•'• 41,
UNIVERSITY OF LONDON
IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY
• DEPARTMENT OF ELECTRICAL ENGINEERING
ON-LINE SECURITY CONTROL AND OPTIMUM DISPATCH
ON A POWER. SYSTEM SIMULATOR.
BY
EDUARDO ARRIOLA.- VALDES
Ing, M.E., M.Sc.(Eng), D.I.C.
Thesis submitted
for the degree of
Doctor of Philosophy
in the Faculty of Engineering. London, February 1977.
To my wife
Rosa Maria
To my sons
Eduardo Jr.
and
Carlos Alberto.
ABSTRACT
A set of algorithms for state estimation, security
analysis and secure optimum dispatch, have been implemented
on a PDP-15 computer to analyze real time data collected
from a power system simulator. This data is processed
first to assess the present operating conditions of a
network set up on the simulator, and then to calculate,
if necessary, appropriate control actions resulting in a
secure operating state that simultaneously optimizes given
dispatch criteria.
The analogue measurements taken from the simulator
are sent via the interface to the PDP-15 computer where
they are transformed into a 10 bit word by the analogue-
digital converter. The digitalized measurements are placed
in the appropriate locations in computer core, and after
conversion to per-unit values they are processed to obtain
the present operating state of the system. The use of a
redundant set of measurements, i.e. more measurements than
system state variables, allows the estimation algorithm to
filter the natural random errors which occur in,the process
of measurement and conversion of variables, and allows the
detection and identification of gross errors, thus
establishing a reliable data base.
To assess the security of the system at the present
operating state, a fast and reliable algorithm is used to
simulate single line outages and check for possible
overloads in the remaining lines of the network. At the
2
end of this simulation, a table containing all the
relevant information pertaining to the lines where
overloads occur is transformed into a set of security
constraints and is made available to the dispatching
algorithm. A linear cost function is associated with
each generating unit in the system and a linear
programming algorithm is used to calculate the corrective
actions which comply with the constraints derived from
the security analysis algorithm, and to simultaneously
minimize a given objective function.
a
3
ACKNOWLEDGEMENTS
.This work was carried out under the supervision
of Dr. L. L. Freris, M.Sc.Eng., Ph.D., D.I.C.,
M.I.E.E., whom I would like to thank for his constant
guidance and encouragement.
I would like to express my appreciation to my
colleagues of the Power Systems Laboratory for their
advice and assistance, in particular Dr. C.B. Giles,
Mr. L. Mogridge, Mr. S. A. Molina and Dr. G. Gonzalez.
I wish to express my gratitude to Comisicin
Federal de Electricidad (C.F.E.) and Consejo Nacional
de Ciencia y Tecnologia (CONACyT) for the leave permit
and financial support which made this work possible.
I also wish to thank the General Electric Company Ltd.
for their financial contribution in the early stages
of this project.
Lastly, I would like to express my gratitude to
my wife Rosa Maria who patiently typed the manuscrit.
4
4
Chapter 1
Chapter 2
CONTENTS
7 Introduction
The Simulator, The Interface and
the Computer 12
2.1. Introduction 12
2.2. The Power System Model 13
2.3. The Interface 16
2.4. The Digital Computer 19
Chapter 3 State Estimation. Theoretical Aspects
and On-Line Implementation 23
3.1. Introduction 23
3.2. The Linear Model and Least Squares
Estimation 25
3.3. Mathematical Model of Electric Power
Systems 33 3.4. The Non-Linear Model. Linearization
by Taylor Series Expansion 37
3.5. The Non-Linear Model. Linearization
through. Transformation of Variables 40
3.6. Detection and Identification of Gross
Measurement Errors 46
3.5. Algorithm for the Solution and On-Line
Tests
Chapter 4 Security Control Via Optimum Dispatching
of Power 60
4.1. Introduction 60
5
a
e • •
4.2. The Fast Decoupled Load Flow 64 4.2.1. Formulation of the Method 64 4.2.2. Line Outage Simulation 70
4.3. Line Outage Simulation Using Fictitious
Injections. 73 The Exact Method. 73
4.3.2. The Method of Sachdev and Ibrahim 77 4.3.3. A modification to the Method of
Sachdev and Ibrahim 78 4.3.4. A New Method for the Simulation of
Branch Outages 81 4.4. Derivation of Security Constraints 99 4.5. Optimum Reallocation of Power for
Security 103
Chapter 5 On-Line Implementation of Algorithms 113
5.1. The Real Time Operating System 113
5.2. Implementation of Algorithms in the
RSX System 118
5.2.1. Module 1. Data Input 121
5.2.2. Module 2. State Estimation \ 122
5.2.3. Module 3. Economic Dispatch 126
5.2.4. Module 4. Security Analysis 129
5.2.5. Module 5. Security Constrained Dispatch 130
5.3• Numerical Examples 134
Chapter 6 Conclusions 147
6.1. Concluding.Remarks 147
6.2. Further Work 150
6.3. Original Contributions 152
Appendices
Appendix 1 Stott's Algorithm for the simulation of
line outages
Appendix 2 Data for Test Systems
Appendix 3 The D.C. Load Flow. Simulation of Line
Outages by Fictitious Injections
Appendix 4 Conversion to p.u. of Measured Line Flows
and Nodal Voltages
Appendix 5 Formation of Task OPTIME
References
153
155
159
162
164
167
7
1 INTRODUCT.ION.
The growth in size and interconnection of power
networks' demands increasingly sophisticated techniques
for operation and control. This fact, coupled with the
greater emphasis placed on security of operation,
requires that reliable information about system conditions
be placed at the disposal of the operator. The wealth
of information required for efficient control is now
handled by a digital computer which receives on-line all
telemetered quantities, and displays at the control
centre only the relevant data enabling the operator to
take quick action during emergencies.
Due to occasional failures of the metering equipment
or communication links the appearance of grossly erroneous
data is inevitable. Recognition of this fact, and of the
need to compute, from measured data, other variables of
interest which for technical or economic reasons are not
explicitly available, has led to the use of computer
algorithms for processing the raw measurements. The use
8
of'Estimation Theory (ref. 1) has been shown (refs. 2-9 )
to make optimal use of the available real-time data in
the calculation of a set of variables representative of
the state of the system. The algorithms, which for
obvious reasons, are called state estimators, have been
accepted as a necessary part of power system monitoring.
Under steady state conditions the voltage magnitude
and phase angles at all nodes of the system are chosen
as the components of the state vector. The reason for
this selection is that the vector of nodal voltages
constitutes the minimun set of variables which completely
characterizes the operation of the system, i.e. once
they are known every other electrical quantity of the
network can be calculated. Since in the process of
estimating the state the algorithms are capable of
detecting and identifying grossly erroneous measurements,
the resulting vector of nodal voltages can be used to
form a reliable data base. Estimated voltages, line
flows, equipment loadings etc., can be readily checked
against their stipulated limits, and any violations
reported to the operator via the display system.
This initial stage in the use of measured,data to
give the operator a complete picture of the operating
conditions of the system has been defined by DyLiacco
(ref. 10) as security monitoring. He defines two
additional functions, security analysis and security
constrained optimization, which together with security
9
.4 monitoring constitute an adequate and viable strategy
which reduces the possibility of the appearence of
dangerous operating conditions in the system.
The security analysis function requires the
observation of the steady state response of the system
to a series of simulated contingencies. The resulting
state under each of these simulated conditions is
required to comply with the operating constraints
of all the components of the system. Failure to do so
indicates the need for corrective actions. The
calculation of these corrective actions is the task of
the third function, security constrained optimization.
The objective here is to determine the optimum allocation
of generation to satisfy the actual demand of the
system observing at the same time the operating limits
of the generating units. The constraint set for the
optimization process is augmented by constraints,
which. during security analysis are found to be violated.
The difference between the calculated schedule and the
present generation schedule in the system constitute
the corrective actions or security control required
to attain a secure operating state.
The objective of this project has been to develop
and implement on a real-time domputer the necessary
algorithms for security monitoring, security analysis
and security constrained optimization. The digital
computer is interfaced to a power system model provided
10
with comprehensive instrumentation. This arrangement
allows the testing of the algorithms in a more realistic,
and hence more hostile, enviroment than would be possible
in a purely digital simulation.
A brief summary of the contents of this thesis is
given as follows:
Chapter 2 describes the power system model, the
interface equipment and relevant hardware of the computer
which were used in this project for the on-line testing
of the algorithms.
Chapter 3 presents the general theory of linear
least squares estimation and two differ3nt aproaches
to its application in the solution of the state estima-
tion problem in power systems. The statistical analysis
of residuals and their use in detection and identification
of gross measurement errors is described. The estimator
was tested on-line under normal conditions and under
the presence of large errors. Samples of results obtained
from these tests are shown in this chapter.
Chapter 4 is concerned with the problems of security
analysis and the calculation of corrective actions. A
flexible and efficient algorithm for the solution of
the contingency analysis problem is discussed, and the
derivation of a new approximate method for line outage
simulation is presented. Results obtained in 3 test
systems using the two algorithms are compared. An
effective way for the calculation of linear security
11
constraints and their subsequent use in the calculation
of corrective actions is described.
Chapter 5 describes the real-time operating system
and the implementation and structure of the algorithms
developed for on-line operation. The results obtained
from on-line tests using the power system model are
reported in detail at the.end of this chapter.
Finally chapter 6 presents a summary of the work
developed in the course of this project, the original
contributions and suggestions for possible future
developments.
12.
THE SIMULATOR, THE INTERFACE AND THE
COMPUTER.
2.1. Introduction.
A power system simulator which can be used in
-conjuction with a PDP-15 digital computer to study
on-line control problems has been built at Imperial
College. The project was started by a donnation by
CEGB of part of the equipment used in a power system
model that was constructed for the Central Electricity
Research Laboratories and described by Bain in ref 11.
The generator units were redesigned by A. Sheldrake
(ref.12,13),eliminating the mechanical parts of the
original model to produce a totally electronic simulation
of the dynamic behaviour of a power plant. The
synchronous machine is represented here as a voltage
behind a transient reactance, but an alternative model
using two axis theory was designed and built by C. Giles
(ref.14), who also designed electronic analogues for the
load units. The network is provided with comprehensive
13
instrumentation and the necessary interfacing hardware
was designed and built by M. Bolton (ref.15,16).
Only a brief description of the different components
of the analogue model, interface equipment and computer
hardware is given in this chapter. The discussion of
the computer software is deferred until chapter 5 where
the implementation and on-line testing of the algorithms
• for state estimation and security constrained dispatch
will be described.
2.2. The power system model.
A generating plant is simulated by electronic analogue
models of the turbine, the speed governor, the synchronous
generator and the automatic voltage regulator. The
frequency of the voltage signal present at the terminals
of the generating unit is continuosly variable between
47.8 and 52.2 hertz. The actual frequency is determined
by the instantaneous balance of power in the system.
The unit is provided with instruments to measure the
real and reactive power generated, the terminal voltage
and current, the rotor angle, the governor position and
throtle valve position. All these quantities are
displayed on the front pannel of the unit.
The loads are also simulated by electronic analogue
models with the advantage that this type of representation
allows for an infinitely variable choice of real and
reactive load settings. The lOad units act as current
sinks, absorbing a given current at a specified pdwer
factor which is virtually unaffected by small variations
in.terminal voltage and frequency.
The transmission lines are represented by nominal
pi lumped models and they are fitted with voltage and
current transducers at both ends, so that the real and
reactive power flows can be measured. As will be shown
in chapter 3 these quantities are essential in the
state estimation process.
The terminals of each genera4;or and load unit are
brought out in the connection pannel where circuit
breakers, represented by small electromechanical relays,
and busbars, are available to assist with the inter-
connections.
The; simulator was designed to incorporate a maximum
of
. six generating units
. five load units
. ten transmission lines
. four tap changing transformers
. sixteen circuit breakers.
The connection of the different components of the
model is made by means of patching cords and thus any
network configuration within the capacity of the model
can be set up.
The base values which were chosen for the simulator
15
of.a single phase voltage of 20 volts r.m.s. and 0.03
amperes r.m.s. were the same values used in the original
simulator developed by CEGB.
An schematic representation of the layout of the
analogue section of the model is shown in figure 2.1.
.
GENERATOR
UNITS
LOAD
UNITS
circuit breakers
• Line
Terminals
Busbars
IGeneratorsl]Loads1
Infinite o BUS
Figure 2.1. Layout of Analogue Section of the model.
16
2.3. The interface.
To link the analogue model to the digital computer
two multicore cables, one carrying analogue and the other
digital signals, run•from level 8 (model) to level 5
(computer) of the Electrical Engineering Department.
The analogue data which is gathered in the model has to
be transformed into digital form before it can be used
by the computer. The inverse transformation is required
to implement in the model control actions derived in
the computer. These functions are accomplished by means
of the analogue-to-digital (A/D) and the digital-to-analogue •
*(D/A) converters, which form part of the computer
peripherals. The input signal to the A/D converter must
be in the range 10 V, this means that all analogue
signals coming from the model have to be conditioned to
meet this requirement. The length of the output word
is selectable from 6 to 12 bits..
Six channels of the A/D converter multiplexer are
connected to the interfacing equipment adjoining the
model in level 8. Each line is multiplexed 16 ways to
provide an input of 96 analogue signals.
Figure 2.2. shows the arragement for the
transmission of these signals.
Com-
puter.
Level 5
MUX
17
A/D
Signal
Condition ing.
96 Analogue Signal,
Level 8
MUX MODEL Analogue
Signals
6 lines to level
5
Figure 2.2. Analog input system.
The four channels of the D/A converter demultiplexer
are connected to an analog demultiplexer located in level
8. Each channel is demultiplexed 16 ways to produce a
total of 64 analogue outputs. To retain the analog
outputs which are used as control signals, the level 8
demultiplexer is followed by zero-order hold circuits.
The arragement for the transmission of analog data from
the computer to the model is shown in figure 2.3.
J Zero order hold cir-cuits
MODEL
level 8
D MUX
level 5
D MUX
Com- puter.
D/A
64 analog signals
4 lines to level
8
18
Figure 2.3. Analog output system.
Multiplexing and demultiplexing facilities are
provided for the digital signals which are carried by a
separate cable. The data is organized as 4 sixteen bit
words giving a capacity of 64 bits of digital input and .
outpUt. To complete the transfer of digital data a
digital input/output unit is connected to the I/O bus of
the computer. This unit and the interfacing hardware in
level 8 are described in references 15116.
It was mentioned earlier that due to the fact
that the A/D converter accepts input signals in the
range of - 10 volts, all analogue data transmitted from
the model to the computer needs to be conditioned.
19
Since power flows and busbar voltages are of particular
importance to this project, a brief comment about their
measurement is not out of place.
All the lines in the model have current transformes
(C.T.'s) fitted at both ends. To measure the real and
reactive power flow in a line the current signal from
the C.T. plus the voltage signal obtained from the
same point in the line are used as inputs to a Watt/
Var meter. The meters are calibrated to give an output
signal of 5 Volts d.c. when measuring 1 p.u. flow.
The voltage meters are adjusted to measure the
voltage deviation from 1 p.u., that is, they have a zero
output when measuring 20 volts r.m.s. The meters are
adjusted to give +10 volts d.c. for an input of 24 V rms
or 1.2. p.u., and -10V for 16V rms or 0.8 p.u.
Further comments concerning the use of the measured
line flows and voltages in the process of state estimation
will be given in chapters 3 and 5.
2.4. The Digital Computer.
The model is connected by means of the interface
to a Digital Equipment Corporation PDP 15/20 computer,
with 24K of core storage and a word length of 18 bits.
The central processing unit is fitted with hardware
20
to perform fixed-point multiplications and divisions,
but floating point arithmetic is performed by software
routines with a substantial reduction of execution speed.
The input output processor of the computer is fitted
with a real time clock and an automatic priority
interrupt (API). These facilities play a most important
role in the operation of the system on a real time basis.
The clock coordinates the computer operation with the
real world's time schedule, while the API allows the
normal flow of execution of instructions to be altered
to permit the computer to attend to some urgent or higher
priority function.
In addition to the analogue-to-digital converter,
the digital input-output unit and the digital-to-analogue
converter all of which are used for communication with
the model, the computer is furnished with the following
peripheral devices:
• A Teletype and an alphanumeric cathode-ray tube
(CRT) for communication with the operator.
. A high speed paper tape read/punch useful for
program and data preparation.
. A line printer.
. Two fixed-head disk units, each with 256K words
of storage capacity.
. A dual DEC tape drive.
Figure 2.3. ilustrates the configuration of the
computer and its peripheral equipment.
t
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Ft6U1ZE • 2.5. THE DIGITAL COMPUTER Nib ITS PEIZIINIERALS CoMmutOctatot/ WITH MODEL
22
To conclude this chapter two photographs of the power
system model and the level 8 interface equipment are shown below.
"MI
Figure 2.4. The Power System Simvaator,
Figure 2.5. The PDP-15 Interface.
STATE ESTIMATION. THEORETICAL
ASPECTS AND ON-LINE IMPLEMENTATION. .
3.1. Introduction.
In order to control efficiently a power system, or
any physical system for that matter, one must have suf-
ficient and reliable knowledge about its present operating
conditions. This essential requirement for proper con-
trol, has motivated Electric Utilities to install wide-
spread instrumentation with data transmission facilities
terminated at a central control center equiped with
digital computers. The information, which is gathered
and telemetered to the control center, is:
a) subject to noise,
b) incomplete because not each system variable is
measured,
c) bccassionally misleading as one or more meas-
urements could be grossly in error due to un-
detected failures of meters or communication
channels.
24
It can be concluded that the assessment of
the state of the system cannot be always made satisfac-
torily through raw measurements, there is therefore a
necessity of algorithms for data processing. The aim.
of these computational algorithms or estimators is to
process the available information obtained from the
system and to deduce a minimum error estimate of the
state of the system by utilizing
. knowlegde of the structure of the system
. assumed statistics of measurement errors
. redundancy of information.
The last item allows the reduction or filtering
of measurement errors and the detection and identifi-
cation of grossly erroneous measurements. The process
is ilustrated in figure (3.1.)
telemetering system
SYSTEM STATE
P 0 w E 0
I
N
T
R
F
A
C
E
Digital Computer
Processing ( Data
Algorith
Meter 1 R
S 0 Meter 2
T E
X 0 0 0
observations
7*rn Meter m
Figure 3.1. Block diagram depicting the technique of
Estimation.
The'estimated state x obtained by processing the
available information ym may then be used to drive dis-
plays and to form the data base from which control
actions could be derived.
3.2. The linear Model and-Least Squares Estimation.
In this section the general theory of least squares
estimation, and its application to the situation in power
systems is described.
Consider the linear model of the form
yam . = A (3.1) •
where:
ym : (m x 1) vector of observations
A : (m x n) matrix of known coefficients
x : (n x 1) vector of unknown variables
• : (m x 1) vector of error random variables.
