation of the

imaginary component is 3.22% of the mean value. Considering that the

imaginary component is the dominant component in the power system Thevenin


137

impedance, the standard deviation of the estimated real component also can be

acceptable. However, both standard deviations can be decreased by improving

the estimation accuracy through increasing the length of the data window, as it

was shown in section 6.3.2.

0.04 0.045 0.05 0.055 0.060

10

20

30

40

50

0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.180

20

40

60

Real component of the estimated thevenin impedance (pu)

Imaginary component of the estimated thevenin impedance (pu)

Num

ber o

f dat

a po

ints

Num

ber o

f dat

a po

ints

Figure 6.14 Histograms of the components of the estimated Thevenin impedance

The system actual impedance in this case is 0.0541 + 0.1640i. The mean values

of the errors in the real and imaginary components of the estimated impedances

are - 4.07% and - 0.61%, respectively. This implies a 1% mean error in the

magnitude of the estimated impedance that can still be decreased by decreasing

the simulation time step.

6.4 Validation of the Proposed “Load Characterization by

Sequential Peeling” Method

This section is devoted to the validation of the proposed load characterization

method in chapter 4. The so called peeling method is applied to the BPA test


138

system. The composite load is connected to bus 10, i.e., the secondary side of

OLTC, but the voltage and current measurements are taken at bus 9, i.e., OLTC

primary side. However, to characterize the load, the voltage and current at bus

10 are required; otherwise OLTC should be included in the load model that is not

intended in this research work. Hence, it is assumed that OLTC parameters are

known.

In this section, the initial tap position in bus 9 is set to 1 and the load is

characterized by applying a disturbance, without a fault, to the system and using

the measured data at bus 9, from disturbance time to the OLTC first operation.

The load connected to bus 10 is simulated as a combination of induction motor,

constant impedance, and constant power loads. The induction motor load is

simulated as a constant shaft power demand load; i.e., the motor dynamics

restores the motor active power consumption to its per-disturbance value. Small

random changes are included in both system loads.

To get the voltage at bus 10, for each data point the voltage drop across the

OLTC transformer is subtracted from the measured voltage. Using this voltage

and the measured current the load power is calculated at each data point. A

simulation of the bus 10 voltage and the measured current magnitudes, for a 0.04

sec time step and 250 sec total simulation time, is shown in Figure 6.15. The

disturbance is considered to be the loss of one of the branches between bus 5 and

bus 6 at time 50 sec. Figures 6.16-6.19 indicate the simulated load total active

and reactive powers and their components. The effects of the disturbance and

OLTC operations on the load voltage, current and power can easily be seen in

Figures 6.15-6.19.


139

50 100 150 200 2500.9

0.95

1

1.05Magnitude of load voltage-bus 10

Vol

tage

(pu)

0 50 100 150 200 2502.8

3

3.2

3.4

3.6

3.8Magnitude of load current-bus 10

Time(sec)

Cur

rent

(pu)

Disturbance OXL (G3)

Figure 6.15 Simulation of the load voltage and current magnitudes

0 50 100 150 200 2502.4

2.6

2.8

3

3.2Load total active power

Per

uni

t

0 50 100 150 200 2501.1

1.2

1.3

1.4

1.5Load total reactive power

Per

uni

t

Time(sec)

Figure 6.16 Simulation of the load total active and reactive powers


140

Figure 6.17 Simulation of induction motor load power

0 50 100 150 200 2502

2.2

2.4

2.6

2.8Constant impedance load active power

Per

uni

t

0 50 100 150 200 2501

1.1

1.2

1.3

1.4Constant impedance load reactive power

Per

uni

t

Time(sec)

Figure 6.18 Simulation of the constant impedance load power

50 100 150 200

0.26

0.28

0.3

Induction motor load active power

Per

uni

t

50 100 150 200 2500.03

0.04

0.05

0.06

0.07

Induction motor load reative power

Per

uni

t

Time(sec)


141

0 50 100 150 200 250-1

-0.5

0

0.5

1

1.5Constant power load active power

Per

uni

t

0 50 100 150 200 250-1

-0.5

0

0.5

1

1.5Constant power load reactive power

Per

uni

t

Time(sec)

Figure 6.19 Simulation of the constant power load

Equations (4.39)-(4.41) are used to estimate the load active power components.

The comparisons of the estimated and actual active powers for the different

compositions of the load are shown in Table 6.6.

To estimate the induction motor load reactive power, the frequency dependent

random changes in the load active and reactive powers are required. A one-step

prediction algorithm and the method described in the section 4.2.3 are used to

extract the frequency dependent components of the random changes in the load

post-disturbance active and reactive power. Changes in the load total power, the

random components after removing the dynamics and the voltage and frequency

dependent components of the active and reactive powers are shown in Figures

6.20 and 6.21, respectively.


