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Durham E-Theses
Real-time power system security assessment
Sha�e-Pour, A. R.
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Sha�e-Pour, A. R. (1989) Real-time power system security assessment, Durham theses, Durham University.Available at Durham E-Theses Online: http://etheses.dur.ac.uk/9303/
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REAL-TIME POWER SYSTEM SECURITY ASSESSMENT
A THESIS SUBMITTED TO
THE UNIVERSITY OF DURHAM
FOR THE DEGREE OF
PHD
BY
A.R. SHAFIE-POUR
SEPTEMBER 1989
SCIENCE LABORATORY
SCHOOL OF ENGINEERING
THE UNIVERSITY OF DURHAM
2 5 APR 1991
A.R.SHAFIE-POUR REAL-TIME POWER SYSTEM SECURITY ASSESSMENT
ABSTRACT
The increasing complexity of modern power systems has led to a greater
dependence on automatic control at a l l levels of operation. Large scale
systems of which a power system i s a prime example, i s an area i n which a
wide gap e x i s t s between t h e o r e t i c a l mathemat ica l ly based research and
engineering prac t ice . The research programme at Durham is directed towards
br idging t h i s gap by l i n k i n g some of the avai lable and new theoret ica l
techniques wi th the prac t ica l requirements of on-l ine computer control in
power systems.
This thesis i s concerned wi th the assessment of security of power systems in
real- t ime operation. The main object ive of t h i s work was to develop a
package to be incorporated i n the Universi ty of Durham On l i n e Control of
E lec t r i ca l Power Systems (OCEPS) suite t o cater f o r network is landing and
analyse the features and the f e a s i b i l i t y of a real-t ime ' secur i ty package'
f o r modern energy control centres.
The real-t ime power systems simulator developed at Durham was used to tes t
the algorithms and numerical resul ts obtained are presented.
ACKNOWLEDGEMENTS
The author wishes to express his sincere thanks and grat i tude to his
project supervisor, Professor M.J.H. S t e r l i n g , f o r his guidance and
encouragement during the development of t h i s p ro j ec t .
I would l i k e to thank my parents without whose support and encouragement
t h i s work would not have mater ia l i sed , and also support of my own f a m i l y .
I would l i k e to dedicate t h i s work to my parents, Nicola , Danie l le ,
Stephanie and V i c t o r i a .
The author also wishes to thank Janet Goodman f o r typing t h i s thes i s .
Hearty thanks go to Dr. M.R. I r v i n g f o r his time and valuable discussions
throughout the p r o j e c t .
LIST OF PRINCIPAL SYMBOLS AND ABBREVIATIONS
1. Network
B ' , B" approximate Jacobian matrix
H, J , L, N Jacobian submatrices of p a r t i a l der ivat ies
I vector of in jec ted nodal current
P active power
Q reactive power
P S | ? speci f ied power
Q S f ) speci f ied reactive power
V vector of node voltages
Y admittance matrix = G + jB
e vector of busbar voltage angles
Z impedance matrix
C connection vector
f modified branch impedance
Quadrature tap postion
P^k Active power through the phase s h i f t e r
PI performance index
Z(p) overload performance index
Z voltage performance index
2. Subscripts
i , j , k nodes kei subset of nodes d i r e c t l y connected by branches to node i t transpose
o o r ig ina l m modified
3. Other Symbols
C a p i t a l l e t t e r s i n d i c a t e a v e c t o r , lower case i n d i c a t e s a seal
quant i ty .
A delta n number of f a u l t s
I sum notation
4. Abbreviations
SCADA Supervisory Control and Data Acquis i t ion
FDNR Fast Decoupled Newton Raphson
ACS Automatic Contingency Selection
AUTOUT Automatic Contingency Selection OUTput
SECASS SECurity Assessment
SECOUT SECurity Assessment OUTput
ASA Automatic Security Assessment
CONTENTS
Page
INTRODUCTION 1
CHAPTER 1 REAL-TIME CONTROL AND SUPERVISION OF POWER SYSTEMS 9
1.1 Energy Control Centres 9
1.2 Supervisory Control and Data Acquis i t ion (SCADA)
1.3 Control Systems 15
1.3.1 Simulation Function 16
1.3.2 Topology determination and State Estimation 17
1.3.3 Automatic Generation control (AGC) 19
1.3.4 Economic Despatch (ED) 21
1.3.5 Unit Commitment 22
1.3.6 Load Forecast 24
1.3.7 Emergency Rescheduling 25
1.3.8 Load Shedding 26
1.3.9 Security Assessment 26
1.4 Discussion 29
CHAPTER 2 REAL-TIME SECURITY ASSESSMENT 42
2.1 Introduction 42
2.2 Power System Operating States 43
2.3 Security Functions in System Control 48
2.3.1 Security Monitoring 48
2.3.2 Security Enhancement 51
2.3.3 Security Analysis 52
2.4 Real-Time Security Assessment 55
CHAPTER 3 THEORY OF NETWORK CHANGES 60
3.1 Introduction 60
3.2 Solution to Linear Networks 61
3.3 Theory of Network Changes 65
3.3a Applicat ion of diakoptics to the theory of branch outages 65
3.3b Development of the algorithm f o r simultaneous
branch outages 70
3.3c Node addit ion and removal 79
3.3d Asymmetry 81
3.3e Applicat ion t o non-linear elements 84
3.3f S p l i t Network 85
3.3g Node merge 90
CHAPTER 4 APPLICATION OF THE THEORY OF BRANCH OUTAGES TO
LOAD-FLOW 91
4.1 Introduction 91
4.2 Summary of fas t decoupled load-f low 91
4.3 Automatic Control and Adjustment 96
4 .3 .1 On-load transformer tap-changing 101
4.4 Transmission-line Outages 102
4.5 Applicat ion of the recursive method to load-f low 105
CHAPTER 5 AUTOMATION CONTINGENCY SELECTION (ACS) 109
5.1 Introduction 109
5.2 ACS Techniques 110
5.3 Line Overlaod Performance Index 117
5.3.1 Analysis of a Single Branch Outage 119
5.3.2 Change of performance index 6Z. due to change
of 68^ - Outage of branch V 121
5.4 Sequence of Compilation 126
5.5 Voltage Vio la t ion Performance Index 128
5.5.1 Change of Z (6Z ) due to changes i n voltages fiV, v 129
5.6 Analysis of a single branch outage 132
5.6.1 Effec t of a single branch outage on system susceptance 132
5.6.2 Calculation of 6V„ 134
5.6.3 Calculation of voltage phase angle 135
5.7 Sequence of Computation 136
CHAPTER 6 NETWORK ISLANDING AND EQUIVALENCING 139
6.1 Introduction 139
6.2 Sp l i t network 140
6.3 Theory of Network Islanding 141
6.3.1 Summary of basic algorithm f o r outage studies 142
6.3.2 Pre-processing f o r System Separation 145
6.3.3 Islanding 147
6.4 Equivalencing 151
6.5 External Network Equivalencing 159
CHAPTER 7
7.1 Introduction to Real-Time EMS Integrat ion 162
7.2 OCEPS EMS Software Integrat ion 165
7.3 Numerical Results 167
7.3.1 Test Network 167
1
INTRODUCTION
The system operator has a v i t a l l y important ro le in successful bulk power
system operations. He manages both the production and the transmission of
e l e c t r i c i t y . His objectives include an uninterrupted supply of power,
minimum production cost, maximum savings on interchange transactions, safety
of f i e l d personnel, power del ivery without damage t o equipment and well
co-ordinated maintenance.
To accomplish these, he must have a thorough knowledge of the
charac ter is t ics of the power system, understand economics, know how to
e f f e c t i v e l y use computer resources, analyse c a r e f u l l y but act quickly and
dec i s ive ly , and survive r epe t i t i ve work. His mistakes can be cost ly and
very v i s i b l e while the pounds he saves and the in ter rupt ions he avoids go
unnoticed.
Power systems are becoming more t i g h t l y in tegrated, complex and the demand
f o r e l e c t r i c i t y i s growing and as a resul t the system operator 's l i f e i s
becoming more complicated and he i s becoming more heavily depended upon.
System operators are not born wi th the necessary capab i l i t i e s and u n t i l now
there have been no comprehensive college programmes to educate them. S t i l l
much of the burden of t r a i n i n g system operators f a l l s on indiv idual control
centres. Excellence i n t r a i n i n g i s essen t ia l .
2
The components of the operator 's i n i t i a l t r a i n i n g programme are classroom
education ( lectures , reading, video tapes, workbooks, discussion covering
theory, pract ica l app l i ca t ion , operating procedures, r e l i a b i l i t y standards
e t c . ) , f i e l d t r i p s ( to hear the operator 's problems f i r s t hand), on-the-job
t r a i n i n g (by working side by side wi th an experienced system operator) ,
f a m i l i a r i s a t i o n and practice using the control computer. The refresher
t r a i n i n g and the upgrading of s k i l l s includes more classroom education,
meetings wi th other control centre operators and work projects such as
forecast ing load and generation.
The increasing size and complexity of modern power systems has led t o a
gradual change in control requirements. The increase i n the cost of energy
has enhanced the need t o seek economic o p e r a t i o n s , w h i l s t the system
complexity has taken the control task beyond the d i rec t a b i l i t i e s of the
operators. Simple indicat ions and controls have given way to more
sophisticated means of display and analysis , and automatic operation of
plant and systems i s more common place. I t fo l lows that i t i s necessary to
i n s t a l l sophist icated, yet extremely r e l i ab l e integrated control systems,
with complex real- t ime f u n c t i o n . One benef i t of integrated simulation
management and control software i s the a b i l i t y to provide a useful operator
t r a i n i n g a i d , which includes power system dynamics, telemetering and energy
management f a c i l i t i e s . Power system simulators are an important operator
t r a i n i n g tool i n today's modern control centres. They provide pract ica l
exerc i se w i t h h y p o t h e t i c a l s i t u a t i o n s such as s i m u l a t i o n of emergency
3
s i tua t ions , p a r t i a l blackout and res torat ion condi t ions; i n understanding
how one's power system q u a l i t a t i v e l y responds under abnormal frequency
conditions including in teract ions of system frequency, governors, prime
movers, load shedding, load res torat ion and t i e - l i n e synchronisation. In
addi t ion the development and implementation of an integrated simulation and
control sui te i s a pre- requis i te of any research establishment wi th in teres t
i n the operational control aspects of generation and transmission systems,
since i t enables the consideration of r e a l i s t i c environments.
Energy Management Systems (EMS) are part of the integrated control system
and t y p i c a l l y include funct ions such as Network Analysis (Topology,
state-est imator, bus load fo recas t ) , securi ty analysis , load-f lows, security
constrained economic despatch, interchange scheduling, system load forecast ,
despatcher t r a i n i n g s imula t ion , network reduction and automatic generation
c o n t r o l . Some of these functions are required to run in real- t ime mode and
some in study mode. Although due to the c o n f l i c t i n g objectives i n operating
the system and the p e c u l i a r i t i e s of generation, load, transmission and
d i s t r i b u t i o n f a c i l i t i e s of d i f f e r e n t systems, a standard a l loca t ion of tasks
and re spons ib i l i t i e s to the levels of control is not p r a c t i c a l , many EMS
systems have s imi la r s t ruc tures .
Energy Management Systems provide means f o r the secure, economic operation
of e l e c t r i c a l power networks and recently wi th the inc lus ion of expert
system f a c i l i t i e s reduces the burden on systems operators.
4
The f i n a l r e spons ib i l i ty f o r the safe, secure and economic operation of the
power system rests wi th the s t a f f i n the system control centre. Their
capab i l i ty to carry out t h e i r control r e spons ib i l i t i e s re l i es heavily on the
adequacy of the operation plans and guidance provided to them by planning
s t a f f . In the event, system conditions may d i f f e r from those expected
conditions on which the plans were based, and there is a need to review and
revise securi ty plans in the control room environment. This i s the funct ion
of the on-l ine security assessment package, which i s one of the functions
car r ied out 1n a modern system control centre.
The general object ive of security assessment i s to ensure that the power
system is operated at the minimum cost necessary to ensure compliance wi th
securi ty standards.
The progression of time in the real- t ime environment means that there is
always a danger of events overtaking the control engineer before a decision
has been made. These two f a c t o r s , r e q u i r e t h a t the o n - l i n e s e c u r i t y
assessment f a c i l i t y be as automatic as possible i n terms of picking up nodal
demand productions, generator schedules and planned network conf igurat ion
and as f a s t as possible i n producing resu l t s .
As an aid to the network control engineer i t i s possible to assess the
e f f e c t of any scheduled or unexpected l i n e or generator outages by means of
a f a s t approximate load-f low method. Unfortunately, even applying the f a s t
decoupled load flow method wi th e f f i c i e n t updating of outage solutions using
the modified matrix technique, the solut ion time required f o r consideration
of every possible outage may be excessive. I t i s , therefore , advantageous
t o perform a s i m p l i f i e d ana lys i s r e f e r r e d t o as automatic contingency
so lu t ion , which determines the subset of possible contingencies predicted to
be most severe.
The security assessment phase then proceeds to analyse the in tac t system and
the preselected outage cases one at a t ime.
Conventional techniques f o r the assessment of transmission network securi ty
normally assume the preselected contingency l i s t w i l l not s p l i t the network
in to is lands. Should such a s i tua t ion arise the approach i s usually to
r e s t r i c t analysis to the largest is land so created. This approach has
serious disadvantages i n power systems which are subjected to frequent major
outages and f o r which islanding 1s a common occurrence.
This d isser ta t ion represents fas t and accurate techniques f o r the
preselection of contingency l i s t s f o r transmission l ine outages. A new
technique f o r the diakoptic solut ion of the network f low when subjected to
c o n f i g u r a t i o n changes i s presented. Based on the t h e o r i e s desc r ibed ,
computer programmes in Fortran language were developed f o r the real-t ime
securi ty assessment of e l e c t r i c a l power networks and were integrated i n the
On Line Control of E lec t r i ca l Power Systems (OCEPS) sui te at the Universi ty
of Durham Science Laboratories. A technical paper based on new techniques
developed f o r the securi ty analysis incorporat ing network is landing was
published.
6
Computer assisted control of e l e c t r i c a l power transmission and d i s t r i b u t i o n
systems represents a large scale real-t ime data processing problem and
demands an i n t e g r a t e d approach t o so f tware design and t e s t i n g . The
a v a i l a b i l i t y of r e l a t i v e l y inexpensive computer hardware and the increasing
importance of energy management has led to the wide-spread adoption of
sophisticated control systems. Objectives of these schemes include the
minimisation of production cost, improvement of the securi ty of supply and
the a b i l i t y t o ensure the c o r r e c t genera t ion of power system p l a n t .
Computer based supervisory control and data acquis i t ion systems have been
widely used i n most power u t i l i t i e s f o r SCADA and energy management (Chapter
1 ) .
Chapter 2 represents a survey of approaches to the estimation of system
securi ty and discusses power system operating states, monitoring and
security enhancement.
The solut ion t o l inear networks and the theory of network modificat ions i s
described i n Chapter 3 and i s f u r the r expanded to non-linear networks i n
Chapter 4, where the appl icat ion of the recursive branch outages technique
to load f low i s invest igated.
The automatic contingency selection problem is concerned wi th developing
computer algorithms f o r quickly i d e n t i f y i n g those contingencies which may
7
cause o u t - o f - l i m i t conditions so as to reduce the number of contingencies
that need to be evaluated when assessing the power system's securi ty i n a
real-t ime environment. Chapter 5 presents an automatic contingency
selection algorithm f o r the determination of a l l the contingencies that give
e i ther voltage or l i n e f low problems when the system is subjected to single
l ine outages.
Power systems are subjected to frequent major outages and network s p l i t i s a
common occurrence. Should such a s i tua t ion arise the approach i s usually to
r e s t r i c t analysis to the largest is land so created. Theory of network
is landing i s described i n Chapter 6. The new technique presented i s based
on diakoptics and overcomes the above disadvantage.
One of the requirements of today's modern power system control centres i s to
determine equivalents f o r extensive power systems external t o an in terna l
system equipped wi th a central control computer. These equivalents are
needed f o r d i f f e r e n t system studies where the in ternal system is represented
i n de ta i l and the external system is represented by t h e i r equivalent, to
simulate the in te rac t ion e f f ec t s of the external system on the in ternal
system f o r disturbances o r ig ina t ing i n the in ternal system. The addit ion of
t h i s f a c i l i t y to an EMS package enhances the security assessment f u n c t i o n .
A survey of techniques available f o r network reduction, i s presented i n
Chapter 6 and a new approach i s discussed.
8
Chapter 7 of t h i s thesis represents the numerical resul ts obtained
throughout t h i s work and reference i s made t o overal l EMS i n t e g r a t i o n .
9
CHAPTER 1
REAL-TIME CONTROL AND SUPERVISION OF POWER SYSTEMS
1.1 Energy Control Centres
The increasing size and complexity of modern power systems has led to a
gradual change in control requirements. The increase in the cost of energy
has enhanced the need to seek economic operations, whi ls t the system
complexity has taken the control task beyond the direct ab i l i t i es of the
operators. Simple indications and controls have given way to more
sophisticated means of display and analysis , and automatic operation of
plant and systems is more commonplace.
E l e c t r i c i t y Authorities operate highly complex generation and
distribution networks with an ever-increasing requirement for energy saving
and permanency of supply. I t follows that i t i s necessary to i n s t a l l
sophisticated, yet extremely reliable control systems to achieve these
objectives.
The role of system control under normal operating conditions is to
maintain e lectr ic i ty demand and generation continuously in balance, whilst
staying within defined levels of frequency and voltage, and whilst
maintaining a defined degree of security against unforeseen hazards. Power
10
station plant must be controlled to cope with a variety of system
requirements such as eff ic iency, frequency regulation, part-load operation
and operation with frequent start-ups and shut-down (2 shift ing) and at the
same time meet such requirements as frequency control, voltage control, real
power demand, line-flow constraints, security against disturbances and
economic despatch.
To meet these needs, the operator requires a range of computing aids
such as load-flow analysis to identify potential constraints and resolve
them, state-estimation to identify the system configuration, data redundancy
to enable bad data to be eliminated, economic load despatch, contingency
analysis, VAR despatch to avoid the risk of voltage collapse.
Electr ical power plants are interconnected and require a control point
from which frequency and generation can be controlled, especially when
control areas are established, and interconnection flows as well as
generation are to be monitored, in order to control the amount of
interchanged power with other control areas. The information is gathered
via telemetry systems.
For control purposes generation and transmission in England and Wales,
managed by the CE6B, is divided into 7 Grid control areas each with i t s own
control centre. In addition control i s co-ordinated nationally by the
National Control Centre. Each area control centre i s responsible for
switching of c i rcui ts within the area, and for meeting demand in the area.
11
It i s also responsible for instructing the switching of c i rcui ts between
areas, and for optimising, as far as possible, economic operation between
areas.
The present method of economic operation is one in which National
Control carries out studies 24, 12, 6, 3 and 1 hour ahead to determine which
plant is l ikely to be in merit and required for loading. This plant i s then
scheduled to be synchronised just before i t i s needed. The schedules for
synchronisation are then issued by Area Control Centres to the power
stations.
The advent of automation, array processors, transputers and, on the
software side, expert systems, have much to offer system control. In recent
years the traditionally manned substation has given way in some locations to
unmanned substations controlled by telecommand from an Area Control Room.
I t i s l i k e l y , as nat ional ly co-ordinated despatch comes i n , that the
economic role of Area Control Centres wil l give way to a switching control
r o l e , in which advanced aids w i l l a s s i s t the control engineer in the
management of the system. Training simulators can be one of enhancing
control engineers' abi l i ty to deal with emergencies.
The CEGB has approximately 130 power stations with a total generating
capacity in excess of 55 GW. Figure (1.1) i l lust ra tes the location of the
major power stations in the CEGB. The control of the power generation
throughout the network is a hierarchical process which commences with a
12
prediction of the load demand at a Central Control Centre and ends with
closed loop controllers which regulate the primary energy source supplied to
the generators in response to variations in the desired and actual values of
frequency and output power. The hierarchical levels in the control sequence
include the long term planning of unit commitment, based on the long term
load forecasts, the short term adjustment of the desired levels of
generation (economic despatch) based on the short term load forecasts and
the desired operating frequency and f inal ly the continual adjustment of the
generator regulations by the closed loop controllers.
The CEGB divide the unit commitment and economic despatch problems
amongst a National Control Centre and Area Control Centres. The National
Control Centre i s responsible for determining the overall operating levels
throughout the network, while the Area Control Centres are responsible for
implementing the leve ls . Figure (1.2) i l lustrates the location of the
Control Centres in the CEGB. Figures (1.3) , (1.4) and (1.5) i l lustrate
CEGB's Super Grid and demands.
1.2 Supervisory Control and Data Acquisition (SCADA) and Data Communication
In order to be able to meet the ever-more stringent operating
conditions with the ever-decreasing levels of equipment redundancy, the
operator requires up-to-date and accurate information about the state of the
entire network. The function of supplying the operator with this
information and additional information on the security of system derived by
13
processing the raw measurements can be provided by an on- l ine d ig i ta l
computer. Analogue data i s scanned periodically in the order of 1 second to
a few seconds. Each scan is triggered by the system control centre at the
prescribed interval by issuing a request to a l l remote stations to send in
data.
The acquired system information together with state estimation methods
provide indication of alarm or abnormal conditions, even with several remote
terminations out of service.
The major features of a SCADA software can be:
- Alarm/event acquisition and processing
- Telemetering
- Telecontrol
- Measured acquisition and processing
The use of the special application functions such as load shedding and
generator scheduling and control i s possible only by virtue of the
establishment of a comprehensive supervisory control and data acquisition
system. The use of microcomputer-based Remote Terminal Units (RTUs),
enables and encourages the acquisition of more alarm information than was
possible with older solid-state and electro-mechanical systems.
Accordingly, careful attention must be paid to the processing and display of
alarms i f the operators are to be able readily to identify the information
during major system disturbances. The provision of several different alarm
14
categories, each with a dedicated display, and the extensive use of colour
to rank alarms in order of importance, would resu l t in a powerful and
comprehensive alarm processing/display fac i l i t y for control engineers.
Operators are made aware of network event within 2-3 seconds of i t s
occurrence.
For telecontrols, the system can employ the common
'check-back-before-operate' feature, or on receipt of a command telegram,
the RTU can energise a command relay associated with the c i rcu i t breaker to
be operated. In the la t ter , before completing the command process, the RTU
ini t iates a current injection test to confirm that only one relay is primed
and on ver i f icat ion, the command process is completed. This interval check
at the RTU coupled with Digital Pulse Duration Modulation (DPDM)
communications provides adequate security.
Data transmission between the control centre and the substations can be
by: Voice Frequency Telegraph (VFT) equipment operating over telephone
cable, Power Line Carrier (PLC) and UHF radio f a c i l i t i e s . Main and standby
channels with ful l duplex point-to-point communication (for example at 200
baud) can be provided for each RTU where pract ica l , standby channels can be
carried on physically separate routes. In the event of a channel fa i lure ,
channel monitoring equipment can switch communication to the alternative
channel.
The communications protocol commonly uses DPDM with information being
15
transmitted as a message comprising an address block, an information block
and a data protection block. The use of DPDM and data protection blocks
provides a communications protocol with a very high probability of error
detection. The control centre can be provided with the fac i l i t y to request
the message to be re-transmitted, following the detection of errors.
The supervisory control function operates only upon specif ic operator
request and may use the same computer interface, communication channels and
RTUs as does the data acqu is i t ion system on a time-shared b a s i s . The
devices being controlled are mostly c i rcui t breakers, motor operated
disconnectors and load tap charging transformers.
1.3 Control Systems
The operational control of electr ical power systems can be divided into
two major functions; simulation functions and control function. Further, as
shown in Figures (1.6) and (1 .7) , each function comprises of a number of
tasks executed in the manner shown in a real-time environment. Throughout
the rest of this chapter reference is made to the Energy Management suite
(OCEPS) developed at the University of Durham under the supervision of
Professor M.J.H. Sterl ing. Figure (1.8) i l lustrates the execution time for
performing different tasks.
16
1.3.1 Simulation Function
If operators are to have confidence in and rely on the data being
presented to them at the control centre, i t is necessary to have some means
of verifying the displayed information. Further, to provide rel iable and
secure operation of the network i t i s desirable to have means of checking
operations in simulated form prior to performing network changes.
Computer assisted control of e lectr ical power transmission and
distribution systems represents a large scale real-time data processing
problem and demands an integrated approach to software design and testing.
The avai labi l i ty of relatively inexpensive computer hardware and the
increasing importance of energy management has led to the widespread
adoption of sophist icated control s y s t e m s . L l ^ 2 - ' Objectives of these
schemes include the minimisation of production cost, improvement of the
security of supply and the abi l i ty to ensure the correct operation of power
system plant. Testing and validation of analysis and control software may i'3T
be achieved with the aid of a real-time simulation system. J In this way
the performance of energy management software can be evaluated against a
rea l i s t i c model of the network under a range of operating conditions. A
secondary benefit of integrated simulation, management and control software
is the abi l i ty to provide a useful operator training aid, which includes
power system dynamics, telemetering and energy management f a c i l i t i e s . The
development of an Integrated simulation and control suite has been found to
be essential in order to conduct research into operational control aspects
17
of generation and transmission systems, since i t enables the consideration
of rea l i s t i c environments.
F igure (1 .9) shows the major database elements and computational
procedures required for simulation. The simulator applies to a non-linear
a lgebraic model of the network in conjunction with a set of low order
differential equations representing generator dynamics to produce telemetry
information. In order to obtain stable numerical integration at speeds
which are consistent with real-time operation, i t has been found that the
implicit trapezoidal technique combined with sparse Newton-Raphson solution
is very suitable. J
1.3.2 Topology determination and State Estimation
A pre-requisite of any power system security monitoring or control
scheme i s a r e l i a b l e database in which the raw observations have been
systematically processed in order to f i l t e r out the effects of uncertainty
by inc lus ion of information on the network structure and measurement
accuracy. Any form of f i l t e r i n g implies a loss of information and
consequently the complete determination of the system state wil l require
additional measurements with inherent extra cost. For example, the 2R-1
independent variables consisting of voltage magnitudes E K (K=l,n) and angles
Gk(k=2,n) at a l l buses, are suff icient to determine the s ta t ic state of an
n-bus power system relative to a reference bus where e^, i s assumed to be
zero. However these measurements are not adequate for dynamic operation in
18
the presence of measurements errors and possible fai lure of a section of the
real-time data collection equipment. Consequently, more measurements than
the number of unknown state variables are needed. The way in which this
redundancy is ut i l ised gives r ise to various techniques for state estimation
which have application in several areas of power system monitoring and
control.
State estimation techniques may be broadly divided into two categories, T 5 T
namely s t a t i c and dynamic est imat ion. J S t a t i c methods mainly have
application in determination of load-flow in a transmission network,
although the system does not remain in a fixed state, the approximation of
steady-state over a short period of time is adequate. This assumption wil l
be valid i f the system is subject to relatively low frequency disturbances.
Measurement redundancy is essential in this type of application.
Dynamic state estimation on the other hand is normally associated with
transient or dynamic s tab i l i ty problem, where the dynamics of elements of
the system must be considered and i s typified by the integrated power plant
problems.
Both stat ic and dynamic estimation only operate successfully provided
limits can be set on the s ta t is t ica l properties of transducer and telemetry
system. Raw data could consequently f i r s t be processed to remove gross
errors which would at least delay convergence of the estimator and, at
worst, corrupt valid measurements. This pre-processing on raw data is known
19
as bad data suppression or data v a l i d a t i o n . The complexity of the
algorithms incorporated in data validation software can range from simple
limit checks on incoming analog data through to ful ly structured bad data
suppression schemes possibly based on linearised system models.'-^-'
In addition to the standard data validation methods of l imit checking,
consistency checking, exponential smoothing, logical f i l t e r i n g , OCEPS suite
includes more sophisticated algorithms based on linear programming which are
able to eliminate the majority of bad data values at the topology
determination s t a g e . ^ This considerably reduces the computational burden
imposed by bad data detection and elimination during the state estimation
process. The data validation program take ful l advantage of network and
linear programming sparsity to achieve the necessary solution and speed. L
I f enough well-placed measurements are available in the network, the
state estimator problem i s well defined and the network i s sa id to be
observable. A valid observability determination algorithm is therefore an
important component of the estimation subsystem.