It is assumed that a redundant set of 'measurements
is available, ie. m>n, and that the errors are uncorre-
lated with zero means and variance cr2. Mathematically
the assumptions concerning the statistics of e are expressed as
E (S) = 0 (3.2)
and covariance matrix
E e E t ) = G2Im (3.3)
where;
• E (e) indicates expected value of e
• superscript t indicates transpose, and
• Im stands for the unit matrix of order m.
The least squares method requires that the scalar
sum of squares
J (x) = (ym-A x)t (ym- A x) (3.4)
be minimized with respect to the variation in the compo-
nents of x. The function J(x) is minimized if the partial
derivatives with respect to xi (i=1,2,...,n), vanish si-
multaneously, i . if
t ,i( x ) for i=112,...n
This requirement leads to
2 At (y m- A x) = 0
which gives for the least squares estimator the vector
(AtA)-1 At y111 (3.5)
. assuming that AtA is nonsingular and can therefore be
inverted.
A desirable property of any estimator is that it
should be unbiased. The least squares estimator given
by eqn. 3.5 is such an unbiased estimator.of x .
Using eqn. 3.1. in eqn. 3.5
(AtA ) t(A x e)
(3.6)
and taking expectations on both sides of eqn. 3.6
E(X) = E [(AtA)-1 x + (AtA)-1At €3 (3.7)
since A is a constant matrix and due to eqn. 3.2 we get
E(X) = x (38)
Eqn. 3.8 states that the expected value of the estimated
quantities x is its true value, showing that eqn. 3.5 is
an unbiased estimator of x .
In order to determine the spread of the estimated
quantities around the mean, the covariance matrix of X
can be calculated from
= E(x-x) (X - x)t3
substitution of X from eqn. 3.6, yields
= E f(x + (AtA)-iAte - x) (x +(AtA)
At A)-1 At E(e et) A (AtA)
-1
(3.9)
which from eqn. 3.3 becomes
LX = 0-2 (AtA) (3.10)
An estimate of the observation vector can now be
obtained using the estimated values of x from the linear
relation
= A
(3.11)
substituting eqn. 3.6 into eqn. 3.11 and taking expecta-
tions we have
E(i) = [A (AtA)-1 At (A x + E)]
27
which yields
E (y) = = y (3.12)
This indicates that the expected value of y is its own
true value, thus eqn. 3.11 is an unbiased estimator of y. A
The covariance matrix of y can be obtained from
28
Zs; = E {(y-Y) (y-Y)t)
which from eqn. 3.11 and the relation y=A x
= E (A(-x) (X-x)t At
E[(DAr-x) (X-x)tlAt
= A 2A At
(3.13)
which from eqn. 3.10 becomes
2A (AtA)-1 At (3.14)
One important aspect of estimation theory is the
analysis of residuals, because they form the basis on
which decisions concerning the validity of the solution
are taken.
The residual vector is defined as the difference
between the measured and the estimated quantities
(3.15)
using eqns. 3.1 and 3.11 and taking expectations we have
E (z) =E (A x 8 - A X) =0 (3.16)
29
To calculate the covariance matrix of the residual
vector z it is convenient to express z as a function of
the measurement noise € .
From eqn. 3.1 and 3.11 the residuals can be written
as
z= ym - y= A x e- A i‘c
and substitution in this eqn. of the value of X given by
eqn. 3.6 gives
z = x E A(AtA)-1 At (A x E)
or
z = (1m - A (AtA)-1 At (3.17)
Since the expected value of the residuals z is zero the
covariance matrix is given by
E (z zt) = Ef(Im - A(AtA)-1 At) et (I .-11(AtA) - 1 At)1
(3.18)
Performing the operations indicated in. eqn. 3.18 we
get
(zzt2 (Ira - A (AtA)-1 At ) (3.19)
One final important result is the value of the sum
of squared residuals obtained by evaluating eqn. 3.4
using the estimated values of x
J (.2) = (ym - A 2)t (ym - A 2) - (3.20)
substituiing in eqn. 3.20 the value for the residuals
given by eqn. 3.17
J-60 = st t(I -A(AtA)-lilt) (I 8
= &t - A(AtA) ) e
Any quadratic form et. B E is a scalar and identical
to its trace, tr (StBE). Since matrices may be commuted
under the trace operator
E (X) = tr (etBS) = tr (BEEt)
(3.21)
where:
30
B = Im - A A(AtA)-1 At
taking expectations on both sides of eqn. 3.21
E (J(2)) r_ttr B E (EEt
SO
E (J(I)) =6-2 tr B
=a42 tr /Im A(AtA) At}
=cr2 ttr Im tr (AtA
AtA
E (J(X)) =G-2 (tr Im tr In)
finally
E (J(X)) =G-2 (m - n) (3.22)
Eqn. 3.22 states that the expected value of the sum
31
of. the squared residuals is equal to the variance times
the number of observations minus the number of unknown
variables. This is indeed a very useful result and, as
will be shown later, will be used to determine whether
the solution is acceptable or not. Values of J(X) which
depart considerably-from its expected value as given by
eqn. 3.22 are suspected as containing observation errors
outside the assumed statistical limits, hence vector X
is not accepted as a good estimate of x.
It is important to note that so far, the results
derived from the least squares theory have been obtained
without any assumptions about the probability distribution
oftheobservationerrors&i that is to say that the
unbiased estimators deduced for the parameters are inde-
pendent of the form of such probability distribution
functions.
For the basic model of eqn. 3.1 it has been assumed
that the observations ym are independent and all have
equal variances G2. This last condition can be relaxed
to give a more general model in which observations are
still assumed to be independent but they may have differ-
ent variances,
Ym = A x 6
E (2) = 0 (3.23)
E (eet) = R
where R is a known diagonal matrix with rii 2
=cre. the
32
variance'of the ith observation. The model of eqn. 3.23
can be reduced to the basic model given by eqn. 3.1, 3.2
and 3.3 through the transformation (Rao ref.20 )
4 Y = R. Y, (3.24)
which gives
R4 A x eB (3.25..a)
with
4 E (R-- 6) = 0 (3.25 b)
and
E (R-1 EEt = Im (3.25 c)
all the results derived for the basic model being valid
for the model given by eqn. 3.25. The most important re-
sults for the model of eqn. 3.23 can be obtained from
those of the basic model through the transformation given
by eqn. 3.24 and they are shown below.
Quadratic form to be minimized
J(x) = (ym-A x)t R- 1 (ym -A x) ,- - •
least squares estimate of x
De‘c = (AtR-1A)-1 -1
(3.26)
(3.27)
Covariance matrix of X
7 (Ati(lA)-1 (3.28)
.Unbiased estimate and covariance matrix of
• 33
A A y =A x
2. = A (Atf" A
Vector of residuals and its covariance matrix
(ym s'r).
(R — A(At -1A)-lit)
(3.29)
(3.3o)
(3.31)
(3.32)
3.3. Mathematical Model of Electric Power Systems.
As described in section 3.2. the problem of estimation
involves the processing of a set of noise corrupted
observations y m1 to determine a least squares estimate
of a set of variables x. Vector x can be chosen so that
it contains the minimum amount of information, ie. number
of variables, that fully characterizes the operating
state of the system; x can then be rightly called the
state vector of the system.
Due to the continous change in load demand patterns,
a power system may never achieve a true static operating
point, nevertheless under normal operating conditions it
is valid to assume that the system remains in steady
state over short periods of time, with sudden transitions
between such short periods. The observations are then
seen as a snapshot of the system representing a static
operating point.
34 •
In power systemsa natural selection of the state
vector is the one containing the complex voltages of all
'n' network buses, as once these are known, any other
.electrical quantity in the system may be evaluated. In
other words the vector of complex voltages fully charac-
terizes the state ofthe system. To provide the angular
reference for the system, the phase angle at one bus
(the swing bus) is conveniently set equal to zero. Thus
in a network with .'n' buses and denoting the swing bus
as bus number one, the components of x are defined as
at 4ellj 0.0), (e2,g2),...1(entifn)1 (3.33)
Measurement of different network quantities can be
made to compose the observation vector ym of section 3.2,
and in the most general case this vector would contain
measurements of
• Real (P) and reactive (Q) power injections
• Nodal voltage magnitudes (VI and angles 6A
▪ Active (T) and reactive (U) power flows in the
lines
so
.[L,t,at , (3.34)
where 13,20,11/ 1,6, T and U are vectors containing the
available observations.
In order to relate the measured quantities with the
state vector as defined in eqn. 3.33, knowledge about the
structure of the transmission network and the value of
35
its paratheters is required. The transmission lines are
modeled as linear lumped RLC elements as shown on figure
3.2.
T. U R. 1jij , ij AMA"
T. ;U. Xij 4 31-±
V.=e.1-jf. 1 1 1
i/11, /l1!/
V.3=e 4.jf
Figure 3.2. Transmission line model using all network
equivalent.
The line parameters R. , X. and Y. are assumed 1j 1j lj
known, and are used in the formation of the bus admit-
tance matrix YBUS. With this information and Kirchoff's
laws, the following mathematical relations between obser-
vations and state variables can be derived:
Real power injection into node i
36
P. = e3. . 2: (e G. - B ) k k ik k=1
f. (f e B. ) I k=1 k ik k
Reactive power injection into node i
CtI . = f 3. . k= 1 Le - f B ) — k k ik
e. (fk Gik ek B. ) k=1
(3.35)
(3.36)
Voltage magnitude at node i
IV.I= (e f )i i (3.37)
Voltage phase angle at node i
01; = arc tan (fi/ei) (3.38)
Real power flow from bus i to bus k
T.
[e.(e. - ek) 1 f. f ik
[ei ei ek)] Bik (3.39)
Reactive power flow from bus i to bus k
Ulk `1!i(ei - ek) . (f.1 - f k B • ik
1 (f 1 ) fi (eiek
J Gik -
(e f) Y. • 1 ?. 1k./2 (3.4o)
where the parameters G and B are elements of YBUS.
As can be readily seen, eqns. 3.35 to 3.40 are non-
37
linear functions of the state variables while the theory
developed in section 3.2. deals with linear models only.
In the following two sections two alternative solution.
methods for the non-linear model will be discussed.
3.4. The non-linear model.* Linearization by Taylor series
expansion.
Mathematically the non-linear model can be written as
7m = f (a) = (3.41)
where:
ym : (mxl) vector of observations containing aiy
combination of nodal injections, voltage magni-
tudes and phase angles and real and reactive
line flows.
x
(nxl) state vector of the system as defined in
eqn. 3.33.
f(x) : (mxl) vector of functional relations given by
eqns. 3.35 to 3.40 depending on the type of
observation.
8 : (mxl) error vector associated to the measurements.
Again it is assumed that each of the components of
vector 8 is a random variable with zero mean, ie. meas-
urements are unbiased, and that the covariance matrix given
38
by
E (se ) = R
(3.42)
is known. The covariance matrix R is diagonal because
the measurements are independent and the elements of this
(mxm) matrix correspond to the variances of each individ-
ual meter.
Due to the non-linear nature of the set of eqns.
3.41 a close solution such as the one obtained for the
linear model of section 3.2. cannot be obtained.
However, a Taylor series expansion of f(x) in 3.41 around
an initial guess x° for the state vector and neglecting
higher order terms yields:
f (x) = f (x°) H (x°)[®x (3.43)
where H(k°) is the Jacobian matrix whose elements are
given by
bfk (5') hko. x. 0
X = X ••■• 1.■11
k=112,...m
i=1,2,...n (3.44)
• Substituting eqn. 3.43 into eqn. 3.41 the following
linear relation is obtained
by ym. —
f (o) =H (x0)6x 1. 6 (3.45) -
The model given by eqn. 3.45, the assumption
E(E)=0 and the error covariance matrix given by eqn.
3.42, have the same form as that of the model given by
. 39
eqn. 3.23. As shown in section 3.2. the quadratic form
to be minimized is
J (Ax) = ( 1SY 11(x°) 1).x) R 1 (Ay-H (x°)A x)t (3.46)
and has a solution given by
Qx 111(x0)] -1 Ht(xo)R-1,iy
411t(x°) R (3.47)
Because of the non-linear nature of the problem,
the final solution has to be obtained by iteration.
The algorithm is as follows:
1. Assume an initial value for the state xo
2. Set iteration count i=0
3. Calculate the Jacobian matrix with the current
state vector x
4. Solve fox 4X the set of linear eqns.
bit (xi )R-111(x9Ax = Rt (xi )R-1 (ym f(xi ))
A 5. Iftix is less than the given tolerance, solution
has been obtained and xi is the estimated state
of the system X . Otherwise proceed to step 6
6. Calculate the new state vector of the system from
xi+1 = Yi +Ax
7. Increase iteration count i4--i+1 and return to
step 3.
The method described in this section was proposed
by Schweppe et al ( refs. 2,3,4 ). Although the method
11.0
produces` optimum estimates of the system state and "pro-
vides a solid basis for bad data detection and identi-
fication' ( 17), it has the following disadvantages for
on line implementation
a) large core requirements due to the dimensionality
of the Jacobian matrix,
b) the Jacobian is a function of the state and
therefore has to be computed and factorized
at each iteration,
c) due to (b) excessive computing time required to
reach the solution.
3.5. The non-linear model. Linearization through
transformation of variables.
Dopazo et al (ref.7,8) developed a different method
of solution in their search for computational efficiency.
In this method only line flow measurements are accepted
as observations and linearization is obtained by trans-
forming them into 'measurements' of voltage drops across
the lines. The later quantities are linearly related to
the state vector of the system and linear estimation
theory as developed in section 3.2. can be applied to
solve the transformed problem. Here again the final
solution has to be obtained by iteration to account for
the non-linearities. A description of the method now
follows.
• In terms of complex line flows and complex nodal
voltages the mathematical relation expressed by eqn.
3.41 is written as
S* =(Vt 1 k + V! v. Y ) (3.48)
ik m,ik I Z I ik/2 Cik
where:
Sm ik : measured complex power flow from bus i to
bus k
Vi Vk : true value of complex nodal voltages
Zik : complex line impedance
Yik : shunt addmitance of line ik at node i
eik : complex error associated with measurement
Sm,ik
from eqn. 3.48 an expression for the true value of the
voltage drop accross the transmission line connecting
buses i and k can be obtained as
(V. -V 1 k
S* Y. Z.- e )z ik k (3.49) 1 rue ' V! I 2
k ' ik Vt
i
1
Defining Vm,ik as the 'measured' voltage accross line
i - k we get
V = (V) u ( mlaJL i-'k'Imeasured' it=
S* y.
1 2 ' ik 1
- V. ik‘z
(3.50)
.and its associated measurement error
ik ik V! #DE (3.51)
we can therefore write
42
V mt ik = (Vi_Vk)true + ik (3.52)
which expressed in matrix form gives
Vm = B V + 411
(3.53)
where matrix B is the (mxn) measurement-bus incidence
matrix with only two nonzero elements in each row. For
instance, if the meters are numbered from 1 to m and meter
1 measures the complex power flow from node i to node k,
all the elements in row 1 of matrix B will be zero except
bi,i = 1.0 and bi,k = -1.0
As mentioned earlier the phase angle at node 1 is
set to zero to provide the angular reference. This method
requires that measurements of the voltage magnitude are
taken at the bus chosen as reference.
This voltage measurement is assumed to be the true
voltage magnitude of this bus. Under these circumstances
the state at bus number 1, ie. V1=e1+jf1 is known and
eqn. 3.53 has to be modified as follows to account for
this
(V -- 121 V1 ) = Br Vr + (3.54)
where:
b1
(mxl) vector corresponding to column 1 of B
E : reduced incidence matrix r
Vr : reduced state vector of the system, ie.
Vt = V3 Vn) r (V 29 9...
The statistical properties of the measurement error
can be derived from the statistical properties of the
original error vector which are given by
E (eik) = 0
E (e* e ) - mi ik ik 'k
(3.55)
(3.56)
where gik is the variance of the meter located in line
ik
From eqn. 3.51 we have
E ) = E _( -
sinceZik isaknownparameterandV.is the true value
of the complex voltage at node i
E ik) = - Zik E ( e ik)
which from 3.55 becomes
E "tik) = 0
(3.57)
Zik
Vi Elk)
44-
with variance
E . en E 4'Zt 1"u o*
Zik
‘tik Ilk' V ik Cik Vi )
1 2 v 1 E (Et e. )
1 V.I2
which from 3.56 becomes
• lk = E (Tik Itik) = I v.1 2 gik 1
(3.58)
As the voltage magnitudelly in 3.58 corresponds
to the true but unknown value of this quantity, a guess
about its value has to be made in order to compute the
variance of the transformed measured quantities. Since
the voltage magnitude at all buses is usually kept around
1.0 p.u. this value constitutes a good approximation to
i=2,...,n and is therefore used in 3.58.
n matrix form
E 01* t) R (3.59)
where R is an (mxm) diagonal matrix with elements obtained
from eqn. 3.58.
From the results derived for the linear model in
section 3.2.1 it can be shown that the corresponding
results for the model given by eqns. 3.541 3.57 and
3.59 are as follows.
Quadratic form to be minimized
45
J(Vr) - ) = t Vm -bAVA ) -Bryn - t R-1 I(Im-21V1) -B * (3.60)
A Estimated state V
-r
-= (Bt R-1Br)-1 Bt R-1 or .1)1111) T r
A
Covariance matrix of V •-"r
Coy r) = (Bt R-1Br)-1
(3.61)
(3.62)
Unbiassed estimate of the transformed observation A
vector V m
A
= B V bA VA -m r -T -•1 I
Sum of squared residuals
.y(11T ^M
) = - M )t R-1 (VM - )* J(Vr ) = -
A
Expected. value of J(Vr)
(3.63)
(3.64)
E (J(Vr)) = (m - n) (3.65)
In order to solve the original non-linear problem
given by a set of m equations with n unknowns and
(eqn. 3.48), an iterative technique is required. The
recursive formula is derived from eqn. 3.61 and given by
(Bt R-1B r V ) i+1 = Bt R (V-m i - b1 V1 ) • -r r - (3.66)
where Vi is calculated from eqn. 3.50 using as an -r
approximation to the true value of V .
46
In exchange for the disadvantage of not being able
to handle different kinds of measurements, this method
offers a fast and reliable algorithm with small core
requirements. This attractive characteristics of the
method are obtained through the fact that matrix
• (Bt R-1Br) is real, symmetric and positive definite,
requiring factorization only once at the beginning of the
iterative process. Furthermore, the matrix is not affect-
ed by changes in the structure of the system involving
lines which are not metered, and being very sparse
allows the writting of very efficient computer programs
for large systems, through the use of optimal ordering
and sparsity techniques.
The detection and subsequent identification of
gross measurement errors is accomplished by means of
the statistical analysis of the estimation residuals.
3.6. Detection and Identification of Gross Measurement
Errors.
One important aspect of the use of estimators for
the processing of measured data is their ability to de-
tect the presence of measurement errors that go beyond
their assumed normal limits. These excessive errors
could be due to failures of the meters or the communi-
cation channels that convey the information to the com-
puter at the control center, and are known as gross
47
measurement errors (18).