142

20 40 60-2

-1

0

1

2x 10-5

20 40 60-1

-0.5

0

0.5

1

1.5x 10-5

20 40 60-2

-1

0

1

2x 10-5

20 40 60-1

-0.5

0

0.5

1x 10-5

Active power changes Random component

Voltage dependen random component

Frequency dependent random component

Time(sec)Time(sec)

Figure 6.20 Changes in the load post-disturbance active power

20 40 60

-5

0

5

x 10-6 Reactive power changes

20 40 60

-4

-2

0

2

4

6x 10-6

20 40 60

-5

0

5

x 10-6

Random component

Time(sec)20 40 60

-10

-5

0

5

x 10-6Frequency dependent

random component

Time(sec)

Voltage dependent random component

Figure 6.21 Changes in the load post-disturbance reactive power


143

Equation (4.46) is used to estimate the induction motor reactive power. The other

components of the load reactive power are anticipated on the similar way to the

components of the load active power. The estimated results and the actual values

for different load compositions are also compared in Table 6.6. As it can be seen,

the estimated results for most of the components are quantified to the actual

values. The induction motor load reactive power estimation, which is based on

the signal processing in the load bus data random changes, is also quantified.

TABLE 6.6 COMPARISON OF THE LOAD ESTIMATED AND ACTUAL POWERS FOR

DIFFERENT COMPOSITIONS OF LOADS, DISTURBANCE IS THE LOSS OF ONE OF THE

BRANCHES BETWEEN BUS 5 AND BUS 6, P1 & Q1: INDUCTION MOTOR LOAD,

P2 & Q2: CONSTANT IMPEDANCE LOAD, P3&Q3: CONSTANT POWER LOAD

1P 2P 3P 1Q 2Q 3Q

EST2.(PU)

ACT3.(PU)

ERR4 (%)

0.4006 1.7338 0.2000

0.4000 1.7344 0.2000

0.1519 -0.0332 -0.0164

0.1463 0.8624 0.0000

0.1414 0.8673 0

3.4563 -0.5650 -

EST.(PU)

ACT.(PU)

ERR (%)

0.2996 1.7435 0.3000

0.3000 1.7431 0.3000

-0.1497 0.0263 -0.0034

0.0726 0.8734 0.0500

0.0747 0.8712 0.0500

-2.8695 0.2448 0.0224

EST.(PU)

ACT.(PU)

ERR (%)

0.5012 2.1248 -0.0000

0.5000 2.1261 0

0.2449 -0.0569 -

0.2377 1.0711 0.0000

0.2458 1.0630 0

-3.3095 0.7625 -

EST.(PU)

ACT.(PU)

ERR (%)

0.6015 1.4998 0.1000

0.6000 1.5013 0.1000

0.2554 -0.1032 0.0182

0.3562 0.7677 0.1000

0.3732 0.7506 0.1000

-4.5676 2.2742 -0.0243

EST.(PU)

ACT.(PU)

ERR (%)

0.2996 1.7435 0.3000

0.3000 1.7431 0.3000

-0.1497 0.0263 -0.0034

0.0726 0.8734 0.0500

0.0747 0.8712 0.0500

-2.8695 0.2448 0.0224

The results of the estimation for a fixed initial load combination and different



144

system disturbance sizes are compared to the actual values in Table 6.7. As it is

seen, those values estimated based on the disturbance data are estimated more

accurate for the larger disturbances. The estimated induction motor reactive

powers are also close enough to the actual values, considering its direct

proportion to the already estimated induction motor active power, according to

equation (4.46).

TABLE 6.7 COMPARISON OF THE LOAD ESTIMATED AND ACTUAL POWERS FOR

DIFFERENT SIZES OF THE SYSTEM DISTURBANCE.

Voltage step (%)

1P 2P 3P 1Q 2Q 3Q

1.5

EST

ACT

ERR

0.3066 1.9704 0.2999

0.3000 1.9768 0.3000

2.1837 -0.3258 -0.0369

0.0682 0.9850 0.0501

0.0650 0.9883 0.0500

4.9212 -0.3305 0.2131

3

EST

ACT

ERR

0.2966 1.9229 0.3002

0.3000 1.9196 0.3000

-1.1392 0.1702 0.0502

0.0679 0.9587 0.0500

0.0671 0.9596 0.0500

1.2443 -0.0887 0.0332

4.5

EST

ACT

ERR

0.3028 1.8590 0.3000

0.3000 1.8618 0.3000

0.9224 -0.1502 0.0106

0.0707 0.9294 0.0500

0.0694 0.9307 0.0500

1.8481 -0.1366 -0.0223

6

EST

ACT

ERR

0.3014 1.8016 0.3000

0.3000 1.8030 0.3000

0.4744 -0.0786 -0.0021

0.0730 0.8666 0.0500

0.0720 0.9014 0.0500

1.3948 -3.8598 0.0345

7.5

EST

ACT

ERR

0.2996 1.7435 0.3000

0.3000 1.7431 0.3000

-0.1497 0.0263 -0.0034

0.0726 0.8734 0.0500

0.0747 0.8712 0.0500

-2.8695 0.2448 0.0224

6.4.1 Statistical Evaluation of the Estimated Power Components

In the proposed method of load power peeling, the induction motor load reactive

power is estimated using the correlation of the random changes in the load

reactive and active power and the other load power components are estimated


145

using the changes in the load total active and reactive power caused by the

disturbance. However, due to the unpredictable random components in the load

voltage and current, the load active and reactive power also contain such

random components, as it can be seen from Figures 6.20 and 6.21. Hence, all

estimated power components are statistically random results. In this section, the

statistical characteristics of the estimated load power components are

investigated. The system loading condition is similar to the last case in table 6.6

and the disturbance considered is loss of a transmission line between bus 5 and

bus 6. The histograms of the 200 estimations of the load power components are

shown in Figures 6.22-6.24. Different parameters of the histograms are show in

Table 6.8.

