Design of Observers for the Swing Dynamics of Power
Networks
by
Paisarn Sonthikorn
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 2002
@
Massachusetts Institute of Technology 2002. All rights reserved.
A uthor ................................................................
Department of Electrical Engineering and Computer Science
May 24, 2002
Certified by .................
George C. Verghese
Professor of Electrical Engineering
Thesis Supervisor
Accepted by .............
Arthur C. Smith
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OFTECHNOLOGY
BARKER
JUL 3 1 2002
LIBRARIES
Design of Observers for the Swing Dynamics of Power Networks
by
Paisarn Sonthikorn
Submitted to the Department of Electrical Engineering and Computer Science
on May 24, 2002, in partial fulfillment of the
requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
This thesis presents a design of swing-state power network observers. The network observer
concept is motivated by researchers' attempt to better understand complex interactions of
power networks in order to achieve efficient fault monitoring processes. Based on the nonlinear DAE swing model and the nonlinear measurement model, the network observer has its
gain computed by using the Linear Quadratic Estimator (LQE) method. Using the classical
nine-bus system as a test system and having two representative system disturbances: a line
drop and a change in power injection at a load node, numerical studies of the observer shows
impressive results in terms of both fast convergence rate and low offsets between the real
state values and predicted ones. The underline reasoning behind the network observer good
performance is that, by using a highly nonredundant set of sensors, this network observer
can exploit its inherited nonlinear models to accumulate over time and to interpolate over
space in order to generate satisfactory numerical state predictions. Later, Two fault isolation methods using the network observer: multiple observer scheme and nominal observer
scheme, give us good results and provide insights into the effect of the disturbances on the
network behaviors.
Thesis Supervisor: George C. Verghese
Title: Professor of Electrical Engineering
2
Acknowledgments
I am deeply thankful to Professor George C. Verghese, my great mentor, who has stimulated
my research interest and has provided invaluable advice to me all along my M.Eng. year at
MIT.
Thanks to Associate Professor Bernard Lesieutre, my supportive and friendly ex-academic
advisor, for your guide and encouragement.
Thanks to Ernst Scholtz for all your supports, help and understanding in both research
and psychological issues.
Thanks to Joshua W. Phinney for his technical advice on LATEX.
Thanks to Vivian Mizuno for all the help she has provided me all along.
Many thanks to P'Teng (Poompat Saengudomlert) and P'Yong (Watjana Lilaonitkul)
for your consistent supports.
Thanks to my oldest brother, Paiboon Sonthikorn, I am deeply in debt to your guidances.
Thanks to my cheerful older sister, Ratchanee Sonthikorn, for keeping my world bright and
joyful.
Thanks to my older brother, Paitoon Sonthikorn, for always making me realize
how to stay strong in this world. And thanks to my little sister, Nong Pond (Thanawan
Kittisuwan), for cheering me up every time I am desperated and doomed.
Forever thanks to my Mom, Saovaluck Sonthikorn, and my Dad, Jane Sonthikorn, for
having nurtured me to be a good person, and for always being proud of who I am. Being
your son is a blessing from Heaven!
Thanks to the people of the Kingdom of Thailand for giving me all these opportunities;
I hope to prove your tax money worthwhile!
Thanks also to the Electric Power Research Institute (EPRI) and the Department of
Defense (DoD) for partial support of my research.
... Here I am, at the Massachusetts Institute of Technology, which used to be my dream
school and has been such a great institute that will shape my life forever.
AND NOW I'M OUT OF HERE!!!
3
Contents
1
2
3
4
Introduction
9
1.1
Motivations for Network Observers . . . . . . . . . . . . . . . . . . . . . . .
10
1.2
Term inology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Modeling for Observer Design
2.1
Nonlinear Swing Model
2.2
Nonlinear Measurement equations
2.3
12
. . . . . . . . . . . . . . . . . . . . . . .
. . . .
12
. . . . . . . . . . . . . . . . .
. . . .
14
Linearizing the Swing and Measurement Models . . . . . . . . . .
. . . .
15
2.3.1
Linearizing the Swing Model
. . . . . . . . . . . . . . . .
. . . .
15
2.3.2
Linearizing the Measurement Models . . . . . . . . . . . .
. . . .
16
2.4
Collapsing the Linearized Swing Model . . . . . . . . . . . . . . .
. . . .
17
2.5
Observer Design Methodology . . . . . . . . . . . . . . . . . . . .
. . . .
18
2.5.1
Linear Observer for Linearized State-Space Swing Model .
. . . .
18
2.5.2
Nonlinear Observer for Nonlinear DAE Swing Model . . .
. . . .
19
Numerical Results of Observer Studies
3.1
The Observer in Action
3.2
Selecting the
21
. . . . . . . . . . . . . . . . . . . . . . .
. . . .
22
and R Matrices . . . . . . . . . . . . . . . . . . .
. . . .
25
3.3
Types of Measurement and Placement . . . . . . . . . . . . . . .
. . . .
27
3.4
Pole Placement VS LQE . . . . . . . . . . . . . . . . . . . . . . .
. . . .
28
Q
Fault Isolation Applications
32
4.1
Multiple Observers for Fault Isolation
. . . .
32
4.2
Understanding Faults on the Network: Line Parameter Disturbances . . . .
35
4.2.1
36
Normal Operation
. . . . . . . . .
. . . . . . . . . . . . . . . .
4
. . . .
4.3
4.2.2
Investigation of the Steady-State Output Error Vectors
4.2.3
Gaining Insights into ey Vectors
4.2.4
Achieving Fault Isolation via a Nominal Network Observer
. . . . . . .
36
. . . . . . . . . . . . . . . . . . . .
40
. . . . .
42
. . . . . . . . . . . . . . . .
45
4.3.1
Discussion about the Analysis . . . . . . . . . . . . . . . . . . . . . .
46
4.3.2
Numerical Results of the Analysis
47
4.3.3
Exploring on the Nonlinear DAE Swing Model:
Analysis of Steady-State Output Error Vectors
. . . . . . . . . . . . . . . . . . .
Complement Tool For Fault Isolation . . . . . . . . . . . . . . . . . .
4.4
5
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary
47
49
50
5.1
Brief Content Overview
5.2
Suggestions for Future Studies
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
50
51
53
5
List of Figures
3-1
Classical 9-bus example studied in this paper. The line parameters are shown
as im pedances.
3-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time plots of 64-61 (system) and 64-61 (observer) when Event 1 occurred
(beginning at t =0.1s, duration 3s).
3-3
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
Time plots of 6 4-6 1 (system) and
. . . . . . . . . . . . . . . . . . . . . .
S4-61
(beginning at t = 0.1s, duration 3s).
3-7
26
Time plots of 66-61 (system) and 66-61 (observer) when Event 2 occurred
. . . . . . . . . . . . . . . . . . . . . .
26
Time plots of 64-61 (plant), Sso-61 (suboptimal) and S4'0-61 (worst case companion) when "event 1" occurred (beginning at t =
3-9
25
(observer) when Event 2 occurred
. . . . . . . . . . . . . . . . . . . . . .
(beginning at t = 0.1s, duration 3s).
3-8
24
Time plots of 6 3-6 1 (system) and 63-61 (observer) when Event 2 occurred
(beginning at t = 0.1s, duration 3s).
3-6
24
Time plots of w 3 (system) and &3 (observer) when Event 1 occurred (beginning at t = 0.1s, duration 3s). . . . . . . . . . . . . . . . . . . . . . . . . . .
3-5
23
Time plots of 65 -6 1 (system) and 65-61 (observer) when Event 1 occurred
(beginning at t = 0.1s, duration 3s).
3-4
22
0.1s duration 3s).
. .
28
Time plots of 62-61 (system) and 62-61 (for both LQE observer and poleplacement observer) when Event 1 occurred (beginning at t =
0.1s, duration
3s ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3-10 Time plots of 67-61 (system) and 67-61 (for both LQE observer and poleplacement observer) when Event 1 occurred (beginning at t = 0.1s, duration
3 s ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-11 Time plots of w6 (system) and
C.,
6
30
(for both LQE observer and pole-placement
observer) when Event 1 occurred (beginning at t = 0.1s, duration 3s). . . .
6
30
3-12 Time plots of 64 -6 1 (system) and
54-
1 (for both LQE observer and pole-
placement observer) when Event 1 occurred (beginning at t =
3s)..............
4-1
Time plots of the residuals
0.1s, duration
........................................