It has already been shown in section 3.2. that the
expected value for the sum of squared residuals J(i)r)
given by eqn. 3.65 is equal to the number of observa-
tions minus the number of unknown variables (m - n) and
it was pointed out that this result is independent of
the distribution function-of the measurement errors.
If it is now assumed that the measurement errors OM.
are gaussian, ie. normally distributed, it can be shown
that the random variable J(ir' r) has a chi-square distri- -
bution with ( m - n ) degress of freedom, and the follow-
ing probabilistic statement about J(171.) can be made
P J(Qr) 72(m-11),c
c (3.67)
where:
2 (m-n)ox is the 100oc percentage point of the
chi-square distribution with (m-n) degrees of
freedom.
Eqn. 3.67 indicates that the probability of J(Vr)
exceeding the tabulated percentage point 1X2- is (m-n)ox equal to O, which is a preselected quantity. In fact
the selection of a value for ac with its associated value
of 'X2 (m-n)oc provide the acceptance criteria for the
solution. For example, if J(V ) were to exceed the
given threshold value X( In...n)loc the solution would be
rejected as this indicates the possibility of one or
more bad data points in the observation vector. It should
be pointed out that there is a close relationship
48.
between effective detection and the degree of measurement
redundancy. Indeed, if it is assumed that m = n, the A
estimated state given by 3.61 would be VT = B r (Vm -b1 V1 ) A
and substiiution of Vr in eqn. 3.10 would
yield a J(17 ) which is identical to zero and therefore
useless for detection of bad data points.
Assuming then that a test has been made and the
solution was rejected due to an unusually high value of
J(Vr), then the problem of identifying which specific
measurement is responsible for this irregular error
behaviour has to be solved.
First let us recall that the residuals have been
defined in section 3.2. as the difference between the
observations and their estimated value, in our case
A z = v
131
and that
E (z) =0
with covariance matrix that is given by
, = (R - Br r (Bt R-1 B r r ) Bt )
(3.67)
(3.68 )
(3.69)
Due to the assumption that the measurement errors
are normally distributed, the residuals are also normal
random variables. Let(rz,i be the standard deviation of
the ith residual ie. the square root of the ith main
diagonal element of 2z and define
. ( Z1 )
gi_ cr • z, 1 (3.70)
49
The variable gi given by eqn. 3.70 is a standard
normal random variable, with mean E(gi) = 0 and unit
variance.
The tests made to identify gross measurement errors
are conceptually similar to the detection test and are
based on probabilistic statements about the value of the
residuals gi, ie.
P t- K 4
GC/2 = gi = KIX12 = cc (3.71)
where cc is known as the level of significance of the test
and kW2 is the 100 0c/2 percentage point of the normal
distribution, Here again cc is selected as a small quantity
and OcK/2
is found from tables (22). For example, if
m=0.05 was chosen, then the absolute value of gi should
not exceed 1.96 if it is to be considered an acceptable
estimated value. If it fails the test, then the particu-
lar gi is a suspected gross measurement error. The phenom-
enon known as smearing, ie. the appearance of large resid-
uals in measurements which are 'good', due to a gross
error is common in this type of application. This means
that several residuals will fail the identification test
given by eqn. 3.71, but experimental results have shown
that in most cases where gross errors have been intro-
duced the largest normalized residual usually appears at
the point where the gross error is located. Once the
gross error is identified, it is deleted from the obser-
vation vector and a new estimate of the system state is
computed.
50
3.7. Algorithm for the solution and on-line tests.
. The American Electric Power, AEP, estimation method
described in section 3.5., together with the techniques
for detection and identification of gross measurement
errors of section 3.6., were implemented on a PDP-15
computer for on-line processing of data obtained in real
time from the power system simulator described in chapter
2. The a priori information required by the method, ie.
network topology, line parameters and location of meters,
is fed to the computer from the hi.gh speed paper tape
reader, while the measured quantities are obtained from
pre-specified locations in the computer core, and converted
to p.u. values prior to their use in the process.
The complete set of data, a priori and measured,
is then used to form the bus addmitance matrix YBUS and
matrix (Bt R-1Br). The fact that YBUS is a sparse matrix
is taken advantage of by storing only its non-zero ele-
ments. Matrix (Bt R-1Br) is formed and factorized
using Cholesky's method (24 ) which exploits very effec-
tively the matrix symmetry and positive definiteness.
Only the lower triangle and the diagonal elements of
the factorized matrix are stored for use in the iterative
process which in algorithmic form can be described as
follows:
1) Set iteration count i=0
2) Give an initial guess to the unknown bus
voltages Vt = j O. 0 k=2,3,...,n
51
3) Calculate 'measured' voltages accross the lines
with eqn. 3.50 and using the current value of
the state vector.
4) Calculate new bus voltages V(i) from eqn. 3.66
5) Test for convergence! V(i) - V(i-1)1 toler-
ance for k=213,...,n.
6) If convergence has been obtained proceed to step
7. Otherwise advance iteration count k k+1
and return to step 3.
7) Check the validity of the solution by comparing
the value obtained for the sum of squared resid-
uals against the selected threshold value
A If the check is positive ie. J (V )= km-n) r
7C2 a valid estimate for the state has been (m-n),cc
obtained otherwise proceed to step 8.
8) Examine individual residuals and eliminate from
the vector of observations the one that contains
the largest normalized residual, modify accord-
ingly, refactorize matrix (Bt R-1 Br) and return
to step 1.
For the on line testing of-the algorithm the network
shown in figure 3.2. was connected on the power system
simulator and several tests were made. The results of
these tests are shown in figs. 3.3, 3.4 and 3.5.
LOAD LOAD LOAD
2 3
Figure 3.2. Network connected on Power Systems Simulator
Figure 3.3. shows the results obtained with the AEP
estimator. The meters are represented by black squares
located at the end of the line Where the measurement of
real and reactive power is taken; the measured value is
shown as a complex number by the side of the meter. Only
ten 13,Q meters are used to estimate the 5 unknown system
complex state variables Vili=2,...16 therefore the degrees
of freedom of the chi-square variable given by J(Vr ) are
5. A level of significance m=0.05 was selected giving
2* 2 05 a threshold value of 11 . 10 forNI5
used in the -.
53
detection test. Only two lines have measurements at both
ends. This is felt to be a stringent and perhaps more
realistic situation as opposed to providing maximum
redundancy with meters at both ends of all the lines.
The measured line flows together with the voltage measurement
at the reference bus are processed to obtain the estimated
system state which in turn is used to calculate the line
flows. For comparison, the estimated line flows are written
in parenthesis next to the measured values, and it can be
seen that they are in close agreement. The performance index
J(V ) is 4.01, this being lower than the threshold value
indicates that an acceptable solution has been obtained.
.104+30.01 measured voltage
.534+3.0889 .2499+3.00555 (°5359+,0815- ) (.2652+3.0089)
(-.2565+3.023) -.2659+3.0501
1
(.2425-j.0221) .2125-3.0224
(.6371+3.2138) .6140+3.2286 (r..6247-3.1751) -.6577-3.1655
(-.254+3.072) -.267+3.068
r .2790+3.1588 (.2823+3.1386)
.3198-3.2404Y -.5102-3.2263
(-.1030+3.022) -.1049+3.042
Performance index 4.014 : solution accepted.
Figure 505. On-line test of State Estimator (no bad data present).
L LOAD
0—
(.6 463+j.2071) .6188+j.2223 —.6339—j.1682) —.6698—j.1527
(—.2643+.0761 —.2760+j.076
5
LOAD
.100-f-j0.0 I
(.2907—j.0145) .2833-j.0055
(—.2801-1-jeo543) —.2850-1-J.064
3
54
Figure 3.4. shows the results obtained when one
measurement is made almost zero ie.
error is present in the observation G
a gross measurement
vector.
(.6223-1-J.0067) bad data1.01954-j.0257 point
(.2521-J.0246 ) .2232- j,029
.3074-j.227 —.3152—.24)
—.1088+j.0481 (-.1062rj.0256
6
LOAD
Performance index•.= 31.97 : solution rejected.
Figure 3.4. On-line test of State Estimator (meter 1 is
d bad data point).
Due to the high value of the perforMance index, the
algorithm detects the presence of gross measurement
errors. The individual residuals are tested, with the
residual at meter 1 having the highest value (-5.22).
This measurement is eliminated froM the measurement set
2194-j.1583 Q(.2831+j.1351) .
55 I'
and the estimation process is performed again. The
results shown in parenthesis are obtained after the bad
data is deleted. As a consequence the performance index
is reduced to3.85and the solution is acceptable.
1.1038
.55324-j.0756" (.55784-j.0686
.2805-j.0002 (.2831...j.0031)
(..27324-J.0343) ....2878+j.0605
3 (.2652-j.0288) .2316-J.0275
(.6,7271-j.1957) 1.0185-j.01621Bad data point
(.6593-j.1538) .6688-j.1572
(-.266 0 j.075 -.2773+j.072
...3b81..j.2449) -.2988-j.2319
(...1088-1-j.033 ) -..10 924-j. 0 261
.28144.j.1561 (.2820+j.1372) ,
5 6
I -1 LOAD LOAD LOAD
Performance Index'= 179. 05: solution rejected.
Figure 3.5. On-line test of State Estimator.
In fig. 3.5. a bad data point in the meter located
at bus 2 in line 2-.4 is present.
the previous test is that this line
The difference from
is measured at both
ends, and it can be seen that in this case the perform-
ance index is about 5.5 times higher compared to the
56
case who're only one end of the line was measured. Here
again the solution is rejected and the identification
subroutine is called to analyze the individual residuals.
As previously, the largest normalized residual is located
at the bad data point with a value of 10.80 .Once the
bad data point is deleted, the estimated flows are cal-
culated and shown in parenthesis. As a consequence, the
performance index is reduced to '2.50.
Table 3.1. shows the results of tests performed
off-line in the CDC 6400 computer. They were obtained
by simulating the presence of a partial error in the meter
located in line 1-2. Both real and reactive power
measurements were multiplied by a factor ranging from
.0.5 to 1.5 and the effect of this on the performance in-
dex was observed. One of the meters on line 2-4 was then
located at bus 2 on line 1-2 wo that this line had meas-
urements at both ends and the experiment was repeated.
The results of these tests are shown graphically in fig-
ure 3.6.
57
Corruption
Factor
Meters on both sides of line
Meter on one side only
J(X) Normalized residual
J(X) Normalized residual
.5 77.32 1.41-j 8.41 25.16 .62-j 4:44
.6 53.36 1.46-j 6.83 18.40 .67-j 3.60
.7 33.54 1.52-j 5.25 13.08 .73-3 2.77
.8 20.42 1.57-j 3.67 9.16 .78-j 1.92
.9 11.44 1.62-j 2.10 6.66 .84-3 1.09
1.00 ' 7.45 1.67-3 .52 5.58 .85-3 .24
1.10 8.45 1.72+3 1.06 5.90 .96+j .58
1.20 14.46 1.774-3 2.64 7.64 1.01+3 1.42
1.30 25.46 1.83+j 4.22 10.79 1.06+3 2.25
1.40 41.46 1.88+j 5.80 15.35 1:13+j 3.09
1.50 62.47 1.93+3 7.38 21.32 1.18+j 3.93
Table 3.1. Tests on the discriminatory power of the
performance index.
. Line measured at both ends
+ Line measured at one end only
70
60
30
20
10
58
Performance index
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Corruption Factor
Figure 3.6. Discriminatory power of performance index.
59
Although the most common types ofgross measurement
errors would be either zero or full scale measurement (18)
A these tests show how the performance index J(V ) becomes
—r
much more sensitive to small errors if meters at both
ends of the line are used.
In this chapter the use of state estimation
techniques for on-line processing of data obtained in
real time in a power system simulator have been discussed.
It should be pointed out that the task of the state
estimator is the formation of a. reliable data base which
can be displayed to the operator and used as input to
other algorithms concerned with the security and economy
of operation of the system. These algorithms will be
discussed in chapter 4.
4 SECURITY CONTROL VIA OPTIMUM
DISPATCHING OF POWER.
4.1. Introduction.
Due to the importance that continuity of supply of
electric energy has in modern society, system security
has become the overriding consideration in the operation
of power systems, and improving the security of a system
is considered in itself a major justification for on-
line computer control. In his work T.E. Dy-Liacco
(ref.25) has defined the operating conditions of a power
system in terms of three operating states
a) preventive or normal
b). emergency
c) restorative
Normal state is defined as the operating state which
satisfies the real and reactive power demand of the sys-
tem without violation of the operating limits of its
component parts, ie. lines, transformers, etc. An emer-
gency condition will be one where the satisfaction of
61
the demand is accompanied by violation "of operating
limits in one or more system components, ie. line or
transformer overloading, etc. This requires fast correc-
tive action to relieve this anomalous situation before
the automatic control of the system .operates to protect
the violated component, causing perhaps further component
violations and leading to a cascade effect that may end
up with a partial or total system shutdown. This latest
condition where ti:e demand is not satisfied is defined
as the restorative operating state.
The normal operating state can be further classified
as either secure or insecure by referring to a list of
contingencies such as line or transformer outages, sudden
load changes, loss of generation, short circuit, etc.,
and stating that the system is secure if it.is able to
withstand the occurrence of any one of the contingencies
in the list without going into an emergency or restorative
operating state. Although the dynamic transition of the
system from its present operating state to the simulated .
post 'outage steady state is ignored, this method is
generally accepted as a useful assessment of security.
Figure 4.1. ilustrates these concepts using broken
lines to represent the effects of contingencies and solid
lines to represent the effect of corrective actions.
INSECURE
REGION
/
EMERGENCY
OPERATING .
STATE
RESTORATIVE
OPERATING
STATE
■■■•■■ •■•■• ••••••• •■• ■■■■• m••••• Oman
SECURE
REGION
/ /
62
N 0 • R
A
0 P E
A T I
G
S T A T E
Figure 4.1. The 3 operating states of a power system.
Effects of contingencies and of corrective
actions.
Clearly the objective of security control (ref.10)
is to maintain the operation of the power system in
the normal state, ie. preventing or minimizing the
departures from the normal state into either the emergency
or the restorative state. From figure 4.1. it can be
seen that to achieve this objective, adequate preven-
63
five control actions should be taken whenever possible
to ensure operation in the secure region of the normal
operating state.
Having obtained a reliable data base, by proce-
ssing the information obtained in real time from the
system by means of the techniques described in chapter
3, an on-line security analysis consists of the simu-
lation of the occurrence of each of the contingencies
on the given list, checking the results of every simu-
lation against the predetermined operating constraints
of the system. It can be easily appreciated that the
time required for the analysis is proportional to the
length of the contingency list and therefore it would
be desirable to analyze only those contingencies which
are known from off line studies or prior experience to
be the most severe and have a high probability of
ocurrence.
Assuming that the present operating conditions of
the system are normal the results of security analysis
would' indicate whether the system is operating in
a) the secure region or b) the insecure region. In case
a) no further action is required. In case b) the
indication of the contingencies which cause the opera-
tion of the system to go into an emergency state together
with the constraints that are violated as a result, are
transformed into a set of security constraints and used
to calculate the necessary preventive control actions
that would enhance system security by leading the state
• 64
of the si6tem into the secure region. The selection of
the preventive control actions is made in accordance
with an appropriate criterion for optimum performance.
Due to time considerations the mathematical models
involved in the calculation of these optimum control
actions, should be as simple an approximation as possible
consistent with the quality of performance desired. For
this reason and because of the time lag between system
conditions input and decision output, the calculated
control actions are strictly speaking sub-optimal.
In the network to be analyzed in the power system
simulator only single line outages are treated as members
of the contingency list. From the computational point
of view this type of outage is more demanding because
it alters the structure of the network. The methods
described in sections 4.2 and 4.3 make use of special
techniques to obtain fast solutions to the security
analysis problem.
4.2. The fast decoupled load flow.
4.2.1. Formulation of the method.
The fact that changes in nodal voltage values affect
mainly the reactive power flows whilst changes in phase
angles affect the real power flows, implies that a
degree of decoupling exists between the real and reactive
65
equations. The exploitation of this natural decoupling
has led to the development of fast, efficient and reliable
methods for the solution of the load flow problem (refs.
26, 33 ). Among these methods the author has found the
"Fast decoupled load flow" (ref.33 ), developed by Stott
and Alsac, together with the use of the ,Therman-Morrison
technique for the simulation of contingencies,a very
effective combination which requires small core storage
with fast running times and reliable convergence. These
characteristics are of course of prime importance in on-
line security analysis where repetitive multiple case
solutions are required.
. The decoupled eqns. are derived from the polar-power
mismatch formulation of the formal Newton method as
applied to the load flow problem by Tinney and Hart
(ref. 34). The linear relationship between small changes
in real arid reactive powers and voltage phase angles and
magnitudes can be written in the following partitioned
form.
AP
AQ
(4.1)
J
where:
APk j/1Qk = complex power mismatch at bus k
LV = voltage phase angle, magnitude correcttions
66
H , N = partial derivatives of real power with
respect.to voltage angle and magnitude
J f L = partial derivatives of reactive power
with respect to voltage angle and magnitude.
In this formulation submatrices N and 0" represent
weak coupling terms and their numerical values are
therefore small compared with those of H and L. The
decoupling can then be achieved by neglecting N and J in
eqn. 4.1 with the result that
AP = H (4.2)
and
Q = L AV/IT (4.3)
The solution can be obtained by iterating with eqns.
4.2 and 4.3 and although core storage has been greatly
reduced by solving two smaller systems of eqns. instead
of one large one the powerful quadratic convergence of
Newton's method is replaced by a weaker rate of conver-
gence. In eqns. 4.2 and 4.3 H and L are formed and
factorized at every iteration just in the same fashion
as the original Jacobian matrix of eqn. 4.1 but a closer
look at the elements of these matrices indicates the
possibility of further simplifications which bring about
the increased efficiency of the method proposed in ref.33
Consider a network with n+1 nodes, numbered from 0
to n. The eqns. relating real and reactive power injected
at node k as function of voltage magnitudes and phase
67
angles are given by
In
P = V 7: tv G cos(e- )iV B sin(a -4 ) k k k=0 m km k m m km k m
k#m
(4.4)
and
Qk = Vk k= iVmGkm sin(e- )-V m Bkm cos(e-k )} (4.5) E
0 k m m icra
and so the components of submatrices H and L are
a aPk Hkm = 38®mm
Vk V m [G.km em sin(9 - )-Bkmcos(9.k-9M) I (4.6) -
"k L t Vm aV
m = Vk Vm IGkm sin(e & k m. - ) -B cos(e
k km km ((4.7)
and
5Pk
H = = -B V2-Q
kk kk k k (4.8)
Qk = V = k aVk
2 -BkkVkk
(4.9)
68
For stability reasons the branch phase angles
(E).k m) are kept small And therefore sin (e.k-eM) is much
smaller than cos (lek-rem). In addition, the series
conductance of the branches Gkm is smaller than the
susceptance Bkm, so that a good approximation to eqns.