1.3.3 Automatic Generation Control (AGC)
The importance of AGC, also known as load frequency control, depends to
a large extent on the size and form of a power system.
In large density interconnected systems such as in the UK, AGC is of
20
less importance than in say the USA, where many separate companies jointly
supply power to widely distributed consumers via a network of
interconnections or t i e - l i n e s . The system regulation coefficient is often a
guide to the relative importance of AGC with values ranging from in excess
of 2000 MW/HZ for systems s imi la r to that of the UK to 200 MW/Hz more
typical of systems in the USA and Afr ica.
Manual load-frequency control rel ies on an operating schedule prepared
perhaps 24 hours in advance, which indicates that the anticipated demand
profile and the corresponding set point changes together with t i e - l ine power
exchanges that should be maintained. The continued alteration of the set
points needed for c lose frequency and t i e - l i n e control i s frequently
replaced by alteration of set points at regular intervals according to the
schedule. Further corrective action may be applied i f the frequency or
clock error deviates from a certain band. Each alteration of the set points
represents an appreciable step change to the set point which i t s e l f
introduces a transient frequency disturbance before the desired result is
achieved. In order to avoid these transients, the set points must be ramped
between their current and desired values over a reasonable time provided
successful manual control of both the frequency and the t ie -Hne power
exchanges therefore necessitates operator vigilance and changes of the
t ie - l ine exchange need special attention.
Computer control schemes have been devised which wil l automatically
compute and implement set point changes and also fac i l i ta te smooth
21
transit ion between t ie - l ine exchange requirements. J The digital
implementation of A6C generally necessitates the sampling of the t ie - l ine
power flows and the system frequency, together with control of the governor
set points at approximately regular i n t e r v a l s . O p t i m a l control approach
to AGC does not prove very advantageous in pract ice. ' - 1 1 ^
1.3.4 Economic Despatch (ED)
The complex optimisation problems associated with the control of a
power system have been the subject of considerable research. The complexity
of the overall control problem, in which the maintenance of supply, quantity
and quality is of more importance than economic factors, has resulted in a
division of the research into two main areas; network control and
operational economics. Network control, including transmission
switching, voltage, power factor, frequency and individual plant control,
have necessarily been of prime importance, for without re l iab i l i ty in this
f i e l d , the consideration of operational economics is of no value. However,
once rel iable system operation has been achieved, large financial benefits
may be accrued from economic allocation of generation.
For a p a r t i c u l a r load and set of network condi t ions, an optimal
combination of generators can be determined by consideration of the
difference in operational characterist ics of the plant. Load variation
consequently necessitates the calculation of new optimum generation
despatches.
22
When an accurate estimate of the load i s ava i lab le the generator
outputs must be allocated so that a l l operational constraints are sat isf ied
and the production cost i s minimised. The time scale of the optimisation
permits the separate consideration of two problems, plant ordering phase
being typical ly 4 hours in advance, plant loading or despatch phase normally
attempted as frequently as possible, typically every 5-10 minutes.
The simplest ED algorithm is the merit-ordering method. Only a l inear
or piece-wise linear cost function, upper and lower generator active power
limits and the load balance equality can be accommodated. The committed
generators are indexed in order of increasing incremental cost, and are
in i t i a l i sed at their lower output l imit . Generators are then considered for
loading to t h e i r maximum l imi t in order of merit unt i l the demand i s
sa t is f ied . The advantage of the merit order is i ts simplicity and that
there i s no di f f icul ty in dealing with large scale problems. The
disadvantage i s that only simple cost functions may be considered. Other
techniques for ED are based on linear programming, quadratic programming,
recursive quadratic programming and dynamic despatch.C12]C53C133C14]^
1.3.5 Unit Commitment
Since the load varies continuously with time, the optimum combination
of units may alter during any period. In practice however, one hour is the
smallest time period that need be considered, as the start-up and shut-down
time for many units i s of th is order. This plant ordering or unit
23
commitment phase of operational control thus represents a course running
cost optimisation in which units are brought into and taken out of service
such that the suff icient generation is always available to meet the system
load and to provide spinning reserve. The loss of economy which can result
from an incorrect unit commitment schedule may exceed the savings obtained
by optimum allocation of generation among synchronised units.
I t might at f i r s t appear that a merit-order approach is sufficient in
which units are brought into service in accordance with their relative cost
eff ic iency, cheaper units being scheduled before less ef f ic ient units.
However, i f an ineff icient unit is shut-down when i t is no longer required,
i t wil l have to be restarted several hours later prior to the next peak
load. The saving gained by shutting down the unit is consequently offset by
the cost of starting up the unit again.
The main factors influencing the plant ordering decision are:
(a) The shape of the hourly integrated consumer demand curve
(b) The relative efficiency of each unit to be shut-down compared with that
of the other synchronised units
(c) The start-up and shut-down costs of each unit .
Unit commitment based on dynamic programming considers standard
24
operation constraints for thermal units and will schedule hydro-plant in
order to sat isfy water usage and conservation requirements. The resulting
generation schedules ensure that hydro-plant contributes to the economic and i"5iri5i
secure operation of the network.L J L J
1.3.6 Load Forecast
The necessity for estimating the power system load expected at some
time in the future is apparent when i t i s remembered that generating plant
capacity must be available to balance exactly any network load at whatever
time i t occurs. In the long term the instal lat ion of new plant and network
expansion is dependent upon an estimate of the future peak consumer demand
up to several years ahead. In the short term the variation of the system
load must be known in order that prior warning of output requirements may be
given to power stations, enabling limitations on boiler fuel feed rates, and
generator rate of change of output constraints, to be observed.
Furthermore, the economic schedule for the start-up and shut-down of plant
i s dependent on an estimate of the network load so that the cost of
providing spinning spare capacity for system security can be minimised.
In a power system under automatic computer control i t is this
short-term projected load that would be used to ca lcu la te a generator
despatch for which a l l operating limits were sat isf ied and the generator
cost a minimum. Unfortunately, the consumer load 1s uncontrol lable ,
although small variations can be effected by frequency control and more
25
d r a s t i c a l l y by load shedding. The var ia t ion of the load does however
exhibit certain daily and yearly pattern variations and the analysis of
these forms the basis for several prediction t e c h n i q u e s . ^ 1 7 ^ 1 8 ^
The method of load prediction using spectra l expansion was f i r s t
proposed by Farmer in 1 9 6 3 . I t has subsequently been the subject of
several publications in which the application of the method to an automatic
on-line load prediction and scheduling technique (implemented on the
South-West region of the CE6B) is discussed in (5) .
1.3.7 Emergency Rescheduling
During emergency conditions in which insufficient generation is
available to meet system loads or where one or more unexpected plant outages
have occurred, i t is important to be able to rapidly reschedule generation
and allocate the degree of load shedding required. It is vital that this
phase of operation should be executed rapidly, and modern computer control
systems are increasingly being used in this area.
Under emergency conditions the economic operation of the system has a
lower priority than the minimisation of load shedding. However, emergency
rescheduling methods which are now avai lable, generally assign a r t i f i c i a l
costs to each load supply point, and therefore allow a priority order of
load shedding and also enable a trade-off between generation rescheduling r ig 1
and load shedding. J
26
The rescheduling problem is mathematically similar to the economic
despatch problem, but the difference in modelling and solution requirements,
indicates that separate software f a c i l i t i e s are necessary. The sparse
quadratic programming algorithm by Chan and Yip, and a second method based
on network flow and the 'out of k i l t e r ' algorithm are commonly used. ' - 2 0 - '
1.3.8 Load Shedding
Although the emergency rescheduling techniques may give an automatic
indication of the required degree and location of load reduction, many
applications require a more detailed analysis and further operator
interaction before any load shedding takes place. Interactive programs have
been developed which assign pr ior i t ies and allowable load shedding fractions
to each consumer demand point. J Load may be reduced in a number of
pre-defined stages and operator confirmation is necessary before any load
shedding programs in real-time environment may be init iated either manually
by operator selection or automatically by the detection of serious under
frequency or load/generation imbalance conditions.
1.3.9 Security Assessment
With the present state of the system established through the state
estimation program, i t i s necessary to investigate whether under present
operating conditions, the system could withstand the outage of any l ine ,
transformer or generator without violation of the loading constraints of the
27
remaining elements. This i s the task of the contingency evaluation program.
The contingency evaluation program accepts a l i s t of outages from either a
fixed l i s t or from an operator entered l i s t of additional outages.
Contingency evaluation is performed in real-time at fixed intervals or at
operator's request. During the outage simulation, a table is compiled of
a l l the elements that are overloaded, this information being vital in the
derivation of the security constraints which are used in the power
rescheduling algorithm.
There are currently two basic approaches to the estimation of future
system security and r e l i a b i l i t y , namely the deterministic approach and the
probabilistic approach. The deterministic technique requires that certain
calculations such as load-flow, unit commitment and stabi l i ty studies be
made for a l l conceivable future contingencies. The results of these studies
are then reviewed and a judgement i s made of whether or not some corrective
action is required to maintain adequate system security. However, this
technique does not take d i rec t account of the unequal probabi l i ty of
occurrence of each system contingency, and furthermore that the number of
ways of reaching undesirable operating s ta tes i s not constant. Thus,
whereas the deterministic approach clearly indicates those operating
conditions that would result in breach of system security, i t does not
supply a consistent criterion for control action, given the results of the
contingency tes ts . Unfortunately, probabilistic methods are not yet widely
implemented, since the fundamental prerequisite is a detailed knowledge of a
performance of the existing system in terms of fai lure rates, repair times,
28
and event probabil it ies. The introduction of such techniques would
therefore normally only be feasible after a prolonged data collection and
analysis period and consequently their implementation would not appear
relevant as part of an in i t i a l power system control package.
Non-linear load-flow techniques are quite time-consuming and costly,
especially i f many cases are to be studied. The d-c load-flow approach is
the fas tes t technique, but i t s accuracy i s low and does not provide
information regarding reactive flow in the system elements. This technique
has the advantage of simple data, speed of execution and reasonable storage
requirements. I t provides the basis for the provision of an outage
contingency l i s t , which would be later accepted by an AC security assessment
p r o g r a n . ^ 2 "
The basis of any on-line security assessment scheme i s an AC power flow
solution. Each cycle of security monitoring and analysis determines the
effects of a selected l i s t of outage contingencies on the steady-state
operation of the system, making use of real - t ime data. The program
ca lcu la tes busbar voltages and power f lows, using a vector of bus-bar
injections and predicted network configuration. In general, the busbar
injections are calculated from s ta t is t ica l data obtained on-line or state
estimation, loading conditions and interchange schedules.£24][25][26]
Modern security assessment packages should cope with network islanding
which could be caused as a result of transmission l ine outages or as a
29
result of real-time topology changes in the power system.
1.4 Discussion
The increasing complexity of modern power systems has led to a greater
dependence of automatic control at a l l levels of operation. The higher
levels of control have tradit ionally relied heavily on human judgement
especially in respect of economic factors. However, recent international
research experience has shown that centralised computer control i s feasible
for small sca le networks but that costs would r i s e alarmingly for the
implementation of schemes on a national basis. The geographical
distribution of the plant presents enormous data gathering and control
problems which, for large scale systems, preclude any direct approach to
centralised control.
Despite the overall problem complexity many national supply boards are
already instal l ing computer control systems for spec i f ic , mostly loca l ,
functions within the network. Generator start-up, shut-down and control was
one of the f i r s t areas to be computer controlled, together with boiler and
other plant controls. These schemes are however, usually only within a
single power station and rely on the manual coordination of the stations to
meet national c r i t e r i a . Their existence does however, fac i l i ta te data
reduction and simplified control implementation for the co-ordinating
control leve l .
30
The primary function of an electr ical power system is to provide a
secure and reliable source of e lect r ic i ty to the consumer. E lectr ic i ty
Authorities operate highly complex generation and distribution networks with
an ever-increasing requirement for energy saving and permanency of supply.
It follows that i t is necessary to insta l l sophisticated, yet extremely
rel iable control systems to achieve these objectives.
In order to be able to meet the ever-more stringent operating
conditions with the ever-decreasing levels of equipment redundancy, the
operator requires up-to-date and accurate information about the state of the
entire network. The function of supplying the operator with this
information and additional information on the security of system derived by
processing the raw measurements can be provided by an on- l ine d ig i ta l
computer.
Throughout this chapter the main functions of a control system package
where highlighted. OCEPS suite developed at the University of Durham, not
only provides the basis for operator training and future research but is a
unique example of an Energy Management System (EMS) package. The
co-ordination of the functions mentioned in section 1.3 of this thesis is
i l lustrated through Figures (1.10) and (1.13).
C Coal t i r e d G Gas t i r e d O o i l f i r e d N Nuclear r pumped >tora(e
31 KNSXI C o a l f i e l d
N
a a
H N
N N
F i g . l . l : L o c a t i o n of the major power s t a t i o n s i n the CEGB.
Leeds
B l r a i n j h a a
St.Aloan
• B r i s t o l national Control
E a s t Grlnstead
FIGURE 1.2 LOCATION OF THE GRID CONTROL CENTRES
To South of Scotland • Substation E l e c t r i c i t y Board
400 kV. l i n o 275 kV. l i n o
Barker S t e l l a
Hawthorn P i t i Norton
11 TCreka Back
Vyl r a Grimsby Vast
t t u Dlnorwlg
' S i l
Travsfynydd Norwich •
t Irunaridze
Feckenhaa i Cllfynydd Braaford
1 Penoroke i Grain i Blnklay Point
T F l a a t A l r a r d l s c o t t Nursling £olney ' Taunton • Ex*tar HV. DC.
cross Indian Dungeness Channal Queans Favley jvadean l i n k
F i g . 1.3: CEGB. Supergrid system
MW.»1000
40
JO
20
10
0 6 12 I S
1. V l n t t r peak deaand day 2. Typical Winter day 3. T y p i c a l Sumner day k. Suoner Sundjy ainiauo deaand
FIGURE 1.4 COMPARISON OF SUMMER AND WINTER DEMANDS ON THE CEGB
33
DE1A?P. R O Y A L (GWJ
20
19
IS
17
16
W E D D I N G 29.7.81
\ Return of Royal Procession to Buckingham Palace
Following Appearances on The Palace Balcony
E n d of Prayers
Solemnisation ' of Matrimony Signing of
Registers
Prince & Princess ol Wales leave Waterloo Station 1800MW increase in demand
X , A U G U S T
B A N K H O L I D A Y 25.8.80
Following Br ide 's Arrival at Altar
J 08.00 09.0U 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 IS.00
T I M E (HOURS)
F R E Q U E N C Y (Hz)
50.5
50 0
49 5
12 00 09.00 10 00 11 00 13 00 08 00 IA 00 15 00 16 00 17 00 18 00 T I M E (HOURS)
FIGURE 1.5 ILLUSTRATION OF THE EFFECTS OF TELEVISED EVENTS ON THE SYSTEM DEMAND AND FREQUENCY
SIMULATION FUNCTION
LOAD VARIATIONS AND OTHER DISTURBANCES
ft PHYSICAL SYSTEM DATA
EXACT TOPOLOGY • DETERMINATION
(EXACT N TOPOLOGY y
i DYNAMIC SIMULATOR
CONTROLS"^
TCLEMETERI DATA
lEm
FIGURE 1.6 OPERATIONAL CONTROL OF POWER SYSTEMS SIMULATION FUNCTION
35
ANALYSIS AND CONTROL FUNCTION
LOAD FREQUENCY CONTROL
GENERATOR OUTPUT TARGETS
ECONOMIC DISPATCH
T
GENERATOR , COMMITMENT TARGETS
UNIT COMMITMENT
T
DATA VALIDATION. AND STATE ESTIMATION
PHYSICAL SYSTEM OAT/ 9 <LOAD "N
PREDICTIONS)-
LOAD PREDICTION
ESTIMATED NETWORK CONDITIONS MONITORING , / HISTORICAL^
LOAD DATA J
SECURITY ANALYSIS AND •—1
1 RAUIT S T l i n i P S
^JCRIT ICAL \ ~~ VCONTINGENCIES/"
EMERGENCY RESCHEDULING AND LOAD SHEDDING
FIGURE 1.7 OPERATIONAL CONTROL OF ELECTRIC POWER SYSTEM ANALYSIS AND CONTROL FUNCTION
Operator power flovy
Load forecasting Short circuit
Level monitoring Execution rate
Contingency analysis
Execution rate State estimation
Loss sensitivity Network monitoring
Execution rate Economic despatch
Execution rate AGC
executionj-ate Display refresh
Status data AGE
Analogue data AGE
-Runson demand
4-12 sec 1
10 sec
10 sec
40 sec
100 msec 1 sec 10 sec
1h
5- 0 min
5-30 min r 1
5-30 min —\ 1
1-10 min
Time 1+
1 min i h
FIGURE 1.8 TYPICAL E X E C U T E RATES
37
PLANT OUTAGES
CONTROL INPUTS
(TELEMETERED \DATA
SIMULA >TOR
EXACT TDPOLO DETERM
GY INATIDN
MEASUREMENT NOISE
PHYSICAL SYSTEM DATA
BREAKER CONTROL
LOAD VARIATION
FIGURE 1.9 SIMULATION SUBSYSTEM
38
OPERATIONAL CONTROL OF E L E C T R I C POWER SYSTEMS
OVERVIEW OF MAJOR FUNCTIONAL ELEMENTS
SIMULATION FUNCTION
U U VARIATIONS AND OfHM MS1IMAKCS
PHYSICAL STSTCN n o
ANALYSIS AND CONTROL FUNCTIONS
£
DYNAMIC SIHULAIC
MOMIEMS DAlA VALIDATION AND STATC CS7MATICH
PHYSICAL IVS7CH DATA 0
K CCXlATOt \ I C O N D N I C / G M J . T C A \ U N I T C C N H I T H C X T
<LCM N •DCICTICNS } -
CIACT I G P O L O G V KICiMIIAIICH
UACt Topaocr >
LOAD ' H I I C T I C M
< MSTCHICAL LOAD DATA
S C C U I I I T ANALYSIS - / AND FAULT STUD1CS
^ { (CHITICAL N CGHTIMGCNCICSI-
CHCKCNCT USOC3U.3d AND LOAD ShCDDlHS
FIGURE 1.10 OPERATIONAL CONTROL OF ELECTRIC POWER SYSTEMS OVERVIEW OF MAJOR FUNCTIONAL ELEMENTS
39
STATE ESTIMATES J
STATE ESTIMATOR
VALIDATED \ D A T A
]
OBSERVABILITY ^DETERMINATION
AREA VALIDATION AREA VALIDATION
/ V A L I D A T E D 1 TOPOLOGY
SUBSTATION VALIDATION 1
( VALIDATED J \ D A T A
SUBSTATION VALIDATION 2
SUBSTATION VALIDATION N
(TELEMETERED^ W)ATA J
?ED\
FIGURE 1.11 STATE ESTIMATION AND TOPOLOGY SUBSYSTEM
40
LOAD PREDICTIONS,
PHYSICAL FUTURE LOADS
PREDICTOR
LOAD LOG
LOAD VARIATION
LOAD MONITORING
/ LOAD MONITORING (
J
STATE ESTIMATES
f PHYSICAL H SYSTEM V^DATA
FIGURE 1.12 LOAD MONITORING AND PREDICTION SUBSYSTEM
41
LOAD PREDICTIONS
/FREQUENCY \MEASUREMEf
UNIT COMMITMENT
/6RDERED ( GENERATION \COMMITMENT:
ECONOMIC DISPATCH
I ORDERED GENERATION
JARGETS
GENERATION AND LOAD RESCHEDULING
\ LOAD FREQUENCY
1 CONTROL
PHYSICAL SYSTEM
JDATA
FIGURE 1.13 GENERATION AND LOAD CONTROL SUBSYSTEM
42
CHAPTER 2
REAL-TIME SECURITY ASSESSMENT
2.1 Introduction
The concept of secur i ty can be expressed as the capab i l i t y to meet
demand i n terms of a v a i l a b i l i t y of enough product ion and t r a n s m i s s i o n
resources to match any load r i s e and credib le but unforecasted outages
F28 I without exceeding acceptable l i m i t s in frequency and voltage drops.
System secur i ty studies s t a r t in the long-term planning stage when the
transmission and generation systems are designed taking into account some
contingency s i t u a t i o n s . However, due to economic cons idera t ions , only so
much s e c u r i t y can be b u i l t i n t o a power s y s t e m . In the s h o r t term
simulations of the future condit ions in the network are car r ied out in order
to assure that the transmission and generation f a c i l i t i e s scheduled to be
operative are adequate to meet pre-estab l ished secur i ty requirements.
However, the planner cannot predict a l l possib le system configurat ions and
system demands. There i s then the n e c e s s i t y of p r o v i d i n g a d d i t i o n a l
secur i ty ana lys is and control in the real - t ime environment.
The purpose of secur i ty assessment i s to detect and a l e r t operators to
insecure and abnormal condit ions so that cor rec t ive strategy may be planned
43
and cor rec t ive act ion taken.
This chapter represents a survey of approaches to the estimation of
system secur i ty and d iscusses power system operating s t a t e s , monitoring and
secur i ty enhancement.
2.2 Power System Operating States
The power system 1s thought of as being run under t h r e e s e t s of
c o n s t r a i n t s , ^ 2 9 ^ which are :
- the load constra in ts
- the operating constra ints
- the secur i ty constra in ts
The load constra ints simply recognise the prime task in power system
operat ion, namely to meet the load demand. The operating c o n s t r a i n t s ,
furthermore, s t ipu la te the object ive of using the system components within
permissible l im i ts and to meet the qual i ty standard of the system.
Overloaded components, bus-voltages outside the to lerances and abnormal
frequency deviat ions are examples of v io la t ions of the operating
c o n s t r a i n t s .
Some of the secur i ty constra ints are magnitude, a c c e s s i b i l i t y and
locat ion of operating resources , l i m i t s in transmission capac i ty , e t c . The
44
system i s secure i f i t f u l f i l s the appropriate secur i ty constra ints in order
to meet some cont ingencies. By means of these three sets of constra in ts
four c l a s s e s of operating s ta tesC 2 8 H29] [30 'J [31 ] c a n b e a s fo l lows:
(a) the normal s ta te
Al l three se ts of constra in ts s a t i s f i e d , the object ive in t h i s s ta te i s
to operate the system so as to remain in t h i s s t a t e and to a l l o c a t e
generation to minimise tota l production cost within s e c u r i t y .
(b) the a l e r t s ta te
Loading and operating constra ints s a t i s f i e d but e x i s t i n g reserve
margins are such that some disturbance would resu l t in a v io la t ion of some
secur i ty c o n s t r a i n t s . The operating object ive in the a l e r t s ta te i s to
return to the normal s ta te as rapidly as p o s s i b l e .
(c ) the emergency s ta te
Operating and secur i ty constra ints not s a t i s f i e d , load const ra in ts not
necessar i ly s a t i s f i e d . The operating object ive in t h i s s ta te i s to prevent
the spread of the emergency and to restore the system to at l eas t the a l e r t
s t a t e .
45
(d) the restorat ion s ta te
Operating constra in ts s a t i s f i e d , load and secur i ty constra in ts not
s a t i s f i e d . This i s the s ta te of the system following an emergency where the
emergency has been s t a b i l i s e d but the establishment of the normal s ta te has
not yet been achieved. The operating object ive in t h i s s ta te i s to restore
a l l se rv ices as rapidly as possible and to return to normal s t a t e .
The four operating s ta tes and the t r a n s i t i o n between them are shown in
Figure ( 2 . 1 ) .
The t r a n s i t i o n s are caused by c o n t i n g e n c i e s and c o n t r o l a c t i o n s .
According to these s t a t e s , secur i ty i s defined with respect to a set of
random events ca l l ed the se t -o f -nex t -cont ingenc ies .
T h i s i s a c o l l e c t i o n of disturbances that could occur . A system i s
sa id to be secure i f i t i s in the normal s ta te and none of the contingencies
could cause i t to enter the emergency s t a t e .
As shown in Figure ( 2 . 1 ) , the function of the preventive control i s to
take act ion to make the system secure . Some centres have programs that can,
when they are act ivated by the operator, produce schedules to el iminate
cer ta in l im i t v i o l a t i o n s . However, most preventive control i s manual.
The funct ion of the emergency control i s to take cor rec t ive action to
46
make the system normal . As ide from load s h e d d i n g , which i s done in
accordance with precalci l iated schedules and implemented by under-frequency
r e l a y s , the operator i s usual ly responsible for a l l emergency c o n t r o l . The
function of the res tora t ive control i s to restore serv ice to the system's
loads. Al l res tora t ive decis ions are made by the operator.
In the power system control centres the main emphasis has been given to
the preventive c o n t r o l .
Restorat ive
Control
NORMAL
«*» | | 1 1
Preventive I t Control
• 1 i I i 1 i I
RESTORATIVE ) ALERT
EMERGENCY
CONTROL
EMERGENCY
t r a n s i t i o n caused by contingencies
- — — t r a n s i t i o n caused by control act ion
FIGURE 2.1 - POWER SYSTEM SECURITY STATES
48
2.3 Securi ty Functions in System Control
Securi ty a s s e s s m e n t ' - 2 8 - " - 2 9 - ' i s the process of checking for v io la t ion of
o p e r a t i n g l i m i t s on the power s y s t e m , in i t s p resent s t a t e or in an
ant ic ipated future s t a t e , and e i ther in the in tac t condit ion or on the
occurrence of a l i k e l y contingency.
The purpose of contingency assessment i s to detect and a l e r t operators
to insecure and abnormal condit ions so that cor rec t ive strategy may be
planned and correct ive act ion taken.
There are three basic functions in a secure control system, which a re :
(a) secur i ty monitoring
(b) secur i ty enhancement
(c ) secur i ty ana lys is
These are i l l u s t r a t e d in Figure ( 2 . 2 ) .
2.3.1 Securi ty Monitoring
Securi ty monitoring i s the on- l ine i d e n t i f i c a t i o n of the actual
operating s ta te of a system by checking real - t ime data to determine whether
i t i s in a normal, a l e r t , emergency or res tora t ive s t a t e .
49
MEASUREMENTS.
FILTERING
NETWORK CONFIGURATION S T A T E ESTIMATION
CHECK SYSTEM CONSTRAINTS VIOLATIONS
ON-LINE LOAD FLOW
EXTERNAL EQUIVALENT
NORMAL 1
EMERGENCY EMERGENCY
CONTROL
SHORT-TERM LOAD FORECAST
SECURITY i ANALYSIS
CONTINGENCY EVALUATION
L .
SECURE INSECURE
DISPLAY
SECURITY CONSTRAINED OPTIMISATION
NO SOLUTION
CONTINGENCY PLAN
SOLUTION
DISPLAY DISPLAY
SECURITY
MONITORING
DISPLAY
I RESTORATIVE
RESTORATIVE CONTROL
SECURITY
ENHANCEMENT
FIGURE 2.2 REAL-TIME SECURITY CONTROL SYSTEM
50
The f i r s t step in secur i ty monitoring of power systems i s the set t ing
up of a real - t ime database. The data acqu is i t ion function s t a r t s with the
measurement of the physical quant i t ies in a power system such as bus voltage
magnitude, l i n e flows and bus power i n j e c t i o n s . In addi t ion , data on the
status (open or c losed) or c i r c u i t breakers, and switches are required. The
measurements data are telemetered from locat ions to the control centre
computer. Glar ingly bad data such as t rans ient excursions in the measured
values are rejected by f i l t e r i n g the transmitted data through a simple check
of t h e i r reasonableness or of the consistency between breaker status and
analogue information, or by some smoothing rout ine . The real - t ime
information about the status of c i r c u i t breakers and disconnect switches i s
systemat ica l ly processed to determine the network conf igurat ion.