0.297 0.298 0.299 0.3 0.301 0.3020

10

20

30

40

50

0.0716 0.0718 0.072 0.0722 0.0724 0.07260

10

20

30

Num

ber o

f dat

a po

ints

Num

ber o

f dat

a po

ints

Induction motor load estimated reactive power (pu)

Induction motor load estimated active power (pu)

Figure 6.22 Histograms of the estimated induction motor load active and reactive

power


146

1.7655 1.7661.7665 1.7671.7675 1.768 1.7685 1.7691.7695 1.770

10

20

30

40

0.885 0.8852 0.8854 0.8856 0.8858 0.886 0.88620

10

20

30

40

Num

ber o

f dat

a po

ints

Constant impedance load estimated active power (pu)

Num

ber o

f dat

a po

ints

Constant impedance load estimated reactive power (pu)

Figure 6.23 Histograms of the estimated constant impedance load active and

reactive power

0.2999 0.2999 0.3 0.3 0.30010

20

40

60

80

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.050

20

40

60

Num

ber o

f dat

a po

ints

Constant power load estimated active power (pu)

Num

ber o

f dat

a po

ints

Constant power load estimated reactive power (pu)

Figure 6.24 Histograms of the estimated constant power load active and reactive

power


147

TABLE 6.8 STATISTICAL PARAMETERS OF THE HISTOGRAMS OF THE COMPONENTS OF

THE ESTIMATED THEVENIN IMPEDANCES

Estimated power component

Mean value (pu)

Median (pu)

Standard deviation (pu)

Coefficient of variation (%)

P-induction motor 0.3000 0.2999 9.7829e-004 0.3261 Q-induction motor 0.0720 0.0720 2.3269e-004 0.3231

P-constant impedance 1.7680 1.7680 8.9899e-004 0.0508 Q-constant impedance 0.8855 0.8855 2.5128e-004 0.0284

P-constant power 0.3000 0.3000 2.5960e-005 0.0087 Q-constant power 0.0500 0.0500 1.1468e-005 0.0229

The coefficient of variation is defined to be: %100mean value

deviationstandardCV ×= .

Referring to Table 6.8, all histograms are normally distributed with very small

standard deviations.

The mean values of the estimation errors in this experiment are shown in Table

6.9. The accuracy of the induction motor load reactive power estimation can be

improved by increasing the length of the data window through decreasing the

simulation time step.

TABLE 6.9 MEAN VALUES OF THE ESTIMATION ERROR IN THE DIFFERENT COMPONENTS

OF THE LOAD ACTIVE AND REACTIVE POWER

Load component Active power (%)

Reactive power (%)

Induction motor load -0.0142 -2.0849 Constant impedance load 0.0051 0.1784 Constant power load 0.0004 0.0002

6.5 Validation of the Proposed “On-line Estimation of the Time

to a Long-term Voltage Instability” Methods

In this section the proposed methods in chapter 5 for the on-line identification of

possible long-term voltage instability are validated by applying the methods to

the BPA test system. The load connected to bus 10 is considered to be a

composite of the induction motor, constant impedance and constant power loads.


148

At time 50 sec, a disturbance is applied to the system and the post-disturbance

load voltage and current phasors are measured at bus 9, i.e., the OLTC primary

side. At time 110 sec OLTC operation starts and the time delay for the

subsequent tap movements is set to be 10 seconds.

6.5.1 Maximizing OLTC Secondary Side Voltage Criterion

Consider the equivalent system in Figure 6.25. It was shown that how the post-

disturbance measured voltage and current at bus 10 can be used to estimate the

system Thevenin impedance and the load power components. Once the load

power is peeled off, the load parameters are identified using the bus 10 voltage,

which is computed using equation (6.1):

InjXnV

V oto

×−= 910 (6.1)

where, on is the tap initial position.

Figure 6.25 Equivalent of the BPA test system with OLTC transformer and

composite load in bus 10.

Following the disturbance, OLTC successive operations tend to restore the

voltage at bus 10 its set point value. Voltage at bus 9 becomes unstable if OLTC

inverse control action starts. One of the proposed methods in this research work

uses the load bus voltage maximizing as the criterion of the voltage instability

starting point.