4.6 and 4.7 with cos (e-e)=1.o is given by
H = L V V km km' k mB km (4.10)
Because of the fact that eqns. 4.8 and 4.9 are
strongly dominated by the term -BkkVkl Qk can be neg-
lected so that an approximate vain::: for the diagonal
elements of the sensitivity matrices is given by
u2n = Lkk.'":7 vekk (4.11)
Let us now assume, without loss of generality, that
the voltage controlled buses, ie. buses where the voltage
magnitude is known and fixed but the reactive power is
unknown, are numbered from 1 to 1 using bus number zero
as the slack. Use of the approximate values for the
elements of the Jacobian matrices H and L given by eqns.
4.10 and 4.11 in eqns. 4.2 and 4.3 yield the following
relationships
k=1,...In (4.12 )
69
EL alr. . = V. Z 1+1 -Bia Va - V ; i=1--1,...,n (4.13)
3
The following additional refinements are required
to obtain the final form of the algorithm.
a) Divide both sides of eqns. 4.12 and 4.13 by Vk
and V. respectively
b) Set Vm in eqn. 4.12 to 1 p.u.
c) Use in eqn. 4.12 the modified susceptance matrix
B given by
B1km = -1/X km
n
Btkk - 2: / xkin
m=0 mik
where:
xkm = series reactance of line k-m
b) and c).have the effect of removing from the calculation
of LW. those elements in eqn. 4.12 which mainly affect
reactive power, resulting in a more stable algorithm
with better convergence characteristics as pointed out
in the discussion of ref. 33. Defining B11 as the negative of the susceptance
submatrix which is used in the calculation of voltage
increments in eqn. 4.13, the final form of the decogpled
eqns. is given by
ATI/V = B' ,se- (4.14)
70
AQ/V = Bit AV (4.15)
Front what 'was said previously is
obvious that matrices B' and WI are constant so that
they are factorized only once at the beginning of the
iterative process and because they are symmetric only
the lower triangle of the factorized matrices need be
stored.
4.2.2. Line Outage simulation.
Although the outage of a line alters the structure
of matrices B' and Btl, for the purpose of simulation
it is inefficcient and unnecessary to represent the
outage by changing B' and B" as this would require a
net• triangularization for each outage. A special
application of a general method for modifying inverted
matrices can be used to simulate the effect of a branch
outage on the solution. This gives the method maximum
efficiency because the matrices B' and B11 are factor-
ized only once at the beginning of the process and used
in their factorized form, storing only their lower
triangles, for the whole solution cycle, ie. the analy-
sis of .a series of branch outages.
The outage of line k-m alters. elements (k,k),
(m,m),(k,m) and (m,k) of matrix B' in eqn. 4.14. In
matrix notation the change in B' can be expressed as
B' = B' 6,13 MMt new
(4.16 )
where:
Ob = -1/xkm
x - series reactance of branch k-m km-
M= an'n'vector that is all zeroes except element
k which is +1 and element m which is -1.
The inverse of the modified matrix Blew
can be shown to be (ref35).
(BtgAbMMt)-1=B1-1(a +MtB1-1M)-1B1-1 MMtB1-1
defining vectorZ as
Z = Bt-1
and the scalar
= ( 4.b
Mt z)
(4.17)
(4.18)
(4.19)
and substituting eqns. 4.18 and 4.19 into 4.17 we get
Bt-1 = -1 C Z Mt Bt-1 new (4.20
The solution for 40 taking into account the outaged
line, would be given from eqn. 4.14 by
-1 • = B tilYIT — new new — (4.21)
where 11P is calculated for the new system conditions,
ie. without line k-m.
Substitution of eqn. 4.20 into eqn. 4.21 gives
— new = -1 AP/V-c Z Mt Bt -1 AP/V
(4.22)
•
which from eqn. 4.14 can also be written as
A. = n6 -c Z Mt 4■G. --new — (4.23)
Equation 4.23 shows that the solution for the base
case problem can be easily corrected to account for line
outages, thus avoiding the time consuming modification
and refactorization of matrix B'. The additional work
required is the computation of vector Z which is obtained
at the beginning of the outage case, by forward and
backward substitution using the factors of B' with M as
the independent vector and other operations indicated in
eqn4.19 to obtain c and in eqn. 4.23 to obtain the final correction of vector thel: This require very few arith- ....
metical operations because of the special form of vector
M.
A similar procedure is required to obtain the nec-
cessary corrections for the voltage magnitude increments
and these are given by
AV = d Y t 61/ -- new — — (4.24)
where scalar d and vectors Y and N have similar meaning
to scalar c and vectors Z and M respectively, but account
for the differences between matrices B' and BII.
The following additional considerations apply to
the reactive equation solution
73
a) If the outage is a transformer-the entry corre'
sponding to bus k is made equal to the off-nominal
turns ratio referred to bus m
b) If the line outage connects a voltage controlled
bus with a load bus,vector N contains only one
non-zero element t 1.
c) If the line outage connects two voltage controlled
buses it is not necessary to make any corrections
to AV as matrix B" is unaffected by the outage.
The algorithm for the sequential simulation of line or
transformer outages is described in appendix 1
A computer program based on this algorithm was written
to analyze sequentially the outage of each of the lines
of a given network identifying as will be shown in section
4.4., those outages which result in the overloading of
other lines of the system.
4.3. Line outage simulation using fictitious injections_.
4.3.1. The Exact Method.
An alternative method for line and transformer
outage simulations can be obtained by injecting at nodes
k and m, connecting the line whose outage is being
analyzed, adequate amounts of real and reactive power.
These fictitious injections have the effect of making
the system behave as if line (k,m) were not present,
without any actual change in the topology of the system.
. 74
This means that the structure of matrices B' and B" in
eons. 4.14 and 4.15 remains intact, thus there is no need to
refactorize them for the analysis of a line outage.
This idea is ilustrated in figures 4.4, 4.5. and 4.6. In figure 4.4. the basic state of the system is shown. In the on-line node this would be the present
operating state of the system, obtained by processing
measured data by means of the techniques discussed in
chapter 3. If the outage of line k-m were to incur,
the final state of the system after the transient
phenomena has died down is shown in figure 4.5.. A sim-
ilar effect, on the state of the system. would be ob-
tained, if the line were retained and adequate amounts
of real and reactive power injected at nodes k and m.
This situation is depicted in figure
Figure 4.4. The basic system state.
Pm+ Qm
P+ Qk A k 6 APm+j AQm
POiqk Pk+i AQk APm+j 4Q1;
V'
75
Figure 4.5. Outage of line k,m.
to system to system
Figure 4.6. Simulation of the outage of line k-m.
76
The-required fictitious injections at nodes k and m
should be equal to the line flow in the final system
state.
This equality is expressed by
2 km - APk =Pkm =17/k g -VIV1mgkm k cos(19.1-83m)+VIVIbmkm sinWk-elm) k k
(4.25)
-V'V'b cos(eJ-e4)-ViVig sin(e°-&1 )-V/2bi k m km km km km k m k km
(4.26)
AP =Pmk m =V/2gmk mk ViVigmk W cos-e1)+ViVibmk sin(e.1-e4k) m mk mk m
(4.27)
4Qm=Qmk =Vehmk-VI;Vpmkcos(%-91)-VIVLgmksinN-911)-V12bLk
(4.28)
where
VI g and Vtan, = voltage magnitudes and phase
angles that would be obtained
at nodes k and m if line k-m
were removed
gkM4Jbkm = complex admittance of line kim
= shunt susceptance of line kola
77
The direct way to the calculation.of the required
injections would be to introduce in the iterative process
of the decoupled load flow described in section 4.2.2.
the following modification to the injections at the end
buses of the line whose outage is being simulated
Pk
= PG
Pm+1 = PG
m-PPmmk (4.29 )
ni+1 nG '4k = k k-'`kt
Qm+1 = QG -QD -141i
m m mk
where Pkno Pmk'km and Qik are the line flows calculated
from eqns. 4.25 to 4.28 with the voltage magnitudes and
phase angles as they stand at iteration i. The iterative
process is continued until the normal convergence criteria
is satisfied. At the solution point the fictitious
injections and the new voltage profile are obtained.
The outage has therefore been succesfully'simulated
without modifying the topology of the system. Although
the results obtained with thib simple method are accurate,
convergence is very poor.
4.3.2. The method of Sachdev and Ibrahim.
In ref. 36 the injections required to simulate the
line outage are calculated from sensitivity relations
obtained by linearizing the power flow eqns. around the
base case state. The sensitivity matrix is given by the
inverse of the Jacobian matrix of eqn.'4.1, and can also
be written in partitioned form as
■•••=11, MEW'S.
C D AP ■■■••
E F /IQ
a
(4.3o)
At the base case state the APs and AQ in eqn. 4.50
have very small numerical values and for all practical
purposes they can be considered equal to zero.
The additional injections required to simulate the
outage of line connecting buses k and m is obtained by
solving in conjuction with eqns. 4.25 to 4.28 the fol-
lowing set of 4 linear eqns. derived from elements of
eqn. 4.30
••••••■•
ckk Ckm
Cmk mm
ONIMIW
D
kk Dkm
Dmk Dmm
■••■■ •••••••
4Pk
AP m
AQk
60,11
••■■
••■••
Immo •INEN.
AE3-
Le- m
(4.31) Ekm
Emk mm ••••••••
Pkk Fkm
Fmk
Fmm ■•■•■
AVk
AVm .6■111,
The voltages are then corrected by means of:
. 79
V +1 = Vi Ami &k 4.1
k = L16'
(4.32) i i+1
Vm
= Vm
AVm 8 = e i zie-±
where i is the iteration count.
Since the relationship between the changes in real
and reactive power injected into buses k and m and the
magnitudes and phase angles of the voltages at these
buses is nonlinear, the final solution is obtained by
iteration. The steps are as follows
a) Set iteration count 1=0.
b) With the present voltage magnitudes and phase
angles at nodes k and m use eqns. 4.25 to 4.28
to calculate the required APs and AaQs.
c) Use the calculations of step b in the solution
of eqn. 4.31. This yields the voltage correct-
ions.
d) Update the voltage magnitudes and angles by
means of eqn. 4.32.
e) With these new voltages calculate the flows in
line k and m and compare them with the injections
calculated in step b. If the differences are
within the given tolerance stop the iterative
procedure. Otherwise return to b.
The real and reactive flows in line k-m obtained
through this process are the approximate fictitious
injections required to simulate the outage of this
line. The new state of the system is then obtained
by using these injections in eqn. 4.30 to perturb the
base case solution. Recalling that originally all Ws
andligs are zero it can be seen that only four columns
of the inverse of the Jacobian are needed to simulate
the outage.
4.3.3. A modification to the method of Sachdev and Ibrahim.
Although the method of section 4.3.2. can be
considered as computationally efficient, i.e. short
running times for approximate solutions, the need to
store the lower and upper triangles of the base case
Jacobian matrix represents a disadvantage for on-line
implementation in small-core process computers.
To overcome this disadvantage the author investigated
the possibility of substituting eqn. 4.30, which uses the
base case Jacobian, by the decoupled eqns.
4669.= [x '] AP/V
• 16 1 = [xi t]ip,/v
(4.33)
(4.34)
where X' = By-1 and X'' = B"-1. Under these
circunstances eqn. 4.31 becomes
APk/Vk
AP /V m m
■•■•■■
4e-
46.(30
(4.35a)
Xt Xt kk km
XI X' mk mm
.1111•••■
Ae
81
0•1•1=111. ••■■•■ OMNI.
X" X" kk km
X." X" mk mm
AVk
AlTm
(4.35b)
Unfortunately this method combining fast solution
times with the small core requirements of the decoupled
load flow gives unsatisfactory results as the calculated
injections are Dlr from the ones required to simulate
the line outage. In consequence the system state that
results from these computations is not an acceptable
approximation to the true numerical solution of the
problem.
j
4.3.4. A new method for the simulation of branch outages.
In a "d.c." approximation to the load flow problem
the outage of the line connecting nodes k andm., can be
simulated by adding to the base case phase angles & the
vector AS,resulting from the solution of the set of
linear eqns.
11111Ma.
X *OAP .11 ik Xlm X111
Xk1 Xkk Xkm X kn . ,
Xmk X • Xmn
m1 "mm
X ... X X 0.0 Xnr
n1 nk nm'
0
APk • 4P
m
6
(4.36)
••■••■ ••••■
X1
.km
X .mm
Xnm
m (4.38)
•••••• •••■•
82
In appendix 3 it is shown that the injections
required in eqn. 4.36 are given by
km (4.37 )
-(Xkk +Xmm -2X )J km
Al' =
where:
Po : base case real power flow in line k-m km
: series reactance of line k-m km
Xkm : element of matrix X'
In this method the injections given by eqn. 4.3?
are used as an initial approximation to the injections
required to simulate the outage of line k-m. They
are introduced as a perturbation to the right hand side
of eqn. 4.33. Since at the base case solution all
power mismatches are sufficiently small as to be
considered equal to zero eqn. 4.33 can be written as
MO.
X1
X .kk
X .mk
Xnk
APk Vk
So. only two columns of matrix X' are used in the
calculation of the phase angle increments, and they can
be obtained by one forward and one backyard substitution
83
each, using the already available triangular factorization
of matrix B'. The amount of computation required at this
step is comparable to that needed in the initial
preparation of the method described in section 4.2.2.,
where the Sherman-Morrison formula is used to modify
matrix B' so that changes in topology caused by the outage
of line k-m are accounted for.
The phase angle corrections obtained from eqn. 4.38
correspond, as mentioned -earlier, to the d.c. solution
to the line outage problem. On some systems or for
certain outages this solution might be a sufficiently
accurate approximation, but if the voltage solution is
required or this approximation is not satisfactory, an
improvement can be obtained by continuing the process,
i.e. iterating towards the solution using two reduced
sets of linear eqns. in the following fashion.
With the phase angles obtained calculate the real
power flow in line k-m and recalculate APk and APm
from
Ap fpsch ▪ p cal) , cal k ` k km ' rk
(4.39) Ap. = (pSCh • pCal ) p cal m ` m mk
where: •
s Pkoh I Scheduled injection at node k
cal : Calculated injection at node k
cal Pkm Calculated real power flow in line k,m.
84
The real power mismatches of eqn. 4.39 are used in
eqn. 4.38 to calculate a new vector of phase angle
corrections.
After updating the phase angles the reactive power
flow is calculated in line k,m and this flow is used to
perturb the right hand side of eqn. 4.34 using
AQk =(<11. ) _ Qcal
AQ ((lsch Qcal) Qcal mk
(4.40)
where:
sch Qk : scheduled reactive power injection at node k
ncal .‘km : reactive power flow in line km, calculated
with the available voltages at nodes k and m.
cal Qk : reactive power injection at node k, calculated
with the available voltages in the system.
As the reactive power mismatches in the remaining
nodes of the system are very small compared with those
of nodes k and m, a good approximation to the required
increments in voltage magnitudes can be obtained from
•■■••• =WIMP
Av. .1+1
.k
AV sm 0
AVn 4111.1.
it
X .1+11k
* .k,k
• • rr .mik .n. Xn,k
6Qk iiQm
.1••■■•
”. .1+1,m • .
X, . ' • H X .m,n
Xn,m
V Vm
(4.41)
wheretSqk andag,m- are calculated from eqn. 4.40.
85
The solution is obtained by iterating with eqns.
4.38 and 4.41 until W3k., Wpm, given by eqn. 4.39, AQk
and 4m, given by eqn. 4.40 are all less or equal the
specified tolerance. This is equivalent to saying that
the difference between the real and reactive flows in
line k,m calculated at two consecutive iterations is
sufficiently small. A block diagram for this algorithm
is shown in figure 4.7.
86 START
Identify nodes connecting line whose
outage is to be simulated. Nodes k and m
Calculate real power flow
in line k and m
Use eqn. 4.37 to evaluate the approX. needed injections
4Pk and 4Pm and add them to the pre-outage injections.
Solve eqn. 4.38 and update
phase angles
Calculate real power flow in line k,m and add it to
the pre-outage injection.. Calculate real power
mismatches using eqn. 4.39
Solve eqn. 4.38 and-update
phase angles
Calculate reactive power flow in line k-m and
add it to the pre-outage injection. Calculate
reactive power mismatches using eqn. 4.40
Solve eqn. 4.41 and update
voltage magnitudes
Figure 4.7. Branch outage analysis using
fictitious injections. END
.87
In order to obtain the voltage magnitude solution
when the outage of a line connecting two voltage-controlled
buses or a V.C. bus with the slack bus, the solution
of eqn. 4.15
= AQ/V
is introduced in the iterative process. Although this
increases the running time the solution time is still
very short as shown in tables 4.8, 4.9 and 4.10
Tables 4.2, 4.3 and 4.4 compare the post-outage
real and reactive line flows obtained with the
approximate method described in this section with the
exact numerical solution obtained using Newton's method
for the 3 test systems shown in appendix 2. Only the
results for those lines which are most affected by the
outage are shown. The selection was made using the
line-line coupling factor to be defined in section 4.4.
In tables 4.5, 4.6 and 4.7 the voltage magnitudes
at load buses obtained by this method are compared with
the 'exact' numerical solution. Tables 4.8, 4.9 and
4.10 compare the solution time per outage needed by the
new approximate method with the time required by the
decoupled load flow described in section 4.2., using
the Sherman-Morrison formula to simulate branch outages.
88
The derivation of the method has been obtained for
the simulation of the outage of a line connecting two
load buses .k and m. In this case two columns of the
inverse of B' and two columns of the inverse of Blt are
needed in the computation. It is a:well known fact
that the real and reactive power injections at the slack
bus and the reactive injections at the remaining voltage
controlled buses are not known until the solution of the
load flow problem has been obtained. The-equations
relating these unknown variables to the state of the •
system are therefore eliminated from the calculations.
For this reason outages of lines connecting different
types of buses have different computational requirements
and they. are summarized in table 4.1. For each
calculated fictitions injection 4P or 4Q only one
column of matrix B' or matrix B" respectively is required
to obtain the final system state.
Type of bus connected
' by the line
Fictitious injections re-
quired in the calculation
bus
V. C.
Load
V. C.
Load
Load
bus m
Slack
Slack
Vs Co
V. C.
Load
X
X
X
X.
X
6Pm
X
X
a Qk
X
X
hQm
Table 4.1. Injections required in the simulation of line
outages. (*V.C. Voltage controlled bus)
89
Outage of Line
Flow in Line
Coupling.