(65-61)
31
(65-61) of different observers when the
-
real system has Line 9 taken out. . . . . . . . . . . . . . . . . . . . . . . . .
4-2
Time plots of the residuals
(4-61) - ( 6 -6 1 )
real system has Line 9 taken out........
4-3
Time plots of the residuals
34
of different observers when the
.........................
34
(68-61) of different observers when the
(68-61) -
real system has Line 9 taken out. . . . . . . . . . . . . . . . . . . . . . . . .
35
4-4
Normal operation of the classical 9-bus sytem: angles and power flows. . . .
37
4-5
Steady-state output error plot of all six simulations.
38
4-6
EY vector plot of all six simulations with all the line parameters perturbed
. . . . . . . . . . . . .
within five percent of the original values. . . . . . . . . . . . . . . . . . . . .
4-7
iy vector plot of all six simulations with all the line parameters perturbed
within ten percent of the original values. . . . . . . . . . . . . . . . . . . . .
4-8
40
ey vector plot of all six simulations with all power injections and extractions
perturbed roughly by ten percent.
4-9
39
. . . . . . . . . . . . . . . . . . . . . . .
41
Steady-state after Line 4 getting cut off from classical 9-bus sytem: angles
and power flows. ..........
.................................
7
43
List of Tables
4.1
The measurement errors for all three sensors of the observers when there is
no line or power variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ey
41
4.2
The direction of
. . . . . . . . . . .
44
4.3
The magnitude changes of Ey vectors of all the studies. . . . . . . . . . . . .
45
vectors of the previous four studies.
8
Chapter 1
Introduction
The concept of state estimators or observers has been prevalent in the control theory area.
However, there has not been sufficient study of its application to the swing dynamics of
power systems.
This thesis has the objective of discovering more about power network
observers in terms of design concepts and later providing application examples in fault
isolation as supporting evidence that the design can achieve promising results.
This thesis shows how to approach dynamic real-time swing-state estimation for power
networks using observer design. The study in this thesis has been developed and expanded
from that in
[1]
to have more types of sensors and to allow the network to include load
buses (or "buses"). The network observer design methodology we developed utilizes three
important components to achieve good state estimates: observer gain based on linearized
and collapsed swing models, with the standard Linear Quadratic Estimator (LQE) technique
applied; a nonlinear DAE (differential-algebraic equation) swing model as the inherited
model of the observer; and the freedom to use four possible measurement types for observer's
inputs: a bus angle, a speed deviation from synchronous associated with generator, a net
power at bus flowing into the network and a power flow on a tranmission line.
Based on the classical nine-bus network in [2] as a test system, we can have the modelbased observer, as will be shown, generate good numerical state predictions by using the
information that the nonlinear model of the observer accumulates over time and interpolates
over space from the measurements of a highly non-redundant set of sensors. The classical
nine-bus example helps illustrate our results and explore how performance varies with the
number, nature, and placement of measurements. Although still needing further investiga-
9
tions, comparative simulations with an observer using a heuristic gain shows that an LQE
observer can achieve good predictions with a fast convergence rate and low residuals.
Next, we briefly investigate possible application of our network observer in fault detection and diagnosis. First, a multiple observer scheme [3] is introduced and implemented
using our observers. Then we examine a possibility of using the steady-state output error
(
y) vectors resulting from complete single line-out faults to achieve fault isolation. After
that we do analyses to gain insights into the relation between faults and their corresponding
ey vectors.
Chapter 2 describes the modeling and design methodology of our network observer.
Chapter 3 presents the numerical results of our observer by using the classical nine-bus
system as a test system and also briefly shows performance comparison study between our
LQE observer with a pole-placement observer.
Chapter 4 provides a few examples in the fault isolation application of the network
observer.
Chapter 5 offers the conclusion of this thesis and suggests possible future studies.
1.1
Motivations for Network Observers
Large interconnected networks, such as power grids, communication networks, and the Internet, sometimes suffer from cascading outages [6].
To prevent future blackouts, it is
essential to understand such networks' dynamic behavior and relate it to their underlying network structures. At MIT's Laboratory for Electromagnetic and Electronic Systems
(LEES), researchers are examining relationships between the graph of a power network (i.e.,
the topology of the network) and the dynamic properties of the system. Here one important
issue is the monitoring of power systems after faults or disturbances. These disturbances
generally give rise to oscillating modal components, which in a worst-case scenario can
go unstable.
Such a phenomenon can pose a serious problem to system reliability if not
detected and damped out.
The concept of network observers then becomes a crucial element in determining the
evolution of network states. By better understanding the network behaviors both before
and after disturbances, we can then design and generate appropriate counteractions against
possible threats from faults. Eventually, we want our network observer to be robust and
10
informative in predicting the states of huge and complex networks in order to help researchers to better understand the complicated evolution of the network characteristics
both electrically and topologically. Thus, we can have a solid foundation to build effective
fault detection and diagnosis schemes for real world systems.
1.2
Terminology
Fault Monitoring Process:
The terminology of the monitoring process can be confusing
since there is no standard. Here, we want to provide definitions from
[4],
which are used
throughout this thesis.
" Fault detection answers the question - Has a fault occured?
" Fault identification answers the question - Which observation variables should we
identify as most relevant to diagnosing the fault?
" Fault diagnosis answers the questions - Which fault occured? What is the cause of
it? And what is the type, magnitude and time of the fault?
" Fault isolation answers the question - Where exactly in the network is the faulty
component?
In this thesis, we focus on studying the application of our network observer mostly to fault
isolation, i.e., determine which component is faulty. Throughout Chapter 4, we assume that
the faults of interest are complete single line-out faults; what we are trying to achieve is
determining which line is cut.
Distinction between ParameterEstimation and Observer-Based Method:
damental analytical methods in various engineering fields.
Both are fun-
However, the reader should
understand clearly the distinction between the two since they are similar in some aspects
and might cause some confusion if one wanted to proceed in the parameter estimation direction. And though they are two different methods, one can certainly combine the two to
do the fault monitoring process. It is noted that the focus of this thesis is the observerbased method. We advise the reader to refer to [5] to familiarize himself or herself with the
concepts of both methods. Also, [6] provides discussions about the combination of the two
analytical methods.
11
Chapter 2
Modeling for Observer Design
In this chapter, the nonlinear swing model that is the basis for our network observer is
first presented. Next, we show the nonlinear models for different types of measurements:
bus angles, generator speeds, line power flows, and power injected into or extracted from
buses. By linearizing and collapsing the nonlinear models, we can then construct a linear
observer for the linear state-space swing model. Using the linear gain computed via the
LQE method, we can integrate it with the nonlinear models to create a nonlinear observer,
which is the network observer we will use throughout this thesis.
2.1
Nonlinear Swing Model
Throughout this thesis, we use the nonlinear swing model to understand the behavior of
the power network. Also, this model is a crucial part of our observer design since it is the
knowledge base of our observer in understanding the electromechanical behaviors of the
network. Therefore, this section intends to help the reader understand the details of the
model and the notations of its associated elements.
It is noted that, for the power systems of our study, we assume that the voltage magnitudes of the network are tightly controlled around 1 p.u., so we take them all to have this
value. Let 6 denote the vector of bus angles, and P(6) denote the real power flowing into
the network, so
P(6) = -F3 sin(F'6) + F|g cos(F'6) - diag(FgF')
12
(2.1)
where: 1) F is the directed bus-line incidence matrix of the network graph (the orientation
of line h can be picked arbitrarily, and F,,h = -1,
bus s to bus
Ft,h
1 if this directed line goes from
t);
2) ' denotes matrix transposition;
3) Depending on the context, diag(-) extracts the diagonal of its matrix argument and
forms a column vector, or forms a diagonal matrix by placing its vector argument on the
diagonal;
4) sin(.) and cos(.) imply taking elementwise sine or cosine of the corresponding vector
arguments;
5) B and
g
are diagonal matrices with line susceptances and conductances as diagonal
elements.
(
Throughout this thesis all vectors are typically ordered as:
scripts g and I indicate generator and load buses. For examples,
it is noted that we define the speed vector as w
=
[2]'
[
=
=
'
[
'',
6'1
,5'
where sub'.
However,
because we are only interested in
determining the speed of generators.
We also define the following: n is the total number of buses in the system; ng, is the
number of generator buses; and n, = n -
n is the number of load buses. We note that
F'6 yields a vector of angle differences across branches of the network; diag(FgF') is an
n-dimensional vector whose elements are the sums of the conductances of lines emanating
from the corresponding buses.