Miss ing and erroneous data occur frequently in the real - t ime
environment. Faced with er rors in the measurements, missing measurements,
and er rors in the transmitted data , the ava i lab le data must be processed to
ob ta in an es t imate of the s t a t e v a r i a b l e s , the v e c t o r of bus vo l tage
magnitudes and phase angles , of the power system; that i s the task of the
s ta te estimate program.
The following function in secur i ty monitoring i s the on- l ine check on
the operating l imi ts of the power system based on the ca lcu la ted va lues.
51
2 .3 .2 Securi ty Enhancement .
Securi ty enhancement determines whether the system can be made secure .
B a s i c a l l y , i t determines what preventive act ion should be taken to make an
insecure system secure , or l e s s insecure .
Prevent ive act ion requires an optimisation rout ine, or optimum power
f low. Let us assume that a prevent ive-act ion solut ion can be found optimal
or not; s ince the power system normally i s operated at minimum operating
c o s t . System secur i ty w i l l have to be obtained at a p r i c e . I f the cost of
the p r e v e n t i v e a c t i o n i s s m a l l , the opera tor can p l a c e the c o n t r o l i n
e f f e c t . Where the cost i s high but the contingent emergency not too severe ,
the operator may decide not to take any act ion to improve system s e c u r i t y .
I f i t i s not possible to f ind a prevent ive-act ion so lu t ion , a
contingency plan would be developed by assuming that the contingency has
occurred. The object ive of t h i s would be to minimise the amount of load to
be c u r t a i l e d , t h i s strategy can be part of an emergency control funct ion.
The de f in i t ion and object ives of real - t ime secur i ty assessment when the
power system i s operating under emergency or s t r e s s f u l condit ions should be
evaluated. Under these condi t ions , the object ives of operation change from
cost minimisation to control of the system to bulk o s c i l l a t i o n and return
the system to the normal operating s ta te and/or i n i t i a t e other control
act ions towards a res tora t ive state to insure against cascading
52
interrupt ions and major blackouts. The computer software used in system
operations should have the f l e x i b i l i t y to r e f l e c t these d i f ferent
o b j e c t i v e s .
2 .3 .3 Securi ty Analysis
The r e l i a b l e and e f f i c i e n t operation of interconnected e l e c t r i c energy
systems i s an important object ive for the nation and a chal lenging problem
for power systems planners and a n a l y s t s .
S e c u r i t y of a power system i s c o n s i d e r e d to be an i n s t a n t a n e o u s ,
t i m e - v a r y i n g c o n d i t i o n tha t i s a f u n c t i o n of the s y s t e m ' s robustness
r e l a t i v e to imminent d is turbances.
S e c u r i t y a n a l y s i s ' - 3 2 - ^ 3 3 ^ ' - 3 4 - ' c o n s i s t s of evaluating the system under
various contingencies in order to obtain a measure of system secur i ty and
providing inputs to enhancements s t r a t e g i e s . So, secur i ty of a power system
i s defined with respect to a l i s t of possib le transmission l i n e outages and
generator outages, c a l l e d cont ingencies. As the system condit ions change,
the contingencies which cause insecure operation may a lso change.
A framework was establ ished for determin is t ic s teady-s ta te secur i ty
assessment, in which, the robustness of the system i s t e s t e d , using a power
f low, on a set of se lected cont ingencies. The major components of the
c l a s s i c a l secur i ty assessment in a modern control centre inc lude: s ta te
53
est imat ion, topology determination, contingency se lec t ion and eva luat ion ,
and secur i ty c o n t r o l . An in-depth survey of the c l a s s i c a l approach was
presented by F.F.Wu and S .N .Ta lukar . ' - 3 0 - '
The determin is t ic technique requires that cer ta in c a l c u l a t i o n s such as
load-flow unit commitment, and s t a b i l i t y s tudies be made for a l l conceivable
future cont ingencies. The r e s u l t s of these studies are then reviewed and a
judgement i s made of whether or not some correct ive act ion i s required to
maintain adequate system s e c u r i t y .
However, t h i s t echn ique does not take d i r e c t account of the unequal
probabi l i ty of occurrence of each system's cont ingencies, and furthermore
that the number of ways of reaching undesirable operating s ta tes i s not
constant . Thus, when as the determin is t ic approach c l e a r l y ind icates those
operating condit ions that would r e s u l t in a breach of system s e c u r i t y , i t
does not supply a consistent c r i t e r i o n for c o n t r o l . Hence, the
p r o b a b i l i s t i c approach was considered.
The p r o b a b i l i s t i c approach i s not independent of the de termin is t ic
technique but suppl ies a consistent c r i t e r i o n for control act ion given the
r e s u l t s of the contingencies t e s t s .
Because of uncertainty inherent in the predict ion "imminent
d is turbances" , a p r o b a b i l i s t i c framework for secur i ty assessment i s more
appropriate. Furthermore, most of the major power system break downs are
54
caused by problems associated with system dynamic response. Hence, dynamic
models of the system should be represented in the secur i ty assessment.
A framework for p r o b a b i l i s t i c dynamic secur i ty assessment has been [351
presented by F.F.Wu and T . K . T s a i . L J The d e v i a t i o n of p r o b a b i l i s t i c
dynamic secur i ty assessment i s conceptually based on a two-level model of
power systems. The f i r s t level of the model descr ibes the evolution of the
system's s t ructura l s t a t e . The second level of the model descr ibes the
t r a j e c t o r i e s of system var iables associated with component dynamics. ' - 3 6 - '
The two l e v e l s are coupled.
In the p r o b a b i l i s t i c dynamic secur i ty assessment approach the secur i ty
of a system i s c h a r a c t e r i s e d in terms of the s t e a d y - s t a t e and dynamic
secur i ty regions. The time to insecur i ty i s used as the measure of system
s e c u r i t y . A l i n e a r v e c t o r d i f f e r e n t i a l equat ion can be d e r i v e d whose
solut ion gives the probabi l i ty that the time to insecur i ty i s greater than
t . The c o e f f i c i e n t of the d i f f e r e n t i a l equations involve: component f a i l u r e
and repair r a t e s , power generation, load demand and secur i ty regions.
Techniques have been developed for obtaining steady-sta te and dynamic
secur i ty regions. In the former case one i s in terested in s i tua t ions where
system configurations are changing. The space of i n j e c t i o n , however, i s
i n v a r i a n t . Hence, the s e c u r i t y reg ions in the space of i n j e c t i o n are 137 "I
charac te r ised . J
55
S i n c e the loss of t rans ient s t a b i l i t y i s a severe breach of s e c u r i t y ,
in the l a t t e r case dynamic secur i ty region a f te r a fau l t i s defined in terms
of t rans ient s t a b i l i t y region. For t rans ient s t a b i l i t y a n a l y s i s , a power
system i s considered as undergoing changes in configurations from p r e - f a u l t ,
f au l t - on , to p o s t - f a u l t . I f the i n i t i a l condition for the pos t - fau l t system
l i e s in the region of asymptotic s t a b i l i t y of a s tab le post - fau l t
equi l ibr ium, the system i s t r a n s i e n t l y s t a b l e . The Liapunov method has been
widely used for t h i s analys is .L ' 3 8] [39] [40]L '41]
The usual philosophy, in ex is t ing central secur i ty control systems, in
performing secur i ty assessment could be character ised as determin is t ic in
real - t ime environment. The OCEPS secur i ty assessment package i s based on
the determin is t ic approach.
2.4 Real-Time Security Assessment
The f i r s t stage in the assessment of secur i ty i s the development of a
s e t of events whose o c c u r r e n c e would be c o n s i d e r e d to be a breach of
s e c u r i t y .
In the course of the operat ion, a power system can be considered to be
in t r a n s i t between d i f fe rent s teady-sta tes with smaller t ime-steps. By
means of a few s e t s of c o n s t r a i n t s t h e s e c o n s e c u t i v e s t a t e s may be
c l a s s i f i e d into a number of operating s t a t e s , charac te r is ing d i f ferent
operating condi t ions . The sets of const ra in ts can be:
56
(a) the load constra in ts
(b) the operating constra in ts
(c ) the secur i ty constra ints
The load constra ints simply recognise the prime task in power system
operat ion, namely to meet the load demand. The operating c o n s t r a i n t s ,
furthermore, s t ipu la te the object ive of using the system components within
permissible l i m i t s and to meet the qua l i ty standard of the system. These
constra in ts can be:
(a) voltage at any bus outside working tolerances
(b) overload of any transmission l ine or other equipment
(c) i n s u f f i c i e n t real or react ive generating capacity
The concept of load and operating constra in ts might further be used to
define the concept of s e c u r i t y . Thus, the secur i ty of the system equals the
capab i l i t y of the system to meet cont ingencies, i . e . unplanned events,
wi thout v i o l a t i n g the load and o p e r a t i n g c o n s t r a i n t s in the remaining
system. Such contingencies t y p i c a l l y are forced outages or deratings of
production uni ts and transmission components or essent ia l deviat ions from
the forecasted load.
To analyse the steady-state secur i ty problem, very e f f i c i e n t models and
numerical techniques must be appl ied . For system operat ions, the Fast
Decoupled Newton-Raphson load-flow program can be used as one of several
57
programs in an automated despatch centre. The program can incorporate the
outages according to a contingency l i s t , thus indicating to the system
operator the constraint violations that wil l result from a particular
outage. The types of outage considered include single and multiple
transmission line outages, load generation or station outage, transformer
outages.
The effect of load changes and generation rescheduling can be easily
evaluated,^2][43][32] b ( j t t ) l e o u t a g e simulation of a line or transformer is
more complex because these contingencies change the system configuration.
The loss of a ' generator or reduction of generator output may
precipitate voltage problems due to an inability to match reactive
generation and load. It is therefore necessary to simulate the effect of
partial or full generator outages on the power system which may be achieved
by means of a load-flow study which has access to the actual state of the
power system.
As the system conditions change, the contingencies which cause insecure
operation may also change. In the real-time environment i t is hardly
feasible to perform on-line load-flows for all contingencies to determine
how well the system, in its present state, can withstand these
contingencies. Methods have been developed to rank contingencies according
to the severity of their effects on bus voltages or line flows. On-line
load-flows are then performed for each case starting at the top of the l i s t
58
and stopped when the case does not give problem.
Some automatic contingency selection methods have been
proposed.^44^23][45] p o r e a c h o utage considered in turn the results of the
load-flow study are checked against the crit ical limits and any violations
are brought to the attention of the operator. Such contingency evaluations
are carried out at prespecified intervals and also whenever there is a
change in the system.
Figure (2.3) illustrates the typical components of a security
assessment software package suitable for the real-time monitoring of
electrical power systems.
STATE ESTIMATES
1 FAULT STUDIES
AUTOMATIC OUTAGE SELECTION
I ( SELECTED I OUTAGES
I SECURITY ASSESSMENT
I CRITICAL OUTAGE UST
I GENERATION AND LOAD RESCHEDULING
FIGURE 2.3 SECURITY ANALYSIS SUBSYSTEM
60
CHAPTER 3
THEORY OF NETWORK CHANGES
3.1 Introduction
The basic equations which can lead to the solution of a given network
are derived from first principles using Ohm's law and Kirchloff's laws. The
object of the theory of network changes is to obtain the solution to an
intact network when i t is subjected to modifications. The solution to a
linear or non-linear network can be easily obtained from the inverse of the
admittance matrix. When the intact network is subjected to branch outages,
the theory of simultaneous outages can be applied which determine the new
solution from the previous or old solution by modification of the original
impedance matrix.
Diakoptics can be used for this purpose where the new solution can be
obtained from the original solution in an efficient and organised manner.
This chapter represents the theory of network modifications. Init ial ly, the
solution to linear networks is investigated. Application of branch outages
to non-linear networks is investigated in the following chapter.
61
3.2 Soluti on to Li near Networks
The solution to linear network equations may involve the inverse of the
bus admittance matrix, Y. There are two methods of approach:
(a) to obtain Z in matrix form
(b) to obtain Z in product form
In the case of the former, the solution is obtained by directly
inverting the bus admittance matrix:
V o = lo ^
Vo = V 1 ! o ( 3 ' 2 >
or V„ = l l n (3.3) O O O * '
There are many ways of obtaining YQ~*. One of the simplest and most
effective ways of calculating the inverse of large matrices in connection with power system analysis is the generalisation of the elimination process.
It is most easily understood by considering two simultaneous equations:
*11V1 + *12V2 = h ( 3 ' 4 )
*21 v l + * 2 2 v 2 = (3.5)
Dividing (3.5) by y 2 2 and re-arranging:
v 2 = y 2 2 _ 1 ^2 " y 2 1 v l * * 3 , 6 )
Substituting into the equation (3.4) for v 2 gives:
( y n ' v i + *v#'\ih = h . ( 3 - 7 )
The above equation can be writen in matrix form with v 2 and
interchanged:
y l l y12 v l h '"12 V l
hi y22
•
v 2
>
»'zi y ' 2 2
•
v 2
where
y ' l l = y l l 'hi^zzhi
y ' l 2 = y 1 2 y *22
y ' 2 2 = y * 2 2
63
The process can be repeated to interchange v and i^. The order in
which the interchange is carried out is unimportant and can be extended to
any number of variables and leads to the following set of rules'-^6-':
-1 dd
(b) y ' r d = y r d y ' d d
(c) y ' r c - y r c - y ' r d y d c
<d> y'dc "y'dd^dc
r = 1 n, r *d
r = 1 n, c = 1 r * d , c * d
c = 1 n c * d
where
y d d = diagonal element in row and column d
y = element in row r and column c rf rc
y' = new element which replaces its predecessor
The process is repeated for all diagonal elements taken in any order, with
the new elements stored in the position of previous values.
When sparsity techniques are used, signif icant savings in storage and
computation time can be achieved i f a programming scheme is used which
stores and processes only the non-zero elements. The inverse of the
admittance matrix is not explicitly available. The best known are
64
triangular decomposition and the product form of the inverse. The
bifactorisation method is a very important recent modification of Gauss
elimination. It is particularly suitable for analysing large, sparse
systems that have non-zero diagonal elements which predominate over
off-diagonal elements that are either symmetrical or, i f asymmetrical, have
a symmetric sparsity structure. The method combines the triangulation and
the product form to express the inverse of a matrix A as the product of 2n
matrices:
L A R = U (3.8) m m
L n L 2 L 1 A Rj R2 R n = U (3.9)
A R, R9 R = L , " 1 L , " 1 L " 1 (3.10) 1 2 n 1 2 n
A R,R9 R L L> L, = U (3.11) 1 2 n n 2 1
R,R9 R L L-L, = A"1 (3.12) 1 2 n n ' l
where
R = right-hand factor matrices
L = left-hand factor matrices
U = unit matrix
n = order of matrix
For computing, the following rules can be developed:
k k 1) R ^ = L k i (only for symmetrical matrices, k column of is identical
to the k row of R )
65
2> L \ k -
3> L k1k • * k \ <>k~\l 1 ' <"•!> "
4> R k k j ' - " " u /"""tt J • < k n » "
5) ^ . . " " ^ - * \ ^ , J > \ , . j W j . . . . . . . . n
Any one column can be obtained from the solution of (3.12) with the
right-hand vector set to zero except for unity in the position corresponding
to the desired column of the inverse matrix.
3.3 Theory of network changes
3.3a Application of diakoptics to the theory of branch outages^46^
Assuming that VQ is known from the solution of the original equations:
Y 0 V0 - I„ (3.13)
where YQ represents the original admittance matrix, i t is required to
find new value after one branch in the network has been modified. The
modification may be addition of branches, removal of branches, or changes in
the branch admittances. The procedure to remove branches or to change the
66
admittance of branches is the same. If a branch is removed which is not
mutually coupled to any other branch, the modified bus impedance matrix can
be obtained by adding, in parallel with the branches, a link whose
admittance is equal to the negative of the admittance of the branch to be
removed. If the admittance of an uncoupled branch is changed, the modified
bus impedance can be obtained by adding a link in parallel with the branch
such that the equivalent admittance of the two branches is the desired
From the solution to the original network, the solution to the modified
network can be obtained by applying diakoptics. Consider the network shown
in Figure (3.1), where only one branch is modified from y = y n to y = o.
I I I I 01
1
I I l ' l (a) original network (b) modified network
67
h *1
,-_> i i i i < f
A 1 I V V,
V l • 'o (c) simulated modified network
V i = V 1 ! (d) simulated modified network
Figure 3.1 Network Changes
For above yj = -yQ
From Figure (3.1c):
(3.14)
or in matrix form:
h=h C l V l (3.15)
where
68
(C) is referred to as the connection matrix and consists of as many elements
as there are nodes. It is established by inspection and has elements +1 (for
sending end i ) , -1 (for receiving end j ) and null everywhere else. Hence,
multiplication by (C) reduces to simple summation only. From Figure (3.1.d):
I x = -C x i x (3.16)
(3.15) + (3.16):
h = " c i y i C i t V l ( 3 , 1 7 )
From Figure (3.Id):
YQ V, - IQ + h (3.18)
. . Vx - Z 0 I Q + Z 0 I L (3.19)
From Figure (3.1a):
\ - Vo ( 3 - 2 0 )
(3.20)-> (3.19) and (3.17) • (3.19):
69
v i = vo " zo c i h C i t V l <3-21>
Multiplying (3.21) by C ^ :
c i t v i = c i t v o " C i t zo c i * i C i t V l ( 3 ' 2 2 )
(1 + ZQ Cx y x ) V1 = VQ (3.23)
Let f x = (y^" 1
Dividing (3.23) by y ^
( f i + C i t zo C i ) h C i t v i = C i t V0 ( 3 * 2 4 )
Let d x = ( f 1 + ZQ C j ) " 1 (3.25)
From (3.24)
y x Cj* - d x VQ (3.26)
(3.26) + (3.21):
v i ' vo " zo c i d i C i t V0 ( 3 ' 2 7 )
Hence, if the solution to the original network ( i .e . VQ) and ZQ are available,
70
the new solution can be obtained without forming ly
It can be shown (as in 3.3f) that if d^ = 0 a split network results. The
reason is because the new impedance Z 1 becomes singular.
3.3b Development of the algorithm for simultaneous branch outages^7^
It was shown in section (3.3a) that after removing a branch from a
network, the new solution would be:
1 Zo C l d l c l * Vo
From Figure (3.1b):
V l " Z l lo (3.28)
From (3.27) and (3.28):
1 1 0 0 0 - Zo C l Zo C l d l C l t Z_ I
" Z0 " Z0 C l d l C l 0 (3.29)
71
Hence, the solution procedure for single fault can be summarised as:
1) X ^ Z ^ j
2) dx = ( f x + Cj* X ^ " 1
3) b - dj Cj* VQ
4) Vj - VQ - Xxb
where:
X is the difference of column i and j of ZQ
Cj* X is the difference of element i and j of X
•G-j* V Q is the difference of element i and j of Vg _
The theory of simultaneous branch outages can be generalised to cater for any
number of faults (modifications). For the second outage, ZQ in (3.27) must be
changed to 1^ to incorporate f irst outage. It is not necessary to evaluate
full Z 1 ; i t is sufficient to evaluate Z 1 C 2 only. From first outage Xj
and d, are available.
Multiply (3.29) by C 2 :
Z l C2 = Z0 C2 " Z0 C l d l C l t Z0 C2 ( 3 - 3 0 )
or X 2 = W - Xxa
where:
X2 = Z l C2 W = Z Q C 2
a = d x C ^ W
V2 can be obtained from V :
2 = 1 "" 1 2 2 ^2 1
. i - V g V - j — X-2 dg-Gg-*- VT - - - (3.31)-
where:
d 2 = ( f 2 + Z X C 2 ) _ 1
d 2 = ( f 2 + C 2
4 X 2) - 1 (3.32)
Hence, the solution procedure for double simultaneous outage i s :
1) xx = z0 c x
2) d x = ( f x + X x ) - 1
3) b = d x VQ
4) V l = VQ - X x b
5) W = Z Q C 2
6) a = W
7) X 2 = W - X x a
8) d 2 = ( f 2 + X 2 ) - 1
9) b = d 2 Cg* V x
10) V 2 = V x - X 2 b
where:
U i s the d i f fe rent of column i and j of Z
74
C^" X 2 i s the d i f ference of element i and j of X 2
i s the d i f ference of element i and j of Vj
A lso :
Z 2 = Z l " Z l C 2 d 2 C 2 t Z l ( 3 , 3 3 )
For the t h i r d fau l t mult iply (3.29) and (3.33) by C 3 :
Z l C 3 = Z 0 C 3 " Z 0 C l d l C l t Z 0 C 3 ( 3 ' 3 4 )
Z 2 C 3 = Z l C 3 " Z l C 2 d 2 C2 l l C 3 * 3 , 3 5 )
From (3 .34 ) :
W = W - X x a
where:
w = zx c 3
W = Z 0 C 3
x = z o C l
a = d x W
From (3 .35 ) :
where:
X 3 = Z 2 C 3
V 3 can be writ ten in terms of (as in (3 .27) ) ;
V 3 = V 2 " Z 2 C 3 d 3 C 3 t V 2
or:
where:
d3 = (f3 + c 3
t z3 c 3 ) - x
or:
75
V 3 = V 2 - X 3 b (3.36)
76
d 3 - ( f 3 + C 3
t X 3 ) - 1 (3.37)
b = d 3 C j * V 2 (3.38)
Hence, addit ional procedures for the t h i r d outage are:
1) w = z Q c 3
2) a = dj_ W
3) • W - W - a
4) a ' = d 2 C 2 * W
5) X - W - X, a '
6) d 3 = ( f 3 + X 3 ) - 1
7) b = d 3 C 3
l V 2
8) V 3 = V 2 - X 3 b
A l s o :
h-h- h c 3 d 3 c 3 * h (3,39)
77
For the fourth outage mult iply ( 3 . 2 9 ) , ( 3 .33 ) , (3.39) by C 4 ; a l s o ,
mul t ip l ica t ion of (3.33) by C i s requi red. Hence, the addit ional procedures
can be shown to be:
1) W = Z Q c 4
2) a = d x W
3) W = W - X x a
4) a ' = d 2 Cg* W
5) W" = W - X 2 a '
6) a" = d 3 C 3
t W
7) X 4 = W" - X 3 a"
8) d 4 = ( f 4 + C 4 * X 4 )
9) b = d 4 V 3
10) v4 = v3 - x4 b
78
I t can be seen from the above that except for the f i r s t outage, the rest
follow a s i m i l a r pat tern . The process i s in a recurs ive manner. The flow
diagram of Figure (3 .2) was developed:
i = 1, n where n i s the number of outages
w = z 0 c .
For i * 1
j = 1, i-1
a = d . W J J
W = W - X. a J
X. -_W - - - ._
d, - ( f , • < : , * x,)- 1
a - d, c, 4 v,-i
v i ' v i - i " X1 a
Figure 3.2 Flow diagram for simultaneous branch modif ications
79
and X. must be stored for next f a u l t , V i can be overwri t ten, a , d, f are
s ing le elements, W, Z Q , C , V, X are vectors .
A numerical example i s solved in Appendix B to i l l u s t r a t e the appl icat ion
of the recurs ive method to l i n e a r networks.
3.3c Node addit ion and removal ^ 4 6-^
The network modification process can be used to modify e x i s t i n g branches
as well as to network extensions. This can be i l l u s t r a t e d from the following
example:
j i j k
T" TT—JT-J"
(a) or ig ina l network (b) Extended network (c) simulated network
I I 1 '1 Y„ V, = 1 + 1 o l o 1
Figure 3.3 Network Extension
80
From Figure ( 3 . 3 c ) :
v l " Z o ! o + Z o h (3*40)
From Figure ( 3 . 3 a ) :
and
: th
therefore from (3 .40) :
V v * \ V V ^ \
where Ij^ i s known nodal current at node k.
Node i s o l a t i o n i s a specia l case which can be dealt with here . Suppose that
81
the or ig ina l network be represented by Figure (3 .3b ) , for which the network
s o l u t i o n w i l l be Y Q V 0 = I Q . I f branch y be removed, node k w i l l be
i s o l a t e d . This can be simulated by in jec t ing a current equal to -1^ to node
j . The new 1 i s :
: t h
Therefore equation (3.40) can be solved except for v^.
3.3d Asymmetry^ 4 6^
I t i s sometimes necessary to analyse problems which have asymmetrical
c o e f f i c i e n t matr ix . Such problems a r i s e , for example, when transformers
with quadrature taps are represented. In genera l , such an asymmetry ocurrs
in only few elements but necess i ta tes the ca lcu la t ions and storage of the
f u l l c o e f f i c i e n t matr ix . I f symmetry i s to be preserved, the asymmetry can
be transformed as an in jec ted current and the problem solved i t e r a t i v e l y . A
b e t t e r a l t e r n a t i v e i s to be s o l v e the symmetr ica l problem f i r s t and
represent the asymmetry as network modi f icat ions. This can be i l l u s t r a t e d
by considering the solut ion of the following asymmetrical problem:
82
1 1 1 1
1 1 1 1
1
I r i
Restoring symmetry
1 1 1 1
1
2V 1 1 1 1
1 1
83
or in general form as:
(3.42)
The solut ion of the asymmetrical problem can be now be wri t ten a s :
V l * Z o lo ~ Z o h (3.43)
10 5 4 2
5 3 2 1
_4 2 2 1
2 1 1 1
21
11
• Z o *o
84
Z I, = 2 Vn 10 Therefore: V, = 21 - 2V, (10) V 1
/ V. » 11 - 2V. (5) V. = 1
»2 *2 » *2
V1 = 9 - 2V. (4) A 3 A 2
V, = 5 - 2V (2) U *2
V = 1 x 3
V = 1 A 4
3.3e Appl icat ion to non- l inear elments ( e . g . d . c . l i n k s )
I f the removed element i s non- l inear , then the current flowing in i t
can be defined a s :
1 i * f < " i \ V (3.44)
Compare with (3.14)
From equation (3 .19 ) :
Vi = Z I + Z n I , 1 0 0 0 1
(3.20) (3.19) and (3.16) (3 .19 ) :
l o o . (3.45)
85
Multiply (3.45) by C t
c * v i s C* V - C* l n CI l o o 1
Writing e x p l i c i t l y the 1 and j equations:
X i ° i V ° i i ° i j 1
V l . " V o . " < Z o.. " Z o . . > h J J J l JJ
The above equat ions, in conjunction with equation (3.44) can be solved as a
set of two simultaneous non- l inear equations in two var iab les (V, , V, J and
hence for 1^ from (3 .44 ) . F i n a l l y solut ion for can be obtained from
(3 .46) :
3.3f S p l i t Network
As the s i z e of the networks i n c r e a s e s , i t i s possib le to tear the
network i n t o s u b d i v i s i o n s . The s o l u t i o n to the untorn network can be
obtained from the solut ion to the subd iv is ions . For each subdivis ion the
theory of network changes can be a p p l i e d and i t can be shown t h a t the
general equation (3.27) withholds.
(3.46)
86
When a s p l i t network a r i s e s as a consequence of a branch outage, each
sub-system formed w i l l , independently of the o thers , cater for the supply of
i t s own loads i f p o s s i b l e . The s p l i t condit ion can be e a s i l y detected from
the c a l c u l a t i o n of a s c a l a r f a c t o r "d" de f ined f o r branch o u t a g e s , as
fo l lows:
Consider Figure (3 .4) which represents two sub-networks connected by the t i e
between buses i and j having the admittance equal to J^y I t i s in te res t ing
to see what happens when the t i e i s removed.
1 \
Figure 3.4 Network Interconnection
88
The equivalent c i r c u i t for the above f igure can be shown to be as in Figure
(3 .5)
i
1/Z, i j
1
1/ZC
Figure 3.5 Equivalent C i r c u i t
where Z A and Z & represent the equivalent impedances of the two sub-networks.