Factor
EXACT SOLUTION APPROXIMATE SOLUTION
I IR Q I P Q
1 - 2 1 - 3 1.078 .700 -.043 .637 .679 -.044 .617 3 - 5 .382 .598 -.134 .561 .600 -.133 .563
1 - 3 1 - 2 1.047 .675 -.100 .619 .679 -.101 - .623 2 - 3 .570F .419 -.051 .384 .418 -.051 .382 2 - 4 .469 .913. .114 .835 .913 .114 .835
• 4 - 5 .468 .457 -.318 .522 .454 -.318 .519
2 - 3 1 - 3 .293 .344 -.042 .314 .34o -.041 .311 2 - 4 .688 .999 .101 .911 .995 .102 .908 4 - 5 .684 .539 -.345 .600 .529 -.343 .591
2 - 4 1 - 3 .292 .517 -.050 .47a .498 -.050 .454 2 - 3 .732 .90.4 -.097 .825 .898 -.097 .820 4 - 5 -1.310 -.433 -.360 -.565 -.488 -.265 -.548 3 - 5 .814 1.079 .056 .990 1.116 -.038 1.032
4 - 5 1 - 3 .242 .359 -.043 .328 .364 -.043 .333 2 - 3 .599 .546 -.069 .499 .565 -.071 .517 3 - 5 .63o .716 -.289 .707 .704 -.253 .685
5'- 6 3 - 6 .949 .607 .305 .622 .621 .235 .608
3 - 6 2 - 4 .224 .876 .221 .820 .884 .172 .817 5 - 6 .975 .610 .307 .641 .621 .215 .612 3 - 5 .827 .811 .046 .744 .823 -.057 .756
3 - 5 2 - 4 .4o5 .998 .146 .915 1.019 .102 .930 4 - 5 .390 .538 -.301 .58o .558 -.325 .605 3 - 6. .522 .682 .109 .633 .687 .083 .633
Table 4.2. Comparison of post outage flows. (6 bus system)
90
Outage of Line
Plow in
line
Coupling
Factor
EXACT SOLUTION APPROXIMATE SOLUTION
P l
Q I P Q I
2 - 1 2 - 4 1.012 3.840 ;-".278 3;670 3:844 -.278 3:674 4 - 7 0.17-3 .586 -.052 0.566 .592 -.054 0.571
1 - 3 1 - 4 '.405 1.853 -.236 1.796 1.865 -.236 1.808 2 - 4 .326 3.408 -.262 3.258' 3.42o -.263 3.27o 4 - 7 .892 1.311 -.166 1.271 1.361 -.173 1.319
1 - 4 1 - 8 .257 3.073 1.189 3.169 3.052 1.186 3.148 2 - 4 .531 3.894 -.280 3.722 3.866 -.279 3.695
2 - 4 2 - 1 .998. 3.840 -.306 3.656 3.873 -.307 3.704 1- 4 .587 3.322 -.179 3.199 3.299 -.181 3.176 1 - 8 .233 3.432 1.231 3.506 3,414 1.227 3.489
3 - 7 1 - 3 -1.043 .388 -.260 .449 .383 -.258 ..445 1 - 4 .379 1.664 -.232 1.616 1.635 -.231 1.587 2 - 4 .304 3.257 -.254 3.114 3.235 -.253 3.093 4 - 7 1.090 .915 .301 .926 .920 .200 .905
4 - 6 2 - 1 .241 1.000 .043 .955 .991 -.041 .946 1 - 8 .641 3.426 1.374 3.549 3.303. 1.321 3.420 6 - 9 -1.676 -.195 .822 -.812 -.147 .76o -.745 8 ..:. 9 .855 1.242 -.417 1.392 1.065 -.419 1.212
4 - 7 2 - 1 .379 ..928 -.031 .886 .935 -.032 .891 1 - 3 .945 1.382 -.498 1.413 1.304 -.484 1.338 3 - 7 1.059 .915 .295. .916 .94o .233 .923
(..
Table 4.3. Comparison of post outage flows. (10 bus system)
Outage of Line
iflow in Line
Coupling
Factor
EXACT SOLUTION APPROXIMATE SOLUTION
P Q 1 P Q I
_.4 - 8 2 - 1 .176 .821 -.014 .783 .829 -.015 .790 1 - 8 .842 2.846 1.628 3.153 2.841 1.603 3.137 6 - 9 .184 .834 .488 .929 .826 .484 .920
6 - 9 2 - 1 .213 .951 -.035 .907 .952 -.035 .908 1 - 8 .596 3.248 1.481 3.432 3.209 1.291 3.326 8 - 9 .710 1.006 -.219 1.100 .946 -.214 1.025
8 - 9 1 - 3 .181 1.524 -.227 1.482 1.526 -.227 1.483 2 - 4 .145 3.145 -.248 3.008 3.147 -.2/18 3.010 4 - 6 .575 1.163 -.172 1.130 1.196 -.175 1.162 6 - 9. .321 .958 .296 .964 .959 .329 .975
•
Table 4.3. (Cont.) Comparison of post outage flows.
(10 bus system).
92
Outage of Line
Flow in
Line
Cou- pling Factor
EXACT SOLUTION APPROXIIIATE ri"
P Q ,t. I P Q
1 - 2 1 - 8 .995 2.605 .330 2.477 2.531 .293 2.404 2 - 6 -.459 .018 .185 -0.178 -.010 .181 -.174 2 - 8 -.512 -.315 .367 -0.462. -.332 .357 -.467 8 - 6 .480 1.369 -.592 1.495 1.327 -.569 1.444
1 - 8 1 - 2 .998 2.400 -.407 2.296 2.330 -.394 2.229 2 - 6 .347 .835 -.063 .801 .828 -.079 .796 2 - 8 .471 .781 -.032 .748 .777 -.048 .745 8 - 6 -.479 .252 -.151 .291 .237 -.144 .274
2 - 4 2 - 6 .470 .936 -.097 .900 .918 -.095 .883 2 - 8 .345 .689 -.058 .661 .674 -.o56 .647 6 - 4 1.012 1.013 -.296 1.039 .962 -.285 .988 8 - 6 .544 1.031 -.263 1.046 1.017 -.259 1.031
•
2 - 6 2 - 8 .436 .:669 -.040 .642 .668 -.064 .642 2 - 4 .283 .900 -.002 .861 .896 -.202 .857 8 - 6 .662 .985 -.206 .991 .995 -.200 .9.96
. 1 - 8 .274 .923 .020 . .871 .921 -.001 .868
2 - 8 1 - 8 .352 .914 .023 .863 .909 .013 .858 2. - 6 .422 .745 -.065 .716 .741 -.076 .712 8 i. 6 -.593 .371 -.154 .395 .362 -.145 .383
6 - 4 2 - 4 .949 .984 -.007 .941 1.000 -.008 .957 2 - 6 -.432 .447 -.047 .430 .446 -.041 .428 8 - 6 -.491 .497 -.133 .501 .494 -.135 .500
Table 4.4. Comparison of post-outage flows (14 bus system)
93
Outage of
Line
.Flow in
Line
Couplin(
.Factor
EXACT SOLUTION APPROXIMATE ffi-JITTON
Q I. P .
Q. .,
I. P.
8 - 6 1 - 2 . .th 1.779 -.282 1.699 1.857 -.299 1.774
2 - 6 .473. .892. -.101 .859 .937 -.113 .903
9 - 3 .242 .603 .160 .610 .616 .149 .621
11-10 .124 .135 -.011 .129 .125 -.020 .120
6 - 7 6 - 9 .487 .310 .014 .305 .305 .007 .299
8 - 3 .585 .576 .139 .582 .584 .158 .591
'11-10 .229 .120 -.018 .114 .104 -.025 .101
6 - 9 6 - 7 .621 .387 -.072 .386 .389' -.076 .387. 8 - 3 .452 .496 .145 .506 .503 .151 .513
7'- 9 .651 .387 .070 .371. .390 .068 .372
8 - 3 8 -. 6 .768 1.011 -.o85 .984 .992 -.157 .. .971
6 - 7 .617 .574 .079 .566 .590 -.042 .575
6 - 9 .373 .329 -.001 .322 .338 .030 .330
3 -11 -.621 -.197 .136 -.224 -.169 .145 -.208
7 -.9 .636 .574 -.008 .537 .592 .072 .560
9 -10 .507 .334 -.031 .314 .307 -.039 • .292
Table 4.4. (cont.) Comparison of post-outage flows.
(14 bus system)
94
Outage
of
Line
B
U S
VOLTAGE MAGNITUDE :EXALT SOLUTION
VOLTAGE MAGNITUDE APPROXIMATE SOLUTION
ERROR
1 - 2 if 1.069038 1.069123 -0.000085
5 1.086235 1.086199 0.000036
6 1.055879 1.055836 0.000043
2 - 3 If 1.067071 1.067074 -0.000003
5 1.084951 1.085046 -0.000095
6 1.055250 1.055336 -0.000086
2 - If If .997787 1.013773 -0.015986
5 1.058362 1.065264 -0.006902
6 1.041073 1.045434 -0.004361
5 - If If 1.053601 1.058242 -0.004641
5 1.096304 1.093609 0.002695
6 1.061018 1.059600 0.001418
Table 4.5. Comparison of voltage magnitude solution
(6 bus system)
95
Outage
of
Line
TYPES OF
NODES
CONNECTED
B
U
S
VOLTAGE MAGNITUDE EXACT
SOLUTION
VOLTAGE MAGNITUDE
APPROXIMATE SOLUTION
ERROR
1 - 2 SLACK-GEN 1 1.029580 1.029585 -0.000005
8 .954884 .954867 0.000017
9 1.032320 1.032320
3 - 7 GEN-LOAD 7 1.002987 1.009492 -0.006505
8 .953957 .954320 -0.000363
9 .997046 .997084 -0.000038
10 1.032320 1.032320 -
GEN-LOAD 7 1'4029608 1.029609 -0.000001
8 .928981 .930340 -0.004419
9 .992971 .993396 -0.000425
10 1.032320 1.032320 -
8 - 9 L0ADL.LOAD 7 1.029607 1.029609 0.000083
8 .942464 .945806 -0.003342
9 1.002859 1.000790 0.002069
10 1.032320 1.032320 -
Table 4.6. Comparison of voltage magnitude solution
(10 bus system).
96
Outage of
Line
TYPES OF NODES
CONNECTED
B U S
VOLTAGE MAGNITUDE EXACT SOLUTION
VOLTAGE MAGNITUDE
APPROXIMATE SOLUTION
ERROR
2 - 4 GEN-GEN 6 1.015373 1.016046 -0.000673
7 1.060071 1.060429 -0.000358
8 1.017509 1.018098 -0.000589
9 1.053919. 1.054325 -0.000406
SLACK- 6 1.012398 1.015789 -0.003391 LOAD
7 1.059446 1.060655 -0.001.209
8 1.054430 1.055083 -0.000653
6 - 4 LOAD-GEN 6 1.024022 1.023058 0.000964
7 1.064516 1.063966 0.000550
8 1.025327 1.024271' 0.001056
9 1.058999 1.058294 0.000705
8 - 6 LOAD- 6 1.017294 1.018018 -0.000724 LOAD
,7 1.058636 1.061670 -0.003034
8 1.023131 1.020406 0.002725
9 1.049254 1.056068 .-10.006814
Table 4.7. Comparison of voltage magnitude solution
(14 bus system).
Outage of Line
' METHOD OF SECTION 4.2 -
NEW METHOD
TOL= 0.1 TOL=0.0001 TOL=0.01 TOL=0.1
1 - 2 .019 .009 .005 .004
1 - 3 .013 .006 .005 .005
2 - 3 .017 .008 .006 .006
2 - 4 .044 ' .036 .014 .011
4 - 5 .042 .022 .007 .006
5 - 6 .049 .026 .011 .009
3 - 6 .049 .027 .010 .007
3 - 5 .049 .018 .006 .005
Table 4.8. Comparison of running time per outage (6 bus system)
Outage of
METHOD OF• SECTION 4.2 NEW METHOD
TOL= 0.1 Line TOL=0.0001 TOL=0.01 TOL=0.1
1 - 2 .023 .018, .013 .006
1 - 3 .032 .019 .009 .008 .
1*- 4 .030 .015 .012 .006
1 - 8 .042 .021 .015 .010
2 - 4 .044 .024 .016 .012
3 - 7 .086 .o46 .021 .013
4 - 6 .056 .029 .011 .011
4 - 7 .082 .041 .023 .012
6 - 9 .075 .058 .035 .015
4 - 8 .053 .028 .020 .013
8 - 9 .058 .029 .012 .010
97
Table 4.9. Comparison of running time per outage (10 bus system)
98
Outage METHOD OF SECTION 4.2 NEW METHOD TOL=0.1
of Line 20L=0.0001 TOL=0.01 TOL=0.1
1 - 2 .142 .069 .041 .035
1 - 8 .076 .044 .031 .011
2 - 4 .093 .044 .038 • .014
2 - 6 .085 .041 .027 .011
2 - 8 .083 .039 .028 .010
6 - 4 .074 .045 .019 .009
8 - 6 .143 .080 .018 ...010
6 - 7 .118 .053 .026 .013
6 - 9 .052 .027 .018 .009
8 - 3 .073 .043 .026 .011
3 -11 .151 .070 .013 .010
3 -12 .150 .063 .011 .008
Table 4.10 Comparison of running times per outage
(14 bus system).
The results shown in tables 4.2, 4.3 and 4.4,give
an indication of the accuracy of the new method. It
can be seen that the currents represent a very good
approximation to those obtained by Newton's method
using a tolerance of 0.0001 for the real and reactive
mismatches.
The solution for the voltage magnitudes shown in
tables 4.5, 4.6 and 4.7 can be seen to have, in most
99
cases, much less than 1% error. The running time
required by the new algorithm shown in tables 4.8, 4.9
and 4.10 indicates that reasonable savings are obtained
when compared with the efficient method described in
section 4 3 2 . It is expected that the application of
this new method to larger systems would prove still
more advantageous.
4.4. Derivation of Security Constraints.
As the security analysis is performed a list of
overloaded lines due to line outages is formed. This
list contains one entry for each overloaded line
containing the following information:
a) Number of line removed.
b) Overloaded line due to outage.
c) Post-outage current in overloaded line.
d) Line-line coupling factor given by .
ij I!. - I. 10 -
I° km
(4.42)
where:
km t. = additional flow in line ij,]. due to outage of 13
line k in per unit flow in line k m.
100
I! = post-outage current flow in line ij . 1j
Iij = pre-outage current flow in line
If the list is empty at the end of the full cycle
of outage simulations, the system is operating in the
secure region of the normal operating state, otherwise
control actions are required to enhance its security.
In the author's work the rescheduling of power generated
in the different nodes of the network is used as a
suitable variable for preventing or minimizing the
departures of the operating state of the system from the
secure region. Hence, knowledge of the sensitivity of
current flowing in thd lines to changes in generation
are required.
The components of this sensitivity matrix are
obtained column by column, introducing a small change
in generation at each generating bus which is available
for rescheduling, and using one iteration of the fast
decoupled load flow to obtain the solution. The elements
of the matrix are given by
I!. - I.. 1,1 1:1
o PGh
' (4.43)
where:
a. Increase in current aow in line ij' per 13
unit change of power in generator h.
I!. : Current flow in line 13 after generation change.
101
o : Current flow in line ij at base case state. Iij
4PGh• Power perturbation introduced at unit h. •
The line-line coupling factors given by eqn. 4.42
and the line current-generation sensitivity matrix are
then used to calculate the security constraints. The
first step is to express the current flow in the lines
as the addition of the base case current flow plus the
changes in current caused in this line by changes in
generation,
1 I = I° 4. 21 a APG km km km h h=i
(4.44)
Now if the outage of line k m is considered, the post-
outage current flow in line ij can be easily calculated
with the aid of the line-line coupling factors as
1 km o h Iij
= (17.4 4- 21 ai. APGJ t. (I 4- a APG ) d h=i 3 " ij km h=1 km h
(4.45)
or rearranging terms
1 km o 1j 11 . a. 3.)APG
Iij = (Iij ? t. Ikm h= (t
k ) m h ij km
+a 2.h (4.46)
Eqn. 4.46 gives the combined effect on the current
flow in line ij of the outage of line k m and of the
changes in the generating units. Assuming that the
102
security list contains m entries, i.e.'the security
analysis algorithm has reported m line overloadings, there
will be m eqns. of the form 4.46 with the Ill APGs as the
unknown variables. The purpose of the security control
is to determine the necessary rescheduling of power so
that the resulting currents in the critical lines, do
not exceed their specified levels.
This condition can be written in matrix form as,
.4 b + C APG = I ma x (4.47)
where the elements of the m component vector b and
those of the mxl matrix C are calculated from the
information contained in the security list and the
line-generation sensitivity matrix as in eqn. 4.46.
Vector Imax contains the current carrying capacity
of each of the lines included in the security list.
The proposed changes in generation can only be
made within the limits of the individual generating
units. Mathematically this condition is expressed as a
set oT constraints of the form
.4 max . Pe1-11 f=PG.4-1113G...=PG. 1=0,11..,1 • (4.48) 1 1
If the change in losses due to the rescheduling
of generation is neglected, the following equality
constraint express the system power balance
APG0 +APG1 +APG2 + +6,PG1 =0
(4.49)
103
In the operation of a power system the allocation
of power among the different generating units is usually
made in a way that the demand is met at minimum cost..
This economic dispatching of power might on certain
occassions be in conflict with the security of the
system as defined in section 4.1. If security is
given priority over economy of operation, adequate
corrective actions must be implemented in order to lead
the system into the secure operating region. Any vector
LPG observing the inequality constraints.(4.47 and 4.48)
and the power balance eqn. given by eqn. 4.49 represents
one of many solutions which satisfy the security criteria.
To choose a unique solution it is necessary to define
an objective function and to select the control vector
in a way that the objective function attains its
optimum value.
4.5. Optimum Reallocation of Power for Security.
If the cost function at node k is given by the
quadratic expression t(kPG-1-pkPGk the incremental cost
of generation at this unit is given as
df dPG
k
2(3k PG (4.50)
and an appropriate objective function ( ref. 39) for
lc*
the selection of the corrective actions would then be
1 = L (Mk + 2(3k PG) APGk
k=1 (4.51)
Once the objective function has been selected the
complete problem can be stated as:
Minimize eqn. (4.51)-
subject to the operating limits of the generating units
PGkin - PGk +41PGk PGA1mr.c k=1 9 2 1 9...
.the limits on the slack bus generation expressed in terms
of the independent variables by using eqn. 4.49
pGrain 0
PGo
ApG - ApG 1 pGmax
and the security constraints
b + C APG = Imax
Since the required changes in generation could be
positive or negative some auxiliary variables are
required in order to use the linear programming
technique. The change in generation is split as
VG. = APG i=1
wherewhere dPG 0 and 4PG7 41 O. Obviously at the solution
point at least one of the two terms should be zero.
This technique was applied to. the 6 and 10 bus
- systems (appendix 2), and the results of these studies
. 105
will be discussed in some detail.
Table 4.11 contains the security list which results
from the line outage simulation in the 6 bus system,
and includes as a precaution those lines which are
loaded at more than 80% of their capacity as a result
of an outage. For comparison, results obtained with
the method described in section 4.2.2. are shown along
with those obtained using the approximate method of
section 4.3.4.
Outage of Line
Effect on Line
Pre- Outage
Current
Post-Outage Current
Capacity .