9V

I

thE

QjP+

thZ

OLTC transformer

1:n tXjjX

LjX

10V

22 jQP +LR

R srR

=

11 jQP +

33

210

3 jQP

VZ

−=

33 jQP +


149

The voltage at bus 10 is a function of the Thevenin voltage, tap ratio and load

impedance. Taking into account the OLTC impedance, some changes are

required in equations (5.25)-(5.27). The system equations are as below:

ts IYYYYV ×+++= 32110 (6.2)

where,

210

12

10

11

V

PRXj

V

PY −= (6.3)

LL jRXY

+=

12 (6.4)

210

32

10

33

V

Qj

V

PY −= (6.5)

tths

XjnZnY 2

2

+= (6.6)

tth

tht

XjnZ

nEI 2+= (6.7)

IZVE thth += 9 (6.8)

Following each tap operation the induction motor active power 1P is recovered

in steady state to the motor constant demand mP . It was shown in chapter 5 that:

m

mP

XPVVR

24 224

102

10 −+= (6. 9)

...,,,,ininn o 3210=Δ−= (6.10)

In equation (6.10), on and nΔ are the tap initial ratio and tap step size,

respectively. It can easily be shown that setting the transformer reactance tX to

zero, equation (6.2) changes to equation (5.25).

Considering different values for i in equation (6.10), equation (6.2) can be solved


150

for 10V until this variable is maximized if the magnitude of thE in equation (6.8)

remains constant through the OLTC operation. However, due to the voltage

control provided by the generators, the magnitude of thE changes, causing

problem to the voltage instability anticipation. A simulation of the thE

magnitude is shown in Figure 6.26.

50 100 150 200

1.2

1.22

1.24

1.26

1.28

1.3

1.32

Time(sec)

Vol

tage

(pu)

Magnitude of Thevenin Voltage

Figure 6.26 Simulation of the Thevenin voltage magnitude

To solve the problem, using the measured voltage and current at bus 9, the

magnitudes of thE , at steady state, are computed for the tap initial position and

its first and second operations. Then, evaluating a curve fitted to the computed

values, the thE magnitude is estimated for the other OLTC operations. The

estimated and actual values of the thE magnitudes are compared in Figure 6.27.


151

2 4 6 8 10 12 141.223

1.224

1.225

1.226

1.227

1.228

1.229

1.23Thevenin Voltage - estimated and actual values

Tap number

Vol

atge

(pu)

ActualEstiamted

Figure 6.27 Comparison of the estimated and actual values of the system

Thevenin impedance

Using the estimated voltage Thevenin magnitudes, the Gauss-Seidel method is

used to solve equation (6.2) for the successive tap positions until the magnitude

of voltage at bus 10 is maximized. The voltage instability at bus 9 will start in

the next tap operation if neither the voltage is recovered to its set point nor tap

has exceeded its limit. In any case, the estimated voltage tracing is limited to the

tap position limit point. The voltage is stable if the estimated voltage is still

increasing at this point.

6.5.2 Impedance Matching Criterion In this section, the magnitude of the estimated load impedance, from the bus 9

view, is compared to the magnitude of the system Thevenin impedance as a

method to determine voltage stability. At the point of instability, the magnitude

of the load impedance is equal to and/or smaller than the system impedance


152

magnitude. Equation (6.11) is used to estimate the load impedance:

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+=321

2 1YYY

jXnZ tL (6.11)

where, 321 YandY,Y are computed using equations (6.2)-(6.10). The comparison

of the load and system impedances is limited to the OLTC limit point.

6.5.3 Simulation Results

Three cases with different pre-disturbance conditions are considered. The

disturbance considered at time 50 sec is a sudden change in the transmission

corridor between bus 5 and bus 6. At time 110 sec OLTC operation starts. In all

cases, the initial tap position and the tap step size are 1, and 1%, respectively.

The reference voltage at bus 10 is set to 1, and the deadband is 010.± p.u. The

tap range is 15± steps for the two first cases and 20± steps in the third case. It is

assumed that the OLTC transformer impedance and the initial tap position are

known.

Figures 6.28-6.30 indicate the time responses of the voltages at bus 9 and bus 10,

and the tap ratio of transformer between the two buses. Referring to Figure 6.28,

in first case the OLTC secondary side voltage is restored to the dead band of the

reference voltage by 8 tap successive movements and OLTC operation has

stopped. In the second case, OLTC operation is limited by the tap lower limit

and the voltage of bus 10 is still out of the reference voltage dead band, as it can

be seen from Figure 6.29. Figure 6.30 shows voltage instability that starts after

tap 13th movement. As seen in the figure, at this point the bus 10 voltage is

maximized.


153

0 50 100 150 200 250 3000.9

0.92

0.94

0.96

0.98

1

1.02

1.04Tap position, OLTC primary and secondary volatges

Time (sec)

TapBus 9Bus 10


tap ratio in case1

0 50 100 150 200 250 3000.8

0.85

0.9

0.95

1

1.05

Time(sec)

Tap position, OLTC primary and secondary volatges

TapBus 9Bus10


tap ratio in case2


154

50 100 150 200 250 300

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

Time(sec)

OLTC primary and secondary volatgesV

olta

ge(p

u)

Bus 9Bus10

Figure 6.30. Simulation of the OLTC primary and secondary side voltages in

case3

The 60 sec post-disturbance measured voltage and current phasors at bus 9 are

used to estimate the pre OLTC operation system Thevenin impedance and to

peel off the load power components. The results are compared to the actual

values in Tables 6.10 and 6.11, respectively.