Using (2.1) we can then construct the nonlinear DAE swing model as follows:
i
M
1
0
0
0
0
0
0
0
Mg
f(x,u,w)
1=1
pe P
pe - P9 (6)
(6)
-
(2.2)
Dgw
where 6g, 61, and w are respectively the generator angles, load angles, and generator speed
deviations from synchronous speed. We use x to denote these internal variables of the DAE
description. Note that 6g and w are state or differential variables, while 61 comprises algebraic variables. The vector PFe denotes power injected at the load buses (and hence typically
has negative entries), while Pg is the net power injected at the generator buses (typically
13
mechanical power input to a generator minus the local real-power load) 1 . Integrating these
two vectors, we define Pe as the vector of external bus power injections. These injections
may be partly or completely known; the known parts are gathered in the vector u, while
the unknown parts "process noise" are gathered in w (i.e., implicitly we have Pe -
+ w).
D 9 and Mg are diagonal matrices whose nonzero entries respectively comprise the damping
coefficients and the (normalized) inertias of the generators. Discussion: The second and
the third rows of the nonlinear DAE swing model in (2.2) provide a useful framework on the
network behaviors 2 . The second row tells us that Pe = P (6), which means that the power
extractions at the load buses equal the power flow from the network into those buses. The
last row is essentially the swing equation 3 , which characterizes the behaviors of the power
at the generator buses.
2.2
Nonlinear Measurement equations
Here, we present the nonlinear measurement models characterizing outputs as nonlinear
fuctions of its corresponding inputs. This thesis allows four possible types of measurements.
I
The ith measurement can be
j
=i
if j E{, -
, n}
if j E {1,...
n}
, n}
Pj (6)
if j C{1,
Pst (6)
if s,tiE III,-
where 6j is the bus angle associated with bus
speed associated with generator
j,
j,
(2.3)
n}
wj is the speed deviation from synchronous
P (6) is the net power at bus
j
flowing into the network
(this is simply the Jth element of (2.1)) and Pt(6) is the power injected at s onto the line h
that connects to bus t. To obtain an explicit expression for this latter measurement, note
'It is noted that the net power injected at the generator buses can vary because although the mechanical
power has to stay the same, the local real-power load can shift to another steady-state value due to any event
happening to the system. However, if there is no direct change in load extraction, Pf needs to maintain its
steady-state value.
2
The first row just says that S9 = W. This row, though not informative, helps us obtain a matrix form of
the swing model.
3
For more information about the swing equation, the reader is encouraged to read [7]
14
first that the vector of flows Pine( 6 ) on the lines of the network can be expressed as
Pi,
e(6)
1
= B sin(F'6) -g cos(F'6) - -g(F' - F1')
2
(2.4)
If the orientation picked for line h when defining F goes from bus s to bus t, then Pot(6) in
(2.3) is simply the hth component of Pie() above.
Gathering all the available measurements into a vector y, we obtain a vector measurement equation of the form
(2.5)
y = g(x) + v
where x is as defined earlier, and v denotes a vector of sensor or measurement noise variables.
2.3
Linearizing the Swing and Measurement Models
In order to use standard state-space observer design techniques, we will work with smallsignal versions of the nonlinear swing model and the measurement equations, linearized
around steady-state. Note that a vector in the nonlinear system can be expressed as
((t) =
(* + ((t), where (* is the steady-state vector and ((t) is the vector of deviations from
this steady-state, assumed small when deriving the linearized model. In order to simplify
notation, we will suppress the time dependence of the variables.
2.3.1
Linearizing the Swing Model
We linearize the nonlinear DAE swing model (2.2) around the steady state (loadflow) solution
6
6* and w = w* = 0. Evaluating the Jacobian
a
= K (where K is
referred to as the spring or synchronizing matrix of the system), one finds the following:
K = -FL3diag(cos(F'6*))F' + \F|gdiag(sin(FJ*))F'
(2.6)
From (2.6), we can partition K into four submatrices, Kgg, KI, Kig and K1, as follows:
K=
K9 g
K9 ,
K9
K,
15
(2.7)
where K 9 g E 7Z9"g is a spring matrix associated with transmission lines connecting a
R" ""' is a spring matrix associated with transmission
generator to another generator; Ki E
lines connecting a load to another load; Kg
E
Xfl"'
is a spring matrix associated with
transmission lines connecting a load to a generator, and K
= K' 1 . It is worth noting
that K is positive semidefinite and has a Laplacian structure, which has the property that
Neglecting higher-order terms of A in the linearization of (2.2), we obtain a linearized
DAE swing model of the form:
A
10
0
0
0
0
0
0
M
2.3.2
G
0
0
I
A6 9
-K 19
-Ku
0
A6
-K
-K 9 1
j
-D
0
+
G
Ape (2.8)
G
Aw
Linearizing the Measurement Models
Linearizing (2.5) results in an expression of the form
Ay = CAX + Av.
where C
(2.9)
[-1
=
Any available angle and speed measurements are already linear functions of x, so their
contributions to (2.9) are clear on inspection.
For measurements of bus power injection
P (6), the linearization is simply yielded by the corresponding row of the Jacobian K. For
measurements of line power flow, we similarly pick the appropriate rows of the linearization
( []
of Pine()
_66, = E)
E = 3diag(cos(F'S*))F' +
diag(cos(F'3*))F'.
(2.10)
Thus C in (2.9) is constructed by gathering a selection of rows from each of the following
matrices:
[ Inxn
surements);
0
]
(direct angle measurements);
[ K Onxn, ]
[
Ong
xn
Ing x n, ] (generator speed mea-
(bus power injection measurements);
[
E
Onxn, ] (line power
flow measurements).
Discussion: After achieving a linearized swing model, we still experience a problem in
16
obtaining a state-space swing model for our observer design. The reason is that we want
to mutiply the inverse of the leftmost matrix to both sides to achieve a state-space swing
model; however, the diagonal matrix has zero diagonal elements, which implies that the
matrix itself is singular. The structure of the matrix results from our networks having both
loads and generators. Therefore, we need to another step that can get around the problem,
but can still maintain all the state information.
2.4
Collapsing the Linearized Swing Model
To form a state-space swing model that can be used for our observer design, we execute a
Ward-type reduction on the linearized DAE swing model (2.8). This will help us suppress
the problem that we had in the previous section, i.e., expressing the load angles as a function
of the generator angles.
However, a crucial assumption for this reduction is that K 1 is
invertible, and this is generally the case for our systems of study (implying that the DAE
model is of index 1). From (2.8) the relationship between load angles and generator angles
is found to be
A61 = -Kij 1(Kj9A6S-
GIAPFe).
(2.11)
Substituting (2.11) back into (2.8) and defining
A
=
Kgg - KgiKj Kig
(2.12)
Gc = Gg - K 9 iKIj Gi
(2.13)
Ke
and S = -M- 1 , the collapsed all-generator linearized state-space swing model is expressed
as:
A
og0
x
I
A69
0e
+
-
AP.
(2.14)
The driving term in the above equation can be written as the sum BAu + GAw, where
a
and w are as defined earlier in Section 2.1.
By partitioning C into
[ C69
Cjl
Cw ] and substituting (2.11) into (2.9), we can achieve
17
the following linearized measurement equation written for the collapsed system:
Ay
[ (C69
- C6,Kj7Ki)
C,
Ix
+ Av.
(2.15)
C
2.5
Observer Design Methodology
This section presents our observer design methodology using the state-space swing model
in (2.14) and (2.15). We first start by following a classical observer design to obtain a linear
observer.
Next, through the LQE method, we can obtain a linear observer gain, which
will later be integrated with the nonlinear models in (2.2) and (2.5) to obtain a nonlinear
network observer.
2.5.1
Linear Observer for Linearized State-Space Swing Model
Given the measurements (2.15), an observer for the linearized system in (2.8) takes the
form of a real-time simulation of (2.14), to which is added a correction term proportional
to the discrepancy between the measured Ay and the observer's estimate of Ay. Denoting
the state of the observer by -, we have
S= A- + BAu + L(Ay - CX)
where L is the so-called "observer gain."
(2.16)
Recall that w and Aw are unknown, so the
observer is missing the process noise term GAw that is present in (2.14).
Similarly, the
measurement noise Av is unknown, so the observer's estimate of Ay is just C5.