Over- Loading
-
Exact Method
Approx Method
1-2
1-3 .2299 *.635 *.617 0.40 0;235 1-3
1-2 .3773 *.618 *.623 o.6o 0.018 2-4 .7252 .833 .835 1.00
2-3 . 2-4 .7252 .911 .908 1.00
2-4 1-3 .2299 *.472 *.454 .4o 0.072 2-3 .263o *.825 *.820 .65 0.175 6-3 .408 .575 .569 .70 5-3 .4171 .989 • *1.032 1.00 0.032
4-5 .1-3 .2299 .328 .333 .4o
5-6 6-3 .408 .620 .608 .70
3-6 2-4 .7252 .820 .817 . 1.00 5-6 .2125 .637 .612 .65
3-5 2-4 .7252 .915 .930 1.00 '6-3 .408 .632 .633 .7o
Table 4.11. Security list after line outage simulation (6 bus system)
c0 = 75 mo 4. 35 PG )
= 85 PG1 47 PG1
= 6C PG2 + 36 PG2 2
(4.52)
106
* Active constraints, i.e.'lines which exceed their
stipulated capacity.
After transforming each of the entries in the
security list into a security constraint the linear
programming problem is solved to obtain the optimum
rescheduling of power for security.
In the solution of the problem it was assumed that
the generating units had the following cost functions
and from eqn. 4.51 the objective function used in the
calculation of the corrective actions is given as
Minimize AF = 147.88 APG1 + 92.72 APG2 (4.53)
Because all other quantities involved in the process
are expressed in p.u., it is necessary to scale the
objective function so that its largest coefficient is
equal to 1.0. This scaling avoids the introduction
of round-off errors.
The results obtained using the exact and the
approximate methods together with their associated
costs are shown in table 4.12.
107
Unit Number
Present Generation
Secure Generation'Schedule Economic Dispatch Exact Method Approx. Method
0 Slack
1
2
.6620
.6689
.3434
.4177
.5657
.6752
.4316
.5842
.6463
.5991
.3610
.6519
Cost C/hr. 170.48 162.89 . 163.58 161.20
Table 4.12. Optimum rescheduling of power for security.
Table 4.13 shows the results of a security analysis
performed after implementation of the generation schedule
calculated with the exact method, and it can be seen
that no line overloadings are reported.
An exact security analysis performed after
implementing the generation schedule calculated with
approximate method yields the security list shown in
table 4.14. It is shown here that when line 1-2 is
removed, a slight overloading of line 1-3 is reported.
This is caused by the error introduced by the approximate
method in the calculation of the security constraints.
It can be seen that the error is quite small and being
within the accuracy of measured data, the solution can
be considered as acceptable.
108
Outage of Line
Effect on
Line
Pre- Outage
Current
Post-Outage
Current Capacity Over- Loading
1 - 2 1 - 3 .1375 .3912 .40 ---
2 - 4 1 - 3 .1375 .3396 . .40 --- 2 - 3 .1481 .6205 .65 --- 6 - 3 .4244 .5747 .70 0--
5 - 3 .5035 .9884 1.00 ---
5 - 6 6 - 3 .4244 .6200 .70 ---
3 - 6 5 - 6 .2019 .6370 .65 ---
5 - 3 .5035 .8429 1.00 ---
3 - 5 2 - 4 .6237 .8460 1.00 --- 6 - 3 .4244 .6998 .70 ---
Table 4.13. Security list after rescheduling using exact
method.
Outage of
Line
Effect on
Line
Pre- Outage
Current
Post-Outage
Current Capacity Over-
Loading
1 - 2 1 - 3 .1447 .4042 .40 .0042
2 - 4 1 - 3 .1447 .3508 .40 _-_
' 2 - 3 .1594 .6410 .65 --- 6 - 3 .4232 .5761 .70 ---
5 - 3 .4938 .9914 1.00 - ---
5 - 6 6 - 3 .4232 .6160 .70 ---
3 - 6 5 - 6 .1985 .6358 .65 ' _--
5 - 3 .4938 .8351 1.00 -_-
3 - 5 • 2 - 4 .6349 .8557 1.00 --- 6 - 3 .4232 .6951 .70 ---
Table 4.14. Security list after rescheduling using
approximate method.
109
Using the CIGRE 10 bus system similar results are
obtained. Here againi.both the exact and the approximate
methods have been used in the line outage simulation,
and the resulting security list is shown in table 4.15.
Outage of
Line
Effect on Line
Pre- Outage
Current
Post-Outage Current
Capacity Over-
Loading Exact Method
Approx Method
2 - 1 4 - 2 2.219 2.886 2.903 2.34 0.546
1 - 3 4 - 2 2.219 2.471 2.462 2.34 0.131
1 - If 4 - 2 2.219 2.734 2.719 2.34 0.394
2 - If 1 - 2 .690 2.889 2.937 2.34 0.549 4 - 1 .975 2.261 2.231 2.34
7 4 - 2 2.219 2.333 2.333 2.34 ---
8 - 4 4 - 2 2.219 2.241 2.241 2.34 --- .
8 - 9 4 - 2 2.219 2.330 2.329 2.34 ---
1 - 8 4 - 2 2.219 2.641 2.640 2.34 0.301
1
Table 4.15. Security list after line outage simulation
(10 bus system)
Each component of this list is transformed into a
linear security constraint and used by the linear
programming algorithm in the calculation of corrective
actions. The results obtained using the exact and
approximate methods are shown in table 4.16.
110
Unit No.
Present Generation
• Secure Generation Schedule Exact Method Approx. Method
0 2.135 2.16 2.270
1 3.00 2.434 2.394
3 1.92 1.920 • 1.920
'3 4.6o 5.114 5.07 4 .960 ..960 .960
5 .6o5 .6o5 .6o5
Objective function used: .231SPG1 + .50 APG2 + .254SPG3 +
1.0 ®PG4 + 1.0 ApG5
Table 4.16. Optimum reschedule of power for security.
The generation schedules obtained by the exact
and the approximate methods are then tested by
performing in both cases an exact security analysis.
The results of this analysis are shown in tables 4.17
and 4.18 respectively. It can be seen that there are
still certain outages causing overloading of other
lines, but it is shown that this overloading is only
marginal. This is due to the error introduced.by the
linearization of the security constraints used in the
calculation of the corrective actions.
Outage of
Line
Effect on
Line
Pre- Outage
Current
Post-Outage
Current Capacity Over-
Loading
2 - 1 2 - If 1.873 2.347 2.34 .007
1 - 3 2 - If 1.873 2.114 2.34 ---
1 - If 2 - If 1.873 2.336 2.34 ---
1 - 8 2 - If 1.873 2.277. 2.34 ---
2 - If 1 - 2 .495 2.351 2.34 .011
If - 1 .881 1.964 2.34
3 - 7 2 - If 1.873 1.976 2.34 ---
8 - If 2 - If 1.873 1.887 2.34 ---
8 - 9 2 - If 1.873 1.978 2.34 ---
i
Table 4.17. Security list after rescheduling using
exact method.
Outage of
Line
Effect on
Line
Pre- Outage
Current
Post-Outage
Current Capacity Over-
Loading
2 - 1 2 - If 1.878 2.308 2.34 ---
1 - 3 2 - If 1.878 2.122 2.34 ----
1 - If 2 - If 1.878 2.358 • 2.34 0.018
1 - 8 2 - 4 1.878 2.289 2.34 ---
2 - If 1 - 2 .450 2.312 2.34 ---
4 - 1 .912 1.999 2.34 ---
3 - 7 2 - 4 1.878 1.987 2.34 ---
8 -'4 2 - 4 1.878 1.894 2.34 ---
8 - 9 2 - If 1.878 1.985 2.34 ---
' b ,
Table 4.18. Security list after rescheduling using
approximate method.
112
Finally table 4.19. shows the execution time in
seconds required by the different segments of the
program. These times were obtained running the program
on the CDC 6400 computer at Imperial College.
Segment
6 bus system. 10 bus system. Method • Method
Exact i,pprox. Exact Approx.
Initial load flow
.016 .013 .022 .023
Security Analysis .183 .109. .331 .172
Calculation of Corrective Actions
.055 .054 .106 .109
Total .254 .176 .459 .304
Table 4.19. Execution times.
• 113
5 ON LINE IMPLEMENTATION OF ALGORITHMS.
5.1. The Real-Time Operating System.
RSX-15 is the real time monitoring system used in
the Electrical Engineering Department's PDP-15 computer.
All operations within the system are controlled and
supervised by the real-time Executive or Monitor. The
Executive is responsible for program scheduling,
supervision of input/output operations and interactive
communication with the operator.
The core memory of the RSX system is divided into
partitions to allow several programs to be in core at
any given time. When the system is initially loaded,
an interactive program, called system configurator, is
used to tailor the RSX system to meet the user
requirements. With the exception of the first 4K words,
which are occupied by the RSX-Executive, the remaining
core is divided into partitions which are defined by
their name, base address and size. In addition to
partitions, certain areas of core can bs reserved for
common blocks which can be used for inter-program
communication.
Due to the fact that RSX does not have compiler
or assembler facilities, all programs have to be •
developed and debugged using a separate operating system,
the Advanced Sofware Monitor (ref.44). Programs can be
written either in Fortran 4 or Macro-15 and after
compilation or assembly, a utility program, called Task
Builder, is used to form executable tasks capable of
running under control of the real time monitor. All
programs or tasks are identified by a name and they are
built to execute in specific partitions. Some tasks may
be fixed in core and have exclusive use of their partition
to ensure rapid response, but normally they are stored
on disk and brought into their core partition only when
requested. Any number of disk resident tasks can be built
to execute in the same partition. This is advantageous
from the storage point of view, but obviously their
response time is delayed by the disk-to-core transfer
required.
Operator communication with the system is accomplished
by means of the Monitor Console Routine (MCR). The MCR
consists of a core resident task, which accepts user
commands from the teletype, and a set of disk resident
functions which are brought into core to actually carry
out the indicated requests. MCR allows the operator
to obtain information about the system, install or
115
API
remove tasks, request the inmediate activation of a task,
fix a task in core etc. A total of 22 f•'CR functions are
available and their use is described in chapter 3 of ref.
45. To carry out its real time functions RSX uses an
Automatic Priority Interrupt (API) system. There are
8 levels of API which are numbered from 0-7. Priority
levels 0-3 are used for hardware Input/output devices
and levels 4-7 are used by the Executive. From level 7,
the Executive derives 512 task priority levels. Figure
5.1. shows the structure of the priority system.
Task
Priority Levels
Executive
Priority Levels Hardware Levels
Exclusive use
by the Executive
6 5 4
Used by all I/O
devices.
3 2 1 0
512
LEVELS 7
>1
increasing priority
Figure 5.1.. Automatic Priority Interrup System.
116
When a task is built, in addition"to its name and
the partition from where it will execute, it is given
a task priority, ie. a number between 1 and 512. The
Executive uses this information when the task is
requested for execution. For instance a request to
activate a task will be executed if its partition is
available and a task with a higher priority is not
currently executing. On the other hand if the task
being run, say TASK 1, has a lower priority than the one
requested, say TASK 2, the Executive will interrupt the
execution of TASK 1 and hand control over to TASK: 2.
The execution of TASK 1 will be resumed when TASK 2 exits
or if, for example, TASK 2 is waiting for an Input/Output
operation to be completed.
The RSX system is provided with a very. flexible I/O
structure. All I/O requests are serviced by I/O device
handler tasks. With the exception of the Disk and the
multi-teletype Device Handler Tasks (DHT's) which are
core resident and cannot be deleted from the system,
the user is allowed to include only those device handler
tasks which are required for his available I/O hardware.
Like any other task DHT's are built to execute in a
specific partition and provisions have to be made• at
configuration time to allow the necessary DHT's to be
installed in the system.
Figure 5.2. shows the core layout or configuration
of the RSX system used in this project. The first
117
(octal locations 0 to 10 000) are occupied by the real
time Executive and includes the resident part of the
monitor console routine and the disk and multiteletype
device handler tasks.
0
OCTAL
ADDRESS
10 000
20 000
30 000
60 000
RSX 7 15 •
EXECUTIVE
MCR FUNCTIONS
10.1 .
10.2 10.3
±o.4
DATIN
INAN . .
COMMON AREA
WORKING
SPACE
. PARTITION P3
Figure 5.2. Core Layout of RSX-System.
-(18 c
The partition labelled ICR functions is the area of
core from which the disk resident MCR functions are
executed. Partitions I0.1, 10.2, 10.3 and 10.4 are
used respectively by the DEC tape, Line printer, paper
tape punch and paper tape reader peripheral handler
tasks.
Partition INAN is occuppied by a MACRO-15 subroutine
called ANIN which was written by L. Mcgridge (ref.46) to
execute at task priority level 5. ANIN reads sequentially
via the Analogue to Digital Converter, the 96 analogue
inputs originating from the power system simulator. The
data is then stored in Common area DATIN from which it
can be accessed by other programs.
The lower part of core storage, labelled working
area in figure 5.2., is used for the execution of the
algorithms described in section 5.2.
5.2. Im.lementation of Al orithms in the RSX system.
A set of Fortran programs consisting of a state
estimator, economic dispatch, security analysis and
security constrained dispatch have been written and
implemented on the PDP-15 computer. They run under
control of the real-time Executive using data obtained
on-line from the power system model. This data is
.119
.
processea to obtain the present state of the system, and
to derive control actions which ensure a secure operating
state. The theoretical aspects of these algorithms have
been discussed in chapters 3 and 4 and this section is
concerned with their structure and other aspects of their
implementation on the RSX system.
OPTIME is the name of the task which was built to
carry out the above mentioned functions and executes in
partition P3 (see figure 5.2.) at task priority 30. It
requires 15,412 octal words of core to run, so that if
necessary, the 30,000 octal words of core occupied at
the moment by partition P3 could bo div:Ided into smaller
partitions.
OPTIME has an overlay structure consisting of a
main program, which becomes core resident when the task
is requested for execution, and a set of disk resident
subroutines which overlay each other as they are
sequentially brought into core by request of the main
program. Important data read or generated during the
execution of the different subroutines is stored in
labelled common blocks and thus available to all
subroutines for internal communication. Inter-task
communication, such as the one required to obtain the
value of measured variables in the system, is achieved
by means of the system common blocks which are defined
when the RSX system is initially configured.
To describe the different functions and the structure
120
of task OPTIME, the program was divided into 5 modules.
Each of these modules contains several subroutines as
shown in figures 5.3. to 5.7., and a brief description
of the different components of each module now follows.
<
Data A/D
'Prom
Simulator
Paper Tape Reader
REDAT
Y BUS
ANIN REANA
.SYSTEM COMMN
DATIN
COMPUTER CORE
Figure 5.3. Module 1. Data INPUT.
121
5.2.1. Module 1. Data Input.
Subroutine REDAT (see figure 5.3.) reads from the
high speed paper tape reader the following system
information
• Number of buses, lines and generating units.
• System topology and line parameters..
. Number and location of P,Q meters.
4, Generator constraints and coefficients of cost
functions.
With this information, subroutine Y BUS forms the
addmitance matrix of the system storing only those
elements which are different from zero.
Measured quantities required for the state estimation
processr ie. voltage magnitude at the reference bus and
all available complex line flow measurements, are obtained
when subroutine REANA requests the execution of the core
resident task ANIN. As explained earlier, this task
controls the conversion into digital form of all analogue
measurements made in the system, and transfers the
resulting digital quantities to specified locations in the
. system common area DATIN where it can be accessed by REANA.
The information stored in DATIN as integer numbers is
transformed to p.u. values (see appendix 4 for details)
and stored in a labelled common block for use by the
remaining modules of the program.
122
5.2.2. MOdule 2. State Estimation.
With all data in p.u. the different subroutines which
are used by the state estimation algorithm are brought
into core for execution. Recall that the object of this
step is to solve for the system state Vr the over . —
determined set of eqns. given by eqn. 3.66 which is re-
written here for ease of reference
(Bt R-1B ) vi+1 Bt R r (5.1)
In eqn. 5.1 each component of Vm represents the
'measured' voltage across a line, which is actually
computed from the corresponding line flow measurement,
the impedance characteristics of the line and the nodal
voltage at the measured end of the line. This transform-
ation, as explained in chapter 3, is necessary to obtain
a linear relation between measured and estimated
quantities.
As shown in figure 5.4., subroutine GAIMA initiates
the process of estimation. Its function is to compute
the matrix (Brt R-1 13') using the information on the
location of the meters to form the incidence matrix B,
and the corresponding line impedances and meter variances
to calculate the diagonal weighting matrix R-1.
FAILED
IDENT
123
GAIMA INVER
VOLTA SOLVE
.DETECT
YES
PASSED
LINFLO •
,Figure 5.4. Module 2. State 7stimation with Detection and
Identification of Gross measurement errors..,
For small systems, such as the one connected in the
simulator, it is more advantageous to invert (Bt R-1Br)1 .
and subroutine INVER is used for this purpose. Since
(Bt R-1B ) is symmetrical and positive definite, Cholesky's
method (ref.47) is used in the inversion process,
124
calculating and storing only the lower triangle and the
diagonal elements of the inverted matrix. For large
_practical systems, triangular factorization using optimal
ordering of the eqns. and sparsity techniques would be
used instead of matrix inversion. In terms of the
inverted matrix, eqn..5.1 can be written as
Vi+1 = (Br Bt R-
(5.2)
where the superindex i indicates the iteration count.
Subroutine VOLTA solves eqn. (5.2), using the latest
calculated value of the state in the computation of the
'measured' voltages across the lines, using an auxiliary
vector y
- Bt R-1 (Vi b V1 ) -m (5.3 )
and calling subroutine SOLVE at each iteration to obtain
the new state vector V1+1 from -r
Vi+1 = (Bt R-1Br )-1 yi
(5.4)
The iterative process is stoped when the difference
in the value of the state vector in two consecutive
iterations is less than a given tolerance..
The validity of the results obtained by VOLTA is
checked in subroutine DETECT by means of the Chi-square
125
test described in chapter 3. If the estimated state
fails this test, it is assumed that bad data is present
and subroutine IDENT is brought into core. IDENT
analyzes each of the normalized residuals ie. the
difference between 'measured' and estimated voltage
accross lines divided by their standard deviation, by
comparing them with a given threshold value. Because of
the smearing effect, the presence of one erroneous
measurement causes two or more measurements to fail the
residual test. Subroutine IDENT uses the Sherman-Morrison
technique to reflect in the iterative process carried out
by VOLTA, the 'absence' one at a time, of each of the
suspected meters. If the elimination of one of these
meters results in an estimated system state that passes
the chi-square test, the solution is accepted and the
meter is reported as erroneous. If, on the other hand,
the chi-square test fails, IDENT assumes that the meter
was actually good but had failed the residual test because
of the smearing effect. Hence, the meter is reconnected .
and the process is repeated using another suspected
measurement.
It could happen that an acceptable solution has not
been found at the end of this process, This situation is
taken to indicate that more than one meter is erroneous.
In our network this condition becomes unacceptable
because eliminating two or more meters could lead to
isolation of certain nodes of the system. So when this
126
happens the state estimation process isstopped and
repeated with a new set of measurements. If the condition
persists, the P and Q meters have to be physically tested
and recalibrated if necessary.