TABLE 6.10

COMPARISON OF THE SYSTEM POST-DISTURBANCE ESTIMATED AND ACTUAL THEVENIN

IMPEDANCE FOR DIFFERENT CASES

thZ (actual) Estimated value

Case 1 0.0437 + 0.1645i 0.0420 + 0.1669i Case 2 0.0455 + 0.1671i 0.0393 + 0.1646i Case 3 0.0948 + 0.1843i 0.0926 + 0.1828i


155

TABLE 6.11

COMPARISON OF THE LOAD POST-DISTURBANCE AND PRE OLTC OPERATION ESTIMATED

AND ACTUAL POWER COMPONENTS. P1 & Q1: INDUCTION MOTOR LOAD, P2 & Q2:

CONSTANT IMPEDANCE LOAD P3&Q3: CONSTANT POWER LOAD

1P 2P 3P 1Q 2Q 3Q

Case 1

EST5.(PU)

ACT6.(PU)

ERR7 (%)

0.3041 1.7273 0.1000

0.3000 1.7314 0.1000

1.3764 -0.2374 -0.0187

0.0735 0.8622 0.1000

0.0704 0.8654 0.1000

4.5274 -0.3720 0.0341

Case 2

EST.(PU)

ACT.(PU)

ERR (%)

0.3001 1.5701 0.1000

0.3000 1.5702 0.1000

0.0333 -0.0064 -0.0020

0.0795 0.7840 0.1000

0.0786 0.7849 0.1000

1.1450 -0.1147 0.0400

Case 3

EST.(PU)

ACT.(PU)

ERR (%)

0.5006 2.6128 0.1000

0.5000 2.6135 0.1000

0.1235 -0.0241 0.0086

0.2342 0.6678 0.1000

0.2486 0.6534 0.1000

-5.7944 2.2038 0.0079

Having the load power components peeled off, the bus 10 voltage computed by

equation (6.1) at time 110 sec is used to calculate the constant impedance load

parameters and the constant reactive component of the induction motor load

impedance in any case.

Using equation (6.8), the system Thevenin voltage is computed for the tap initial

position and its first and second operations for three cases of simulation, and

then, the results are used to estimate the system Thevenin voltages for the other

tap positions, using the method described in section 6.5.1.

All estimated powers, voltages, and impedances are used to solve the equations

of sections 6.5.1 and 6.5.2 for the bus 10 voltages, and the load impedance from

the bus 9 view point, for different cases and at different tap positions. The

impedance of the compensation equipment in bus 9 is also included in the load



156

impedance. The comparisons of the system Thevenin and load estimated

impedances, and the estimated bus 10 voltages to the actual values are shown in

Figures 6.31-6.34.

2 4 6 8 10 12 14 16 18 200.95

1

1.05Load estimated and actual volatges

Vol

tage

(pu)

EstimatedActualReference

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3

0.35

0.4

Tap number

Impe

danc

e(pu

)

Estimated load and thevenin impedances

LoadSystem

Tap limit

Figure 6.31. Estimated OLTC secondary side voltages, system Thevenin

impedance, and load impedance in case1

Referring to Figure 6.31, the 6 initial estimated load bus voltages are very close

to the actual values. As a general assessment, the estimated voltages up to the tap

limit point are increasing, i.e., voltage is stable. The estimated system Thevenin

Impedances are also above the load impedance for all those points and thus the

same conclusion of voltage being stable is obtained using the impedance

matching criterion. The particular outcome is that the estimated voltage for the

8th tap movement exceeds the voltage reference, and hence, tap movement stops

at this position. The result agrees with the simulation result.

In the second case, the estimated voltage is of reasonable accuracy, however, as


157

it can be seen from Figure 6.32, there is a consistent overestimation which gives

rise to conservative estimation of stability. The estimated voltages are increasing

and the estimated system Thevenin impedances are above the estimated load

impedances, i.e., voltage instability will not happen. All the estimated voltages

are below the reference voltage, i.e., tap reaches the limit and stops. The results

agree with the simulation results.

2 4 6 8 10 12 14 16 18 200.9

0.95

1

1.05


Vol

tage

(pu)

EstiamtedActualReference

2 4 6 8 10 12 14 16 18 200.15

0.2

0.25

0.3

0.35

0.4

Tap number

Impe

danc

e(pu

)


LoadSystem

Tap limit

Figure 6.32 Estimated OLTC secondary side voltages, system Thevenin

impedance, and load impedance in case2

Figure 6.33 shows that in case 3, there is a small difference between the

estimated and actual voltages; however, this creates less difficulty for the voltage

stability assessment in this case, because in such cases, that the system is facing

large disturbances, the post-disturbance voltages are too far from the reference

value and the risk of the estimated voltage to be above the reference value;

implying stable voltage, is too low.