Defining the observer error to be e =
x
- X, we see on subtracting (2.16) from (2.14)
that
(A - LC)e + GAw - LAv,
with e(0) = x(O) - -(0).
(2.17)
It is noted that the state matrix A - LC determines the stability of the error dynamics.
If it has eigenvalues with negative real parts, the observer is stable and the error will
eventually converge to zero if Aw and Av are zero, which is the desired outcome (i.e., the
states of the system are correctly mimicked asymptotically by the states of the observer).
18
If Az and Av are nonzero but bounded, then a stable observer will end up with a bounded
error. Therefore it becomes obvious that we want to achieve L that can make our observer
stable and can guarantee that the error will converge to zero or at least stay within a certain
bound.
To be able to move all the eigenvalues of A - LC, The technical condition required
for variations in L is observability of the pair (A, C) [8]; under this condition, a proper
choice of L can make A - LC have any self-conjugated set of eigenvalues. However, getting
A - LC such that all its eigenvalues have very high negative real parts, for rapid decay of
transients in (2.17), requires large values in L, which in turn accentuates the effects of the
measurement noise Av in (2.17) and of any modeling error. This is the basic tradeoff in
choosing L.
Imposing only the requirement of stability does not sufficiently specify L, so we can look
for a stabilizing L that minimizes some measure of e, given appropriate characterizations
of Aw and Av. If Aw and Av are modeled as zero-mean white noise processes, and if we
ask for the L that minimizes the error variance, we arrive at a special (steady-state) case
of the Kalman filter, also called the Linear Quadratic Estimator or LQE filter [8]. The
corresponding optimal observer gain L, is obtained by first solving the Algebraic Riccati
Equation (ARE) in (2.18) below, and then using the expression in (2.19):
PA' + AP - PC'R-1 CP + GQG'
0
L, = PC'R 1
The matrices
(2.18)
(2.19)
Q and R represent the spectral powers of the white noises Aw and Av
respectively. More generally,
Q and R may be seen as design parameters that can be varied
to yield different observer gains. Roughly speaking, increasing R relative to
Q results in less
"aggressive" observers, with the L, in (2.19) being correspondingly smaller; this tradeoff is
further discussed in Section 3.2.
2.5.2
Nonlinear Observer for Nonlinear DAE Swing Model
The linear observer designed above may be expected to do a reasonable job of tracking
deviations of the swing model from steady state, provided these deviations are small enough
to be reasonably captured by the linearized model. To track larger deviations from nominal,
19
we can try replacing the real-time simulator in our observer with the full nonlinear DAE
swing model, rather than using the linearized model. The internal variables of this simulator
are denoted by
2.
The correction term is now made linearly proportional to y - g(I) rather
than Ay - C , but uses the gain computed via the linearized model, adjusted to feed in
to the appropriate equations of the nonlinear DAE swing model. Specifically, partitioning
the observer gain matrix L, as L, =
defined as L =
[ L'
0
L'
M
where M,
f, u, w
1'.
[ L'
L' ]', the DAE model's observer gain matrix is
The nonlinear observer then takes the DAE form
=
f ( , u = u*, w = 0) + L (y -
),(2.20)
are as defined in (2.2), y is the measured data, and j
g(2) is the output
predicted by the observer in accordance with the model (2.5).
So far, we have obtained a methodology of our nonlinear network observer design. Our
observer has a both nonlinear models and a linear gain computer using the LQE technique.
We next want to test our observer in numerical studies to make sure that the observer
performs well in different scenarios before using it in fault isolation.
20
Chapter 3
Numerical Results of Observer
Studies
After demonstrating the observer design methodology, we want to present numerical results
from our observer performance tests using a classical nine-bus example shown in Figure 3-1.
In the performance study of this chapter, we will use the following two events to represent
faults that usually occur in a general network.
Note that this small classical nine-bus
example can become unstable easily if any big perturbation is applied since we do not have
a governor implemented here. Therefore, we carefully choose events that will give us mild
perturbation since we are interested only in illustrative examples of how well the observer
performs anyway.
" Event 1: A 0.05 p.u. change in the real power load at bus 8. We assume that this
(unscheduled) load change is not known to the observer.
" Event 2: Losing one of the two lines between bus 8 and 9. We assume that the line
loss is not known to the observer.
To make our investigations more realistic, we want to take information inaccuracy in our
knowledge about the system's parameters into account. Therefore, it is assumed that the
actual inertias of the generators and the electrical characteristics of the lines are within ±1%
of the nominal parameters given in Figure 3-1. The generators' damping coefficients, which
usually cannot be measured directly, are assumed to be within ±10% of the applicable
nominal values. For our studies, the observer uses the nominal parameters as shown in
21
230kV
'Event
" P8 = 100MW
230kV
7
jO.0625
9
625
jO.0586
0.0119+j0.1008
0.0085+jO.072
8
3
13.8kV
M3 =0.0 16
D3 =3*M3
"Event 2"
2
18kV
M2 =0.034
D2 =2*M2
5
6
P5 =125MW
P6
90MW
230kV
4
MI= 0.1254
jO.0576
16.5kV
Figure 3-1: Classical 9-bus example studied in this paper. The line parameters are shown
as impedances.
Figure 3-1, while the system parameters are fixed for each set of simulations at randomly
selected values within the above ranges.
3.1
The Observer in Action
This section demonstrates the working of a particular observer in the noise-free case, in
response to the occurrence of either Event 1 or Event 2 on the system. Here, the observer
uses measurements from three sensors: a direct angle sensor at bus 5; a power injection
sensor at bus 1; and a power flow sensor on the line directed from bus 7 to bus 8. (The
choice of observer gain and sensors will be discussed in Sections 3.2 and 3.3, respectively.)
We have the system and the observer start with their respective steady states for both
simulations. The applicable event occurs during
0.1s < t < 3.1s in each simulation. For
the ensuing discussion, we will examine the angle at a bus relative to the angle at bus 1; so
we define
jl
--
61 and 6i1 =6i-
6
where i
C {2,...
n}.
Simulation of Event 1: The real state evolution of the system and the state prediction of our
observer for Event 1 are shown in Figures 3-2, 3-3 and 3-4. In Figure 3-2, 641 approximates
64 1 fairly accurately throughout the simulation. Note that the estimate follows the real state
closely even after the event occurs. It is worth emphasizing here that the good estimate of
22
Event 1 Simulation: System's
-
and Observer's
4
-0.039
S-ystem
-Obs1
- --
-t
- ---
-
-
-0.041 -
--
-
-
-0.04 - -
-
- --
I
-0.042
-0.
42
-
-I.43
r
-
0
--
..
1
2
. . .. .
...
1..
. .
-0 .04 3 - -.
-0.045I
-
3
. .. . . .. . . . .. . .. .. . .. . .
4
5
6
Figure 3-2: Time plots of 64-61 (system) and 64-61 (observer) when Event 1 occurred (beginning at t = 0.1s, duration 3s).
641 is obtained despite not having any direct measurements at bus 4. The ability to obtain
this spatial interpolation is directly the result of using the dynamic model to complement
the measurements.
From Figure 3-3 we see that 651 tracks the general form of the variations in 651 but
settles by the end of the event period to a constant angle offset, caused by the observer's
ignorance of the deviation of the load at bus 8 from its scheduled value during this interval.
After this event, this offset abruptly decreases but gradually returns to the same offset as
that in the beginning of the simulation due to the initial condition difference.
Figure 3-4 shows the convergence of the speed estimate
W3
to the actual
W3.
Again,
note that there are no direct speed measurements taken at bus 2, or in fact anywhere in
the network, so the dynamic model plays a key role in providing the speed estimate. All
these figures demonstrate that the observer converges asymptotically within 2 seconds to
the system variables.
Simulation of Event 2: Results for Event 2 are shown in Figures 3-5, 3-6 and 3-7. Figure 3-5
shows the relative angle 631 at the generator bus nearest to the affected line, and its estimate
631.
Notice that
631
reflects the correct general form of 631 during the event period, but a
constant angle difference is evident, mostly due to the observer's lack of the information
23
Event 1 Simulation: System's 8 - 8 and Observer's ^-5^
-0.073
-System
--
-0.074
Obs1
-0.075
-0.076
q
I
-
-0.077
-.
---
.
-
-
- --
-..
-0.078
-0.079
t
-0.08
-- -
-
~ ~-.-- -.
~ - -...
~ -........
- -.
I
-
-
-- - -
- -- -
- ..
---
-
- -.
...-.
..--- .....