When an acceptable solution is obtained, subroutine
LINFLO calculates the real and reactive power and current
flows in all the lines, and the teal and reactive power
injections at all nodes of the network. The estimated
state of the system, line flows and nodal injections are
output to the line printer via subroutine WRIT.
5.2.3. Module 3. Economic Dispatch.
The purpose of this module of the program, is to
determine the most economic allocation of power that
satisfies the present estimated demand of the system. The
production cost at each generating unit is assumed to be
a quadratic function of the form
V . =0(f. pG +g Pi -c•xi (5.4)
where:
C. : the total cost of generation at unit i
e< is cost coefficients which are read by REDAT
PG.: generation in p.u. at node i.
In a system with 1 generators and a total demand
PD the optimum power dispatgh is obtained by minimizing
127
the total cost of generation given by
OX.PG.13.Pe) a. 3. 1 3. 3. (5.5)
observing at the same time the stipulated limits on the
generating units
4 ax
Amin . - PG. - PG.m I i=1,...1
1 1 1 (5.6)
and the static balance of power in the system
i=1 - = 0
(5.7 )
where PL are the total real losses in the network.
Subroutine DISPA in figure 5.5., solves the economic
dispatch problem neglecting in the equality constraint
given by eqn. 5.7 the real losses in the system PL.
I DISPA.
Figure 5.5. Module 3. Economic Dispatch.
128
DISPA uses the method of Lagrange multipliers adjoining
the equality constraint given by the power balance
equation to the cost or objective function given by eqn.
5.5, as shown in eqn. 5.8.
1. 1 C* = C i PG.43 Pe) -1A( )1 PG -PD) i=1 i=1
(5.8)
and obtaining the minimum of C* by solving for the '1'
generated powers PG and the lagrange multiplier the
set of 1+1 linear eqns.
= OC.1+2.PG.3.
+ )■= 0, i=1,21 .11 1 DEG.
1 (5.9)
pc* = L PG. - PD = 0 i=1
Following the solution of the set of eqns. 5.9.,
check is made to ensure that the inequality constraints
given by eqn. 5.6 are not violated. If they are, the
generation of those units is set to the violated limit
and the problem is reformulated in terms of the remaining
units.
With the economic schedule of generation obtained
from DISPA, and information derived from the estimated
state of the system, a load flow is solved to examine
the proposed optimum operating state. The method used
is the fast decoupled load flow described in section
4.2. which requires the iterative solution of two sets
of eqns. which are reproduced here for easy reference.
129
= (BI )-1 W/V (5.10)
and
AV = (13 1' )-1 6.Q/V (5.11)
The first step in the solution-is the calculation
and inversion of matrices B1 and B''. These functions are
carried out by subroutines NATRX and INVER respectively.
Here again because of the symmetry of the matrices only
the lower triangles and diagonals of their inverses are
stored.
The iterative solution of eqns. 5.10 and 5.11 is
then carried out in subroutine FASFLO.
Subroutines LINFLO and WRIT are also part of module
2 and their function has already been described. The
difference here is that the line flows and currents which
are now displayed on the line printer correspond to the
optimum operating point.
•
5.2.4. Module 4. Security Analysis.:
In this module the operating state resulting from
the economic allocation of power is analyzed from the
security point of view. With the exception of CONFLO
the remaining subroutines in the module of figure 5.6.
have already been described as they are also used in
the economic dispatch program.
SOLVE CONFLO FASFLO
MATRX INVER
1.30
Figure 5.6. Module 4. Security Analysi.l.
Subroutine CONFLO organizes the simulation of line
outages performing the necessary calculations for the
use of the Sherman-Morrison formula, which as explained
in chapter 4 avoids the recalculation and reinversion
of the matrices involved in the process.
When the line outage simulation is completed, CONFLO
forms the security list. This list contains all the
relevant information of those lines which are overloa-
ded as a result of an outage. Finally it computes, by
introducing small perturbations in the generated powers,
the line-generator sensitivity matrix.
5,2.5. Module 5. Security Constrained Dispatch
The last module in Task OPTIME. (see figure 5.7.)
131
is concerned with the calculation of a..generation schedule
which results in a secure operating state for the system.
It is brought into core only when the security analysis
detects that the optimum operating state is not a secure
one. In this case subroutine TABLO uses the information
generated in the security analysis program to produce a
set of linear constraints. An additional set of
constraints related to the maximum and minimum limits
of the generating units is also formed in this subroutine.
This information is used by subroutine LINPG*, a linear
programming algorithm using the simplex method, to
determine the optimum rescheduling of generation required
to lead the system into a secure operating state.
TABLO
LINPG
RESCII CONTROL REQUIRED
FOR SECURI
Figure 5.7. Nodule 5. Security Constrained Dispatch.
* Subroutine LINPG was obtained in coded form from ref.48.
132
Finally Atbroutine RESCH is used to interpret the results
of LINPG outputing all relevant information via the line
printer.
The complete structure of task OPTIME is shown in
block diagram form in figure 5.8. Appendix 5 contains the
actual task building process for the whole program.
It can be seen in figure 5.8., that once the
measurements are processed to obtain an estimate of the
present operating state, of the system, the program can
follow two different paths. The selection is made by
testing the value of a variable which is part of the
data supplied from the paper tape reade::,
When path 1 is chosen the security analysis is
performed on the present operating state of the system.
If the system is found to be secure no control action
is required. If on the other hand the system is found
to be in a potentially dangerous state, the program
proceeds to the calculation of the necessary corrective
actions, and displays the relevant information to the
operator. When the program branches to path 2, an
economic dispatch is performed with the information
supplied by the state estimator. The security analysis
is then carri-ld out on the optimum operating state. If
it is found to be secure, the difference between the
present and the economic schedule of generation is the
control required to satisfy at minimum cost the present
demand of the system. If the security analysis reports
< A/D Paper Tape
;1- Reader
Prepare Data
Base
BEGIN
measurements
Security Constrained Dispatch
State
Estimation
present operating, state
no
yes (Path 2)
optimum operating state
no yes action
required
no
Optimum Dispatch Available for Implementation
yes
Secure Dispatch Available for implementation
Figure 5.8. Block diagram of task OPTIME
(Path 1).
System secure ?
Secure operating State
D/A
P
0
E
R
S
S
T
E
N
N
'0
D
E
133
134
line overloadings,the security constrained dispatch
calculates the minimum departure from the most economic
schedule that satisfies the security criteria.
There are certain situations when the security
constrained dispatch program fails to find a feasible
solution, ie. a generation schedule resulting in a
secure operating state. In this-case, the program
tries to find a schedule of generation resulting in an
operating state that is less vulnerable than the one
that has been analyzed, 'and reports those line outages
whose occurrence, in spite of the new calculated schedule,
would result in other lines becoming overloaded.
5.3. Numerical Examples.
The network shown in figure 5.9. was connected in the
power system model and used to test task OPTIME. There
are 6 nodes in the network, 8 transmission lines and 10
line flow meters. The voltage at the end of the two lines
connected to bus 1 is measured, and the average of the
two readings is taken by the state estimator as the
voltage magnitude of the reference bus.
LOAD LOAD
4
LOAD
135
0 P,Q measurement
Figure 5.9. Diagram of test system.
In the early stages of the project a task-called
LOGOS 4 was written to sample, via subroutine ANIN, other
analogue measurements wich although related to the network
are not used by the state estimator. LOGOS 4 takes 20
samples'of all nodal voltages and power injections at
load buses and calculates an average value for those
quantities. Since the system is considered to be in
136
steady state, running LOGOS 4 before requesting the
execution of OPTINE provides useful information for a
post-morten analysis of the results obtained from the
estimation process.
Some numerical results are shown in tables 5.1. to
5.13. to ilustrate the performance of task OPTINE.
Numerical Example No. 1.
meter 1 1.06213 Average of 20 samples
meter 2 1.06735
meter 1 1.06278 Sample used for S.E.
meter 2 1.06796
Reference voltage for State Estimation 1.06537
Table 5.14. Measurement of Reference Voltage.
137
Line measured-estimatee, measured flow Estimated flow from to iReal Reactive Real Q ' Real Q 1 2 -.00759 .00749 .39037 .03246 .39796 .02497 1 3 .00016 -.00459 .'.20566 ' .00563 .20550 .01022 2 3 -.00229 .00605 1 .19779 .01587 .20008 .00982
2 4 -.00074 .03367 .62156 .28802 .62230 .25435
3 6 -.01038 .02334 .28148 .18429 .29186 .16095
5 3 -.00739 .01681 -.33797 -.20365 -.33058 -.22046 6 5 -.00835 .02300 -.11538 .01390 -.10703 -.00901
5 4 .00343 -.00453 .18498 .02634 -.18841 .03087 4 2 .00415 .02865 .60514 -.18487 -.60929 -.21352
3 1 -.02364 .02784 -.22352 .03882 -.19988 .01098
Table 5.2. Measured and estimated line flows.
In addition to the measurements shown in tables 5.1.
and 5.2. voltage and injection measurements which are not
used An the state estimation process are also available.
They provide an independent check on the results obtained
with the state estimator and are shown in table 5.3.
138
Node ,Type of
measurement Average of 20 samples
Calculated value
Discrepancy in p.u.
2 Voltage 1.05156 1.05210 .00054
3 it 1.03286 1.03649 .00363
4 It .9996 1.0104 .0108
5 tt .99823 1.0057 .0075
6 11 .99040 .9984 .0080
4 Real Power .39921 .4193 .0201
5 It .4446o .4113 .0333
6 II .41222 .3938 .0184
4 Reactive .22857 .2403 .0117 Power
5 it - .19962 .1787 .0209
6 It .14022 .1532 .0130
• Table 5.3. Voltage magnitude and injection measurements.
It can be seen from table 5.3. that the estimated
values for the voltage magnitudes are indeed very
similar to the measured values, with the largest
difference occurring at node 4 where an•error of .0108
p.u. is registered.. Similarly the estimated values
for the real and reactive power injections at nodes 4,
5 and 6, ie. the load buses of the network, are shown
139
to be quite close to their measured values with the
largest discrepancy of .0333 p.u. occurring in the real
power injection at node 5.
It has already been mentioned that the measurements
shown in table 5.3., are not used as data for the state
estimation process and therefore they have no influence
in the calculation of the estimated state of the system.
Although they are also subject to errors, just like any
other measurement, the fact that they all are in
reasonable aggreement with the value calculated-using
the estimated state of the system gives an additional
indication of the validity of the solution obtained
with the estimator.
At the estimated operating point the total power
demand and the loading of the generating units can be
calculated. In this particular example a total load
of 1.2244 is being supplied by the 3 generators whose
participation and cost coefficients are shown in table
5•1+0
Present Operating Point Economic Dispatch Generated UGenerates,Power Cost
Generate Power Cost
1 75 35 .60346 58.00 .48305 44.40
2 85 47 .42863 45.07 .23252 22.30
3 68 36 .23037 17.58 .53773 46.98
Total cost of Generation 120.65 . s/hr. 113.68 •
Table 5.4. Present -and optimum generation schedules.
14o
The present operating point is the one which results
from the initial setting of the simulator and hence the
distribution of power among the generating units is an
arbitrary one. A quadratic cost function has been
attached to each generating unit so that the scheduling
of generation can be made in an optimum way by minimizing
the total cost which satisfies the given power demand.
The resulting optimum scheduling or economic dispatch is
also shown in table 3.4., where it can be seen that a
reduction of about 7/hr. can be made in the operating
cost of the system if the generating units are loaded to
the levels suggested by the E.D. A security analysis
performed at this optimum operating point gives the
results of table 5.5.
Outage
of line
Effect •
on line
Pre-Outage
current
Post-outage
current Capacity Comment
- 2 1 - 3 .1239 .4794 ek000 OVERLOAD
3 - 5 .4534 .9304 1..000 •
3 - 5 3 - 6 .3399 .5886 .700.
Table 5.5. Results of security analysis.
Due to the overloading of line 2-3 on occurrence
of the outage of line 1-2 the system, although in a
normal operating state, is termed insecure and control
141
actions are required to lead it into the secure region.
As explained in chapter 4 the security list shown in
table 5.5. is transformed into a set of security constraints.
In addition to overloaded lines, lines whose loading
exceeds 80% of their stipulated capacity are also
included in this list. This enables the calculation of
the security control to take into account those lines
which, on occurrence of certain outages, are only
marginally secure and thus prevents any changes in
generation aimed at relieving the overloaded lines to
result in the overloading of others.
The selection of the security control actions is
made in such a way that the departure from the most
economic operating point is minimized. This results
then in an optimum operating point, from the economic
point of view, which is also inside the secure region.
For this example, the necessary rescheduling of
generation and the related costs are given in table 5.6.
Secure Economic Dispatch Economic Dispatch
Unit Generated Power Cost
Generated Power Cost
1 .4014 35.74 .4831 44.4o
2 .3106 30.94 .2325 22.30
3 .5406 47.28 .5377 46.98
Total Cost of Generation 113.96 4/hr.. 113.68
Table 5.6. The secure economic dispatch.
142
A security analysis is performed to make sure that
the proposed schedule of generation corresponds in fact
to a secure operating point. The results of such
simulation are shown in table 5.7.
Outage
of line
Effect
on line
Pre-Outage
current
Post-outage
current Capacity Comment
1 - 2 1 - 3 .1114 .3937 .4000
2 - 4 3 - 5 .4505 .9311 1.000 • .
3 - 5 2 - 4 .5566 .8015 1.000
3 - 5 3 - 6 .3390 .5858 .7000
Table 5.7. Security analysis using the proposed generation
schedule.
From these results it can be seen that with the new
dispatch there are no violations of security reported.
It can also be seen that line 1-3 which was originally
overloaded when line 1-2 was taken out is now marginally
below its capacity, thus the resulting operating state
is termed as secure.
The running times required by the different segments
of the program can be obtained by requesting via the
monitor console routine the clock time at the beginning
and end of the execution of the programs. The time,
which includes the transfer from disk to core, is given
in hours, minutes and seconds so only a rough idea can
143
be obtained. They are shown in table 5.8. where the
running times required by the same program in the CDC
6400 are also shown for comparison.
PROGRAM Time on PDP-15 Time on CDC 6400
State Estimation
(VOLTA, DETECT, LINFLO) 1 secttf-2•sec .031 sec.
Economic Dispatch
(MATRX, DISPA, FASFLO., LINFLO)
1 sec=t=2 sec • .022 sec.
Security Analysis
(CONFLO/FASFLO/SOLVE) ' 5 sec=t6 sec .180 sec.
Security Control
(LINPG, RESCH) - 1 sec`-tt2 sec •
.034 sec.
Table 5.8. Program runing times.
Numerical Example No. 2.
This is a similar analysis to that of example No. 1
except that a higher loading is used. Tables 5.9. and
5.10. show the measurements of reference voltage and
those of the line flows used in this example.
. 144
Average of 20 samples meter 1 1.05913
meter 2 1.06396
meter 1
1.06012 Sample used for S.E.
meter 2 1.06513
Reference voltage for state estimation 1.06263
Table 5.9. Measurement of reference voltage.
Line measured-estimated measured flow Estimated flaw from to Real Reactive Real Q Real Q
1 2 -.00791 .00948 .57077 -.00128 .57868 -.01076 1 3 .00205 -.00497 .27468 -.01116 .27263 -.00619 2 3 -.00141 .00885 .23915 .00226 .24056 -.00659 2 4 .00098 .03651 .68080 .25364 .67982 .21713 3 6 -.01006 .02488 .31988 .16330 .32994 .13842 5 3 -.00731 .01821 -.28949 -.16845 -.28218 -.18666 6 5 -.0078 -.01639 -.17837 -.02043 -.17057 -.00404 5 4 .00353 -000518 -.18732 .07074 -.19085 .07592 4 2 .00630 .03082 -.65882 -.14020 -.66512 -.17102 3 1 -.02058 .03574 -.28328 .07935 -.26270 .04361
Table 5.10. Measured and estimated line flows.
145
A security analysis performed at this operating point
yields the results shown in table 5.11.
Outage
of line
Ftfect
on line
Pre-Outage
current
Post-outage
current Capacity Comment
4 - 2 1 - 3 .25663 .88386 .400 OVERLOADED
1 - 3 1 - 2 .54467 .81469 .600 OVERLOADED
2 - 3 1 - 3 .25663 .32881 .400
2 - 3 2 - 4 .67901 .83042 1.000
2 - 4 1 - 3 .25663 '.47716 .400 OVERLOADED
2 - 4 2 - 3 .22897 .7390o .65o OVERLOADED
2 - 4 3 - 5 .33443 .89283 1.000
3 - 6 1 - 2 .54467 .5757o .600 •
3 - 6 5 - 6 .17034 .53937 .65o
3 - 5 1 - 2 .54467 .58710 .600
3. - 5 2 - 4 .67901 .84930 1.000
Table 5.11. The Security list.
In this example the economic dispatch step is bypassed
and the linear objective function, which as explained earlier
is related to the incremental costs of generation, is
calculated at the present operating point. After scaling
• the objective function is given by •
minimize AF = 1. APG2 .1- .6475 liPG3
The resulting generation schedule calculated using
this objective function and the corresponding linear
security constraints derived from table 5.11 is shown
in table 5.12. The present allocation of power and that
resulting from an economic dispatch are also shown for
comparison.
146
Present Operation Economic Dispatch Secure Dispatch Generated Generated Generated
Unit Power Cost Power Cost Power Cost
1 .8513 89.21 .5002 • '46.27 .3800 33.56
2 .3506 35.58 .2448 23.62 .3516 35.70
3 .1165 8.41 .5529 48.61 .564o 49.81
Total Cost 133.20 118.50 119.07 £/hr.
Table 5.12. Cost of different schedules of generation.
A security analysis using the scheduling of generation
given in table 5.12. under the heading secure dispatch,
confirms that all line overloads which were reported in
table 5.11. have been succesfully eliminated. The results
of this analysis are shown in table 5.13.
Outage of
Line
Effect on
Line
Pre- Outage
Current
Post-Outage
Current ... Capacity Comment
1 - 2 1 - 3 .1073 .3728 .400
2 - if 3 - 5 .41092 .8927 1.00
3 - 6 5 - 6 .1406 .5395 .65
3 - 5 6 - 3 .3702 .5985 .7o •
4
Table 5.13. Security analysis after optimum rescheduling
of generation.
147
6 CONCLUSIONS .
6.1. Concluding remarks.
The use of estimation theory for on-line processing
of measured data in power systems has become a necessity
in view of the common occurrence of gross measurement
errors. The estimator therefore provides a systematic
method of.checking the quality of data and ensuring that
reliable real-time information is available for control
decisions.
'For on-line implementation,' the AEP algorithm was
selected after analysis of various solution approaches.
Off-line tests using simulated data showed that this
algorithm is fast, has reliable convergence and is most
economical in core usage.
The initial testing of the algorithm in the on-line
mode proved useful in detecting various hardware faults
in the model, the Watt/Var meters and the interface.
. 148
This phase of the project proved to be very time
consuming, but once all. hardware faults were cleared,
succesful on-line operation of the estimator was obtained.