158

2 4 6 8 10 12 14 16 18 200.92

0.925

0.93

0.935


Vol

tage

(pu)

Tap number

EstimatedActual

Figure 6.33 Comparison of the estimated OLTC secondary side voltages to the

actual values in case 3

120 140 160 180 200 220 240 260 280 300

0.18

0.2

0.22

0.24

0.26

Time (sec)

Impe

danc

e (p

u)

Actual load and thevenin impedances

systemload

2 4 6 8 10 12 14 16 18 200.16

0.18

0.2

0.22

0.24

0.26

Tap number

Impe

danc

e(pu

)


loadsystem

Figure 6.34. Estimated system Thevenin impedance and load impedance and

their comparison to the actual values in case 3


159

In Figure 6.33, the estimated voltage is maximized at the tap 13th movement and

the voltage is still below the reference value and tap limit has not been

reached, i.e., voltage instability is anticipated at the next tap movement. At the

same point of tap position, the estimated load impedance becomes smaller than

the estimated system Thevenin impedance, as it is seen in Figure 6.34.

Considering the above examples, the OLTC secondary side voltage tracing is a

better index, comparing to the impedance matching criterion, because of the

capability of anticipation of the stopping point of the OLTC operation.

6.6 Summary

In this chapter, the developed methods for the voltage instability anticipation

were applied to the BPA standard test system. In the first part, the validity of the

system Thevenin impedance estimation method was investigated. The estimated

results were confirmed by the simulation results. The load peeling method was

also applied to the test system. The estimated results were of reasonable

accuracy.

For a chosen load composite and disturbance size, the different estimation results

were statistically evaluated using the histograms. All results were normally

distributed and the mean values and the standard deviations were also

acceptable.

Finally, these tools were combined to infer voltage stability by applying them to

the test system in three different cases. It was confirmed by simulation that the

proposed methods are capable for the anticipation of a long-term voltage

instability caused by the OLTC operation following a large disturbance in the

system.

Chapter 7

Conclusions Motivated by finding local measurement based on-line methods, the work

presented aimed to estimate the remained time to a possible long term voltage

instability caused by the on-load tap changer operation, following a large

disturbance in the power system.

In order to develop and implement a true on-line voltage stability analysis

method, the on-line accurate modeling of the supply system and the load, based

on the local measurements, is required. The shortcomings of the current

approaches in the system equivalent identification and load modeling were

discussed in chapter 2.

Based on the mentioned objective and the requirements, novel methods were

developed to:

• Estimate on-line the supply system Thevenin impedance using signal

processing on the random changes in the load bus voltage and current

caused by the load.

Chapter 7. Conclusions

161

• Characterize load in an on-line peeling process using changes in the load

bus measured data caused by the disturbance and the random changes in

the supply system.

• Anticipate on-line the expected time to a long-term voltage instability

caused by the OLTC operation.

7.1 Summary of the Results In this section, the main conclusions of the thesis are summarized.

On-line System Thevenin Impedance Estimation

In chapter 3, the idea that the load bus voltage and current changes caused by the

local load can be used to identify the supply system was used to introduce a

method to estimate the power system Thevenin impedance.

Using a one-step ahead prediction method, the dynamic components in the load

voltage and current were removed and the residuals (the random changes) were

then used to estimate the system Thevenin impedance. In order to cancel the

effect of the supply system changes in the load bus voltage and current changes,

they were correlated with the load admittance changes. It was shown that the

ratio of the correlations of the random changes in the load voltage and load

admittance with respect to the correlation of the random changes in the load

current and load admittance can be used to estimate the system Thevenin

impedance.

The proposed method was applied to a four bus test system in different cases and

it was concluded that the estimation result improves with an increase in the

portion of the local load related changes in the bus voltage and current changes

and/or increases in the length of the data window.


162

The invalidity of the current approach in literature, i.e., the ratio of the changes

in the load voltage with respect to the changes in the load bus current, when the

changes are present in the supply system, was also investigated.

The proposed method was also applied to the real data from Brisbane load

centre. The results were consistent across subsequent time slices; however, the

judgment on the results is still open to the availability of the system actual

Thevenin impedance.

Load Characterization by Sequential Peeling

In chapter 4, considering the requirements of a load model in the on-line voltage

stability study, a method was introduced to characterize the behaviour a

composite of an induction motor, constant impedance and constant power loads,

following a large disturbance in the supply system.

The changes in the load total active power, caused by the disturbance, were used

to identify the active power of the different components of the load. Then, the

electromechanical dynamics and the voltage related random changes in the post-

disturbance load active and reactive power were removed and the residuals, i.e.,

the random changes caused by the load bus frequency changes, together with the

induction motor estimated active power were used to estimate the post-

disturbance induction motor reactive power, and then peel it off from the load

total reactive power. Finally, the reactive powers of the constant impedance load

and the constant power load were estimated using the changes in the load

measured reactive power, caused by the disturbance.

The proposed load peeling method was applied to a four bus test system in two

different cases:


163

• Different pre-disturbance load compositions with the same disturbance

size

• The same pre-disturbance load composition with different disturbance

sizes

The estimated values showed less than 1.61% error in both cases. However, it

was found from the second experiment that the larger the disturbance, the better

the estimated results. The reason is that, in the proposed method, the estimation

of the five out of the six load power components is based on the load power

changes due to the disturbance. However, they are also affected by the random

changes contribution in the load power changes, and the larger the disturbance

size, the lesser the relative share of the random changes in the load power

changes.