---.....
-.
-.
-.
-0.081
- V
- ---
...
-- -
....
-.
....
-- - -- --.--.--
-0.082
-uuua 0
1
2
3
Figure 3-3: Time plots of 65-61 (system) and
ginning at t = 0.1s, duration 3s).
4
13-SI
6
5
(observer) when Event 1 occurred (be-
Event 1 Simulation: System'su 3 and Observer's
.1
0.1
Syste
-- Obs 1
0.05
0
IiIt
:
-0.05
i
-i -
-0.1
-l- --
- it-.-.
~
--
I~
It iIt
-0.15
-
I.
-
-
-
-- -
-
-
--
-
i
-.III.
t
-
ti-
-
- --
-.- -
-t
--
.- . --. -t
I
- -
-
.
. . . . . . . . ...
-0.2
-
.1.
.
. .
. . . . .. 1
.
. . . . . 1 .
-0.25
0
1
2
3
4
5
6
Figure 3-4: Time plots of w3 (system) and w3 (observer) when Event 1 occurred (beginning
at t = 0.1s, duration 3s).
24
. .
-
Event 2 Simulation: System's
3
-8 8 and Observers
8^_-A
0.105
System
-- Obs 1
0.1 -
0.095 -
0.09 -
- ..- .
.-.
0.085 -
0.08
-
-
-
-
0.075I-
-
-.
.
-t-
*-
-
--
......................-
-.
. .. .. .
-.... .. .-..-.-.-.-.-
t.
-
-
- -
...-.- -.
0.07
Figure 3-5: Time plots Of 63-JI (system) and 3-S1 (observer) when Event 2 occurred (beginning at t = 0.1s, duration 3s).
regarding the change in network topology. After the event the observer estimates
J31
well,
although some steady-state discrepancy persists due to model mismatch between the system
and the observer. Similar explanations are also applicable to the 64, and
3-6 and to the 6, and
661
Selecting the
plot in Figure
plot in Figure 3-7. In Figure 3-7, the algebraic nature Of
evident in the instantaneous angle
3.2
J41
J6 is
jumps
Q and R Matrices
The process noise and measurement noise intensity matrices,
Q and R respectively, influence
the observer gain matrix L through (2.18) and (2.19). If the process noise is significantly
larger than the measurement noise, the observer will preferentially weight the measurement
information relative to the dynamic model. On the other hand, if the measurement noise
is significantly larger than the process noise, the observer will not trust the measurements
and will essentially run in an open loop, i.e., as an uncorrected real-time simulator.
Even in the absence of process and measurement noise,
Q and
R may be used as design
jumps are only possible for the algebraic variables, not for the differential variables such as the
angles or the speeds of the generators.
isuch
25
Event 2 Simulation: System's
1 and Observer's 4 ^
4-
-0.038
-- System
Obs I
-0.039
- --
-.-.-.-
-0.041
-.
....
-. ..
-.-.-.
...............
-0.041
.-.
..-
.......
.............
............ .........
*
It
-0.042
-0.044
1
0
2
3
4
5
6
Figure 3-6: Time plots of 64-61 (system) and 34-61 (observer) when Event 2 occurred (beginning at t = 0.1s, duration 3s).
and Observers 6^-1
Event 2 Simulation: System's86 _.n
net.
-System
-
Obs1
-0.066
-0.068
-0.07
-t-- --
I
-
-0.072
... ... ...
-I - -
-0.074
-t
-0.076
-0.07f
0
-
- --
-
---
-
...... . . . .
1
.. .
-
.. . . . . . . . . . . . . . .
.. . . . .
2
3
Figure 3-7: Time plots of 66-61 (system) and
ginning at t = 0.1s, duration 3s).
16-11
26
4
5
. . .
6
(observer) when Event 2 occurred (be-
parameters for the observer. Adjusting their relative values allows one to weight the observer
towards heavy reliance on the measurements, or heavy reliance on the model, or a range of
intermediate compromises.
For our investigations we chose
Q
and R to both be diagonal matrices. The results 2
simulate as shown in this thesis are for the case in which each diagonal entry of
Q
is
25 x 10-8, while the diagonal entries of R are 64 x 10-10 in the positions corresponding to
direct angle measurements, and 64 x 10-8 in the positions corresponding to the remaining
three types of measurements. A more comprehensive examination of the effects of varying
Q
and R is needed, but left for future study.
3.3
Types of Measurement and Placement
The choice of sensor types and of their placement plays a key role in designing good observers. However, comparing and rank-ordering different choices of sensors and placements
is not straightforward, because several factors are involved.
If we commit to choosing the observer gains through the LQE methodology, and if
Q
and R indeed represent the intensities of white process and measurement noises, then P in
(2.18) represents the error covariance matrix of the state estimate. One could then use, for
example, a weighted sum of the state component error variances of P -
i.e., a weighted trace
as a measure 3 of observer quality for each choice of sensor types and placement. A
more thorough study of this possibility is left to future work, but we present an illustration
here.
The plots in Figure 3-8 show the results obtained under the same Event 2 scenario
considered in Figures 3-5-3-7. Observer a is the observer used in all the results shown so
far, while Observer b uses the same three types of sensors, but placed differently: a direct
angle sensor at bus 9; a power injection sensor at bus 5; and a power flow sensor on the
line directed from bus 4 to bus 5. The (unweighted) trace of P for Observer b (7 x 10-6)
is one order of magnitude larger than that for Observer a (7.7 x 10-7), indicating that
Observer b is poorer than Observer a. The waveforms compare the actual W, (full line) with
2
We achieved this results by adjusting the values of Q and R until we saw obvious but mild effects of the
noises on the states of the system.
3
One difficulty we are experiencing here is a physical interpretation of this measure because the state
vector has both angles and speeds as its components; therefore, the weighted sum actually combines both
units.
27
Event 2 Simulation: System's ,, Observer a's
o
and Observer b's W^
0.02
-
0.015 0
System
-- Obs I
Obs 2
-01
-
0.005 -
01CL-0.005
-
-
-0.01 -
-
-.
--
-0.015 -~I
-0 .02 -0.025
-0.03
. . .. ..
..... ... . ..
1
'
0
1
2
3
time (s)
4
5
6
Figure 3-8: Time plots of 64-61 (plant), so-61 (suboptimal) and 37 -61 (worst case companion) when "event 1" occurred (beginning at t = 0.1s duration 3s).
the estimates W
(dashed line) and W1 (dot-dash line) provided by the two observers. The
differences in performance are clearly visible.
3.4
Pole Placement VS LQE
This section presents results from a simulation comparing the performance of the network
observer with the LQE linear gain to the performance of the network observer with a gain
computed using pole placement. Pole placement is another filter design technique [9] used
in control. This technique provides an easy-to-understand but powerful way to control the
steady-state convergence rate of the observer although the choice of desirable pole locations
is not always clear.
Figure 3-9, Figure 3-10 and Figure 3-11 present the state evolutions of the classical 9-bus
system and two observers for Event 1. The first observer using the LQE technique is the
same one we used in previous sections, but the other one is based on the pole placement
technique and still uses the same three sensors that the first observer does. Note that the
gain of the second observer is designed so that all the eigenvalues of (A - LC) are equal to
28
System's 82-5 and Obs's 6^-S^
0.1
0
-System
- - LQE observer
Pole-Placement Obs
0.175
- -
0.17
-. -.
-.-.-.-.-.- -.
..
--. .
-
-a
0.165
V
0.16
0.155'
0
I
I
I
I
I
1
2
3
time (s)
4
5
Figure 3-9: Time plots of 62-61 (system) and 62-61 (for both LQE observer and poleplacement observer) when Event 1 occurred (beginning at t = 0.1s, duration 3s).
A's eigenvalues 4 (in (2.14)) shifted by -1
and multiplied by 10 to give an observer that is
around 10 times faster than the system.
Figure 3-9, Figure 3-10 and Figure 3-11 show that our LQE observer outperforms the
pole-placement observer in predicting the real state evolutions even during Event 1; however, Figure 3-12 shows that the pole-placement observer and the LQE observer can also
perform equally well. Although the results in this section suggest that our LQE observer
usually performs better than the pole placement observer, there should be more comparative
research on their performances. We leave this for future study.
4
A always has a zero eigevalue due to its structure
29
System's 87-81 and Obs's
^
076
System
observer
Pole-Placement Obs
-
- LQE
0. 0740. 072 --
0.07 -
-
0.068
C1.