Figure (3 .5) i s representing the admittance network for which the admittance
matrix i s :
Y =
0
- y i j
1
0 1
- Z B
1
h
(3.47)
89
Hence the impedance matrix would be obtained from the inverse of Y above,
as:
Z =
ZA h h
ZA h + i
y i j
h y i j J 1 J
(3.48)
However, 'd' as in the theory of branch outages is defined as:
1/d = (f + C b ZQC)
where:
f + c Z o C = f + Z o i i + Z o i i " 2 Z o i j (3.49)
Applying (3.49) to (3.48) y ie lds:
1/d = ( - J _ ) + Z A + Z A + ( J _ ) - 2 Z A
. . 1/d = 0 where for the above i = 1, j = 2, f = - l / y ^
Hence, a sp l i t network results when 1/d = 0 (see Appendix 2 for a numerical
example).
90
3.3g Node Merge
When a network is subjected to branch modifications, the effect would
be simulated in ' f where:
1/d = f + C t ZQC
' f represents the modified branch impedance as was explained in section
(3.3a)
Under short c i rcui t conditions between 2 buses, one bus would reside on
the top of another, i . e . the e f fect can be simulated by adding a l ine
between the shorted buses of impedance f = o.
Hence, when considering short c i r c u i t s , ' f in the above equation must
be set to zero. The theory of branch modifications withholds, however,
equal voltages wil l be obtained for the corresponding shorted nodes (see
Appendix 2 for a numerical example).
91
CHAPTER 4
APPLICATION OF THE THEORY OF BRANCH
OUTAGES TO LOAD-FLOW
4.1 Introduction
Load-flow calculations are performed in system planning, operational
planning and operation/control. For steady state security assessment of the
existing and future system, a load-flow solution method capable of computing
a large number of s ingle and multiple contingency studies reasonably
accurately is necessary, and particularly in the system operations context,
i t must be fast and ef f ic ient . This makes the fast decoupled AC load flow
solution ideal for this purpose.
This chapter represents the application of the recursive branch outage
routine to load-flow and investigates i t s speed and accuracy.
4.2 Summary of fast decoupled load- f low 1 - 4 8 - 1 1 4 9 3
I f an approximate root x r of a simple algebraic non-linear equation i s
known, then a better approximation can be obtained from:
92
r+1 f ( x r ) / f ' ( x r )
or:
r+1 r x = x - A x
where:
A x = f ( x r ) / f ( x r )
Above is known as Newton-Raphson method and can be extended to a set of
simultaneous non-linear equations, in which case
A x = J " 1 F ( x r )
The square matrix J i s the Jacobian matrix of F(x) whose ( i , k ) element is
defined as 3 f i / 3 x k .
To use the algorithm for load-flow solution i t is only necessary to write
the equations defining the load-flow problem as a set F(x) = 0.
The well-known polar power mismatch Newton method is the formal application
of the above algorithm for solving non-linear equations (see equations 4.4,
4 .5 ) , and constitutes successive solutions of the sparse real Jacobian
matrix equation:
93
For PV and PQ nodes
For PQ only
A Q
AV
-1 AP
AQ (4.1)
where A G . and A V . are the corrections for angle and voltage magnitude at
node i . The Jacobian matrix of partial derivatives can be divided into four
parts as:
AP
AQ
H N
J L
A G
A V (4.2)
The elements of the Jacobian matrix can be made value symmetrical by
re-defining equation (4.2) as:
AP
AQ
H N
J L
A G
A V / V (4.3)
where the elements of this Jacobian matrix can be obtained by
differentiating the mismatch equations given by:
94
*1 " P 1 S P " V1 ( G i k C 0 S 6 l k + B ik s i n 9 i k ) Vk = 0 < 4' 4>
= ^ i S P " V i l e . <Gik s i n 8 i k " B i k c o s 6 i k ) Vk = 0 <4-5>
For PV-node only (4.4) is required and for slack node no equations are
required. However, as an inherent characteristic of any practical power
system is the close dependence of active power flows on bus voltage angle
and the reactive power flow on bus voltage magnitude, equation (4.3) can be
approximated as:
A P
AQ
H
L
A 0
A V / V (4.6)
where:
H . . = 3 A P . / 3 6 . = Q . S P + V . 2 B . .
H i k = 3 AV3^ = " V i ( G i k S i n 6 i k " B ik C 0 S Q i k > Vk
L „ =V. 3 A V 3 V . = -Q- SP + V . 2
B i .
L i k = Vk 3Q./3V k + H i k (4.7)
Further approximations in the decoupled Jacobian matrix (4.6) can be made:
95
C O S 0 j k 2 1
Q 1 « B-. V . 2
G i k s i n Q i k « B ik c o s Q i k ( 4 - 8 )
from which the elements of the (4.7) reduce to:
Hik • v i Bik vk hk - vi Bik \ <4-9>
or as:
tf./V. - V i B.. AQj • I B . , Vk
k ei k#i
* V V i • B i i *1 +
kIJik k ( 4' 1 Q) k*j
The f i r s t equation relates /f to therefore any terms which predominantly
affect changes in tQ can also be omitted. For example, by setting the
voltages on the right-hand side to unity results i n :
96
A P / V = B' A ©
A Q / V = B" AV (4.11)
The matrix B' and B" is the negative of the imaginary part of the nodal
admittance matrix Y, without the column and row correspondence to the slack
node. Faster convergence can be obtained by neglecting the resistance when
calculating the terms of B ' , hence:
B ' i k = 1 / x i k B " ik = x i k / ( R i k + X i k )
B ' i i = - n / x i k B " i i = - i B " i k M k*i ^ * L £ )
Figure (4.1) i l lustrates the flow chart for the fast-decoupled
Newton-Raphson.
T491 4.3 Automatic Control and Adjustment1- J
A general purpose load-flow program should include the e f fec t of
automatic control devices such as on-load transformer taps, phase-shifter,
area interchange, as well as upper and lower l im i ts on react ive power
generation and voltage l imi ts . Conventionally, adjustments are made at each
iterative step by easi ly programmable logical c r i t e r i a . The process of
correction can also be made using sensit iv i ty factors which relate the
97
change in a controlling variable to the change produced in the controlled
variable.
A transformer with off-nominal tap setting may be represented by an ideal
transformer in series with the transformer nominal admittance, as shown in
Figure (4 .2) .
I I
1 I 1:T I
1
Figure (4.2) Transformer Representation
where:
T - t + jq (4.13)
T t T
_ * / » Z I _ * (4.14)
therefore:
v t - T v i
also T k - y ( V K - V t)
substituting for VI:
hence:
T. = T* y ( T ? . - V k )
Equations (4.18) and (4.19) can be expressed in matrix form as:
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
( t 2 + q 2 ) y - (t+jq) y
- ( t + jq) y y (4.20)
99
Hence, the elements of the nodal admittance matrix can be modified according
to the above. With the complex turns rat io, the Y matrix wil l no longer be
symmetrical. When the off-nominal taps are in-phase, that is q=0, a simple
equivalent * c i rcui t can be deduced from equation (4.20) and is shown in
Figure (4.3) .
I I ty
V / 14 t f c-t)y ( l - t )y
Figure (4.3) In-phase off-nominal taps
For phase shi f ters , the asymmetry in Y-matrix can be eliminated by
representing the effect of asymmetry by an equivalent injected current as
follows:
100
* T T y - T y
-T Y 7
v i 2jqy V.
(4.21)
which can be represented by a symmetrical ir network as shown in Figure
(4.4) :
(TT*-T)y (l-t)7
Figure (4.4) Phase Shifter representation
The additional injected current can be treated as additional power source:
3 / = -2jqy*V. V k * (4.22)
The phase shifter can be incorporated in the fast decoupled load-flow method
as follows:
101
(a) The calculation of mismatch functions is carried out exactly with
admittances as shown by equation (4.21) and adding to the specified power
S , . s ' ) correction for the asymmetry of the phase-shifter given by equation
(4.23).
(b) Representing the phase-shifter in the constant matrix B 1 and B" by:
B ' i i - B ' k k - - 1 / X
B ' i k = B ' k . - 1 / X
B " i i = B"kk = - x / ( r 2 + x 2 ) * - ! / x
B" . k = B" k . = X/(R 2+X 2) « 1/X (4.23)
4.3.1 On-load transformer tap-changing
An automatic on-load tap-changing transformer usually controls a local
bus voltage. Sometimes discrete adjustment in tap steps i s used, but more
usually the continuous approach is preferred. A conventional adjustment
feedback error for a continuous in-phase off-nominal tapping t.. P.U.
controlling the voltage of bus k to V k
s p P.U. i s :
t .^w . t . o l d i 0 ( ^ r _ ^ s P ) ( 4 2 4 )
102
for the tap in quadrature, i f present:
new old + B (P ik ^i k (4.25)
where:
t.. = in-phase tap position
q.. = quadrature tap position
P i k = active power through the phase-shifter
a =1 for fast decoupled Newton-Raphson load-flow
= 0.2 for Gauss-sidal load-flow
For the PV-node i i t i s usual to calculate Q^r to maintain V at V . s p up to
l imit , before adjusting the transformer in-phase tap position i f both
controls are available.
4.4 Transmission-line Outages
From (4.11) angles and voltages can be obtained for each i terat ion:
A G = B -1 AP/V (4.26)
103
AV = B ii -1 AQ/V (4.27)
Line outages can be simulated by the adaption of the inverse-matrix
modification technique (as in Chapter 3) to B' and B" as follows. In the
most general case, the outage of a l ine can be reflected in B' by modifying
two elements in row k and two in j . The new outage matrix is then:
b = line series admittance
c = column vector which is null except for C^ = 1 and Cj = -1 .
Depending on the types of the connected buses, only one row k or j might be
present in B' (or B"), in which case either C k or Cj above i s zero, as
appropriate. It can be shown that:
B new = B' + b a (4.28)
where:
B -1 = B -1 - X d C B -1 (4.29) new
where:
d = (f + C- X)"
and:
X = B ' " 1 C
f = b ' 1 (4.30)
The new angles can be found from:
A ® n o ^ " B ' n a w _ 1 A P / V ( 4 . 3 1 ) new new
Hence, from (4.29) and (4.30), the new angles are found as:
A G n e w = A 0 - B' C d C* A© (4.32)
compare with (3.27)
Similarly,
A V N Q 1 1 = A V - B " C d' C * A V (4.33) new
where:
d' = ( f + C* X ' ) " 1
105
f = b , - i
Also:
b = 1/X. k , b' - X . k / ( R i k
2
+ X . k
2 )
Mismatch equations also need modifications before the tolerance i s checked.
Depending on the type of buses involved, the following rules must be obeyed:
(a) I f the outage occurs between a slack and a PV bus or between two PV
buses, B" requires no modifications (see equation 4.1) .
(b) The connection vector C has elements +1, -1 and null everywhere e lse .
When modifying the angles i f the outage include slack bus, zero is placed in
the corresponding element of the connection matrix. The same happens when
modifying the voltages i f the outage occurs between a PV and a PV or slack
bus.
4.5 Application of the recursive method to load-flow
As i l lustrated above, the angles and voltages need to be modified after
each iteration and the modifications required, ref lect the mismatch
equations and B' and B". Equations (4.32) and (4.33) are just as (3.27);
therefore after each i terat ion, the old angles and voltages can be modified
106
by applying the recursive rout ine. The flow diagram of Figure (4.5)
i l lustrates how this is done. When the convergence is achieved the active
and reactive powers at each bus are obtained from:
p i " V i 2 G i i + V i = ( Q ik c o s 0 i k + B i k s i n 0 i k ) vk < 4 ' 3 4 > K el k*i
Q. - - V . 2 B.. • V. (G . k s i n e . , - B 1 k cose. , ) V, (4.35)
k*i
See Appendix 2 for a numerical example.
107
START WITH INITIAL: V r =1, er=o
FACTORISE B', B" using sparsity
ITER = ITER+1
1 i = i+1
SLK BUS = ? >
PV or PQ
Find AP 1"/V r from (4.4)
Solve for A 8 from ( 4 . 2 0
8 - c + A c
4 BUS = ?
PV >
PQ
Find A Q r / V r from (4.5)
Solve for AV r from (4 .£ l )
V r + 1 = V r + AV r
1 CHECK for convergence JZ>
not converged converged
Calculate Q for PV node (4* ' * )
P and Q for SLK (4.22), (4 .3S)
t
output results
FIGURE 4.1. FLOW DIAGRAM FOR FAST DECOUPLED LOAD FLOW
START WITH INITIAL: V r =1, 8r=0
I
108
FACTORISE B' , B" using sparsity
ITER = 1TER+1
input data for outage study and do corrections for mismatch
equations X
i = i+1
" SLK < BUS = ? > PV or PQ
Find A P r / V r from (4.4)
I Solve for A8 r from (4.2*$} Modify A 6 r using . recursive routine
e r + 1 = e r
+ A e r e r + 1 = e r
+ A e r
BUS = ? PV
> PQ
Find A Q r / V r from (4.5)
Solve for AV r from (4.27) Modify AV r using recursive routine Modify AV r using recursive routine
v r + 1 = V r + AV r , 1 t
CHECK for convergence not converged
converged >
Calculate Q for PV node (4.3*/)
P and Q for SLK (4.22), (4.25)
>
output results
FIGURE 4 .5 FLOW DIAGRAM FOR FAST DECOUPLED LOAD FLOW FOR CONTINGENCY STUDIES
109
CHAPTER 5
AUTOMATIC CONTINGENCY SELECTION (ACS)
5.1 Introduction
Modern energy control centres ut i l ised in e lectr ic u t i l i t y control and
monitor schemes are required to provide real-time security assessments. If
the potential outage of a l ine or a generator would result in the overload
of another l ine , then the system is said to be vulnerable, a condition which
should be quickly detected for possible corrective rescheduling actions in
operation, or for system redesign in planning.
The Automatic Contingency Selection (ACS) problem i s concerned with
developing computer algorithms for quickly identifying those contingencies
which may cause out -o f - l im i t conditions so as to reduce the number of
contingencies that need to be evaluated when assessing the power system's
security in a real-time environment.
This chapter presents an ACS algorithm for the determination of a l l the
contingencies that give either voltage or l ine flow problems when the system
is subjected to single line outages. The ACS captures a l l the contingencies
110
that give out-of-limit conditions and ranks the l ines according to the
severity that they might cause i f the network 1s subjected to their outage.
I n i t i a l l y , a survey of some ACS approaches is presented.
5.2 ACS Techniques
An accepted procedure for on-line steady-state security assessment in
present Energy Control Centres is by the evaluation of a large number of
contingency cases. However, the computational burden that this procedure
places in even the most advanced of present day computer instal lat ions has
prompted the need for studying procedures for the automatic selection of
meaningful contingency c a s e s . The object ive i s the reduction of the
possible cases for consideration and at the same time the determination of a
ranking or ordering of these cases according to severity, for further study.
Traditional contingency analysis uses a simple logical rule to c lass i fy
contingency case results. This logical rule asks i f any system components
wil l experience an out-of-limit condition from a particular contingency. If
the answer to this question is yes , the case is alarmed, or otherwise noted,
for the user; i f the answer i s no, the case is ignored. The drawback to
this technique is the time which must be consumed in calculating the system
conditions for each case before a limit check can be made.
To overcome t h i s , contingency selection techniques were developed which
attempt to rank the cases by severity so that detailed system conditions
I l l
need only be calculated for those cases at the top of the ranking.
The development of a contingency selection technique has two aspects.
F i r s t , one must develop a straightforward way to compare and rank the
severity of one contingency versus another. Past research has concentrated
on the use of a Performance Index (PI) which, when calculated for each
contingency case, indicates i t s relative severity. The second aspect in the
development of a contingency ranking algorithm i s the creat ion of an
algorithm to ef f ic ient ly calculate the value of the PI for each contingency
case. It i s desirable to combine these two stages. However, such attempts
in the past have resulted in misranking. For ACS, a system performance
index is defined. The sensit iv i ty of performance index i s used to rank the
contingencies according to the severity of their effects on the system
performance. This ranked contingency l i s t is used to reduce the number of
contingencies which require ful l contingency analysis solution.
One measure of quality of a performance index is i t s capture rat io ,
which is the ratio of the number of truly severe contingency cases to the
total number of cases in the top portion of the ranked case l i s t . Other
measures can also be used to show the quality of a particular PI .
As to computation, a good contingency se lect ion i s not simply a
repeated calculation for each contingency case. Instead, i f one took as a
maximum the time required for execution of a ful l AC load-flow for each
contingency case, then a good contingency selection algorithm ought to be
112
able to rank a l l cases in one to five percent of this maximum time.
Most of the existing ACS procedures ^ 5 0 " 5 1 ^ are based on l ine overload
limit violations where PI is expressed as a function of normalised line
flows and the use of DC load-flow (P-e) equations. The voltage l imi t
violations are included by calculating the voltage changes from the
linearised load-flow (Q-V) equat ions . ' -^ All these methods are useful for
analysing single branch outages only. In particular, two approaches based
o n ^ 5 ^ and'- 5 1-' are useful for on-line security analysis. These methods are
based on DC load-flow analysis and derive a closed form solution for the
s e n s i t i v i t y of performance index for an outage of a l i n e . The two
algorithms are different in the way they calculate the sensit ivi ty of PI .
Identification of the contingencies that warrant further study by means
of ful l AC-load-flow solution, may be achieved by predicting the values of
PI for each line outage and subsequently ranking the contingencies from the
most important to the least important (largest and smallest value of P I ) .
The predicted value of PI can be computed e i ther from a Taylor s e r i e s
expansion of the PI as a function of the susceptance of the outaged l ine , or l'54'l
using the DC-load-flow approach as reported i n . J The above approaches i'511
are summarised in L .
Ejebe and Wallenberg defined two Pis for voltage analysis . One of the
performance indices was defined as a non-linear function of bus voltage
deviations. The second index was a modification of the f i r s t index to
113
include the effect of generator MVAr limit violations. They described a
procedure for ranking contingencies by computing the s e n s i t i v i t i e s of
performance indices with respect to outages. A gradient technique was used
to predict the change in the performance index by using Tellegen's theorem
to calculate the terms of the gradient. The imperfections showed up as
misrankings of some contingency cases. That i s , some contingency cases
which should have ranked higher in p r i o r i t y were low on the l i s t and T551
conversely, others which should have ranked low were too high. J
Medicherla and Rastogi developed a contingency solution technique which
uses the load curtailment required to raise bus voltages to the 1551
pre-contingency level as an indication of the seventy of a contingency.
The load curtailment approach can provide a measure of the effect of a
contingency leve l . In this approach the PI for a given contingency is
defined as an aggregate sum of load required to be curtailed at the buses to
raise bus voltages to the pre-contingency leve l . The performance indices
for a l l contingencies are computed, and then the contingencies are ranked in
order of the decreasing values of P i s . Whereas the performance index
defined i s of very high quality the calculation of this performance index is
computationally not attractive.
One common approach used to obtain the contingency l i s t in today's EMS
packages i s by considering the base-case solut ion for a power system
network. By considering the violations occurred following the base-case,
l i n e s and units w i l l be ranked and subsequently f u l l solut ions can be
114
obtained for further inves t iga t ion . Whether one obtains the l i s t by
considering ful l AC power-flow or by performing the f i r s t iterations of a DC
load-flow, neither of these approaches identif ies a l l those contingencies
which may cause out-of-limit conditions.
Working in dependent variable spaces, Wasley and Daneshdoost'^-'
presented a reliable technique for identifying and ranking line outages
which result 1n the violation of various l imits related to power system
s e c u r i t y . Security l im i ts defined in terms of real-power l ine f lows,
generator reactive powers and demand bus voltage magnitudes were considered
in normalised sub-spaces and c r i t i ca l contingencies identified by a
f i l ter ing algorithm using the inf ini te norm as a performance index.
Cr i t ica l contingencies were then ranked using Pis which were defined in
accordance with the sub-space of interest . In the case of real-power line
flows, ranking is based on a measure related to distance from the centre of
the corresponding security region, whereas in the case of generator reactive
powers and demand bus voltage magnitudes, ranking is based on measures
related to distances from the boundaries of the corresponding regions. The
above approach is not suitable for real-time purposes due to the execution
time required to obtain the ranking.
f B O 53 1
In L J U » , ' J J i t was mentioned that the performance index in most cases
provides a good measure for determining the severity of transmission line
contingencies. However, in some instances, particularly when a single line
becomes overloaded and at the same time the loading of other l ines
115
decreases, the value of the PI decreases and the overload may not be
recognised. This pheonomenon has been termed "masking" by the authors of i"50 531 / 54 . \
references 1 ' J . I r isar r i and Sassonv ' developed an eff ic ient method
for automatic contingency selection. Their method is based on the
predict ion of the value of a system wide performance index for each
transmission l ine outages and the subsequent ranking of the severity of the
outages according to the predicted values of the PI . Although their method P5Q 441
is based on the work previously reported in references, W , " J i t overcomes
the l imi ta t ions encountered with those approaches. This was achieved
through the eff icient implementation of the DC-load-flow based selection T541
method proposed i n . J To ef f ic ient ly compute the value of the PI after a
l ine outage, i t was shown that only one forward-backward substituting is
required per execution of the ranking algorithm, and the storage of two
vectors. These lines in the network, need only be updated whenever there is
topology changes in the network configuration. [ 5 1 1
The above approach provides a rel iable method for ACS. In L J , the
authors provide means for avoiding the PI masking effect .
Four algorithms most suitable for real-time purposes were tested and
compared with the ranking produced by a ful l AC solution for each
contingency, in reference L % i l - J . These were: one full iteration of the fast
decoupled load-flow, f i rs t -ha l f iteration of DC load-flow, l ine outage
distribution factors based on DC load-flow, sensit ivi ty analysis . It was
concluded that from a performance viewpoint the line outage distribution
116
factors base on DC load-flow is superior. In reference L , the authors
proposed a new alogrithm for secondary outages. The authors investigated
the f e a s i b i l i t y of including the secondary outages of generation/load
outages and simultaneous outages of two l ines in the overall ACS analysis.
This approach is not eff icient for real-time Automatic Security Assessment
of power systems due to the required execution time, however, as will be
presented in la ter chapters, inc lus ion of secondary outages could be
analysed by providing f a c i l i t i e s for the operator to interrupt the Automatic
Security Assessment (ASA) and requesting what i s termed as mixed outage
selection. Subsequent to this study ASA would resume i t s normal task. None
of the existing algorithms investigates the contingency selection for a
radial network or even when the outage of a single l ine could result in a
sp l i t network, most algorithms automatically assign a high value of PI for
the particular l ines , which in fact is not true and further the AC load-flow
which follows the ACS ignores the l ines causing s p l i t . Hence no
steady-state solution could be obtained, except by performing separate
solut ions for each sub-network which i s not at present an e f f i c i e n t
approach.
The ACS algorithm implemented in the OCEPS package incorporates network
is landing and i s based on the s e n s i t i v i t y of the performance index as
in l 5 0 - 1 . i t i s most su i tab le for real - t ime secur i ty assessment. The
algorithm is explained in the next section.
117
5.3 Line Overload Performance Index
The method described here is based on l i n e a r i s e d load-flow ( d . c .
load-flow), which is then known to be numerically stable and ef f ic ient .
Here the line overload performance index is defined f i r s t .
W Z P )
m
W
O 1 m 2P 1
P , . P P. w IV 2 m2 2P
W m2 2P
W (5.
where:
P , = line flow in line ' A '
P. - maximum capacity of l ine ' A '
W = a weighting factor for l ine ' z '
118
L = total number of lines
Alternatively, the performance index can be expressed in terms of e , the
phase angle across line .
W W w z e) m
4=1
W 81 1 /
o 2K 1
w / 2K 62
O w /
2K
(5.2
where
b = susceptance of l ine t
k = P m
0 A = phase difference between i t s sending bus and receiving bus
phase angles.
119
thus:
p
t • V * <5-3>
5.3.1 Analysis of a Single Branch Outage
For a change in the susceptance of branch k ( i . e ; Ab^), the change in
( i . e . the inverse of system susceptance matrix B) i s as follows:
fix = (B + Abk n^ 7 ) " 1 - x = ( B n e w ) _ 1 -x (5.4)
where:
i s a Nxl incidence vector for l ine k, having 1 and -1 at the two
elements corresponding to the sending bus and receiving bus of l ine V , and
zeros for other elements in the vector. Note that x = B"*.
i'581 By the matrix inversion lamma
6x = c^J (5.5)
where
120
" k -Abi,
1 + A b k x k k
x k k = 15k a k
(scalar)
(scalar)
ak = rn^ (Nxl vector)
The corresponding change in e i s :
(5.6)
where:
6 0 N = (fix P J J ) is the change in nodal phase angle.
is the nodal power injection,
Substituting equation (5.5) into equation (5.6)
_ m T T D
SjL ~ nLA n k a k otk
= n k ( m j a*) (X !Dk>T
• \ («LT «*> ("!kT
p k X * k ^ (5.7)
where:
e N = — f - H : ~ d , c * l o a d - f l o w equat ion r e l a t i n g bus power i n j u n c t i o n
and bus phase ang le .
\ = ID|<T ®N i s t n e P n a s e d i f f e r e n c e across l i n e ' K ' .
(Q)
5 .3 .2 Change o f performance index SL^ ' due t o change o f 60^ and Ab^ -
Outage o f branch V
From equat ion ( 5 . 2 )
Z n e w (0 ) = L I 4=1
k new ft new
( 5 . 8 )
where:
122
new b . when 4*k
b k + Ab k when 4=k
and
new 4
7new
0 4 + 6 0 4
Once Z n e w (0) i s found oZ k ( 9 ) = Z n e w (e ) - Z o l d (0) can be c a l c u l a t e d
w i t h Z o l d (0) = =i w v 2 [ * % p ;
2 L
E W 4 0 4 ( 5 . 9 )
s u b s t i t u t i n g b A
n e w and 0 A
n e w i n t o equat ion ( 5 . 8 )
Z n e w (0) = £ w. 4=1 */<
b 4 < 0 4 + aQl]
m L 4
2\ new* 2 + w k ( 2 b k &b k + A b k ' ) ( e ^ ' " " )
( P u m ) 2
4=1 "f l l 4=1 f 9
2 k 4 2 k 4
= I . w , e t
2 + I . w £ 2 0 A C0, + 6 0 / + R
2k
Z o l d (0 ) + I W 2 T V °K XAk 0 4 + ^ i W * \ 2 0 , 4=1 / 0 * _ i / o / o
2 X. 2 + R
2k, ? ' 2
2 K 4 2 k 4
Z 0 l d (0 ) + 2 . 0 , I ™ 4=1
W, (m T a ) T 0 + * / o 4 k
2 k 4
\ Z e k 2 I WA (n>T a ) T (m T a ) +R 1=1 / , k k
2k * *•
Z o l d (e ) + 2r^ ft \ W (m T X m ) T (m T i
2 k /
+ n k 2 6k 2 i \ akT " i " J °k + R
k 2
Z 0 l d (0 ) + Zy 0 k H m T
k X (m, m / )
2k,
V 0 k 2 1 , \ f \ c t %T> < +
2k
z o l d (0 ) + 2 ^ m k
TX I W A = 1 V
m * m * °N
2 k /
2 « 2 T v +nk 0k \ L V m * " l "k + R
£=1 * / 2K,
Z o l d (0 ) + 2 n k 0 k m k
T x (aij m 2 . . . m L ) ' I /
2k
124
+ ^ ^ \ J < m l m 2 m L )
2k, o 2k.
o v,
2k,
nit
m,
. . Z n e w (0 ) = Z 0 l d (G) = 2r^ ^ m k
T X A W 0 A T ^
(5 .10 )
+ \ 2 \ Z \ T A W 0 \ + R
= Z o l d (0 ) + 2 ^ 0 K m k
T X W ^ ^ 0 / c^ T W + R
= Z o l d (0 ) = 2 ^ 0 , 0, + 1 ^ ^ 2 t k k + R (5 .11 )
where
0 k = m k 0 N GN = X P N
P N = W 0N N x 1 vec to r
*kk = \ W °k
W = A W_ A T
0
Scalar
125
A = (m^ nig . • •
o / 2k 1
W / W
e 2k
O V)
/ 2k (5 .12 )
Wk ( 2b k Ab k + A b k
2 ) (Q^™)2
R = — • 2 ( P k
m ) 2 (5 .13)
V " \ + ° \ = ^ + X k k
= 1 + Ab k X k k ( 5 . H )
P k = b k ^ aZ k (0 ) = Z n e w (0 ) - Z o l d (0 )
126
2 2 2 2 * 7 l n \ * k P k ° k H k ^ k P k W k P k " k A b k A b k 6 Z k » ' = — + + — —
b k b k 2 2 P k m A b k b k b k (5 .15)
where l i n e V i s o u t , s b k = - b k
. * k p k e k t k k " k 2 p k 2 W k p k 2 n k « z
k b ) = + — _
b k b k
2 2 P m
k b k (5 .16 )
Equat ion (5 .16 ) expresses the change o f performance index ( l i n e over load
index) i n terms o f the parameters t k k , n k , P k
m , w k and b k o f t he outage l i n e
V .