Steady state security analysis requires, by
definition, the solution of a sequence of load flow
problems. In the case of line outages, each solution
involves the modification of the topology of the
system. For on-line applications it is of prime
importance to use an efficient method to reduce the
required computation time. As far as speed of solution
is concerned, a linear model or D.C. load flow is the
most efficient. However, in systems where a voltage
solution is required or the accuracy of the D.C. load
flow is not adequate, a.c. models have to be used.
An efficient method for the solution of the non-linear
model was presented in section 4.2. The method takes
full advantage of the weak coupling existing between
phase angle and reactive power and voltage magnitude-
real power to derive two sets of linear equations, the
matrices involved being constant. These matrices are
triangularized only once at the beginning of each
contingency evaluation cycle. This desirable
characteristic where the triangularization of the
matrices is avoided is obtained by using the Sherman-
Morrison technique in the simulation of line outages.
As an alternative an algorithm which simulates
a line outage by injecting adequate amounts of real
149
and reactive power at the nodes connected by the line,
was developed by the author. Off-line tests on this
method showed that fast solutions with adequate
accuracy in both real and reactive power flows were
obtained. •
One of the main objectives in the operation of
a power system is the satisfaction of the power demand
at minimum generation cost. However, when security
of operation becomes the prime concern, the optimum
allocation of power is made to comply with a set of
security related constraints. The exact formulation
of the optimization problem results in a complex and
computationally demanding algorithm unsuitable for
on-line implementation. The approach used in this
project was to formulate the economic dispatch algorith
using only the constraints related to the generating
units and one constraint equating the generation to
the demand. A security analysis performed on the
optimum dispatch point detects violations and near
violations of line loading limits and forms the set of
linear security constraints. These constraints are
used in a linear programming algorithm where the
objective function is to minimize the departure from
the optimum allocation of generation. The resulting
point could be appropriately called the optimum
secure operating point although strictly speaking,
because of the inaccuracies in data and the
150
approximation introduced in the linearization of the
constraints, the absolute optimum can never be achieved.
Although the performance of the simulator, interface
and computer was found to be very satisfactory, the
present arrangement in which the computer, is shared
with other sections'of the Elec. Eng. Dept., imposes
severe restrictions on the availability of the computer
for simulator use. The fact that the computer is
physically remote from the simulator also adds to the
complexity of the operation of the model.
With the exception of the linear programming
algorithm, all other programs described in this project
were developed by the author. A great deal of attention
was given to the development of modular algorithms
which are easy to implement and modify.
6.2. Further work.
A computer controlled implementation of the
calculated corrective actions to close the security
control loop would be desirable. To give a more
realistic operating enviroment the load units could
also be controlled from the computer using a suitable
program to simulate typical variations of load.
The availability of other types of measurements,
151
e.g. nodal injections and voltage magnitudes, would
allow other state estimation algorithms to be tested
and their on-line performance compared with the one
developed in this project.
The on-line determination of the topology of
the network is a further routine which could be easily
added to the present set of algorithms.
The work described in this project has been
concerned with the on-line determination of preventive
control actions which are calculated when the system
is operating in steady state. Useful research work
can be pursued in the development and on-line
implementation of fast control actions required when
the system is found to be in an emergency condition
which is likely to lead to dynamic instability.
In interconnected systems the representation of
neighbouring systems by equivalents is an area of
great interest to electric utilities. The ability
of the model to simulate accurately the dynamic
behaviour of a power system could be used to investigate
the problem of on-line identification of dynamic
equivalents of external networks.
The development and on-line implementation of
local dynamic estimators is another area in which
useful research work could be pursued with the aid
of the power system simulator.
152
6.3. Original Contributions.
. The development and off-line testing of a new
algorithm for fast approximate simulation of
line outages.
. The development of an efficient method for the
calculation of linear security constraints
which are obtained as a by product of an A.C.
line outage simulation.
. The use of the Sherman-Morrison technique in
the identification process. This provides a
fast check of all meters failing the residual
test. This is of particular relevance to
this project because due to the limitations
of the size of the model and of the
measurements available, only one gross
measurement error is acceptable
•
153
APPENDICES
Appendix 1. Stott's Algorithm for the simulation of
line outages.
In the Past Decoupled Load Flow the two sets of
linear eqns.
our
B 14V = AVIT
are solved iteratively until the real and reactive
power mismatches are smaller than a given tolerance.
The algorithm which uses this method combined with
the Sherman Morrison formula for the simulation of
line outages as as follows.
1). Form and factorize matrices B' and B'' storing
only their lower triangle.
2) Initialize nodal voltages to the values obtained
'by the state estimator for the present operating
conditions.
3) Identify nodes connecting the line whose outage
154
Is to be simulated.
4) Form vector M and solve for Z the eqn. BIZ=M,
using the elements of B' obtained in step 1.
5) Calculate the scalar C from eqn. 4.21.
6) Form vector N and solve for Y the eqn. B"Y=N
using the elements of B" obtained in step 1.
7) Calculate scalar d.
8) Set iteration count i=0.
9) Calculate AP/V taking into account the line
whose outage is being simulated.
10) Solve eqn. 4.14 forAO-using the factorized form
of B' obtained in step 1.
11) Correct A8 using eqn. 4.23.
12) Calculate AQ/V taking into account the line
whose outage is being simulated.
13) Solve eqn. 4.15 for AV using the factorized
form of B'' obtained in step 1.
14) Correct AV using eqn. 4.24.
15) If convergence has been obtained proceed to
step 16. Otherwise increase iteration count
by 1 and revert to step 9.
16) Print relevant results and if no more outages
are to be simulated terminate the analysis.
Otherwise return to step 2.
155
Appendix 2. Data for test systems.
A.2.1. DATA for 6 bus system.
NODE NI PGEN QGEN PLOAD QLOAD * *
1 1.10214 .0.0 0,0 0.0 0.0 *
2 1.10153 .6689 0.0 0.0 0.0 *
3 1.09197 .3434 0.0 0.0 0.0 * * *
4 1.0 0..0 0.0 0.4330 .3634 * * *
5 1.0 0.0 0.0 0.5956 -.5462
6 1.0 * o.o*
o.o*
0.5882 .2427
* initial value for unknown quantities.
Table A.2.1. VOLTAGE1 GENERATION AND LOAD DATA
LINE FROM TO YC/2 R
•
XL CAP.
1 1 2 0.0 .0300 .15010 .600
2 . 1 3 0.0 .1508o- .56830 .400
3 2 3 0.0 .07538 .28650 .650
4 2 4 0.0 .03188 .10000 1.00
5 4 5 0.0 .04275 .11480 .800
6 5 6 0.0 .05325 .15881 .65o
7 6 3 0.0 .04875 .16181 .700
8 5 3 0.0 .3300 • .09015 1.00
Table A.2.2. LINE DATA.
3
8 4
3
7 8
2
3 4
5 6
7 8
9 10 11 12 13
10 5 9 8
1 1 1 2
7 6 If 4 4
9 6
9
From To Line
2
156
A.2.2. DATA f or 10 bus system.
NODE
PGEN QGEN PLOAD
QLOAD
1 2
3 4 5 6
7 8
9 10
1.05 1.05 1.05 1.05 1.03 1.05 1.0*
* 1.0 1.0 1.0*
0.0 3.00 1.92 4.6o .96
.6o5 0.0 0.0 0.0 0.0
0.0*
0.0*
0.0*
0.0*
0.0*
0.0 0.0 0.0 0.0 0.0
0.0 0.0
2.3o 8.50 .90 .8o .90 .6o 1.o
-1.92
0.0 0.0 1.30 4.90 .30 .3o .4o .25 .5o .8o
*.Initial value for unknown quantities.
Table A.2.3. VOLTAGE MAGNITUDE, GENERATION AND LOAD DATA
.05062
.05062
.07594
.05062
.02531
.07594
.02531
.07594
.05062
.05062
.07594
.05062
.05062
.00988
.045o4 .01185 .01136 .00988 .01630 .00741 .01630 .04879 .00395 .01185 .01877 .04879
.0484o
.12365
.07802
.05531
.0484o
.0638o
.04889
.06519
.19161
.01975
.07802
.06281
.19161
2.34 2.34 2.34 2.34 2.34 2.34 2.34 2.34 2.34 2.34 2.34
2.34 2.34
R
XL CAP
Table A.2.4. LINE DATA.
157
A.2.3. DATA for 14 bus system.
NODE IVI PGEN QGEN PLOAD QLOAD
* * 1 1.06 O. 0. 0. 0.
* 2 1.045 .40 0. .217• .127
* 3 1.070 o. O. .112 .075
4 1.010 0. 0. .942 .19 *
5 1.090 0. 0. 0. 0.
6 lio 0. o. .478 -.039 *
7 1.0 0. 0. 0. o., *
8 1.0 0. 0. .076 .016
9 1.0 0. o. .295 .166 * lo. 1.0 0. o. .090 .058 * 11 1.0 0. o. .035 .018 *
12 1.0 O. O. .061 .016
13 1.0 O. O. .135 .058 * 14 1.0 0. o. .149 .050
* initial value for.unknown quantities.
Table A.2.5. VOLTAGE MAGNITUDE, GENERATION AND LOAD DATA.
NODE SHUNT SUSCEPTANCE
9 CAPACITOR 0.19
Table A.2.6. Shunt elements.
158
r-
Line From To YC/2 R XL .
1 1 2 • 0264 .0194 .0592 2 1 8 .02l .. 6 .0540 .2230 3 2 4 .0219 .0470 .1979 4 2 6 .0187 .0581 .1763 5 2 8 .0:170 .0570 .1739 6 4 6 .0173 .0670 .1710 7 6 8 .0064 .0133 .0421 8 6 7 o. O. .2091 9 6 9 o. o. .5562
10 8 3 o. o. .2520 11 3 11 o. .0950 .1989 12 3 12 o. .1229 .2558 1.3 3 13 o. .oh61 ~1303
14 7 5 o. o. .1761 15 7 9 o. .0. .11 16 9 10 o. .0318 .0845 17 9 14 o. .1271 .2704 18 10 11 O. .0821 <' .1921 19 12 13 o. .2209 .1999 20 13 14 o. .1709 .3480
'. Table A.2.7. LINE DATA •
• Transformer from to tap.
1 6 7 .978
2 6 9 .969
'3 8 3 .932
Table A.2.8. TRANSFORHER DATA.
159
Appendix 3. The D.C. Load Flow. Simulation of line
outages by fictitious injections.
The active power flow between two nodes at voltage
V.LS.1 and VJ IS. connected by an impedance Z. for 3.
which x..*.X./ 13 r3.. 3 . is approximately
P. = ij sin (S.-S.)
7.: bij 1 (S.-S.) (A.3.1.)
when V1.,V j z i 01 and (S -S.) is small
This approximation is used in the d.c. load flow.
Circuits are represented by their reactances and nodal
transfers by the active power components. The result
is an estimate of active power flows.
In a network with nil nodes where node 0 is
selected as angular reference, i.e. the phase angle So
is set to zero, the power injection at node i is given
by
P. = big Sib (S.-8 ) ... + bin (S. 3.0 1 i1 1 1-S - n
= -b S bi -1-b )S....-b 11 1 o i1 in zinS n
with i=1,2,...n4
This set of n eqns. can be written in matrix form
as
[pi [B] Cs] = CP]=PHP1
(A.5.24)
••■•••••
where:
bkm
(Xkkl-Xmm- 2Xicm)bkm El
160
The outage of a line, say the line connecting nodes
k and m, produces changes in elements (kon),(kl k),(m,k)
and (m,m ) of matrix B, eqn. A.3.2. becomes under this
circumstances
[81] = [B] [I] [pit [p] (A.3.3.)
where:
M = vector of all zeroes except positions k and m
which are 1 and -1 respectively,
which by the matrix inversion lemma becomes
[81 km L
H(A.3.4.)
performing the operations indicated in eqn. A.3.4. and
using eqn. A.3.2. we have that .
X .1k - X 1m
X - X .kk km
X X - • .mk mm
Xnk -.Xnm
(Skim) m' (A .3.5.)
4111111•101,
where:
Xkm = is the k,m th element of matrix X.
The change in phase anglesBS] given in eqn. A.3.5.
can be interpreted as being caused by the injection at
161
nodes k and m of a fictitious amount of real power as
shown in eqn.
[Ad
A.3.6.
X ... 1k
Xkk ...
X mk
Xnk
X ... 1m
• km ...
X ... mm
Xnm ..0
•■■
0
P
-P
0
MOM.
(A.3.6.)
X ... .11
Xk1 ... .
X ... .m1
Xn1
X 1n,
Xkn
X mn
Xnn ■■•••
the injection P required to simulate the outage of line
k-m without having to modify the elements of the X matrix,
can be obtained from eqns. A.3.5. and A.3.1 as
P= xkm
'r x(Xkk+Xmm-2Xkm km
where:
xkm : series reactance of line k-m
: pre-outage power flow'in line k-m.
(A.3.7.)
- 162
Appendix`!. Conversion to p.u. of measured line-flows
and nodal voltages.
The A/D converter has an input voltage which must
lie in the range t 10 volts D.C. which is transformed
into a 10 bit word. The PDP-15 computer uses 18 bit
words and twors complement arithmetic, hence the
largest positive analogue input, i.e. 10 volts,
corresponds in integer representation to:
analogue octal . - decimal
10 volts = 377400 130816
Therefore floating the integr quantities and
dividing them by 13081.6 gives as a result the input
voltage seen by the A/D converter. Since the Watt/Var
meters are calibrated to read 5 Volts D.C. when
measuring 1. p.u. flow the transformation to p.u. is
obtained from
P -(p.u.) = float (I.N.)/(13081.6 x 5)
where:
P :is the measured line flow in p.u.
I.N.:is the integer stored in core by subroutine
ANIN after A/D conversion.
The voltage meters were designed to read voltage
deviations from 1. p.u. and calibrated so that a deviation
of 0.1 p.u. is read as 5 volts D.C., henCe the required
transformation is given by
163
V(p.u.)=1.04.float(I.N.)/(13081.6 x 50.)
where:
V : is the measured reference voltage in p.u.
I.N. : is the integer stored in core by subroutine
ANIN after A/D conversion.
164
Appendix 5. Formation of Task OPTIME.
TKB
TASK BUILDER VI A
LIST OPTIONS >SZ NAME TASK >OPTIMES T IMPROPER BREAK CHAR -- >OP TI SPECIFY DEFAULT PRIORITY >30 DESCRIBE PARTI TI ON >P3 (30000,30000) DESCRIBE SYSTEM COMMON BLOCKS >DA TI N(1 7000,200), NENA (24000,4000) DUI NE RESIDENT CODE >MAI NZ DESCRIBE LINKS & STRUCTURE >LKI = GAI MA/I NVER >LK2 =I DENT, VOLTA/SOLVE >LK3=CONFLO, FASFLO/SOLVE >LK4:-.MATRX/I NVER >LKI : LK2: LK3: REDAT: YBUS: REANA: LI NFLO: WRI Ti : LK4 >LK4: DETECT: LI NPG: RESCH: TABLO: DISPA
165
LINK TABLE 30020-30302
RESIDENT CODE MAI N2 30303-31117 EXU. 5 31120-3 1255 TIMF.2 31256-31302 EXIF.1 31303-31305 .DAA .4 31306-313 67 .SS 313 70-31447 GOT 0 31450-31475 ST0.3 31476-31507 OTS.5 31510-3163 6 .SP .4 3163 7-3175 6
002 63
00615 0013 6 00025 00003 00062 00060 0002 6 00012 0012 7 00120
I NTEAE 31 75 7-32072 00114 R ELEA E 32073-33134 01042 .CB 33135-33154 00020 GRL 33155-33170 00014 BUS 33171-333 72 00202 YBU 33373-34017 00425 LI N 34020-34251 00232 OBS 34252-311605 00334 GAIN 34606-35115 00310 ELM 35116-35145 00030 FLU 35146-35625 00460 COM '35626-36046 00221 TIMES 36047-3 6107 00041 XXX 3 6110-3 6123 00014 GEN 36124-36217 00074 IDE 3 6220-3 6220 00001
LI LK1 GA I MA 3 6221-3 7253 01033 INVER 40020-40565 00546 SQRT 3 7254-3 73 45 000 72
LI NK LK2 IDENT 40020-42153 02134 VOLTA 42154-43 743 015 70 SOLVE 36221-36425 00205 ABS 3 6426-3 6443 00016 SQRT 3 6444-3 6535 00072 ATA N 3 653 6-3 6550 00013 .ED 3 6551-3 663 7 00067 .EC 3 6640-3 6703 00044
LI NK L K3 CONFL 0 40020-43165 03146 FASFLO 43166-45411 02224 SOLVE 36221-36425 00205 ABS 3 642 6-3 6443 00016 IABS 3 6444-3 645 7 00014 SQRT 36460-3 6551 00072 BAS 36552-36671 00120
LI NK -- REDAT REDAT 3 6221-3 7560 01340 BCDI 0 40020-43055 03036 FI 0.5 43056-43543 00466
.166
a.
LI NK YBUS
YBUS 36221-3 7601 013 61
LINK R EA NA R EA NA 3 6221-3 7162 00742 I NHENB 371 63-3 71 70 00006 REQF.2 • 3 7171-3 721 7 0002 7 FTS .3 3 7220-3 7307 00070 FLOAT 3 7310-3 7320 00011 BCDI 0 40020-43055 0303 6 FI 0.5 43056-43543 00466
LI NK -- LI NFL 0 LI NFL 0 36221-3 7444 0122 4 SQRT 3 7445-3 753 6 00072
LI NK WRI T1 WRI Tl . 3 6221-3 7254 01034 BCDI 0 40020-43055 0303 6 FI 0.5 3 7255-3 7742 00466
LINK LK4 MATRX 3 6221-3 7221 01001 I OVER 3 7222-3 7767 00546 SQRT 40020-40111 00072
LINK -- DETECT DETECT 3 6221-3 7771 01551 FLOAT 40020-40030 00011 SQRT 40031-40122 00072
LINK -- LI NPG LI NPG 40020-42 775 02 75 6 ABS 3 6221-3 623 6 00016 FL OAT 3 623 7-3 62 47 00011
LI NK R ESCH RESCH 3 6221-3 6705 00465
LINK TABL 0 TABL 0 3 6221-3 7654 01434 BCDI 0 40020-43055 0303 6 FI 0.5 43 056-435 43 00466
LINK DISPA DI SPA 3 6221-3 6641 00421
CORE R EQ'D 30000-45411 15412
KM9 -15 V5 A
$P1P
PIP V13 A
>T DT24-DK1 OPTI ME TSK (B)
167
REFERENCES.
1. R. Deutsch, "Estimation TheoryV Prentice Hall,
Englewood Cliffs, N.J., 1965.
2. F.C. Schweppe and J. ti ildes,"Power System Static
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58. F. Ariatti, "System Telemetry and Estimation of
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59. E. Arriola-Valdes, "Static Economic Dispatch in
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October 1975.