---
---
0.066 0. 064
-
- -
-
-
-
--
-
0.062-
0.06 -
-
-
0.058-
0
1
2
3
time (s)
4
5
6
Figure 3-10: Time plots of 67-61 (system) and 67-61 (for both LQE observer and poleplacement observer) when Event 1 occurred (beginning at t = 0.1s, duration 3s).
System's w and Obs's ^
-System
- - LQE observer
Pole-Placement Obs
0.6
0.4
0.2
t--lt -- --. -
- -
---
--
-.
t-t
..
.- -..
.
0.2
.
.
-
.
-0.4
............
... . .. -
....
...
..
.........
..
..........
-0.6
-0.8
1
1
2
3
time (s)
4
5
6
Figure 3-11: Time plots of w6 (system) and C26 (for both LQE observer and pole-placement
observer) when Event 1 occurred (beginning at t = 0.1s, duration 3s).
30
System's
54-
and Obs's 8^-5^
-
S
-0.04
-
System
LQE obs
Pole-placement obs
-0.0405
-0.041
-5q-
-0.0415
-~~.
-.-.
-.
a
----- - -.- -.
..
. .. .
-0.042
-.
-.-.
-.
-.
- -.
.. -...--.... --.. - --- -
-0.0425
..--...-- -...--
-0.043
- .
. - -.
-.-.-.-.-.
.-.-.-.-.-.-.
.-.-.-.-.-
-..-.-.-..- ...-. -.
-
-0.0435
-0.044
-0
I44
0
2
3
time (s)
4
5
6
Figure 3-1 2: Time plots of 64-61 (system) and 64-61 (for both LQE observer and poleplacement observer) when Event I occurred (beginning at t = 0.1s, duration 3s).
31
Chapter 4
Fault Isolation Applications
The observer studies we did in the previous chapter showed that our nonlinear observer
can give us satisfactory numerical results. This chapter provides two simple case studies
showing that the nonlinear observer is also capable of achieving effective fault isolation. The
first study concerning the multiple observer scheme is a simple fault isolation method that
functions effectively with small networks. The other study instead uses a single nominal
observer to achieve the task; however, the diagnosis process is possible only through understanding the pattern of the steady-state output error (
y) vectors. We then proceed to
gain insights into the relation between the EY vector and line perturbations through Taylor
series expansion analyses based on (2.2) and (2.5).
4.1
Multiple Observers for Fault Isolation
Fault isolation can be achieved via many ways; however, here we implement a simple but
accurate and robust method of multiple observers
[3].
The main idea is to have several
observers work concurrently in real time. Since our focus in this study is only on a complete
single line-out fault, the model assignments to the observers are such that one observer has
the nominal model of the system, and each of the rest has the nominal model with one unique
line taken out. With such a set up, ideally one would expect that the observer whose model
matches the current model of the real system will outperform the other observers in terms
of predictions at the steady-state.
The scheme prompts us to further investigate via simulations its applicability in fault
isolation of power systems. Therefore, we put the method of multiple observers into tests
32
on the classical 9-bus system.
Multiple Observers in Simulation Tests: Here, we examine the performance of the multipleobserver scheme via simulations on the classical 9-bus system. In these simulations, the real
system with the nominal model has Line 9 for the line numbering of the classical nine-bus
network. taken out at the start1 , and there are ten different observers running independently
and concurrently. Note that although having different models, each observer consists of the
same three sensors that we used before in the previous chapter. Also, both the real system
and the observers start off with the same initial condition, which is the loadflow solution
of the nominal system. Therefore, we can see the transients of the real system and all the
observers right after the simulations start.
From the above setup, one would expect that the observer whose model is nominal but
without Line 9 will best predict the real states because its inherited model matches that
of the real system. Our simulation results substantiate the claim. Figure 4-1, Figure 4-2
and Figure 4-3 clearly show that the model-matching observer outperforms other observers
since its residual always converges to zero at the steady state, but those of the others do
not.
Despite the simplicity and the accuracy of the multiple observer scheme, the approach
is computationally intensive and unscalable for bigger and more complex systems.
For
example, in order to deploy the scheme to isolate a single-line fault of a power network, we
need to have n
+ 1 different observers for n lines. The linear growth of the computational
burden with the number of lines in the network poses a serious problem in real-time state
estimations.
Although constrained by its requirement for large computational power, the multiple
observer scheme does not only have a virtue of being a good method for fault isolation
of small networks, but also illuminates the significance of the model inherited in the fault
detection observer to achieve the optimal state predictions. In other words, failing to have
the right model, the predictions of an observer will never be able to converge to the correct
steady states and will either fall short by a constant offset at the steady-state because of
the model discrepancies between its model and that of the real sytem, or will diverge away
from the steady state.
Ideally, one would want to have an economical and efficient algorithm for the fault
'Refer to Figure 4-4
33
System without line 9 + different observers: (8^ - 8^) - (8 - 8)
0.15
0.1
--- ------
0.05
-
- ---- ---- --
-
-
--- - - --
-
---
-
-
-
0
A?
-0.05
-0.1
- - -- -
-
-- -- -- --
obs w/oline2
-
line6
lineB
line9
-obs w/o
--w/o
-obs w/o
scomplete
obs
-.
-0.15
-0.2'
0
1
2
3
4
-.-..-..-
5
time (s)
6
8
7
9
10
Figure 4-1: Time plots of the residuals (05-61) - (65-61) of different observers when the real
system has Line 9 taken out.
System without line 9 + different observers: (8^- 8) - (8 _ )
0.15 r
- -
0.1
obs complete
- --
-.- - -
obs w/o line6
obs w/o line8
obs w/o line9
-
0.05
0
.. . . . . . . . .
.. . . . ...
. ..
co
c
Cd
.
. . . .
-0.05
-
-
-
-
--
- ------------
----
-0.1
-0.151-.-
-0.
2(
-
--
1
2
3
4
5
6
7
8
9
10
time (s)
Figure 4-2: Time plots of the residuals (66-61) - (66-61) of different observers when the real
system has Line 9 taken out.
34
System without line 9 + different observers: (6^ - ^
0.5
(
-8
1
0.4
-
obs w/o line2
obs w/o line6
obs w/o line8
obs w/o line9
obs complete
-
- -- -
-
03..
0.2
.-
-- --
- - -
-
- . - --
0.1.
-
-
- - -
-
-
--
--
0-
-
aa
- 0.1
1
2
3
-.. .
4
5
time (s)
6
7
--- .
-.-.-.-- .-.-
-
~- ~
~~ ~ ~ ~
-
-0.2
-0.3,0
-.-..
-..
.- .-.
-..
.. . --..
-.
.
- ..
-- --.. ...
.- --.
-.
. .-.
8
9
1
10
Figure 4-3: Time plots of the residuals (68-61) - (68-61) of different observers when the real
system has Line 9 taken out.
isolation of power networks. However, to achieve such a scheme, we had better well comprehend the properties of the faults of interest. Having a solid understanding of the fault
characteristics, we hope to use those properties as our fault identification criteria.
4.2
Understanding Faults on the Network: Line Parameter
Disturbances
This section investigates the effects of line-parameter disturbances on the classical 9-bus
power network. The motivation is to shed light on interesting characteristics originating
from line parameter disturbances in the hope that these observations will suggest useful
insights for fault isolation application.
First we need to know what information we can use as inputs for our investigation. It
is worth emphasizing that in practice the output of the actual system, the output of the
observer (or the output prediction of the observer) and the predicted state evolutions are
all the information we know. Therefore it is reasonable and obligated for us to focus on
using this information to form a comprehensive framework for understanding faults in the
real system.
35
However, before getting deeper into details about faults, we need to know about the
normal operation of the system, so that we have a basis for comparisons in future studies
about faults.
4.2.1
Normal Operation
In order to understand faults well, we need to realize what happens at the normal operating
point. Such knowledge can be helpful when we want to have a base for comparison and
for suggestions about the behavior of the network after a fault occurs. Here we show the
steady-state angle values of all the nodes and the steady-state loadflow values for all the
lines in the classical 9-bus example in Figure 4-4, under the operating conditions labeled
in the figure. Note that the direction of the arrow on each transmission line indicates the
power flow direction.