Among these parameters , t k k a n d n k needs p r e l i m i n a r y e v a l u a t i o n before k
a z k $ ) can be c a l c u l a t e d . Parameters t k k and n k ( two vec to rs of l eng th L)
a r e s t o r e d and r e c a l c u l a t e d o n l y when t h e base t o p o l o g y c h a n g e s . S ince
topo logy changes are i n f r e q u e n t , t h i s s to rage scheme saves the c a l c u l a t i o n o f
t k k andn k each t ime ACS i s r e q u i r e d , hence speeding up the response of the
program. *
5.4 Sequence o f Computation
The c o m p u t a t i o n s t e p s o f t h e p rog ram e m p l o y i n g ACS a r e d e s c r i b e d as
f o l l o w s :
127
1. C a l c u l a t i o n
X = B"1
W = A W Q A T
^ - W %
2. For l i n e k = 1 t o L, c a l c u l a t e t h e f o l l o w i n g
d ) k = - b k s ca la r
^ - X i N X I vec to r
IR = (00..1..0.. -1.0) NXI vec to r
p o s i t i o n p o s i t i o n IS IR
128
X kk = Hl^J % s ca la r
*kk = \ * \ s c a l a r
( s t o r e d up as LXI vec to r f o r a l l )
-Ab k s c a l a r TV = ( s t o red up as LXI vec to r f o r
U S 5 k T k - k a l l )
3 . Evaluate 6 Z k ( e ) us ing equat ion (5 .16 ) f o r a l l k,
4 . Ranking 6Z k i n descending order accord ing t o i t s v a l u e , f o r a l l k.
5. Ou tpu t rank ing r e s u l t s . Since 6Z^ i s an over load i n d e x , the f i r s t
ranked cont ingency corresponds t o the most severe l i n e outage ' k ' .
Vol tage V i o l a t i o n Performance Index
The vo l tage v i o l a t i o n performance index i s de f i ned as f o l l o w s :
v =
NB \ii
I I A=l *
V* ,SP
(5 .17 )
129
Where:
W = a we igh t i ng f a c t o r f o r bus ' £ ' .
= vo l tage magni ture o f bus ' A ' .
SP s p e c i f i e d vo l tage magnitude o f bus ' i'.
A V , = maximum a l l owab le d e v i a t i o n of vo l t age magnitude from the s p e c i f i e d va lue f o r bus ' A ' .
I t i s c l e a r t h a t t he vo l t age v i o l a t i o n performance increases va lue as
more buses have vo l tage magnitude d e v i a t i o n s which exceed the maximum
a l l owab le v a r i a t i o n .
5 . 5 . 1 Change of Z y ( 6 Z v ) due t o changes i n vo l tages fiv"A
I f a s i n g l e branch outage cont ingency takes p l a c e , a l l t he bus vo l tages
of t he system would be changed ( $ V A ) . The cor responding change i n Z y can be
de r i ved as f o l l o w s :
Let
new V . + 6V (5 .18 )
v o l d
SP 2 NB w
£=1 2 (5 .19 )
130
and
new NB W r
= 1 r 4 = 1 IT
v new _ v SP.,2 4 4
~AT
"1
( 5 . 2 0 )
S u b s t i t u t i n g equat ion ( 5 . 1 8 ) i n t o equat ion ( 5 . 2 0 )
NB W„ _.V + 6 V 9 - V . new y 4 T 4 4 4
v " I
SP 2
f " 4 r " 4 1 U ' 4 " 4 - |
4 = 1 T~ "~AT! 4 = 1 2 Z T 3 4 = 1 2 A V ?
•• I NB W „ o l d , v 4 SP\ j . W 4 6 V 4 + I X
2 26V ( V - V O R ) + I K t 4 = 1 2 A V / 4 4 4 A = 1 2
z = z new _ z o l d = ( 2 ( v v SP } + v } T V V V V —4 —4 4 '
W ,
2AV, o
w 2
2 A V 2
2
O w NB
2AV NB
131
6ZV = (2(V^ - V £
S P ) + 6 V £ ) T (W b) ( 5V A ) ( 5 . 2 1 )
where
(V n V 9 . . . V^g)
( v SP v SP „ SPv
1 ' 2
f S P . 1 "2
{(N1 AV2 . . . «V N B )
W L
= bus we igh t i ng m a t r i x
By i n s p e c t i o n o f equat ion (5 .21) i t i s c l e a r t h a t :
6ZV - 6ZV < « g
Thus 6Z„ i s a f u n c t i o n o f t h e unknown v e c t o r 6 V „ . Th i s v e c t o r 6V, v i i
represents changes i n the bus vo l tage magnitudes as a r e s u l t o f a s i n g l e l i n e
outage con t ingency . I t remains t o de r i ve fiV i n terms o f a l i n e ou tage .
132
5.6 Ana lys is o f a s i n g l e branch outage
5 . 6 . 1 E f f e c t o f a s i n g l e branch outage on system susceptance
L e t _B_def ine t h e system susceptance m a t r i x , and denote the outage l i n e as
l i n e ' k ' having s e r i e s susceptance b^ and shunt susceptance (charg ing
capac t i ve susceptance) S^.
The outage of l i n e V would i ncu r t h e f o l l o w i n g changes i n B_.
B n e w = B_ + 6b M R
t (5 .22 )
where
M. = 0 0 1. .0 0 0 0 0 " * 0 0 / 0 . . 0 0 -1 0 0
t
NB x 2 m a t r i x ( 5 .23 )
sending r e c e i v i n g bus l o c a t i o n bus l o c a t i o n
b o
k " k / 2
" b k " S k / 2 2 x 2 m a t r i x (5 .24 )
Let
X = B -1
133
then
xnew = ( B n e w } - l
i . e .
X N E W = (B + 1 ^ SD V ) t x - 1
By the m a t r i x i n v e r s i o n lamma (5 .18)
,new B " 1 + B ' ^ ^ V 1
X + X M ^ M ^ X
where:
= X M NB x 2 m a t r i x (5 .25 )
^ = - ( n + 6b x k k ) " 1 A 2 x 2 ™ t r i x (5 .26)
n = 2 x 2 u n i t m a t r i x 1 0
0 1
(5 .27)
2 x 2 m a t r i x (5 .28 )
w i t h Sx de f i ned as:
6x = B k X k 0 k
t (5 .29 )
134
5 .6 .2 C a l c u l a t i o n o f 6V A
The c a l c u l a t i o n o f 6V^ i s based on t h e decoupled r e a c t i v e power equat ion
(5 .19 )
- B N E W ^ ( 5 . 3 0 )
Expressing 6V more e x p l i c i t l y
6VA = - ( B ^ ) " 1 A Q A
- X n e w A Q A
(X + 6X) AQ
= - X A Q £ - 6X A Q A ( 5 . 31 )
where AQj, i s the mismatch c a l c u l a t i o n exc lud ing the e f f e c t o f outage l i n e
V .
Q A
S P NB
AQA = _ + = (Sjy s ince 0 A J . - B | j cos 0 ^ ) V j ( 5 .32 )
V, j = l
135
where + J i s the {l, j ) t h element o f the system bus admi t tance
m a t r i x .
A t f i r s t s i g h t , i t m i g h t seem t h i s i s an i t e r a t i v e a l g o r i t h i m , w i t h
c a l c u l a t i o n s i t e r a t i n g between equat ions ( 5 . 3 2 ) , ( 5 .31 ) and ( 5 . 1 8 ) , i n t h a t
order u n t i l AQ - 0 . However, the exact s o l u t i o n f o r fiV^ i s not r e q u i r e d , but
r a t h e r t he c o r r e c t rank ing o f SL = 6Z ( 6 V a ) . I t has been found t h a t the V V X*
rank ings o f 6Z us ing fiV c a l c u l a t e d a t f u l l convergence and the rankings o f V ™~ At
6W us ing 6V c a l c u l a t e d by a s i n g l e i t e r a t i o n , are p r a c t i c a l l y i d e n t i c a l .
Hence, f o r the present a p p l i c a t i o n , equat ions (5 .32 ) and ( 5 . 3 1 ) need on ly be
c a l c u l a t e d once.
5 .6 .3 C a l c u l a t i o n o f vo l t age phase angle
I n t h e c a l c u l a t i o n of fiV^, t he r e a c t i v e power mismatch AQ^ i s computed
from equat ion ( 5 . 3 2 ) . I t i s obvious t h a t t h i s i nvo l ves the c a l c u l a t i o n s o f
and 0. i n order t o eva lua te 0 „ . = 0„ - 0,-. The phase angle across the l i n e
connect ing l and j .
Now d e f i n i n g
(5 .33 )
136
where:
0 . i s t h e base case value known p r i o r t o the commencement o f the present a l g o r i t h m
60 „ i s t h e change i n bus 1 1 ' vo l t age angle due t o the outage o f * l i n e V .
R e f e r r i n g t o the d . c . l o a d - f l o w equat ion (5 .17 )
SP
thus
T h i s g i v e s the changes i n 0^ due t o changes in_X_as a r e s u l t o f outage
l i n e ' k ' where &X i s d e f i n e d as equat ion ( 5 . 2 9 ) .
5.7 Sequence o f Computation
The sequence of c a l c u l a t i o n can be o u t l i n e d by t he f o l l o w i n g s teps .
1 . Ca lcu la te susceptance B_ and i t s i nve rse X .
2 . For every l i n e k = 1 t o L, c a l c u l a t e the f o l l o w i n g :
6b " b k" S k/2
~ b k S k/2 2x2
"k 001...000
o). . . -ooq location of IS. sending bus of l ine V .
1000 NBx2
location of IR. receiving bus line V .
k
*kk V *k -1 - (n + fib x,^) fib
2x2
2x2
2x2
Store for a l l k as an L x 4 matrix,
3. For each outage line k = 1 to L calculate the following:
(a) Update 0
fi9A= fiXp/P = PA
SP
„ new _ . . .„ 0, - e, + fieA
(b) Calculate sin 0.. and cos 0.. for a l l lines = 1 to L I J I J
. „ .„ f n new _ new\ sin e i j = sin (0. - 0j )
cos ©-j j = cos (e^ew _ 0 new j
138
(c) Calculate A Q b mismatch
Q S P N B = Y~— + J ( G i j s i n e i j - B i j c o s e i j ) V j
For a l l buses
(d) Calculate 6V£
« V X = - X A Q . A - 6X A Q 4
(e) Calculate 6Z v k
« v k • (2(V t - V / P ) • « / (WJ
4. Rank 6Z u f o r a l l outage l i n e ' k ' in descending order of i t s
magnitude.
5. Output ranking results. Since 6ZvJ< is a voltage violation index,
the f i r s t ranked contingency corresponds to the most severe line
outage V .
139
CHAPTER 6
NETWORK jSLANDING AND EQUIVALANC1NG
6.1 Introduction
Conventional techniques for the assessment of transmission network
security normally assume that the network to be studied w i l l only be subject
to a relatively small number of contingencies simultaneously and furthermore
that those contingencies w i l l not sp l i t the network into islands. Should
such a situation arise the approach is usually to restr ic t analysis to the
largest island so created. This approach has serious disadvantages in power
systems which are subjected to frequent major outages and f o r which
islanding is a common occurrence.
In interconnected systems, the results obtained from contingency
analysis studies depend strongly on the representation of the neighbouring
but unobservable networks. As a possible solution to this f ac t , at the
control centre the observed system, consisting of the internal system,
t ie - l ines and boundary nodes, is represented in i t s complete form while the
unobservable external system is modelled by network equivalents.
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This chapter represents a unique approach for the diakoptic solution of
the network flow when subjected to configuration changes.
Based on Gauss - e l imina t ion , a new method f o r obtaining network
equivalents is presented.
6.2 Split Network
Security assessment analysis has evolved from load-flow techniques and
provides a measure of system performance following the occurrence of any of
the pre-specified set of contingencies. The selection process is normally
implemented by an automatic contingency selection algorithm (ACS) commonly
based on DC load-flow. The ACS ranks the outages in order of decreasing
severity and those above a certain threshold, which w i l l not sp l i t the
network into islands, are passed for a detailed network flow analysis.
As the size of the network increases i t is possible to tear the network
into subdivisions. The solution to the untorn network can be obtained from
the solut ion to the subdivis ions. For each subdivision the theory of
network changes can be applied and i t can be shown that the general equation
(3.27) withholds.
When a s p l i t network arises as a consequence of a branch outage, each
subsystem formed w i l l independently of the others cater for the supply of
i t s own loads i f possible. The sp l i t condition can be easily detected from
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the application of diakoptics to network solution.
The conventional method for simulation of branch outages which cause a
sp l i t network 1s based on the introduction of a new reference busbar (slack)
and redis t r ib i t ion of active power among the specified generators in each
subsystem formed and f i n a l l y refactorisation of B' and B" matrices taking
into account the existence of a second reference and branch outages.
Section 6.3 of the present chapter describes a new algorithm for the
diakoptic solution of the network flow when a s p l i t network occurs as a
result of a branch outage. The new technique avoids the need to re-order
and re-factorise the above matrices. I t has been tested using a range of
networks and compared with an alternative approach in which the network
islands are formulated and solved as separate load-flow problems. From the
numerical results obtained i t was concluded that the new technique has
advantages fo r application in real-time security assessment.
(27^ 6.3 Theory of network islanding* '
When a s p l i t network arises as a consequence of a branch outage,
conventionally a new reference busbar is chosen in the island which is not
connected to the original system. This busbar can be introduced as an input
by the operator or automatically chosen as the busbar which had the biggest
active generation prior to the outage. I t must be among the generators
spec i f ied f o r r e d i s t r i b u t i o n of active power. The drawback of such
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approaches is the need to introduce the existence of the second reference
and the outaged branch in (B 1 ) and (B"). In the proposed approach the new
reference is chosen automatically, but i t s existence is not reflected in
(B 1) and (B"), hence the new technique avoids the need to refactorise.
6.3.1 Summary of basic algorithm for outage studies
The application of dlakoptics to linear and non-linear networks is
summarised:
Assuming that VQ is known from the solution of the original linear equation:
I (6.1)
Where YQ represents the original admittance matrix. The new value V
after one branch has been completed can be obtained from: new
V = V -XC^V "new vo A ° o (6.2)
where C = row vector which is null except for
C. = +1 and C, = -1 I J
X ZQC is the difference of columns i and j of 1Q
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Z„ = Y ~ stored in factored form o o
C^)IQ = is the difference of elements i and j of VQ
d = (1/b + (^X)" 1 a scalar factor
b = l ine or nominal transformer series admittance.
The above approach is fas t , accurate and avoids refactorising YQ.
Load-flow is well known as a non-linear problem. The fast decoupled
Newton-Raphson (FDNR) load-flow has proved fas t , su f f i c ien t ly accurate and
rel iable. I t is already well documented in chapter 4 and therefore no
detailed description w i l l be presented.
The basic model is described by:
L A S ] • [ B T 1 C A P / V ] (6.3)
L A V ] = [ B ' T 1 L ' A Q / V ] (6.4)
The numerical technique based on diakoptics developed for line outages
can be applied to FDNR af ter each i terat ion process in order to obtain new
angles and voltages:
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Step 1: [A0 Q ] = L B ' ] " 1 [ t f / V ]
Step 2: X' = [ B ' ] " ^
Step 3: 1/d' = l / b , + C lX'
Step 4: LA9 n e w ] = [A0o'J - d' x' (6.5)
where: b' = 1/X^
For voltages, similar steps to those above can be implemented and the
new voltages are:
= - D " X " M T A V O { 6 , 6 )
where:
X" = [ B " ] ' ^
d" = (l/b' + cV)" 1
b- = V^ 2 ik + x 2 i k )
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6.3.2 Pre-processing for system separation
Consider Figure (3.4) which represents two islands interconnected by
the t i e - l ine between buses i and j . The Thevenin equivalent c i rcu i t for the
above is shown in Figure ( 3 . 5 ) . In Figure ( 3 . 5 ) , Z f t , and Z B are the
equivalent impedances of the two islands. The impedance matrix for this
c i rcui t is given by:
ZA h h
h h + I h + I
h Z. + 1 ZB + I
y i j
I f now the t i e - l i ne between buses i and j is to be removed, from equati
(6.2) we have:
Step 1: X = ZQC
o
•1/y
Step 2: I = (1/b + ^ X ) ci
therefore: l / d = 0
Hence a sp l i t network results when l / d = 0. Without further calculation by
inspection of 'X' obtained from step 1, the buses in each island are
ident i f ied:
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Island 1 (with the original slack)
Island 2
When diakoptics is applied to FDNR, following the steps above 1/c becomes very
small but not identically zero, and fur ther , vector 'X' would not hold
absolute zeroes. A sensi t ivi ty factor should therefore be specified which is
a function of the computer on which the study is implemented.
6.3.3 Islanding
By removing the branch between buses i and j in Figure (3.4) , the
original network divides into two islands. The original admittance matrices
are the following form:
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Submatrix [Zo^j is the impedance matrix of the connected part and [Zo^]
should be modified to obtain the impedance of the second island. In this
work, YQ is assumed symmetrical and since the row elements in submatrix [B]
have equal co-factors, the row elements in [ZOg] are equal.
I f 1 j , i 2 . . . . i m represent the injections at the buses of the intact
network, the original solution is given by:
Vn = Z I O 0 0
o r \ = z l l 1 1 + z12 1 2 + z13 h + * + z l i * j + Z1W V "+zlm \
V o 2
= z 21 h + z22 h + z23 *3 + *+ z 2 j *J + z2k V "+ZZm %
\ = z i i 1 1 + z i 2 h + z i 3 1 3 + ' + Z i j 1 j + Z i k V " + Z i m \
Where column j , k, n, m are referred to as the buses in the second island.
When the t i e - l i ne is removed, the solution to the connected part can be
obtained from:
neWj -\ - z l j " z l k i k
new^ -\ " Z 2 j " Z2k i k
new^ -\ " Z i j " Z i k i k . . . " " zim 1m
A column in llo^j can be chosen, corresponding to the new reference
busbar and hence equation (6.8) can be represented at:
neW l
= \ - z l r ( i j + \ + i m )
= v ^ new2 o 2 2r v j
In matrix form the above can be repesented as:
- CV - C l t C V C 2 t l< (6.9)
[ V n e w J gives the solution to the connected part.
where:
LV 0] = U „ ] C l 0 ]
1 1 1 1
is zero every but ' 1 ' for the new reference busbar. Hence, C 1
t Z £
a column vector corresponding to the new reference busbar.
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C 2 is zero for island 1 and ' 1 ' for island 2. I t is set when
calculating [ x ] as in the previous section. C ^ o i s a s c a l a r factor . I f i n
the subsystem without the original reference bus, there is no provision for
the introduction of a new slack bus, then the reference is taken as the busbar
of the new subsystem connnected to the outaged branch.
The advantage of applying equation (6.9) to obtain the new solution for
the connected network is that i t avoids refactorising.
To obtain the solution to the subsystem without the original reference in
parallel with the solution of the connected part, the following equation can
be applied.
tVnew ] = [ V " c l * C 2 t *o + C l * C2 lo ^
where:
Zm = f ( z 3 , z 1 § z 2 )
Z l = l " r o
Zo = f ( V ^ ^ i / C o ^ L ) y„„ is the shunt admittance connected to the c r l r l L o ro original reference bus.
151
r Q refers to the original reference bus and r^ refers to the new reference
bus.
The new technique has been applied to many linear and non-linear networks
and shown to be e f f i c i e n t , accurate and rel iable, and most suitable for the
real-time security assessment of electrical power networks. I t avoids the
need to refactorise [B'T. and LB"] and furthermore, modifications to angles are
only required after each i terat ion since PV buses are not represented in [ B " j .
I f automatic tap changers are included in the system, the tap positions on the
island with no reference should be set to the i n i t i a l tap positions, before
any modifications are performed.
6.4 Equivalencing
One of the requirements of today's modern power system control centres is
to determine equivalents for extensive power systems external to an internal
system equipped with a central control computer. These equivalents are needed
for d i f fe ren t system studies where the internal system is represented in
de ta i l , and the external system is represented by their equivalent, to
simulate the interaction effects of the external system on the internal system
for disturbances originating in the internal system.
The continuing increase in the size and complexity of electr ic power
systems has demanded that a larger number of cases, involving far bigger
networks, be analysed in regard to the ab i l i t y of the system to provide
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rel iable, secure and economical service. In most cases the system of concern
(internal) is a part of a larger interconnection of power pool. I t is rather
impractical to analyse the entire system in applications involving local
control of an area. To overcome this d i f f i c u l t y two approaches have evolved.
1. To break the power system into a reasonable number of subsystems that can
be handled separately while the interaction between the subsystems is
accounted f o r . <»><»><75)
2. To isolate the area of concern from the rest of the system and construct
an equivalent that provides a f a i t h f u l means for representing the
interconnected network beyond the boundaries of the area of concern,
rather than providing a way to obtain the overall solution of the entire
systen,.< 7 6>< 7 7>
I t is thus desirable, through equivalencing, to develop a representation
of the system outside the area of study (extended system) of as low an order
as possible while maintaining the essential impact upon the area under study
(internal system). I t is also required in cases of operational or real-time
mode, that these equivalents be ident i f ied without knowledge of configuration
and state of the external s i t e .
The construction of the equivalents would have to depend mostly on a good
model of the internal system and data telemetered from locations within and at
the boundaries of the internal system. Furthermore, i t is important that the
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equivalents be constructed in a minimal amount of time so that they can
provide for on-line study modes.
Extended simulations of large power systems with long transmission lines
are needed. If such simulations are to be carried out without consuming
unreasonable computer resources then better solution algorithms must be
available, highly restr ict ive assumptions must be made regarding system models
and/or the effective size of the system must somehow be reduced for solution.
The general approach is to reduce the passive portion of the network.
Such a reduction must provide for an accurate representation of real power
transfer, and at the same time conserve the reactive response of the internal
network. Basical ly , the interconnected power system is divided into internal
and external systems, and the boundary buses are defined as shown in Figure
(6 .1) .
INTERNAL
SYSTEM
1 I EXTERNAL
SYSTEM
INTERNAL
SYSTEM
1 1 EXTERNAL
SYSTEM
INTERNAL
SYSTEM
EXTERNAL
SYSTEM
INTERNAL
SYSTEM 1 1 1
EXTERNAL
SYSTEM
INTERNAL
SYSTEM
I 1
EXTERNAL
SYSTEM
Boundary
Buses
Figure (6.1) Separation between local and external systems
154
The choice of the boundary buses is such that any large disturbances to
be investigated in the internal system would have small effects on the
external system.
A good equivalent of the external system was suggested by Ward based
on a generalised Norton equivalent theory where the current sources are viewed
as constant active and reactive power sources. A similar equivalent can be
derived using Thevenin's equivalent theory leading to constant power and
reactive power sources behind the equivalent internal impedances at each
boundary bus. Either of these equivalents i s a stat ic equivalent made of:
1. Equivalent internal source admittances (or impedances) at each boundary
bus. For a Norton equivalent these admittances are grounded; while for a
Thevenin's equivalent source, creating a f ic t i t ious internal equivalent
bus.
2. Equivalent transfer admittances (or impedances) between the boundary
buses.
3. Equivalent power sources - their sum equals the exchange power scheduled
to flow between the internal and external systems.
Ward reduction is based on Gaussian elimination and has been incorporated
in most of f - l ine and on-line equivalencing approaches, such as extended Ward
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and simplified extended Ward.
The Ward method suffers from poor accuracy when performing security
analysis mainly due to problems in the designation of the boundary bus types.
Unless a bus actually has a unit , i t i s not correct to designate i t as a PV o
PQ bus. What i s lacking is a satisfactory way of representing the external
system's reactive power response to changes in the study system. References K ' l i s t the commonly recognised d i f f i cu l t ies of the Ward approach
experienced mostly in of f - l ine implementation.
Figure (6.2) i l lust rates the study system with Ward equivalent.
INTERNAL
SYSTEM
— 1 INTERNAL
SYSTEM
— 1 INTERNAL
SYSTEM
— 1 INTERNAL
SYSTEM
— —
— 1 INTERNAL
SYSTEM
— —
Boundary buses
Figure (6.2) Ward Equivalent
156
In Figure (6 .2) , the equivalent injection for each boundary bus is given
by:
pi e q • v 1 ° U,i \° ( f i ik c°s V + B i k
s , n V>
" I 6 " - V 1 ° Ik.1 »k° <61k S 1 " °1k° + B i k COS 0 1 k ° ) (6.12)
In practical applications, there are various versions and interpretations
of the Ward method. In some cases, boundary buses are designed as their
original types ^ 8 3 ^ , in others they are considered to be a l l PV types ^ 7 6 ^ , or
are PQ types ( 8 ^ , whereas in others, more complicated heuristic boundary
techniques are used
In the extended Ward equivalent each boundary bus is designated as PV or
PQ as i t actually i s . A new f ic t i t ious PV bus, with P =0 and Vm = V.° is ^ J ' m m l
attached to each PQ boundary bus i , via a f ic t i t ious branch of admittance
[Y^] , as in Figure (6 .3) . The new f ic t i t ious buses contribute no active power
to the system and no reactive power in the base case. However, whenever the
study system conditions change, these buses respond with the supply or
absorption of reactive power. These can be eliminated altogether i f desired
by placing an additional injection ^ 8 3 ^:
AP + jAO = (E. E m * - V ^ ) . Y . * (6.13)
at each relevant boundary bus.
157
If Y. i s susceptive equation (6.13) simplifies to:
AQ - V . ( V . - v i ° ) . Y i (6.14)
INTERNAL
SYSTEM
P. e c* + j Q . e q
Boundary Buses
/ /
Pm = o m
V m = V , m l
Fict i t ious Buses
Figure (6.3) Extended Ward Equivalent
If in the extended Ward equivalent method the resistances in the external
system be ignored, this gives view to simplified extended Ward equivalent.
Though, this method does not provide suff icient accuracy, i t guarantees good
X/R ratio in the equivalent network; and the process can be performed more
economically on a rea l , rather than, complex matrix.
158
The reduction technique can be directly applied to fast decoupled
load-flow which is used extensively in system security assessment. Basical ly ,
the reduction process can be applied to [B'!l and [B"] matrices to obtain B'eq
and B"eq and the equivalent injections at boundary buses are computed from
(6.11) and (6.12) using the admittances j B ' ^ 6 0 1 and j B " i k
e q respectively for
the equivalent branches Then in decoupled load flow solutions, [ B 1 ^ ]
and L B " e q ] becomes incorporated into the internal system [B'3 and [B"J
matrices. The equivalent branches susceptances contained in [ B , e q ] and [ B " e q ]
are used in computing [ A P / V ] and [ A Q / V ] . The decoupled equivalent retains the
advantages of the simplified extended Ward equivalent, however, i t requires
the storage of two separate susceptive equivalent networks.