Anticipation of the Expected Time to a Long-term Voltage Instability

In chapter 5, methods were introduced to anticipate voltage instability for a

system consisting of a constant impedance load case, and/or, a composition of

the constant power, constant impedance and induction motor loads behind an

OLTC. The OLTC transformer assumed to be ideal and the changes in the load

bus voltage and current due to the OLTC first operation were used to anticipate

possible voltage instability and the remained time to such instability. It was also

assumed that the system Thevenin impedance and the load parameter/s are

already identified using the proposed methods in chapter3 and chapter 4,

respectively.

Two criteria were used to anticipate the instability: impedance matching between

load and system criterion, and OLTC secondary side voltage maximizing

criterion. The changing of the sign of the changes in the load power due to the


164

tap operation, at the instability point, was also used as a second index in the

constant impedance load case.

The effect of the load composition was also determined. It was shown that how

the larger induction motor component in the load composite makes the system

voltage safer in the long-term, but increases the risk of the short-term voltage

instability.

To investigate the effect of the measurement uncertainties in the estimation

result, random changes were added to the system and the results were still of

acceptable accuracy. The methods were confirmed by simulation.

Simulation Results on the Standard Test System

To validate the proposed methods in this thesis in voltage stability prediction,

they were applied to the BPA test system. In the simulations, the generators were

presented by their third order models and excitation system and the

overexcitation limiters were also included. The local load and the other load

were modelled as a composite load and constant impedance load, respectively.

A disturbance was applied to the system and the measured load bus voltage and

current from disturbance time until the start of OLTC operation were used to

estimate the system Thevenin impedance and to peel the load power components

off. The estimated results for the different system and load conditions showed

less than 4% and 5% errors for the components of the Thevenin impedance and

the components of the load power, respectively. It was found that the larger the

disturbance, the better the estimated results for the components of the load

power.

The histograms of the different estimations for a chosen load composition were

used to evaluate statistically the estimation methods. The investigation indicated


165

that all results were normally distributed and the mean values of the estimation

errors were less than 4.1% and 2.08% for the components of the Thevenin

impedance and the components of the load power, respectively. The standard

deviations for the estimated values for the components of the load power were

less that 0.33% of the mean values. The standard deviation of the estimated

imaginary component of the system Thevenin impedance was 3.2% of the mean

value. This coefficient for the estimated real component was around 7.5% which

is acceptable, considering the small value of the real component of the system

Thevenin impedance. However, the statistical characteristics can be improved by

using larger data windows in the estimation process.

Finally, all methods were validated by applying them to the test system in three

different cases. It was confirmed by the simulation results that the proposed

methods are capable of anticipation of a long-term voltage instability caused by

the OLTC operation following a large disturbance in the system. The voltage

tracing criterion was found a better index because of the capability of

anticipation of the stopping point of the OLTC operation.

7.2 Future Research

The proposed load characterization method was confirmed by simulation on the

test systems. The first objective in the continuing work is to apply the method to

a composite load in laboratory. A disturbance can be simulated by inserting

series impedance in the circuit. The random changes in the load voltage and

current caused by the supply system can then be used to investigate the accuracy

of the method and also to quantify the similarities and differences with the

results obtained in this thesis.


166

A second objective is to apply the peeling method to the real data obtained from

a large disturbance in the system, followed by the OLTC operation. The

accuracy of the estimation result can be investigated by calculating the load bus

voltage at different tap operation points, using the estimated load parameters, and

comparing them to the measured voltages.

The induction motor load in this thesis is considered to be the aggregation of a

group of induction motor loads. Even though there are aggregation methods

applicable to the dissimilar motors, but it is not recommended in general [89].

Hence, a suggestion for future work is to consider different types of motor load

and to try to peel the load off. It is possible that the peeling can be achieved

using frequency domain analysis methods. The theory provided in this thesis

may be extended to the frequency domain to find the load transfer function and

then to break it into single time constant portions, each representing a different

induction motor model.

Estimation of the post-disturbance dynamic status of the system, determination

of the stability margins, the sensitivity of the transfer limits to those margins, and

implementation of a control process to ensure security in the system are the main

parts of an on-line security assessment package. On-line security assessment

packages are commercially available today but their accuracy is limited by the

load model uncertainties [33]. A further suggestion for future work is to improve

the system security margins by developing new control strategies and

determining accurately the system transfer capacity based on the achievements in

this thesis.

My final suggestion for the future work is more focusing on the OLTC role in

the system dynamic behaviour. Sitting between load and system, OLTC


167

transformer has a dual role. From the primary side it is a part of the load and

from the secondary side it is a part of the supply system. Hence, the changes of

the OLTC primary side and the secondary side voltage and current caused by its

operation may be used to identify the system and the load, respectively.