Note that we are assuming that all transmission lines of the network are lossless. As a
result, the power injected into one end of a line equals the power coming out at the other end
of the line. Another interesting consequence is that the power injections of all generators
(e.g., node 1, node 2 and node 3) into the network equal the power extractions of load
buses (e.g., node 5, node 6 and node 8) from the network. Also, from our observation, the
absolute angles of all the nodes always keep decreasing right after the simulation starts 2 , but
the absolute speeds of the generators will first slightly fluctuate downwards and eventually
reach steady-state values.
4.2.2
Investigation of the Steady-State Output Error Vectors
The seminal thesis of Beard [10] ;suggests that each fault will affect the network behavior to
generate a certain geometric pattern of the corresponding steady-state output error 3 vector
(iy) used in the observer; therefore, we begin our study by doing simulations to scrutinize
the pattern.
2
This phenomenon will not be problematic to us because we are interested only the evolutions of relative
angles. The reason is that, by looking at (2.2), we can see only relative angles, not the absoluate angles,
play a role in determining to values of f(x, u, w).
3
Here error means the predicted values by the observer minus the real values of the system.
36
7 o,
Line 2
Sno"m =
= 0.0128pu
'7
Line 9
0.7627 p.u.
0.2430 p.u.
-0.0179pu
0.8505 p.u.
8
94
.1149pu
2
nom =
9 Line 3
Line 8
1.6307 p.u.
nom 2
-0.0427put
3
Line 6
Line 7
0.8680 p.u.
0.6075 p.u.
5
PO" = -0.1313pu
6
f
P"'
= -0.1255pu
Line 4
Line 5
0.4083 p.u.
0.309 p.u.
4P'
= -0.0962pu
0.7173 p.u.
4
Line 1
no " = -0.0548pu
Figure 4-4: Normal operation of the classical 9-bus sytem: angles and power flows.
37
63"' = 0.032pu
Steady-state e vector when different lines got cut off
x
*
line 4 off
line 5 off
line 6 off
* line 7 off
* line 8 off
o line 9 off
0.12
0
0.05
E
0.
(D
0
005,
ai -0..
0.2
0.1
-
-
0
0.1
-0.1
-0.2
0
-0.3
steady-state e
of the flow sensor
0.2
-0.4
-0.1
steady-state e of the angle sensor
Figure 4-5: Steady-state output error plot of all six simulations.
Setup
This study uses the information obtained from six simulations. Each simulation has both
4
the observer and the system run together. The observer's model is nominal , and it consists
5
of the same three sensors we used in the previous section . The system's model in the ith
6
simulation is the nominal model without line (i + 3). Both the system and the observer
7
have the loadflow solution of the nominal model as the initial conditions. We then use the
information from the simulations to plot Figure 4-5, which shows the steady-state output
error vectors of all the simulations.
we
This setup is reasonable because usually we do not know which fault is going to happen; therefore,
system.
the
in
behavior
hope that our nominal observer can detect any abnormal
it is a
5
Note that in the case that we are only interested in the steady-state values and we know that
values
steady-state
their
that
know
we
because
measurements
injection
bus
load
use
not
shall
we
fault,
line
always stay the same
the
6
We do not deal with simulations whose systems do not have lines connected to generators because
steady
us
give
not
will
generator
a
to
connecting
a
line
on
perturbation
a
system of study is so small that
4
state.
7
We want to see the transients of the real states and those of the predicted states after the simulation
begins.
38
Steady-state e vector when different
lines got cut off of the system with a 5% line perturbation
-
x
*
*
*
o
0.1
line 4 off
line 5 off
line 6 off
line 7 off
line 8 off
line 9 off
005
-0.05,
02
-01
0.10
01
--
0.2
0.1
-- --
0
-0.3
0.2
.-00.1
-0.2
.
-
0
-0.3
steady-state e of the flow sensor
-0-4
-0.1
steady-state e of the angle sensor
Figure 4-6: EY vector plot of all six simulations with all the line parameters perturbed within
five percent of the original values.
Line-Parameter Sensitivity Study
The pattern in Figure 4-5 suggests a promising application in fault isolation since we can
use the directions of the steady-state output error vectors, which are always available, as
a fault identification criterion.
However, we need to make sure that the exhibited pat-
tern in Figure 4-5 is not too sensitive to perturbations on all the line parameters of the
system because practically we do not usually have completely accurate information about
the line parameters of the network. Therefore, line-parameter sensitivity studies should be
performed to ensure us that the pattern of the Ey vectors is not too susceptible to small
line-parameter variations.
To ensure consistent results, we do additional simulations similar to those we did in
the previous section.
However, this time we have all the line parameters of the actual
system perturbed within five percent and ten percent of the nominal values. The results
are encouraging: Figure 4-6 and Figure 4-7 respectively show similar patterns of steadystate output error vectors to that in Figure 4-5 when all line parameters are perturbed
within five percent and ten percent.
39
Steady-state e vector when different lines got cut off of the system with a 10% line perturbation
line 4 off
line 5 off
line 6 off
- line 7 off
* line 8 off
0 line 9 off
0
x
*
0.1
-0.05,-02
00
-01
0.1
01
-0.02
-0.3
0
0.2
.--
-0.1
.--0.2
0
-0.3
steady-state
e of
the flow sensor
1
-.
-0.4
0.1
steady-state e of the angle sensor
Figure 4-7: eY vector plot of all six simulations with all the line parameters perturbed within
ten percent of the original values.
Effects of Power Variations
Other possible variations that can happen in power network include changes in power injections and power extractions. Thus we further investigate the sensitivity of the steady-state
error vector pattern to such changes.
The motivation, besides ensuring the consistency
of the error pattern, is to understand the effects of possible unknown variations such as
changes in power to Ey vectors.
Here we change the power injections of generators or the power extractions of loads by
plus or minus ten percent. Figure 4-8 shows that such variations have a bigger effect on the
directions of E. vectors than line-parameter variations of the same magnitudes do.
4.2.3
Gaining Insights into E, Vectors
This section attempts to show how to interpret a component of a steady-state output error
vector in Figure 4-5.
Understanding the reasoning behind the pattern of the plot will
provide us a more solid comprehension of the network behavior after a fault. Table 4.1
presents the values, from different scenarios, of all steady-state output error components:
angle measurement error; flow measurement error; and bus-injection error.
40
Steady-state e0 vector when different lines got cut off plus 10% change in Pbus injection
0 line 4 off
*
line 5 off
line 6 off
l ine 7 off
*
line 8 off
x
o line 9 off
005
0
0
-0.05
02
-
0.1
0.3
0
01
-0.1
02
-0.2
0
-0.3
steady-state e
of the flow sensor
-0.4
-0.1
steady-state e Yof the angle se nsor
Figure 4-8: eY vector plot of all six simulations with all power injections and extractions
perturbed roughly by ten percent.
Table 4.1: The measurement errors for all three sensors of the observers when there is no
line or power variation.
Line-out
angle measurement error
4
5
6
7
8
9
0.2599
0.0284
0.0763
-0.0536
-0.0667
0.0212
flow measurement error [bus injection error
0.1513
-0.1147
-0.3086
0.2160
0.2690
-0.0857
41
-0.0464
0.0335
0.0870
-0.0671
-0.0843
0.0252
To illustrate the way we understand the values in Table 4.1, we consider the example
of the flow measurement error value when Line 4 is taken out of the system. In this case,
it is worth noting that the flow measurement is on Line 8, which is the line connecting
node 7 and node 8 of the classical 9-bus network. We notice that node 5, which is originally
connected to both Line 4 and Line 6, requires a constant power extraction from the network
at all times. Therefore, if Line 4 is cut off from the real system, there must be more power
flow from line 6 to node 5. However, knowing that the power supplied to both Line 6 and
Line 8 is from the generator at node 2, we expect that the increase in power on Line 6
will consequently decrease the power flow on Line 8 since the power injection at node 2
has to be kept constant. Thus, the real system should have a lower flow on Line 8 than
the corresponding prediction of our observer, since our nominal observer model did not
have any information about the fault on Line 4. Therefore, our reasoning concludes that
the flow measurement error value for this case must be positive. Such a reasoning in the
previous paragraph is especially helpful in understanding the system behavior after a fault.
However, we usually find it difficult to reason the signs of the other components such as bus
injections and angle evolutions without refering to the DAE swing model. For example, it is
difficult to know whether the angle of node 5 decreases after the fault on Line 4. Figure 4-9
shows the steady-state values of the angles of all the nodes and the flows of all the lines
after Line 4 gets cut in from the classical 9-bus system. We can see that all the angles
at the steady-state decrease from their nominal values. This is because the variables are
affected by the second-order factors, which make it harder for one to mentally picture the
mechanism affecting the values of the components.