From the survey done based on a l l the above techniques, i t was concluded
that the extended Ward version is the most accurate ( 8 3 ) . The simplied
extended Ward method is the next most accurate. This and the decoupled
versions sacr i f ice a l i t t l e accuracy for certain computational advantages.
The non-decoupled versions can be used with any load-flow algorithm.
The method presented in section 6.5, uses Gauss elimination to perform
reduction of the passive external network and is an extension to the extended
Ward. It provides accurate and reliable resul ts .
159
6.5 External network equivalehcihg
As mentioned in the previous section, the extended Ward method of
reduction provides the most accurate and reliable results. In this method,
following a reduction on the passive external network, equivalent branches
wil l be introduced which connect the boundary buses. In addition, f ic t i t ious
buses and f ic t i t ious admittances connecting the f ict i t ious buses to the
boundary buses wil l be introduced as shown in Figure (6.3) .
The f ic t i t ious buses can be removed i f equivalent injections calculated
and added to that at the boundary buses. It was decided to use the Gauss
elimination as the basis for the reduction process (see 3.2) . In the new
approach however, the reduction is not solely performed on the external
passive network. Basical ly , from the solution to the original network
(external + internal ) , the injections (Generation-load) at each bus which is
to be reduced wi l l be replaced by i t s equivalent admittance, except for the
boundary buses. Hence, the external network wil l be transformed into a
passive network. At this stage following the rules of Gauss elimination (see
3.2), the equivalent admittances wil l be obtained. However, the shunt
admittances calculated in the reduction process, which are connected to the
boundary buses wi l l contribute to the system losses when performing load-flow
on the reduced network. Hence, once the equivalent shunt admittances are
obtained, these wil l be replaced by fixed injections, which in terms are added
to the original injections at the boundary buses. Figure (6.4) i l lustrates
the steps taken for obtaining solutions to a reduced network.
160
The advantage of this approach is that a l l the non-linear loads in the
external network can be well represented in the reduction process. To obtain
accurate reactive responses, bus matching is performed at the slack bus i . e .
keep the active and reactive generation at slack bus constant. This approach
was tested on a number of networks and provided accurate results as in
extended Ward technique. It i s more suitable for power flow analysis, since
the contribution of shunt admittances at the f ic t i t ious buses are eliminated
and in addition the non-linear loads can be more accurately modelled.
Following the reduction process, in order to obtain a better convergence,
a test could be added to remove the branches with bad X/R rat ios.
INPUT DATA FOR INTERNAL
EXTERNAL
SOLE BASE CASE P.F.
INPUT REDUCTION DATA
REPLACE INJECTIONS IN THE EXTERNAL NETWORK BY
EQUIVALENT ADMITTANCE
> i
PERFORM GAUSS ELIMINATION
>
REPLACE EQUIVALENT SHUNTS AT BOUNDARY BUSES WITH
FIXED INJECTIONS
t
PERFORM LOAD-FLOW ON THE REDUCED NETWORK
FIGURE (6.4) FLOW CHART FOR SOLUTION TO REDUCED NETWORK
162
CHAPTER 7
RESULTS
7.1 Introduction to Real-time EMS Integration
The extensive growth of interconnected power systems together with the
operational complexities and the requirement for improved system management,
has led to a greater dependence on automatic control at a l l l eve ls of
operation and the need for control systems with complex real-time functions.
Typical ly an integrated control system consists of two basic
components; Supervisory Control and Data Acquisition (SCADA) function and
power system applications function. These functions impose dist inct ly
different processing requirements on the computers in which they
operate.'- 7 0^
The power system application's function requires mostly a high speed
heavy numerical calculation environment which is best handled by a powerful
floating point processor and large arrays of contiguous, random access
memory. This requirement i s generated by such functions as production
planning and power network security analysis . The recent inclusion of
163
operator training simulators imposes an even heavier calculation load on the
processors.
Two conceptually different hardware configurations have been developed
to handle the processing requirement of an EMS; Centralised Systems and T311
Distributed Processing Systems. J
Centralised systems consist of a main computer or computers which
handle a l l of the EMS functions. Generally, a l l of the processors in a
Centralised System have to handle both SCADA heavy interrupt load and power
systems numerical calculations. Distributed processing systems contain
multiple processing l eve ls where each level i s assigned to a s p e c i f i c
machine or set of machines which are optimised for the individual
requirements, that i s , SCADA heavy interrupt loads and power system
numerical calculations can be sp l i t into different types of machine.
Throughout the evaluation of SCADA systems, attempts have been made to
use distributed processing architectures. These experiments have not been
totally successful due to the lack of appropriate technology. But, in
recent years, the trend toward broad and diversif ied processing requirements
of u t i l i t i e s has created an even stronger need for the development of
effective distributed architecture. As electr ical networks grow and become
more complex, operators need more sophisticated tools to control them. The 1721
technology to distribute the EMS functions is now available.
164
Due to the nature of power systems, the information management system
is usually hierarchical with data passing from the substations and
generating stat ions through area control centres up to the top of the
hierarchy, where global functions such as generation control and network
analysis are performed. The downward flow of t ra f f ic from system control
centres to substations and generating stations is much less in volume than
the upward flow and contains the operator commands and system co-ordinating
and optimisation controls. The power system application functions normally
divide into two areas of real - t ime and study. The real - t ime set run
automatically on a cyc l ic basis or on a power system event.
EMS has three major sources of data; telemetered data, network
parameters and generation parameters. Conventionally the databases in the
system are built using the f a c i l i t i e s of a Database Manager and database
compiler, which are s p e c i f i c to individual appl icat ion a r e a . ^ - ' The
defined databases each form a logical independent description, from which
any appl icat ion may draw the required data . For example the network
database describes the power network and supports a l l the network
appl icat ions such as power flow, and state est imator , the generation
database describes the generation characteristics and data and supports the
generation applications such as AGC, SCADA database for analog and status
obtained from RTUs. The correspondence between the different databases is
also built in an off - l ine mode but updated in real-time i f necessary.
During the integrated operation of the system many aspects of the
165
operation of the system and performance of the individual algorithms are
dynamically tested using the simulation f a c i l i t i e s . The job of managing and
integrating an EMS system is large and complex and requires good software
support tools. Once the integration phase i s accomplished the performance
of the system as a whole (SCADA and EMS) i s tested. ' - 7 2 - '
7.2 OCEPS EMS Software Integration
Large scale systems of which a power system is a prime example, i s an
area in which a wide gap exists between theoretical mathematically based
research and engineering practice. The research programme at Durham is
directed towards bridging this gap by linking some of the available and new
theoretical techniques with the practical requirements of on-line computer
control in power systems.
Testing and validation of analysis and control software is achieved
with the aid of a real-time simulation system developed at Durham
University. In this way the performance of Energy Management software can
be evaluated against a rea l i s t i c model of the network under a range of
operating conditions. A secondary benefit of the integrated simulation,
measurement and control software is the abi l i ty to provide a useful operator
training a id , which includes power system dynamics, telemetering and energy f 3 |
management f a c i l i t i e s . J
The energy management suite OCEPs developed at Durham divides the
166
operational control of electr ical power systems into two major functions;
simulation function and control function. Further, as shown 1n Figure
(1.10) each function comprises of a number of tasks executed in the manner
shown. The simulation and control algorithms can either be executed on a
single Perkin-Elmer 3230 computer main frame using multi-tasking operation
system or the simulation function can be executed on a dedicated
Perkin-Elmer 3230 with simulated telemetry and control information
transmitted s e r i a l l y to a separate control computer. In general the
simulation imposes the major computational load on the system, special
multi-microprocessor hardware is being developed to allow real-time response
with large systems.
OCEPS security analysis subsystem is divided into the following
modules: AUTomatic OUTage selection (AUTOUT), SECurity Assessment program
(SECASS), manual OUTAGE selection (OUTAGE), and SECurity assessment OUTput
program (SECOUT). In the remaining sections of this chapter numerical
results obtained during the course of this research together with the ful l
integration of the security subsystem to the OCEPS suite are presented.
A technical paper was published based on the network islanding 1*271
technique developed during the course of research at Durham. J
167
7.3 Numerical Results
7.3.1 Test Network
The test network is i l lustrated in Figure (7.1) and the associated
network data i s given in Appendix 3. The Universi ty of Durham OCEPS
project thirty substation test systems, as the name suggests, includes 30
substat ions . This network i s bus and node or ientated. Each bus in a
substation is connected via links to other buses in the same substation. By
switching operations the number of electr ical nodes in a substation can
vary. The above network is composed of 72 buses, 41 branches of which 7 are
transfomers (which include f ixed tap, automatic tap changer and phase
shifter transformers), 23 loads, 6 generators and 2 shunts.
Outage of transmission lines 16, 13 or 34 wil l sp l i t the above network
and hence the contingency selection algorithm gives them the lowest rank
( i . e . most harmful).
Network topology changes can be achieved by opening or closing
appropriate breakers in the system. The above process (scenario) can also
be achieved via a prespeci f ied program. Hence l i n e s , u n i t s , loads,
transformers, shunts, nodes and buses may be removed or added to the
network. The above operation is executed by the system manager via an
interactive program.
168
Protection is an inherent part of the simulator and as a result of an
overload, the simulator will t r ip the appropriate breaker and hence a l ine
or equipment may be taken out of service automatically.
7.3.2 Automatic Contingency Selection (ACS)
The ACS problem is concerned with developing computer algorithms for
quickly identifying those contingencies which may cause out-of-limit
conditions so as to reduce the number of contingencies that need to be
evaluated when assessing the power system's security in a Real-time
envi ronment.
Most of the existing ACS procedures are based on l ine over load limit
violat ions, where the Performance Index (PI) i s expressed as a function of f C Q K I T
normalised l ine flows and the use of DC load flow equations.1- ' J The
voltage limit violations are included by calculating the voltage changes
from the linearised load flow equations.'- 5 2-' All these methods are useful
for analysing single branch outage only.
In Chapter 5, the techniques commonly used for forecasting the PI and
the development of ACS algorithms were compared. Four algorithms most
suitable for Real-time purposes were tested and compared with the ranking T571
produced by a ful l A.C. solution for each contingency. J These were: one
ful l iteration of the fast decoupled load flow, f i r s t half iteration of DC
load flow, l i n e outage d is t r ibut ion factors based on DC load flow and
169
f i n a l l y an algorithm based on sensi t ivi ty analysis. In the above reference,
i t was concluded that from a performance view point the l ine outage
distr ibution factors based on DC load flow is superior.
Two algorithms have been implemented in OCEPS for ranking l ine outages
in terms of their impact on the real power flows throughout the network, and
according to their impact on the bus voltages. The essential components
needed in developing each algorithm (as described in Chapter 5) are: a
scalar performance index (PI) for ranking the contingencies and a
computationally e f f i c i en t method for evaluating the PI for each 187 881 contingency.1- * J
The ACS algorithm developed in Chapter 5 has been applied to IEEE 30
bus test network. The results of ACS-were compared with the f u l l A.C. load
flow rankings obtained by considering each outage case in turn .
Figure (7.2) i l lus t ra tes the ACS for l ine over load contingencies. I t
can be seen that the ACS ranking is very similar to the f u l l A.C. load flow
ranking. All the top 10 contingencies, have been ident i f ied correctly by
the ACS. Considering the top 20 contingencies, 18 of the A.C. rankings are
included, however, l ine 30 and 37 outages are not included. Instead less
severe lines 12 and 35 outages are included. The ACS is nevertheless shown
to be entirely satisfactory for most operational purposes.
The voltage violation ACS algorithm discussed in Chapter 5 was then
170
applied to the test network. The results are compared with the f u l l A.C.
load flow rankings obtained from a complete load flow for each l ine outage
with performance index PI evaluated using bus voltage magnitudes (see Figure
7.3). I t can be seen that the detailed A.C. load flow ranking p ro f i l e is
very s imi la r to the voltage v i o l a t i o n ACS ranking. In the top 10
contingencies only line 41 is missed by ACS. For the top 20 contingency
cases, 16 A.C. rankings are included, but lines 4, 7, 28 and 33 are missed
out where as the less severe lines 8, 10, 12 and 19 are included.
Detailed studies of voltage magnitudes showed that only l ine 36 outage
involved voltage v i o l a t i o n . This l i n e has already been ranked by the
voltage violation ACS. Voltage violation ACS consequently proved to be
e f f i c i en t and satisfactory fo r operational purposes.
The above results confirm the findings in references 50, 53 and 57.
Table (7.1) presents the over load and voltage contingency l i s t s
obtained when the above techniques were applied to the 30 bus test network.
In table (7.1) , the f i r s t column 'RANK' gives the contingency p r i o r i t y .
Those with the lower rank have the highest p r io r i ty for f u l l A.C. solution.
The few at the top of the l i s t s (blinking on the terminal screen) are the
most c r i t i c a l contingencies and are those which could s p l i t the network
(lines 13, 15 and 34 as in Figure (7 .1 ) ) . Further, information about the
islands, bus and substation names are provided.
171
The results obtained from f u l l A.C. power flow solutions (for
individual line outages) are shown in tables (7.2 to 7.5). The f u l l A.C.
solutions confirmed that the only contingency which could cause voltage and
power violations was l ine 36, which was already in the ACS l i s t s .
The ACS algorithm was enhanced to provide the contingency l i s t s for
multiple islands. I f due to topological changes, the system is sp l i t into
mul t ip le is lands , the ACS w i l l provide the contingency l i s t s f o r the
multiple islands in paral le l . Before building the system admittance matrix,
the program auotmatically assigns references in the islands without the
original system reference. The advantage of this approach is that there is
no need f o r re-order and r e - f ac to r i s a t i on of matrices and hence less
computations. Table (7.6) provides the ACS results when the o r ig ina l
network was s p l i t into two islands.
To simulate the above case, lines 33, 10, 41 in Figure (7.1) were
removed out of service prior to the ACS solution.
The column 'ISLAND' in Table (7.6) gives the island number for each
branch. From the results shown in this table, i t can be seen that i f l ine
37, rank 8 in island 2 is to be completed, i t could be more harmful than
l ine 5 rank 9 in island 1, as far as abnormal voltage contingency l i s t is
concerned. These informations are v i ta l to the secure operation of the
power system.
172
The information provided by the ACS 1s also very valuable for the
system operators and engineers who monitor the system and evaluate i t s
securi ty con t inua l ly . The system engineers are normally involved in
performing a large number of power flows in order to obtain the answers to a
huge number of 'what i f situations. The ACS by far reduces the
computational e f for ts because i t highlights the most c r i t i c a l contingencies
and ranks them accordingly.
ACS is now an important tool in the contingency analysis subsystem of
today's modern Energy Management Systems (EMS).
5 S
in
I in in
uu
9 I
I
I
8 I I
PS 9 -
in I In I
in I 5=fi I V I
V I I I
C4 in in
in I A
in in
fT in
HP 8
I I
in
FIGURE (7.1) UNIVERSITY OF DURHAM OCEPS THIRTY SUBSTATION TEST SYSTEM
174
Automatic Contingency Selection of Line Overload
\ Note: Outage of Line 13,16 and 34 split the system. They are excluded from the analysis.
KEY
ASCA RANKING
Load Flow Rankir.a
10 15 20 25 30 35 40 LINE OUTAGE
FIGURE (7.2) ACS RESULTS FOR OVERLOAD CONTINGENCIES
w c -S 'Si
^ 5 w "3 S co w cu to
a; OJ <u
2 <u £ >» 6
o
3H3A3S SS31 aasAas SHOW
FIGURE (7 .3) ACS RESULTS FOR VOLTAGE VIOLATION CONTINGENCIES
176
OCEPS OUTAGE INPROGRESS R3.0.0
THE MOST RECENT LISTS
RANK OVERLOAD CONTINGENCY LINE ISLAND BUS SUB
1 13 1 25-36 9-11 2 16 1 38-39 12-13 3 34 1 55-56 25-26 4 36 1 63-61 28-27 5 19 1 37-45 12-16 6 20 1 40-42 14-15 7 29 1 50-51 21-22 8 33 1 53-55 24-25 9 32 1 52-54 23-24 10 12 1 10-34 6-10 11 22 1 43-47 15-18 12 11 1 11-25 6-9 13 30 1 41-52 15-23 14 17 1 37-40 12-14 15 23 1 47-48 18-19 16 2 1 1-4 1-3 17 28 1 30-51 10-22 18 6 1 2-12 2-6 19 15 1 5-38 4-12 20 7 1 6-14 4-6
TABLE (7.1)
08/02/1985 01:27:17
VOLTAGE CONTINGENCY RANK LINE ISLAND BUS SUB
1 13 1 25-36 9-11 2 16 1 38-39 12-13 3 34 1 55-56 25-26 4 36 1 63-61 28-27 5 14 1 25-35 9-10 6 15 1 5-38 4-12 7 5 1 3-9 2-5 8 25 1 33-49 10-20 9 26 1 26-46 10-17 10 31 1 51-54 22-24 11 24 1 48-49 19-20 12 4 1 4-5 3-4 13 37 1 59-66 27-29 14 9 1 12-16 6-7 15 35 1 55-62 25-27 16 38 1 60-71 27-30 17 18 1 37-44 12-15 18 27 1 29-50 10-21 19 41 1 15-63 6-28 20 2 1 1-4 1-3
CONTINGENCY LISTS
177
OCEPS AUTOMATIC SECURITY ASSESSMENT RESULTS 08/02/1985 00:26:11
SUMMARY OF SECURITY ASSESSMENT RESULTS
VOLTAGE VIOLATION REPORT
LINE OUT FROM BUS-SUB TO BUS--SUB AT BUS-SUB VOLTAGE(PU) LIMIT (PU)
36 63- 28 61 - 27 56- 26 0.89362 0.90000 36 63- 28 61 - 27 60- 27 0.88910 0.90000 36 63- 28 61 - 27 65- 29 0.87148 0.90000 36 63- 28 61 - 27 73- 30 0.86162 0.90000
POWER VIOLATIONS REPORT
LINE OUT FOR LINE FROM BUS-SUB TO BUS-SUB POWER (MW) LIMIT (MW)
36 31 51 - 22 54- 24 16.17978 16.00000
TABLE (7.2) SECURITY ASSESSMENT VOLTAGE AND POWER VIOLATION REPORTS
OCEPS AUTOMATIC SECURITY ASSESSMENT RESULTS 08/02/1985 00:26:34 SECURITY ASSESSMENT SUMMARY THIS IS THE MOST RECENT LIST **TASK USESE COMBINED OVERLOAD AND VOLTAGE LIST**
RANK LINE BUS SUB OVERLOAD UNDERVOLTAGE OVERVOLTAGE
1 36 63-61 28-27 Y Y N
2 7 6-14 4- 6 N N N
3 14 25-35 9-10 N N N
4 12 10-34 6-10 N N N
5 37 59-66 27-29 N N N
6 33 53-55 24-25 N N N
7 38 60-71 27-30 N N N
8 20 40-42 14-15 N N N
9 30 41-52 15-23 N N N
10 32 52-54 23-24 N N N
TABLE (7.3) SECURITY ASSESSMENT SUMMARY
179
OCEPS AUTOMATIC SECURITY ASSESSMENT RESULTS 08/02/1985 00:26:34
LINE FLOW REPORT FOR THE OUTAGE OF LINE 36 ** RANK 1 CONTINGENCY**
LINE BUS SUB LIMIT FLOW LINE BUS SUB LIMIT FLOW
1 1- 3 1- 2 130.00 32.69 2 1- 4 1- 3 130.00 19.30 3 2- 6 2- 4 65.00 13.82 4 4- 5 3- 4 130.00 17.18 5 3- 9 2- 5 130.00 34.58 6 2-12 2- 6 100.00 13.71 7 6-14 4- 6 90.00 0.07 8 7-16 5- 7 70.00 -13.68 9 12-16 6- 7 130.00 32.16 10 13-18 6- 8 32.00 -20.25
11 11-25 6- 9 65.00 -2.70 12 10-34 6-10 32.00 7.92 13 25-36 9-11 65.00 -47.78 14 25-35 9-10 65.00 45.09 15 5-38 4-12 65.00 24.61 16 38-39 12-13 65.00 -7.59 17 37-40 12-14 32.00 6.64 18 37-44 12-15 32.00 14.35 19 37-45 12-16 32.00 2.06 20 40-42 14-15 16.00 1.57 21 45-46 16-17 16.00 -0.91 22 43-47 15-18 16.00 2.23 23 47-48 18-19 16.00 -0.41 24 48-49 19-20 32.00 -7.86 25 33-49 10-20 32.00 9.72 26 26-46 10-17 32.00 8.07 27 29-50 10-21 32.00 19.46 28 30-51 10-22 32.00 10.63
29 50-51 21-22 32.00 5.64 30 41-52 15-23 16.00 6.87 31 51-54 22-24 16.00 16.18 32 52-54 23-24 16.00 4.21 33 53-55 24-25 16.00 13.31 34 55-56 25-26 16.00 2.70 35 55-62 25-27 16.00 10.19 36 63-61 28-27 65.00 0.00 37 59-66 27-29 16.00 4.67 38 60-71 27-30 16.00 5.31 39 65-73 29-30 16.00 2.74 40 24-63 8-23 32.00 3.12 41 15-63 6-28 32.00 -3.17
TABLE (7.4) SECURITY ASSESSMENT LINE FLOW REPORT FOR OUTAGE OF BRANCH 36
180
OCEPS AUTOMATIC SECURITY ASSESSMENTS RESULTS 08/02/1985 00:26:34
VOLTAGE REPORT FOR THE OUTAGE OF LINE 36 ** RANK 1 CONTINGENCY**
BUS SUB HIGH LOW VOLTAGE BUS SUB HIGH LOW VOLTAGE
1 1 1.100 0.900 1.061 2 2 1.100 0.900 1.047 4 3 1.100 0.900 1.934 5 4 1.100 0.900 1.028 7 5 1.100 0.900 1.012 10 6 1.100 0.900 1.021
16 7 1.100 0.900 1.011 17 8 1.100 0.900 1.011 25 9 1.100 0.900 1.023 26 10 1.100 0.900 0.998 36 11 1.100 0.900 1.080 37 12 1.100 0.900 1.027 39 13 1.100 0.900 1.070 40 41 1.100 0.900 1.010 41 15 1.100 0.900 1.002 45 16 1.100 0.900 1.009 46 17 1.100 0.900 0.997 47 18 1.100 0.900 0.991 48 19 1.100 0.900 0.987 49 20 1.100 0.900 0.989 50 21 1.100 0.900 0.984 51 22 1.100 0.900 0.983 52 23 1.100 0.900 0.979 53 24 1.100 0.900 0.955 55 25 1.100 0.900 0.909 56 26 1.100 0.900 0.894 57 27 1.100 0.900 0.889 63 28 1.100 0.900 1.020 64 29 1.100 0.900 0.871 69 30 1.100 0.900 0.862
TABLE (7.5) SECURITY ASSESSMENT VOLTAGE REPORT FOR THE OUTAGE OF BRANCH 36
181
OCEPS OUTAGE IN PROGRESS
THE MOST RECENT LISTS
OVERLOAD CONTINGENCY RANK LINE ISLAND BUS
1 33 2 63-61 2 40 2 24-63 3 35 2 55-62 4 34 2 55-56 5 16 1 38-39 6 13 1 25-36 7 32 1 52-54 8 30 1 41-52 9 18 1 37-44
10 20 1 40-42 11 19 1 37-45 12 23 1 47-48 13 24 1 48-49 14 22 1 43-47 15 11 1 11-25 16 31 1 51-54 17 29 1 50-51 18 15 1 5-38 19 27 1 59-66 20 39 2 65-73
SUB RANK LINE
28-27 1 36 8-28 2 40
25-27 3 35 25-26 4 34 12-13 5 16 9-11 6 13
23-24 7 26 15-23 8 37 12-15 9 5 14-15 10 15 12-16 11 14 18-19 12 18 19-20 13 31 15-18 14 4 6-9 15 24
22-24 16 6 21-22 17 9 4-12 18 2
27-29 19 38 29-30 20 27
10/04/1985 01:27:17
VOLTAGE CONTINGENCY ISLAND BUS SUB
2 63-61 28-27 2 24-63 8-28 2 55-62 25-27 2 55-62 25-26
38-39 12-13 25-36 9-11 26-46 10-17 59-66 25-29 3-9 2-5 5-38 4-12
25-35 9-10 37-44 12-15 51-54 22-24 4-5 3-4
48-49 19-20 2-12 2-6
12-16 6-7 1-4 1-3
2 60-71 27-30 1 29-50 10-21
TABLE (7.6) CONTINGENCY LISTS (NETWORK SPLIT INTO 2 ISLANDS)
182
7.3.3 AC Security Assessment (SECASS)
The formulation of a mathematical model of a transmission network is a
pre-requisite for power systems analysis and control by d ig i ta l computer.
I f the Interconnections of the n buses of the system, the appropriate model
of lines and transformers, and nodal inject ion data are known, a load flow
solution gives a l l the unknown voltages, power injections and l ine flows.
Mathematically the problem is one of solving a set of simultaneous [2211321
non-linear algebraic equations. J L •
The analysis of securi ty assessment has evolved from load f low
techniques and is best thought of as a measure of satisfactory performance
of the system following the occurrence of any one of a pre-specified set of
contingencies.
Three numerical methods have commonly been applied to the solution of
the load flow problem: the Gauss-Seidal method, the Newton-Raphson method
and the Fast Decoupled Newton method.
Gauss-Seidal algorithm is ine f f i c ien t for large networks. The Fast
Decoupled load flow method shares many of the properties of the sparse
Newton-Raphson approach and has the further advantage of a reduction in
computer time and memory loading. This is achieved by neglecting the
(weak) coupling effects between phase angles and reactive power and between
voltage magnitudes and active power. J
183
The Fast Decoupled method i s recommended f o r appl icat ion to the i 321
majority of power system networks. J However in cases with very high
resistance to reactance rat io lines (R/X), as in low voltage distr ibution
networks, i t is beneficial to resort to the Newton-Raphson method.
Given a l i s t of hypothetical plant outages assembled either manually
or automatically (by the ACS algorithm), i t is necessary to execute a
series of AC load flow studies to determine whether any l i m i t violations
would result from the occurrence of an outage. In order to achieve more
rapid solutions where a l i s t of similar networks must be analysed, the Fast
Decoupled algorithm has been revised to include the method of modified
matrices (Diakoptics). This techinque allows solutions for single or
mul t ip le branch outages to be obtained without the need f o r matrix
refactorisation when sparsity techniques are applied (see Figure 7.5).
For contingency studies, two methods of approach were considered:
( i ) Repeat load flow solution
( i i ) Application of the recursive Diakoptics method as explained in
Chapter 4.
Figures (7.4) and (7.5) represent the above solution methods based on
Fast Decoupled Newton-Raphson load flow technique.
INPUT DATA
order
factorise
• ITER = ITER +1
Find F ( X ^ )
Find F U 2
r )
X / * 1 - B ' " 1 F ( X ^ )
x 2
r + 1 = B " " 1 F ( x 2
r )
^ Check convergence ^ > Y
OUTPUT RESULTS
FIGURE (7.4) LOAD FLOW SOLUTION
185
INPUT or ig ina l network data
also set X , R , X ?
R
I -> order
f ac to r i se —> ITER = ITER +1
Find F ( X ^ ) Find F ( X 2
R ) X ^ * 1 = B ' " 1 F ( X ^ )
X 2
R + 1 = B " _ 1 F ( X 2
R )
-Check convergence
I OUTPUT RESULTS
L _ CONTINGENCY
METHOD?
load f low Diakoptics
Input data f o r modified network Also set X , \ X . /
OUTAGES INPUT
1 ITER = ITER +1
Find F ( X / ) Find F ( X 2 )
X L
R + 1 = B M F ( X X
R )
apply Diakoptics method to
modify X . R + 1
r+1 B " _ 1 F ( X 2
R )
N
apply Diakoptics method to
modify X 2
R + 1
Check convergence OUTPUT RESULTS
FIGURE ( 7 . 5 ) FLOW DIAGRAM FOR CONTINGENCY STUDIES
186
As can be seen from the f low diagram of Figure ( 7 . 5 ) , when applying
the Oiakoptics technique there i s no need f o r re-ordering and
r e - f a c t o r i s i n g B' and B". The t o t a l execution times required f o r the two
methods of approach are compared i n table (7.7) based on the 30 bus test
network and mul t ip le branch outages.