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Appendix A

One-step-ahead Prediction for Removing Dynamic Component The one-step-ahead prediction method is calculating the response of the system

one step in the future to an input sequence while the process outputs are known

up to some time instant k-1 [90]. Consider the process

)k()k()mk(ub)k(ub)nk(ya)k(ya)k(y Tmn θϕ=−+⋅⋅⋅+−+−−⋅⋅⋅−−−= 11 11

(A-1)

where u and y are the system input and output, respectively and

pT R,)mk(u)k(u)nk(y)k(y)k( ∈ϕ−⋅⋅⋅−−⋅⋅⋅−−=ϕ ]11[ M , ;mnp +=

and T11 ][ mn bbaa ⋅⋅⋅⋅⋅⋅=θ M , pR∈θ .

To identify y(k) the unknown parameter θ should be identified which can be

estimated by least squares (LS) technique. For N available data samples the

model can be written in vector/matrix form with N-m equations for k = m + 1; …

; N. Assuming n=m, it can be shown that the estimation error is minimized if:

YXXX TT 1-)(=θ∧

(A-2)

where T)]( )2( )1([ NymymyY ⋅⋅⋅++= and the regression matrix X is :

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⋅⋅⋅−−−⋅⋅⋅−−

⋅⋅⋅+−⋅⋅⋅+−⋅⋅⋅−⋅⋅⋅−

=

mNuNumNyNy

umuymyumuymy

X

11

212111

MMMM (A-3)

This is an autoregressive moving average (ARMA) process that is used in this

thesis to estimate the noise free dynamic component of the signal using the

original signal as output and an arbitrary signal as input. The random component

of the signal is extracted by subtracting the estimated signal from the original

one.

178

Appendix B

Frequency Relation between Buses In the three bus system in Figure B.1 we can write:

0331232131 =++ VYVYVY (B-1) Thus:

( )23213133

31 VYVY

YV +−= (B-2)

1V = 1V ∠ 1δ 3V = 3V ∠ 3δ 2V = 2V ∠ 2δ

1Z 2Z I LZ

Figure B.1 Three bus system

Putting 222δ= jeVV and 333

δ= jeVV in Equation 2 we will have:

=33

δjeV ( )223213133

1 δ+− jeVYVYY

(B-3)

Taking the derivatives of both sides in Equation (B-3) with respect to time and considering that bus 1 is an infinite bus, supposing 2V is constant, and

neglecting the changes in 3V ( 03 =dtVd

); we will have:

=ω δ333jeV ( )22322

33

1 δω− jeVYY

(B-4)

Where, dtdδω = . Now it can be shown simply that:

23

232

333

1ω⎟⎟⎠

⎞⎜⎜⎝

⎛−=ω

VV

YY

(B-5)

Considering L

3

ZV

I = and using the current division law, we can write:

179

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=LZ

VZZ

ZZVV 3

21

1232 (B-6)

Replacing 2V in Equation (B-5) from Equation (B-6) and considering that 2

321

ZY −=

andLZZZ

Y 111

2133 ++= ; we will have:

( ) 221

21

21

23 1

111

1

ωω ⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

=L

L

ZZZZZ

ZZZ

Z (B-7)

After some simplification it can be shown that:

221

13 ωω ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=ZZ

Z (B-8)

Equation (B-8) is valid just when both impedances; 1Z and, 2Z have the same

arguments. For the transmission lines normally the resistive component of

impedance is ignored in comparison to the reactive component. Putting 11 jXZ =

and 22 jXZ = in Equation (B-8) we will have:

221

13 ω⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=ωXX

X (B-9)

If there is also any changes in 1δ (i.e. 011 ≠

δ=ω

dtd

), then in can be easily shown

that:

21

21123 XX

ωXωXω

+

+= (B-10)

180

Appendix C

BPA Test System Data

The BPA test system data are as below [19]: Transmission lines (R, X & B in pu on 100 MVA Base) R X B 4-5 0.0000 0.0040 0.0000 5-6 0.0015 0.0288 1.1730 8-9 0.0010 0.0030 0.0000 Transformers (R & X in pu on 100 MVA) R X 1-4 0.0000 0.0020 2-5 0.0000 0.0045 3-6 0.0000 0.0125 6-7 0.0000 0.0030 6-8 0.0000 0.0026 3-6 0.0000 0.0010 Machine Parameters Machine 1: Infinite Bus Machine 2: H=2.09, MVA Base = 2200 MVA Machine 3: H=2.33, MVA Base = 1400 MVA Stator and rotor parameters (Machine 2 &3) Ra=0.0046 Xd=Xq=2.07 X’

d=0.28 T’do=4.10 Exciters Both machine 2 and 3 have ALTHYREX exciters and the data are based on Unit F18 in Appendix D of Anderson and Fouad [88] KA=200 TA=0.3575 KE=0.0 TE=0.0 Field current limiters Data are provided from Kundur [7] Ifdmax1=3.02 pu (or 1.05 x full load current) Ifdmax2=4.60 pu (or 1.6x full load current) K1=150 K2=0.248 K3=12.6 K4=140 ILIM=3.85 Induction motor load data Rs=0.0 Xs =0.092 Xm =2.14 Rr =0.059 Xr =0.075 H=0.342 MVA Base=2440

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Doctor of Philosophy · method, the on-line accurate modeling of the higher voltage (supply system) and the lower voltage level (aggregate load) based on the local measurements is