From the results presented i t can be seen that the Oiakoptics approach
provides the same degree of accuracy as the conventional Fast Decoupled
Newton-Raphson method. However, when the number of simultaneous branch
outages exceeds three, the Fast Decoupled Newton-Raphson method is f a s t e r .
I'32l The above resul ts confirmed the f ind ings of Stot t and Alsac L J and
provided the basis f o r f u r t h e r development of the Diakoptics technique i n
order to obtain network solutions when network s p l i t s occur as a resul t of l'27"l
a branch outage. J
187
30 BUS TEST NETWORK
LOAD FLOW DIAKOPTIC
Time (sec) No. of i t e r a t ions Time (sec) No. of i t e ra t ions
Base solut ion 1.101 6 1.101 6
Base solut ion + single outage 2.061 8 1.903
Base solut ion + double outage 2.092 9 2.073
Base solut ion +
t r i p l e outage 2.090 9 2.124 9
Base solut ion + 5th outage 2.135 12 2.805 12
TABLE (7.7) COMPARISON OF RESULTS BASED ON PERKIN-ELMER 3230
188
7.3.4 Islanding
I n Figure ( 7 . 6 ) i f the l i n e shown w i t h an a s t e r i s k (*) i s t o be
completed, then the network s p l i t s i n to two is lands. The conventional
approach to obtain the so lu t ion to t h i s network i s to r e s t r i c t the solut ion
to the largest is land created. This approach has serious disadvantages in
power systems which are subjected to frequent major outages and f o r which
is landing is a common occurrence.
The common method to solve the above problem i s by solving two separate
power f lows , one f o r each is land created. The disadvantage of t h i s approach
i s that i t requires the formation of B' and B" matrices f o r each is land i n
tu rn and then t o execute two solut ions . In add i t ion , as a f i r s t step to
i d e n t i f y which buses, branches, loads e tc . belong to which i s l a n d , the input
data stage requires a considerable amount of data processing.
The theory of network changes (Diakoptics) as commonly applied to the
load f l o w could not be used d i r e c t l y t o ob ta in the s o l u t i o n t o the
indiv idual is lands. The reason is because during the Diakoptics solut ion a
scalar becomes zero when network s p l i t takes place and hence numerically the
inverse of t h i s scalar i s not obtainable (please re fer to Chapter 5 ) .
The new technique developed during the course of research at Durham,
overcomes the above problem.
189
As mentioned e a r l i e r the new technique i s based on the appl icat ion of
Diakoptics to Fast Decoupled Newton-Raphson, and hence i t i nhe r i t s a l l the
associated advantages.
During the development of t h i s new method, i t was real ised that as a
resul t of one simple mathematical operation, buses, branches etc . i n each
island w i l l be easi ly i d e n t i f i e d (please re fe r to chapter 5 ) .
Tables (7.8) and (7.9) provide the comparison of resul ts obtained using
the new approach based on Diakoptics, and separate power f low solu t ions ,
when the network i n Figure (7.6) was s p l i t due to completion of the branch
shown with an as ter isk .
The results confirmed that the so lu t ion accuracy was s imi la r to Fast
Decoupled method, however, the speed of execution was improved by more than
50%. As the size of networks grow, the advantages of u t i l i s i n g the proposed
technique became more obvious.
In rea l - t ime , topology processing i s usually included i n the state
estimation package in order to construct network connect ivi ty based on the
telemetry. I d e n t i f i c a t i o n of the elements i n each is land ( i f is landing
occurs) requires a large number of data processing act ions. The new method
presented in Chapter 5 reduces the required execution time by a large amount
and i n addi t ion has the advantage of being simple to implement.
n r m c F I G U R E ( 7 * 6 ) UNIVERSITY OF DURHAM OCEPS THIRTY SUBSTATION TEST SYSTEM
191
NO. OF COMPACTING SOLUTION TOTAL ITERATIONS ORDERING TIME TIME
FACTORISING Tl (sec) T2 (sec)
ISLAND 1 wi th o r ig ina l slack
0.513 0.189 0.702
ISLAND 2
DIAKOPTIC TECHNIQUE
6 0.698
6
0.425
0.712
1.123
0.712
TABLE (7.8) RESULTS BASED ON PERKIN-ELMER 3230
GENERATION
BUS NUMBER
1 2 5
13 8
11
SI ack
New Slack
MW
FDNR DIAKOPTIC TECHNIQUE TECHNIQUE
MVAR
40.91 40.00 00.00 00.00
218.90 00.00
40.90 40.00 00.00 00.00
218.89 00.00
FDNR
28.86 •13.08 79.13 27.03
5.47 26.62
DIAKOPTIC TECHNIQUE TECHNIQUE
28.85 -13.17 79.12 26.96
5.44 26.62
TABLE (7.9) COMPARISON OF RESULTS
192
7.3.5 Outages Selection (OUTAGE)
The OUTAGE program allows the operator to ed i t the contingency l i s t
provided i n advance by the ACS algorithm (AUTOUT program). In addit ion
OUTAGE program provides f a c i l i t i e s f o r requesting simultaneous outages of
branches, shunts, loads and generators output.
The operator can request mixed outages of the above components. As an
example, he can request the fo l lowing scenario: simultaneous outage of l ines
A and B, addit ion of l ines C and D ( i f they are already out of se rv ice) ,
reduction of loads E and F ( i n per-uni t or percent) , and s i m i l a r l y f o r a few
generator outputs.
With respect to u n i t s , the operator 's request should be wi th in the
capabi l i ty l i m i t s of ind iv idua l u n i t ' s output, otherwise appropriate alarm
messages w i l l be issued. Loss of u n i t ' s outputs w i l l be red is t r ibu ted
amongst the other units only i f the "unexpected" option i s used. I f the
reduction in un i t outputs are "expected" (according to a defined schedule),
then the r e d i s t r i b u t i o n w i l l not take place.
Loads and uni ts are ranked wi th respect to t h e i r MW/MVAR l eve l s . Table
(7.10) i l l u s t r a t e s the resul ts f o r the case when the output of uni t 2 was
unexpectedly increased by 0.1 per u n i t . As can be seen, the output of other
uni ts decreased. The r e d i s t r i b u t i o n i s done by applying the pa r t i c ipa t i on
factors f o r the uni ts calculated by the economic despatch program. Table
193
(7.11) provides the MW, MVAR ranks f o r loads i n the system. Following a
load r educ t ion ( i n per u n i t or pe rcen tage) , the loss i s r e d i s t r i b u t e d
amongst the u n i t s .
The outage program i s i n e f f e c t the in te r face between the operator and
the securi ty package. Once the operator selects an outage scenario, via the
f a c i l i t i e s provided by the OUTAGE program, he can send a message t o the
SECASS program and request a f u l l AC analysis . During t h i s t r a n s i t i o n
AUTOUT w i l l be paused temporari ly , SECASS stops i t s current execution and
performs the study f o r the operator. At the completion of the analysis a
message w i l l be sent d i r e c t l y from SECASS to OUTAGE to inform the user that
the study i s f i n i s h e d .
Once the results are ava i l ab le , the operator can invest igate them by
inspecting various displays which w i l l be made available to him by the
OUTAGE program. The OUTAGE program is h ighly in te rac t ive and requires no
special operator t r a i n i n g .
The important funct ion provided by the OUTAGE is simultaneous and mixed
outages selection of system components.
194
OCEPS OUTAGE INPROGRESS R3.0.0
GENERATOR BUS-—SUB ACTIVE POWER
08/02/1985.01:27:17
2 3 2 0.674 1 1 1 0.638 4 22 8 0.381 5 36 11 0.378
PLEASE ENTER : GEN.NO., CODE, CHANGE
— CHANGE : IN % OR PU — CODE := 1 EXPECTED
= 2 UNEXPECTED
PRESS RETURN TO STOP
2,2,.1PU
OCEPS OUTAGE INPROGRESS R3.0.0
GENERATOR MAX. GEN MIN. GEN ORIGINAL -P 1 2 4 5
2.00000 1.50000 0.50000 0.50000
0.50000 0.50000 0.10000 0.10000
0.638 0.674 0.381 0.378
08/02/1985.01:27:38
MODIFIED-P
0.618 0.774 0.341 0.338
TABLE (7.10) GENERATOR OUTAGES
OCEPS OUTAGE IN PROGRESS R3.0.0
LOAD-NO. BUS SUB ACTIVE-LOAD LOAD-NO.
08/02/1985.02:10:43
BUS SUB REACTIVE-LOAD
4 9 5 0.272 8 20 8 0.227 5 7 5 0.236 1 2 2 0.112 8 20 8 0.222 7 16 7 0.091
1 2 2 0.167 18 50 21 0.079
6 7 5 0.166 4 9 5 0.061
7 16 7 0.164 10 38 12 0.056
18 50 21 0.123 5 7 5 0.053
10 38 12 0.080 20 53 24 0.045
23 72 30 0.072 14 46 17 0.042
16 48 19 0.065 6 7 5 0.040
14 46 17 0.064 2 4 3 0.026
20 53 24 0.059 3 5 4 0.026
3 5 4 0.056 16 48 19 0.024
12 44 15 0.055 12 44 15 0.018
9 32 10 0.043 9 32 10 0.017
11 40 14 0.043 21 56 26 0.016
13 45 16 0.024 13 45 16 0.014
21 56 26 0.023 23 72 30 0.014
15 47 18 0.023 11 40 14 0.012
19 52 23 0.022 19 52 23 0.011
2 4 3 0.018 22 67 29 0.007
22 67 29 0.016 15 47 18 0.006
17 49 20 0.016 17 49 20 0.006
PLEASE ENTER THE LOAD CHANGES IN % OR PU PRESS RETURN TO STOP
ENTER LOAD-NO., ACT-LOAD, REAC-LOAD
TABLE (7.11) LOAD LIST
196
7.3.6 Security Assessement Output (SECOUT)
The securi ty assessment ouptut program (SECOUT) provides AC solu t ion
results as obtained from the securi ty assessment program (SECASS). I t has
a summary l i s t which provides a quick re fe rence t o h i g h l i g h t those
contingencies which caused voltage or overload v i o l a t i o n s . Consequently,
the operator can select the appropriate contingency and study in de ta i l the
results of securi ty assessment.
I f the contingency resul t l i s t i s outdated, the program automatically
sends an alarm to inform the operator. The operator can then request the
most up to date l i s t . Tables (7.12) to (7.14) i l l u s t r a t e t yp i ca l SECOUT
program contingency output r e su l t s .
OCEPS AUTOMATIC SECURITY ASSESSMENT RESULTS 08/02/1985.00:26:34 SECURITY ASESSMENT SUMMARY THIS IS THE MOST RECENT LIST **TASK USESE COMBINED OVERLOAD AND VOLTAGE LIST**
RANK LINE BUS SUB OVERLOAD UNDERVOLTAGE OVERVOLTAGE
1 36 63-61 28-37 Y Y N 2 7 6-14 4-6 N N N 3 14 25-35 9-10 N N N 4 12 10-34 6-10 N N N 5 37 59-66 27-29 N N N 6 33 53-55 24-25 N N N 7 38 60-71 27-30 N N N 8 20 40-42 14-15 N N N 9 30 41-52 15-23 N N N
10 32 52-54 23-24 N N N
TABLE (7.12) SECURITY ASSESSMENT SUMMARY
198
OCEPS AUTOMATIC SECURITY ASSESSMENT RESULTS
LINE FLOW REPORT FOR THE OUTAGE OF LINE 36
LINE BUS SUB LIMIT LINE 1 1- 3 1- 2 130.00 32.69 2 3 2- 6 2- 4 65.00 13.82 4
5 3- 9 2- 5 130.00 34.58 6 7 6-14 4- 6 90.00 0.07 8
9 12-16 6- 7 130.00 32.16 10
11 11-25 6- 9 65.00 -2.70 12
13 25-36 9-11 65.00 -47.78 14
15 5-38 4-12 65.00 24.61 16
17 37-40 12-14 32.00 6.64 18
19 37-45 12-16 32.00 2.06 20
21 45-46 16-17 16.00 -0.91 22
23 47-48 18-19 16.00 -0.41 24
25 33-49 10-20 32.00 9.72 26
27 29-50 10-21 32.00 19.46 28
29 50-51 21-22 32.00 5.64 30
31 51-54 22-24 16.00 16.18 32
33 53-55 24-25 16.00 13.31 34
35 55-62 25-27 16.00 10.19 36
37 59-66 27-29 16.00 4.67 38
39 65-73 29-30 16.00 2.74 40
41 15-63 6-28 32.00 -3.17
08/02/1985 00:26:34
** RANK 1 CONTINGENCY**
BUS SUB LIMIT FLOW 1-- 4 1-- 3 130.00 19.30
4. - 5 3-- 4 130.00 17.18
2--12 2-- 6 100.00 13.71
7--16 5. - 7 70.00 -13.68
13--18 6' - 8 32.00 -20.25
10--34 6 -10 32.00 7.92
25> -35 9> -10 65.00 45.09
38 -39 12 -13 65.00 -7.59
37. -44 12--15 32.00 14.35
40--42 14. -15 16.00 1.57
43 • -47 15--18 16.00 2.23
48 -49 19 -20 32.00 -7.86
26 -46 10. -17 32.00 8.07
30 -51 10 -22 32.00 10.63
41 -52 15. -23 16.00 6.87
52 -54 23 -24 16.00 4.21
55 -56 25 -26 16.00 2.70
63 -61 28 -27 65.00 0.00
60 -71 27 -30 16.00 5.31
24 -63 8 -23 32.00 3.12
TABLE (7.13) SECURITY ASSESSMENT LINE FLOW REPORT FOR
OUISGE OF BRANCH 36
199
OCEPS AUTOMATIC SECURITY ASSESSMENTS RESULTS
VOLTAGE REPORT FOR THE OUTAGE OF LINE 36
BUS SUB HIGH 1 .OW VOLTAGE BUS
1 1 1.100 0.900 1.061 2 4 3 1.100 0.900 1.934 5
7 5 1.100 0.900 1.012 10
16 7 1.100 0.900 1.011 17 25 9 1.100 0.900 1.023 26
36 11 1.100 0.900 1.080 37
39 13 1.100 0.900 1.070 40 41 15 1.100 0.900 1.002 45
46 17 1.100 0.900 0.997 47
48 19 1.100 0.900 0.987 49 50 21 1.100 0.900 0.984 51
52 23 1.100 0.900 0.979 53
55 25 1.100 0.900 0.909 56
57 27 1.100 0.900 0.889 63
64 29 1.100 0.900 0.871 69
08/02/1985 00:26:34
** RANK 1 CONTINGENCY**
SUB HIGH LOW VOLTAGE
2 1.100 0.900 1.047 4 1.100 0.900 1.028 6 1.100 0.900 1.021
8 1.100 0.900 1.011 10 1.100 0.900 0.998
12 1.100 0.900 1.027 41 1.100 0.900 1.010 16 1.100 0.900 1.009 18 1.100 0.900 0.991 20 1.100 0.900 0.989 22 1.100 0.900 0.983
24 1.100 0.900 0.955 26 1.100 0.900 0.894 28 1.100 0.900 1.020 30 1.100 0.900 0.862
TABLE (7.14) SECURITY ASSESSMENT VOLTAGE REPORT FOR THE
OUTAGE OF BRANCH 36
200
7.3.7 Integrat ion of Security Assessment Package to OCEPS
OCEPS' contingency analysis solut ion approach i s divided in to two
major processes. Contingency screening (AUTOUT program), and f u l l AC
contingency processing (SECASS program).
As an aid to the network control engineer, i t i s possible to assess
the e f f e c t of any scheduled or unexpected branch outages by means of a f a s t
approximate load f low method. Contingency screening i s the f i r s t stage of
the a n a l y s i s . This module p r e d i c t s the number and s e v e r i t y of any
overloads or abnormal voltages which may resu l t from the switching
operation or breaker status change under considerat ion. I t ranks the
harmful contingencies by assigning p r i o r i t y orders. Those wi th the lower
rank order are more l i k e l y to cause v io la t ions than those wi th higher rank
order.
Screening i s a quick way to determine which contingencies are
d e f i n i t e l y harmless. At present, the l i s t of f i r s t ten contingencies w i l l
be processed by f u l l AC s o l u t i o n .
The second stage, secur i ty assessment, examines those contingencies
t ransfer red f o r f u l l AC so lu t i on . The secur i ty assessment phase, hence,
proceeds to analyse the i n t ac t system and the preselected outage cases.
The securi ty assessment program is based on f a s t decoupled Newton-Raphson
and f o r outages simulation i t applies Diakoptics as discussed in previous
201
chapters. This program runs automatical ly based on a p re -spec i f i ed
execution cycle .
I f the network topology changes, the security assessment program will
be paused until the new contingency l i s t i s evaluated by the screening
program.
I f the network configuration changes and the intact networt sp l i t into
two, the program assigns automatical ly a reference bus in subnetwork
without the original reference bus. The screening process is also capable
of selecting contingency l i s t s under such circumstances.
The new technique developed by the author for network islanding is
incorporated in the OCEPS security analysis package. If as a result of a
branch outage the network sp l i ts into two, then by the application of the
new technique the security assessment provides ful l AC solution for both
subnetworks in para l le l .
Figures ( 7 . 7 ) through ( 7 . 9 ) i l lus t ra te OCEPS coordination of
subsystems and the integration of the security assessment package.
202
STATE ESTIMATES
\
FAULT STUDIES
AUTOMATIC OUTAGE SELECTION
SELECTED OUTAGES
i
SECURITY
/CRITICAL ( OUTAGE \ L J S T
GENERATION AND LOAD RESCHEDULING
FIGURE 1 7 . 7 ) SECURITY ANALYSIS SUBSYSTEM
$ ACCESS
WRITES INTO
A U T C M A TIC
ID sBcunv ormrr cot, H j O C X
5 cumty mou l t s
c u w u t
/CONT VCOM.
CONTINGENC BLOCK D OUTAGES
ASSESSMENT
AUTO. DATA
OUTAGE COM. BLOCK
S
UAMUAL DATA
FIGURE ( 7 . 3 ) OVERALL OCEPS SECURITY ASSESSMENT PACKAGE
204
SIMULATION F U N C T I O N I A N A L Y S I S AND C O N T R O L F U N C T I O N S
S V M I M I W S O M I
rutuKci
HTM SAT*
. f CIHWTOl . m [ ourrai
MT1 VM.IDAIICH WD S U T C UTtMIIOK K nntKM. "N
I T I H H DATA J
c c o o i i c I I S P A I C H
~*—(
/
GKUTa CMtnaNT TAOTITJ
CI tuei CMIMATKX
I ACT gPOLOCT > Q
t u i w i n W 1 V O K > 4
f M iTOt lCM. \ -I LOAI &ATA I
ItCMITr AHALT1II
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M l 13A)
FIGURE ( 7 . 9 ) OPERATIONAL CONTROL OF ELECTRIC POWER SYSTEMS OVERVIEW OF MAJOR FUNCTIONAL ELEMENTS
205
CHAPTER 8
CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEACH
8.1 Conclusions
In this work consideration has been given to the Real-Time Electr ical
Power Network Security Assessment which is a prerequisite of any on-line
control and monitoring scheme and which in i t s e l f gives a significant saving
of operational cost by providing an indicat ion of system secur i ty and
information about the power system response after equipment outages have
been introduced.
D ig i ta l computers are used widely in modern power systems in order to
carry out Supervisory Control and Data Acquisition (SCADA) and Automatic
Generation Control (AGC). The SCADA function provides a basis for the
monitoring and control. Montioring of l imit values, trends and simple means
for post-disturbance analysis such as sequence of event recording/replay
action and post mortem review are a few examples of the SCADA function.
Beyond the SCADA category, a lot of functions deal with system security.
The security of a power system is generally judged in terms of the
system's abi l i ty to withstand the impact of sudden changes due to the loss
of transmission l ines , transformers, generators, loads and compensators.
206
Histor ical ly , the load flow is used as a simulation tool when analysing
system security. For steady state security assessment of the existing and
future system, a general purpose load flow solution method based on the Fast
Decoupled Newton Raphson was presented in Chapter 4. The program
development takes f u l l advantage of spars i ty techniques and simulates
automatic tap changers and phase sh i f ters .
The conventional method for simulation of branch outages is based on
load flow techniques and requires the refactorisation of system admittance
matrices for individual branch outage cases. This method is ineff icient for
operational purposes in Real-Time environment.
The solution to a l inear or non-linear network can be obtained from the
inverse of the system admittance matrix. Theory of network changes has been
studied in Chapters 3 and 4. As demonstrated in these chapters, the new
solution to a network subjected to topological changes, can be obtained from
the original solution in an eff icient and organised manner via the
application of Diakoptic techniques. This theory was further developed to
simulate simultaneous branch outages. The technique proved to be eff icient
(up to 3 simultaneous outages) and most suitable for the Real-Time
applications and was successfully applied to the OCEPS security assessment
package.
The conventional techniques for assessment of transmission networks
secur i ty normally assume that the network to be studied w i l l only be
1
207
subjected to a relat ively small number of contingencies simultaneously and furthermore that those contingencies w i l l not s p l i t the network into islands. Should such a situation a r ise , the approach is usually to restr ic t analysis to the largest island so created.
The above approach has serious disadvantages in power systems for which
islanding is common occurrence. In Chapter 6 a unique approach for the
Diakoptic solution of the network flow when subjected to configuration
changes was presented. The algorithm proved to be accurate and ef f ic ient
and most suitable for the online security assessment of power systems.
The above technique was successfully implemented in the OCEPS security
assessment subsystem and based on the results obtained, a technical paper
was published in the IEE second international conference on power system
monitoring and control at Durham 1985.
The common procedure for the security check is to evaluate meaningful
contingency cases. If the potential outage results in limit violat ions,
then the system is said to be vulnerable. This system state should be
reliably detected for possible corrective rescheduling actions.
In an on-line environment, the computational burden that the above
task imposes, does not allow the simulation of a l l conceivable
contingencies. Therefore, security analysis must be restricted to
presumable c r i t i c a l cases defined in a contingency l i s t .
208
In order to reduce the number of contingencies that need to be
evaluated when assessing the power system's security in a Real-time
environment, an Automatic Contigency Selection (ACS) algorithm based on the
fast DC power flow was presented in Chpater 5.
The ACS as implemented in the OCEPS sui te , provides the voltage and
overload contingency l i s t s when network i s subjected to configuration
changes (mult i - islands). In OCEPS implementation, the ACS is executed
automatically according to a pre-selected time step. This package is also
event driven and as a result of a configuration change i t is automatically
triggered to execute.
As discussed in Chapter 7, once the ACS results are available, those
above a certain threshold are passed for a detailed network flow analysis
which is also executed automatically. The contingency l i s t provided by the
ACS can be edited via an interactive program developed and integrated into
OCEPS.
The program allows selection of mixed equipment outages and via system
messages to the security assessment package, provides AC solutions for
further investigation.
The general concept of the organisation and the integration of the
network security assessment software package to the OCEPS suite has been
209
described in Chapter 7. A useful degree of f l ex ib i l i t y has been achieved by
adopting a modular structure of the package in which the programs
communicate with others via task common blocks.
With the development of e lectr ical power industry and the
interconnection of isolated power systems, very large power systems are
formed. The expansion of the system brings not only the great economical
and technical rationality into the system, but also the complexity into the
analysis and the computational simulations of the system. The complexity i s
a great challenge to the s ta t ic and dynamical analysis and computationally
impossible in terms of Real-Time monitoring and control of large scale power
systems.
One way to solve the above problem is by using network equivalent
techniques. Conventional techniques developed to perform electr ica l network
reduction have been studied in Chapter 5. A new approach for obtaining
network equivalents in an off - l ine environment has been presented in Chapter
6.
210
8.2 Suggestions for further research work
The following topics related to the Real-Time security analysis
discussed in this thesis require further investigation:
1 - The theory of network islanding could be further developed to
cater for cases when simultaneous outages of branches cause a
sp l i t or multiple is lands.
2 - The ACS algorithm could be further developed to include
appropriate ranking of loads and generators in the system.
3 - Inclusion of bus and station outages in the security package.
4 - Simulation of c i rcu i t break outages.
5 - Detection and provision of appropriate alarms i f bus sp l i t occurs
as a result of switching.
6 Development of one line diagrams (graphics) in order to select
contingencies via the one l ine diagram and also graphical
representation of the security assessment resul ts .
211
7 - Development of security assessment algorithms to include the
dynamic/algebraic equations of system components in order to
provide dynamic response of the system when subjected to
equipment outages. With the advance in computer technology such
algorithms could soon be implemented in Real-time environments.
Such approaches wil l be most valuable in systems where islanding
is common place.
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69. H.DURAN and N.ARVANTIDIES, "S impl i f i ca t ion for Area Secur i ty A n a l y s i s : a New Look at Equivalence", IEEE Trans. Power App, Vol . PAS 91, pp 670-679, MAR/APR. 72.
70. B.BARAZESH, J.W.LLOYD and W.D.WILSON, " Integrat ion of Modern Real-t ime Energy Management Systems", Second Internat ional Conference on Power System Monitoring and Cont ro l , Durham 1986.
71. A.CREW, C.A.LYNCH "The Development of a Distr ibuted Archi tecture Energy Management System", IEE Second Internat ional Conference on Power System Monitoring and Control , 8-11 JULY 1986, Durham, pp 211-216.
72. M.J.STERLING, M.R.IRVING, "Real-t ime Simulation for the V e r i f i c a t i o n of Advance On-l ine Control Algorithms", IEE Second Internat ional Conference on Power System Monitoring and Cont ro l , 8-11 JULY 1986, Durham, UK.
73. SASSON, A.M.. 1970 "Decomposition Techniques Applied to Non-linear Programming Load-flow Method", IEEE Trans, on Power Apparatus and Systems, Vol . 89, pp 78-82.
74. UNDRILL, J.M. and HAPP, H.H., 1969 "Multi-computer Configurations and Diakopt ics: Real Power Flow in Power Pools" , IEEE Trans on power App and System Vo l . 88, pp 78-796.
75. UNDRILL, J.M. and HAPP, H.H., Automatic S e c t i o n a l i s a t i o n on Power Systems Networks for Network Solut ions" 1970 T-PAS 71 JAN-FEB., pp 46-53.
76. PARSON, E .K . JULY 14-19, 1974 "Network Equivalents for On-l ine Systems", presented at IEEE PES Summer meeting and Energy Resources Conference, Anaheim, CA.
77. DURAN, M. and ARVANTICIDES, N.V. JAN. 31, FEB. 5, 1971 "S impl i f ica t ions for Area Securi ty A n a l y s i s : a New Look at Equ iva lents" , paper 71 TP54-PWR, presented at the IEEE Winter power meeting, BY.
78. WARD, J .W. , 1949, "Equivalent C i r c u i t s for Power Flow S tud ies" , Trans, A I E E , Vo l . 68, pp 373-383.
79. G.CONTAXIS and A.S.DEBS, " I d e n t i f i c a t i o n of External Equivalents for Steady-state Secur i ty Assessment" presented at 1977 IEEE PAS Summer meeting, Mexico C i t y , JULY 1977.
80. T.E.DY LIACCO, K.A.NAMARAO and S.C.SAVULESCU, "An On-l ine Topological Equivalent of a Power System", i b i d .
81. W.F.TINNEY and W.L.POWELL, "The REI Approach to Power Network Equiva lents" , PROC. IEEE PICA Conf. Toronto, pp 314-320 MAY 1979.
82. F.L.ALVARDO and E.H.ELKONYALY, "Reduction in Power Systems", presented at 1977 IEEE PAS Summer meeting, Mexico C i t y , JULY 1977.
83. MONTICELLI, S.DEKMANN, A.GARCIA, B.STOTT, "Real Time External Equivalents for S t a t i c Secur i ty A n a l y s i s " , IEEE Trans on PAS Vo l . PAS-98, No. 2, MAR/APR. 1979.
84. D.DENZEL, N.6RAF and J.VERSTEGE, " P r a c t i c a l User of Equivalents for Unobservable Networks in On-l ine Secur i ty Monitoring", Power Systems Computation Conf. Cambridge, SEPT. 1975.
85. L.W.EMARK, "Computational Implementation of On-l ine Load Flow" presented at 1977 IEEE PAS Summer meeting, Mexico C i t y , JULY 1977.
86. A . C . Securi ty Assessment, F.G.VERVLOET, A.BRAMELER, PROC, I E E E , Vol . 122, No. 9, SEPT. 1975.
87. M.J.H.STERLING and D.T.Y.Cheng, "Automatic Contingency S e l e c t i o n " , Department of Engineering, Univers i ty of Durham, 1982.
88. M.J.H.STERLING and D.T.Y.Cheng, "Voltage Violat ion Automatic Contingency Se lect ion" Department of Engineering, Univers i ty of Durham, 1982.
APPENDIX 1
TECHNICAL PAPER
41
TRANSMISSION NETWORK SECURITY ANALYSIS INCORPORATING NETWORK ISLANDING
A R Shaf1e-Pour, M J H Sterling and M R Irving
University of Durham, UK
ABSTRACT
Conventional techniques tor assessment of transmission network security normally assume that the network to be studied will only be subject to a relatively small number of contingencies and furthermore that those contingencies will not spilt the network Into Islands. Should such a situation arise the approach Is usually to restrict analysis to the largest Island created. This approach has serious disadvantages In power systems which are subjected to frequent major outages and for which Islanding Is a common occurrence.
This paper presents a new technique' for the dlakoptlc solution of the network flow when subjected to configuration changes. The. technique Is shown to be applicable to - the. - automatic selection of system contingencies and the subsequent a.c. security assessment for these contingencies. In the dlakoptlcal method the solution for the Individual Islands Is found using the sparse factors of the Intact network matrices by means of additional forward and backward s u b stitutions and since It avoids the need to reformulate and tactorlse the network matrices It Is most suitable for reai-Ume security assessment.
INTRODUCTION
Security assessment analysis has evolved from load-flow techniques and provides a measure of system performance following the occurrence of any of a pre-speclfled set of contingencies. The selection process Is normally Implemented by an automatic contingency selection algorithm (ACS) commonly based on d.c. loadflow (1.2). The ACS ranks the outages In order of decreasing severity and those above a certain threshold, which will not spilt the network Into Islands, are passed for a detailed network flow analysis. The conventional method for simulation of branch outages which cause a spilt network Is based on the Introduction of a new slack busbar and redistribution of active power among the specified generators In each subsystem formed and Anally refactorlsatlon of [B'l and IB"! matrices taking Into account the existence of a second reference and the branch outaged.
The present paper describes a new algorithm tor the dlakoptic solution of the network flow when a spilt network occurs as a result of a branch outage. The new technique avoids the need to reorder and refactorlse IB') and IB"] matrices. It has been tested using a range of networks and compared with an alternative approach In which the network Islands are formulated and solved as separate load flow problems. From the numerical results obtained It can be seen that the new technique has advantages for application In reel time security assessment
SUMMARY OF BASIC ALGORITHM FOR OUTAGE STUDIES
Tne application of dlakoptics to linear and non-linear networks Is summarised.
Assuming that V Q Is known from the solution of the
original linear equation
0 0 o ( 1 )
where Y Q represents the original admittance matrix. The
new value V n 0 w alter one branch has been completed can
be obtained from
new o o
where
(Ml = row vector which is null except for l i > +1 and M j » - l 1
X • Z M Is the difference of columns I and J o *
0 , 2 o
• Y _ _ 1 stored In factored form o
• Is the difference of elements I and | of V o
0 M V
C - (1 /b • M S O - 1 a scalar factor
b • line or nominal transformer aeries admittance.
The above approach Is fast, accurate and avoids refactorlslng Y .
Load flow Is well known as a non-linear problem. The test decoupled Newton-Raphson (FDNR) loadllow has proved fast, sufficiently accurate and reliable [31. n is already well documented and therefore no detailed description will be presented. The basic model is described by:
(A«) • I B 1 ] " 1 I A P / V 1
fAVl - I B - f 1 IAQ/V)
( 3 )
(4)
The numerical technique based on dlakoptics developed for line Iteration voltages:
outages process
can be applied to in order to obtain
Step 1: - I B ' l " 1 I A P / V 1
Step 2 : X' - I B * f N M
Step 3: i / c - - w • M V
Step 4: , A S n e w ' - fA* 1 - C ' X ' M ' A S . 0 0 where b' - 1 / X | k
<5)
For voltages, similar steps to those above can be Implemented and the new voltages are:
| A V N E W ) - A V ° - C ' X - M ' A V 0 ( 6 )
where
42
PRE-PROCESSING FOR SYSTEM SEPARATION
Consider Fig ( la) which represents two Islands Inter connected by the tle-llne between buses I end ]. The Thevenin equivalent circuit lor the above Is' shown in Fig ( lb) .
in Fig ( lb) . Z ( and Z b are the equivalent Impedances of the two
Islands. The Impedance matrix lor this circuit Is given »y:
V
z. 2. Z a V l 2. V z i ( v ,
(7)
tub-matrix ( 2 ^ ] is me Impedance matrix 01 tne
connected part and (Z 4 ° ) should be modified to obtain
the Impedance ol the second Island. In this work. Y Q is
assumed symmetrical and since the row elements in suD-matrlx IB) have equal eo-tactors. the row elements in
' [ Z 2 ° ] are equal.
It l v Ig. .. I m represent the injections at the buses
ol the Intact network, the' original solution Is given by:
V - Z I o o o
or:
if now tne tie-line between buses I and J is to be removed, from equation ( 2 ) we have:
Step 1: X » Z M
0
-1 /y. I
Step 2 : 1 / C » ( l / b * MSO
(- l /y ( j • 1/y,,)
Therefore
1 / C - 0
¥ l mWi2W3* - • 4 V | * V i / ' " ^irn'm
where column J. k ... m are referred to as the. buses in the second island.
When the' tie-line Is removed, the solution to tne connected part can be obtained from:
new o V l " v l " Y ' u ' k " ' i m ' r n » 2
n e W - v 2 ° - « 2 ) l f z 2 k l k ^ 2 m . m
Hence a split network results when 1 / C - 0 . Without lurther calculation and by Inspection of 'X' obtained from step 1. the buses In each Island are detected:
bus
I
I
t— island 1 (with the original slack)
island 2
wnen diakoptics is applied to FONR. following the steps above 1 / C . becomes very small but not Identically le ro and furtner. vector 'X' would not hold absolute zeros.
• A sensitivity factor should therelore be specified which is a function of the computer on which the study is implemented.
new o *l " y l - V f V . — ^ i V m
(8)
A column in | Z 2 ° ] can be chosen, corresponding to the
new reference busbar and hence equation (8) can be presented as:
now o * 2 - * 2 S r V
ISLANDING
By removing the branch between. buses i and j In Fig ( 1 ) . . the original network divides Into two islands. The original admittance and Impedance matrices are in the following form
V . Z . I
1 1 i •
C | .
: H * -
D ; ; : :
1 '
s;
Z 2 °
v . n e W " V . ° - I . r V ' l i * - V
In matrix form the above can be represented as:
i v n e w i - cvoi - c / aoi c 2 ' i 0
| v n e w ) gives the solution to the connected part,
where: IV 1 « H Ml 1
O 0 0 C/-LTE • i i ; 1 I I I
ct»-crDZSGEori
(9)
43
C , ' it lero everywhere but * r for the new reference
Duiotr. Hence C.'aV l i • column vector corresponding to i o
me new reference busbar. C ?
! Is w o tor Island 1 and
' V tor island 2. It Is set when calculating (X] as In
the previous section. C j ' ts • scalar factor. H
in ine subsystem without the original reference bus. mere Is no provision for tne Introduction of a new slack bus. then the reference Is taken as the busbar of
the new subsystem connected to the outaged branch.
The advantage of applying equation (9) to obtain the new solution for the connected network Is that It avoids refactoriting
To obtain the solution to the subsystem without the original reference in parallel with the solution of the connected part, the following equation can be applied:
where
(10)
l " - f U y Zy V
ro is the shunt admittance
connected to the reference bus
original
Z 3 " , < V r l r , / c 2 V
* S * r l r l
iQ refers to the original reference bus and r , refers to
me new reference bus. The above can be applied directly to automatic contingency selection algorithms which are based on dc loadflow.
APPLICATION TO AC LOAD FLOW
When a split network arises as e consequence of a branch outage, conventionally a new reference busbar Is chosen in the Island which Is not connected to the original system. This busbar can be Introduced es an Input by the operator or automatically chosen, as the busber which had the biggest active generation prior to the outage. N must be among the generators specified for redistribution of active power. The draw back of such approaches is the need to Introduce the existence of the second reference and the outagad branch In IB') and IB'). It is consequently proposed that the new reference is chosen automatically, but as explained In the previous section, its existence Is not reflected In IB'l and IB'] . Hence, there Is no need to retactorlse.
The new technique was applied to FDNR according to the flow diagram shown In Fig (2). Since PV buses are not Included in IB') , no modification Is required to be performed after voltages are obtained In each Iteration.
NO o r i ITERATIONS
COMPACTING' SOLUTION TOTAL ORDERING TIM1 • TIME fACTORISINO T l IMC) T l M C I T I M I f CI !
ISLAND 1 wl l l l or ig inal • M e *
4
i
0 . M 1 t i n I o ioi
ISLAND | t O . t M 0 4 » 1 I N
OMKOPTK TCCHNIOUI
t o . n i o . n i
TABLE 111 - R M I I I W U l M On PE RUIN -ELMER S I M T l and T t ara a t • * » « • M H a O l
GENERATION
BUS NUMBER MW MVAR 1
BUS NUMBER FDNR OWKOPTIC TECHNIQUE TECHNIQUE
FDNR OUMOPTIC TECHNOJUE TECHNIQUE j
1 BMCt t 1
IS 1 Naw Slack
1 1
4 0 . 1 1 4 0 . M 40.00 40.00 00.00 00.00 00.00 00.00
111.10 t l t . t t 00.00 00.00
M . M M . l l -1S.00 - 1 1 IT
T t . 1 1 T i l l I T U M . M . '
S.4T 1 4 4 N . U f t - B I
TABLE tt> - Compan ion t l m u l l s
CONCLUSION
When a split network arises es a consequence of a single branch outage, the above technique based on tne dlakopilcal solution of network flows has been shown to be efficient, accurate, fast and most suitable for tne real time* security assessment of electrical power networks. It avoids the need to refactorise (B'l and IB") and furthermore, modifications to angles are onry required after each Iteration. If automatic tap changers ere Included In the system, the tap positions on the Island with no reference should be set to the Initial tap positions, before any modifications are performed.
R E F E R E N C E S
T.A. Mlkollnnas. B.T. Wollenberg. 'An advanced contingency selection algorithm*. I E E E Trans, on Power Apparatus and Systems. Vol. PAS-100. No. 2. Feb. 1981. pp. 608-617.
S . Vermurl. R.E. Usler. 'On-Une automatic contingency selection algorithm*. I E E E Trans, on Power Apparatus and System. Vol. PAS-102 . pp. 346-354. Feb. 1983.
B. Scott. 0. Aisac. 'Fast decoupled load tiow ' I E E E Trans, on Power Apparatus and System. P A S -93. 1974. pp. 8S9-B67.
The new technique has been applied to many linear and non-linear networks and shown to be accurate and reliable. In Fig 13) If the tie-line between bus 2 and bus 6 is removed, the Intact network divides Into two Islands. Tne new technique has been applied to obtain the network solutions for both Islands in parallel and compared witn those obtained using two separate loaonows The results are given in Table (1). It should be noted that the times shown for tne .onventionai approach eiciude the required time (or data input. The loadtlow results are compared In Table (2). Table(3)and t a b l e ( 4 ) i l l u s t r a t e the t e s t data network.
44
4
(a) - Network interconnection
<b) - Equivalent Circuit
1NPU1 DATA IWIIIAUSC; v • i « • o
1 FACTORISE B'. B '
I INPUT DATA FOR OUTAGES COMPUTE CORRECTIONS FOR MISMATCH EQUATIONS
rmo A P/» FOR PV AND PO BUSES
SOLVE IB ) • IB'IIAP/VI FOR PV AWO PQ BUSES
MOOiFt —H USING EQUATION I I011
SOLVE V • IB'IIAOSVI FOR PO BUSES
IV > V « « V
X H CHECK FOR CONVERGENCE I
Icowvenogp" I
I OUTPUT RESULTS |
ire CI) - Network interconnection Figure (2) - Flow diagram lor the Implementation Islanding In FDNR.
Lj LINE DESIGNATION • P V » p» SUSC p»
1 2 0 01B2 0OS7S 0 0000 I S 0.0452 0.1552 0 0204 f ' 4 0 0570 o . i r i r 0 0154 I 4 0.0132 0 01T5 0 0042 2 * 0.0SS1 0 1751 0 0167 t 004SO 0.1150 0 0102 S » 0.020? 0.0520 0 0005 * I 0.0120 0.0420 0 0000 0 t 0.0000 0.2050 00000
• 10 0.0000 0.5560 0 0006
• 11 0.0000 0.2060 0 0000
• 10 0.0000 0.1100 0 0000 4 12 0.0000 0.2)50 0 0000
17 12 0.0000 0.1400 0 0000 12 14 0.1251 0.2SS5 0 0000 12 IS 0.0552 0.1104 0 0000 12 1 * 0.0545 0.166T 0 0000 14 IS 0.2210 0 1557 0 0006 IS 10 0.1070 6.216) 0 0000 11 I f 0.00S6 0 .11 f t 0 0000 I f 20 0.0140 0.0660 ooooo 10 IT 0.0524 0.064) 0 0000
!• 21 0.0145 0.074t ooooo 10 22 0.0T2T 6.1466 ooooo 21 22 0.0115 6.0216 ' ooooo IS 23 0 1000 0.2020 o o o o o 22 24 0.1150 61760 ooooo 24 25 0.1505 M i l ooooo 25 26 0.2544 0 1100 . ooooo 2 t 27 0 1005 0.2067. ooooo 2S . n 0 0000 0 2666 ooooo 2» tt 0.2195 - 0 4151 ooooo 27 so 0 1 2 0 ! " 0 5027 0 0000 2 * so 0.2100 0 4 ) 1 1 0 0000
• 2 t 0.0550 0.2000 0 0214 « 2 t O O l S t 0.0)06 0 006)
10 10 0.0000 -5.200 ooooo 24 24 0.0000 - 2 5 00 ooooo
re (3) - Te»t Network TABLE (3) - Impedance and line charging data.
BUS NuMBE* VOLlAGE ipui OENEMTIOH LOAD
MW MVAR MW MVAR
i 1 M O O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J 1 04SO 4 0 0 0 0 0 0 0 2 1 TO 1 1 TO 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 2 0 4 0 0 0 0 0 0 0 0 0 ' • 0 0 0 T 6 0 1 0 0 5 1 0 1 0 0 0 0 0 0 0 0 . 0 0 . • 4 . 2 0 1 ( 0 0 « 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 T 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 2 2 . 0 0 1 0 . 0 0 t i . o i oo 0 0 0 0 0 0 . 0 0 J O 0 0 3 0 . 0 0
• 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 5 0 0 2 0 0 1 1 i oa?o 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 1 1 ooooo 0 0 . 0 0 0 0 0 0 1 1 . 2 0 r .so 1 3 1 0 7 1 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 ooooo 0 0 0 0 0 0 . 0 0 0 2 0 1 6 0 I S ooooo 0 0 . 0 0 0 0 . 0 0 1 . 2 0 2 SO I t 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 a. so 1 . 0 0 IT ooooo 0 0 . 0 0 0 0 . 0 0 • 0 0 S . 0 0 11 0 . 0 0 0 0 M O O 0 0 . 0 0 3 . 2 0 o . w I f ooooo 0 0 . 0 0 0 0 0 0 9-SO . 9 4 0 t o ( 0 0 0 0 0 0 0 0 w o o 2 . 2 4 • TO 2 1 ooooo 0 0 . 0 0 0 0 0 0 I T SO 1 1 . 2 0 2 2 • 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 2 3 • 0 0 0 0 0 0 . 0 0 0 0 . 0 0 3 . 2 0 1 6 0 1 4 • 0 0 0 0 0 0 0 0 • 0 0 0 • TO • TO 2 S ooooo 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 2 0 ooooo 0 0 . 0 0 0 0 . 0 0 • so 2 . 3 0 2 T 0 0 0 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 0 0 1 0 ooooo 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 0 0 M 0 . 0 0 0 0 0 0 . 0 0 0 0 . 0 0 2 4 0 o t o M • 0 0 0 0 0 0 . 0 0 0 0 . 0 0 1 0 . 0 0 1 »
TABLE (4): Operating conditions.
APPENDIX 2
NUMERICAL RESULTS
APPENDIX 2
NUMERICAL EXAMPLES
The Appendix i s devoted to a few numerical examples and deals wi th:
A2.1 appl icat ion of the recurs ive method to l i n e a r networks
A2.2 s p l i t network
A2.3 node merge
A2.4 appl icat ion of the d i r e c t method to l i n e a r networks.
A2.1 Applicat ion of the recurs ive method to l i n e a r network
Consider the network shown below, where the in jected real currents are
shown at each node and the real part of the admittance i s a lso given:
1
1
Figure (a) - Simple real l i n e a r network
For the above network the or ig ina l impedance matrix Z Q can be shown a s :
5 2 1
2 4 2
1 2 5
and the or ig ina l voltages as v Q .
5 2 1
V o = Z o ] o = 8 2 4 2
1 2 5
(1)
3/2
4/2
5/2
(2)
Now suppose that network of Figure (a) i s subjected to three simultaneous
branch outages as fo l lows:
1) remove the branch between node 1 and the reference (node zero)
2) then add a branch between node 1 and node 3
3) then remove the branch between node 3 and the reference (node zero)
Hence the connection matrix and the modified branch impedance for each of
the above fau l ts would be:
( i ) for the f i r s t f a u l t C, 1
f l = " 1
( i i ) for the second f a u l t
f 2 = +1
( i i i ) for the t h i r d f a u l t
1
V - 1
see Figure ( b ) .
We want to f ind the new voltages a f t e r each f a u l t .
Fault number 1 - the procedure i s as fo l lows: 1.
a) Find Z c , i . e . d i f fe ren t of column 1 and zero of Zn -o l o
To simulate column zero of Z Q we add a row and a column to
1Q with a l l elements set to zero , i . e .
column 1
5/8 2/8 1/8
2/8 4/8 2/8 Z o =
1/8 2/8 5/8
. . • . . J 0
. - I
' 0
column zero
X l s Z o C l = 8 5 0 5
2 0 1 2
1 0 " 8 1 ) i
1 0 ! i t L l i
b) Find X 1 , i . e . d i f fe rent of elements 1 and zero of X 1 -
X, =
5/8
2/8
1/8
o :
element zero 0
. . X t = 5/8 - 0 = 5/8
c ) Find = ( f j + X j ) ' 1
d,_ = (-1 + 5 / 8 ) " 1 = -8 /3
d) Find C j * V Q , i . e . d i f ference of element 1 and zero of V Q -
To simulate element zero of V Q add a row to V Q set to zero:
V o =
3/2
4/2
5/2
element zero ! 0 I
. . VQ = 3/2 - 0 = 3/2
e) Find b 1 = d 1 VQ
b1 = ( - 8 /3 ) (3 /2 ) = -4
f ) Hence
V l = V o " X l b l
2.
a)
3/2
4/2
5/8
2/8
1/8
_o_ . j Q !
Fault number 2
( -4) = 3
I o !
W = Z C 0 = d i f ference of column 1 and 3 of l n
O L 0
b)
c )
d)
e)
W =
5/8 1/8 4/8
2/8 2/8 0
1/8 5/8 -4 /8 r • i 0 > 0 ! ' 0
Find a = d x W
where W i s the d i f ference of elements 1 and zero of W
C j 1 W = 4/8
a = ( - 8 / 3 ) (4 /8 ) = -4 /3
Find W = W - X 1 a
x 2 = W =
4/8 5/8
0 2/8
-4 /8 1/8 I 1 1
! o 1 1
0 l
( -4 /3 ) =
Find d 2 = ( f 2 X g ) " 1
X 2 = d i f ference of element 1 and 3 of X 2
X 2 = 4/3 + 1/3 = 5/3
d 2 = (1 + 5 / 3 ) " 1 = 3/8
Find b = d 2 C 2 V x
C 2
r Vj = d i f ference of element 1 and 3 of
C 2 V x = 4 - 3 = 1
b = ( 3 / 8 ) ( l ) = 3/8
4/3
1/3
•1/3
f ) Find V 2 = V x - X 2 b
3.
a)
b)
c)
4/3
1/3
•1/3 3 r i . . ! o ; ; o ;
(3 /8 ) =
Fault number 3
W = Z0C3 =
1/8
2/8
5/8
0
W =
28/8
23/8
25/8
: 0 : I I
Find a = d 1 C y W
W = 1/8
a = ( - 8 / 3 ) ( l / 8 ) = -1 /3
Find W = W - X x a
1/8 5/8
2/8 2/8
5/8 1/8
0 0 :
( -1 /3 ) -
1/3
1/3
2/3
1 0 ! ' 1
d) Find a = d 2 CyW
C^W = 1/3 - 2/3 = -1 /3
a = ( 3 / 8 ) ( - l / 3 ) = - 1 / 8
e)
f )
g)
Find X3 = W - X 2a
1/3 4/3
1/3 1/3
2/3 -V.L 0 o !
• Find d, = ( f 3 + C 3
t X 3 ) _ 1
C y X 3 = 5/8
d 3 = (-1 + 5/8)_ 1 = -8/3
Find b = d 3 V 2
V 2 = 25/8
b = ( -8 /3J25 /8 ) = -25/3
Find V3 = V 2 - X 3 b
28/8 1/2
23/8 3/8
25/8 5/8
0 ! •
o !
(-1/8) =
1/2
3/8
5/8 1 i
o !
(-25/3) =
23/3
18/3
25/3
3fL
3 «•
Figure (bl) After f i r s t fault
3/8 24/8 8/8
25/8 23/8 28/8 2/8 5/8
25/8
23/8
Figure (b2) After second fault
< j25/3'T
9/3
18/3
r , r r r
Figure (b3) Afer the 3rd fault
Figure (b) Network Changes
A2.2 Spl i t Network
I t was explained previously that i f 1/d = 0, i t indicates a sp l i t
network. To show this i s true consider the following network:
l l i 2 : 3
1
figure (c)
If now the branch between nodes 1 and 2 be removed (which causes sp l i t )
we investigate what happens to d.
For the above network:
1 1 1
1 2 2
1
CM 3
But:
1/d = f + C* X
1/d = f + C* ZQ C
1/d = f + + X22 " X^2 " ^21
1/d - - 1 + 1 ^ 2 - 1 - 1 - 0
A2.3 Node Merge
Consider Figure (a) for which Z Q and VQ are as in (1) and (2) . If a
short c i rcu i t occurs betweens nodes 2 and 3, then the network reduces to
that of Figure (d).
1 L- 1 i 5
1 i
1 f t * * * * • * * * * AS
1
Figure (d) Node Merge
For the above network i t can be shown that:
Xm = 1/5
and Vm -
Vm can be obtained from the application of the recursive method to
8/5 1
where 1 -where 1 -
11/5 5
the original network, i . e . Figure (a ) , by setting f=0 as follows:
+1
X = Z Q C = 1/8
CTV° = - 1/2
C 1 X = 5/8
d = (f + C T X ) _ 1 = (0 + 5 /8)" 1 = 8/5
Vm = V° - d C T V° X =
3/2
4/2
5/2
- (8/5M-1/2) I
-3
Vm =
8/5
11/E
11/5
as obtained previously. As can be seen, voltages
at nodes 2 and 3 are given as equal amounts.
A2.4 Application of the direct method to l inear networks
The following formula holds in general:
(A + XUY)' 1 = A" 1 X (U" 1 + Y A" 1 X ) " 1 Y A" 1 (Al)
This formulate can be considered as a method for building in the
inverse of (A + X U Y) from the inverse of A.
In applications to networks equation (Al) i s applied in the following
form:
New Z = Z - Z C (f + C* ln C ) ' 1 C* ln (A2) 0 O x O ' 0 * '
where
Z Q = impedance matrix of the orignal network
f = diagonal m by m matrix of impedances of the m branches added to the network (of -ve sign i f the branch removed).
C = connection matrix (mxn) of elements +1 or -1 or zero.
A numerical example is solved to i l lus t ra te the application of the
equation (A2) to l inear networks. It can also be applied to non-linear
problems.
I f the current injections at each node is known, then the new voltages
would be obtained from the original voltages:
newV = VQ - Z Q C (f + C* 1Q C ) " 1 C* VQ
Hence, the procedure is as follows:
1) X = XQ C
2) L = C t X
3) d = (f + L ) " 1
4) M - C* V 0
5) b = d.M
6) V = V - X.b ' m o When the outaged branch is a shunt element, there is no need to add rows and
columns to X and ZQ and VQ having elements of zero as shown below. This is
done only for computer programming purposes.
1 i i Nk 0 ! i *
1
0 ' • — - —I
Nk i i • • 0 1
So j o o ; 1
1 n 1
Number of outages
X =
Nk Nk r
10 I
Nk is the number of nodes.
Equation (c) i s basically that used for simultaneous branch outages dealt
with already, but with minor changes. C is the matrix of outage
representations, f i s a diagonal matrix.
As an example consider Figure (c) and suppose i t is subjected to 2
simultaneous branch additions, between nodes 2 and 3, each branch of
admittance = 1. The original Z Q was shown in section B2 as:
Xo =
1 1 1
1
CM 2
1 ro
3
The new impedance matrix Z m can be shown to be:
1 1 1
1 2 2 (4)
1 2 7/3
Thus Z can be obtained from: m
X = Z - Z C (f + C* Z„ C ) " 1 Ct I m o o o ' o (as in Appendix A)
where:
(5)
c =
0 0
c = +1 +1 c =
-1 -1
and f is diagonal ( network:
f =
1 0
f =
0 1
Hence:
i) z 0 c =
2) C* Z Q C
0 0
0 0
-1 -1
1 1
1 1
3) (f + C* 1 C ) " 1 = 2/3 -1/3
-1/3 2/3
4) C ' 2 0 = 0 0 -1
0 0 -1
5) (f + C* Z Q C ) " 1 C* Z Q 0 0 -1/3
0 0 -1/3
6) ZQ C (f + C* Z Q C ) _ 1 C* Z Q = 0 0 0
0 0 0
0 0 2/3
7) From equation 5:
1 1 1 0 0 0
1 2 2 - 0 0 0
1 2 3 0 0 2/3
1 1 1
1 2 2
1 2 7/3
which is as that shown in (4) ear l ie r .
APPENDIX 3
TEST NETWORK DATA
m
C
111
O
J o a o o o c o o a Q O O o o a o o o o o D o o a Q a a Q a o o o o
j j J j - » O O 0 0 0 O 0 - ' O N a 0 - » O 0 0 O 0 0 O 0 0 0 0 O O O O O iC3W-1O-'SLrtW>0W&'OCcM'0arJOOOOCO-'iM^——»un*—»
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7 3 m o
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H ii -n-* 1 On a » o o 1 t-t l ~ M O 1 - H z
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O I I | T : :E c-i c
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a o o o a a n a o a o o n a o o r j o o o o o D i o o o o Q w o
o o o c o o o o o a o o c o o o o o o o o o o o o o o o c o
o o o o o o a o a o o o o o o a o o o a o o o o c i O Q o o o o o o o o Q O a c j O Q Q o a o o o o o o c o o o o o o o o a
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