4.2.4
Achieving Fault Isolation via a Nominal Network Observer
Here we propose a way to do fault isolation for complete single line-out faults on the
classical 9-bus system. Using the information that we have about the steady-state output
error vectors in terms of directions and magnitudes, we hope to identify the line fault that
makes the system provide the steady-state output error vector we have.
First it is useful to process the information about the directions of the steady-state
output error vectors into a numerical format. To quantify the directions of steady-state
output error vectors, we need to have reference vectors. In this case, we decided to have
vectors
[1 0 0]', [0 1 0]' and [0 0 1]' as our reference vectors. Note that the values in
42
4
Line 2
= -1.0673pu
'1
8
1.0915pu
-
.96
Line 9
0.3321 p.u.
0.6743 p.u.
0.8552 p.u.
8
8pu
3
Line 6
0.1809 p.u.
5
= -1.2835pu
S
Line 7
1.3053 p.u.
4
= -1.0228pu
9 Line 3
Line 8
1.6374 p.u.
S4= -
S
6
S4 = -1.0551pu
Line 4
Line 5
0 p.u.
0.7296 p.u.
=
-0.9861pu
0.7296 p.u.
4
Line 1
4= -0,
9440pu
Figure 4-9: Steady-state after Line 4 getting cut off from classical 9-bus sytem: angles and
power flows.
43
= -0.9727pu
Table 4.2: The direction of
2,
vectors of the previous four studies.
Line-out
no perturbation
5% line perturb.
4
5
6
7
8
9
(0.547,1.051,1.724)
(1.338,2.776,1.294)
(1.337,2.783,1.304)
(1.803,0.378,1.864)
(1.803,0.380,1.866)
(1.338,2.775,1.293)
(0.532,1.065,1.721)
(1.306,2.756,1.297)
(1.334,2.782,1.305)
(1.805,0.38,1.864)
(1.799,0.379,1.868)
(1.343,2.778,1.293)
10% line perturb. [10% power variation
(0.55,1.047,1.723)
(1.37,2.796,1.293)
(1.324,2.772,1.302)
(1.826,0.39,1.858)
(1.81,0.384,1.865)
(1.308,2.76,1.300)
(0.498,1.119,1.767)
(1.343,2.869,1.424)
(1.338,2.826,1.36)
(1.807,0.474,1.973)
(1.806,0.438,1.932)
(1.345,2.886,1.453)
Table 4.2 are in the (a, b, c) format, which denotes that the corresponding ey vector is
a radians referenced to [1 0 0]' (representing the x-axis), b radians referenced to [0 1 0]'
(representing the y-axis) and c radians referenced to [0 0 1]' (representing the z-axis).
By looking at Table 4.2, we can make a major observation about the directions, namely
that there are only three main directions that any ey vector can fall into for our example.
Here, we use
eg
to denote the steady-state output error vector resulting from line a getting
cut off. From the result we have, one direction has just 1'; another has both E and E8, the
other direction has
5
,7
and
E9. Therefore, using just the direction information we have,
we can feel comfortable distinguishing between the
Ey's from different directions. However,
in order to detect the faults that have their steady-state output error vectors aligned well
together, we still need more information. In this case, the information about the magnitudes
of the vectors can significantly improve our situation.
Using the information about the magnitudes of the steady-state output error vectors
in Table 4.3, we can better distinguish faults whose 2, vectors align in the same direction.
For example, to distinguish between Line 7 getting cut and Line 6 getting cut, we compare
the magnitude of the
e from our sensors to the magnitude of E and that of e8. If the
magnitude of the vector that we get from our sensors is closer to that of E7, then the fault
is originating from Line 7 getting cu. The same idea can also apply to the identification of
the faults orginiating from Line 5 out, Line 6 out and Line 9 out.
44
Table 4.3: The magnitude changes of Cy vectors of all the studies.
Line-out
no perturbation
4
5
6
7
8
9
0.3043
0.1228
0.3296
0.2325
0.2897
0.0918
4.3
5% line perturb. [10% line perturb.
0.3130
0.1237
0.3302
0.2331
0.2897
0.0918
10% power variation
0.2975
0.1230
0.3298
0.2325
0.2900
0.0923
0.3394
0.1464
0.3219
0.1743
0.3034
0.1196
Analysis of Steady-State Output Error Vectors
Here we present an analysis of the steady-state output error vector pattern in Figure 4-5.
The objective of this analyis is to gain some insights into the relations between the steadystate output error Ey and the perturbation on Line 9 by applying the Taylor series expansion
to the nonlinear measurement model in (2.5).
In this section, we assume that the value of the admittance of Line 9 is zero in order to
simplify our analysis; therefore, we can focus on using only the change of the susceptance of
m
Line 9 as a representation of the magnitude of a fault on Line 9 (B9 = B act - Bno~
). This
assumption is reasonable since it is usually the case that a line has its susceptance much
greater than its admittance.
Besides the above assumption, it is worth noting that we use nom and act to denote the
nominal observer model and the actual system model respectively throughout.
We first begin by defining the steady-state output of the actual system
steady-state predicted output of the observer model
g (x, u, w, B)
=
(x
g(
,w
=
g)
_=go(,U*
Bq) L2z,B9=Bnom
(g)
and the
(g) in (4.1) and (4.2):
at
g(:,
no
u *, 0, B ct)
X
0, Be ""m)
(4.1)
(4.2)
Using Taylor series expansion up to the second order, we can reveal another relation
between the steady-state output error Cy and the change in the susceptance of Line 9 as
45
follows. Note that we define 5 =
Y
=X
_ gnom(
-
x.
*,* 0, Bn
m
)
4.3.1
BB9
S±a 2 gnom(,
U*, 0, Bnom) -2
2
B
B9
a 2 gnom(iU*, 0, Bnom
u*, 0, Bnom
agnom(5,
X +
1
2
gnom (, u*, 0 , Bg O )OX2
2
2
+B9
(4.3)
Discussion about the Analysis
Equation (4.3) illuminates useful insights into the relation 8 between the steady-state output
error and the perturbation of the susceptance of Line 9.
However, one may notice a possible problem hidden in (4.3). The problem is that t used
in the calculatation for x is usually not available in practice. In this thesis, we assume that
all the possible
's can be substituted by using the loadflow solution of the corresponding
actual system. Those values are prestored in a table and ready to use. This assumption is
reasonable for the small example system of this thesis.
By further investigating the structure of (4.2) in the case of our observer, the vector
gnom(x7,
u*, 0, Bnom) has three rows; each of which contains the output from its correspond-
ing sensor:
gnom(X, u*, 0, B90 m )
G1 - B 1 sin(61
=
- 64)
- G 1 cos(61
G8 + B 8 sin(68 - 67) - G 8 cos(6
8
- 64)
(4.4)
- 67)
With this measurement vector, the second, third and fifth terms in (4.3) will be zero. We
can then automatically simplify the relation from (4.3) to that in (4.5) below. Although
the ensuing equation is simple and obviously useful in fault detection, a crucial assumption
is that each possible ; and its corresponding ey have a one-to-one correspondence:
9 _agnom(5 ,u*,
0, Bnom
aOX
+
(,1
x
(4.5)
Furthermore, the price we pay here is the irrecoverably vanished direct connection between
8
The fourth and fifth terms are written in a suggestive notation, but because x is a vector, the detailed
expressions have to be written more carefully. We omit the details here, in order to keep the notation simple.
46
the e vector and B9 . However, the indirect connection between the 0 and F (when Line 9
is out) still exists. Therefore, fault isolation is still possible, but to solely depend on (4.5) for
fault isolations, we take a possible risk of relying too much on the one-to-one correspondence
assumption that cannot be confirmed, especially if the system itself is highly evolving or
volatile to small perturbations.
4.3.2
Numerical Results of the Analysis
We now want to demonstrate that the preceding analysis can help us achieve good numerical
estimation. First, we use (4.5) to generate the first order approximation.
Here we have
09nom(7-, u*, 0, B
ax
0m
1y = [-0.0212
- 0.0252 0.0857]'
(4.6)
) 7 = [-0.0212
- 0.0252 0.0857]'
(4.7)
,so the first-order analysis is already satisfactory. We also do a similar approximation using
the same approach to get the relation between
E8
y
and its corresponding B 8 , to rule out any
doubt that the good result we got from the approximation for the perturbation on Line 9
was because the last two terms of (4.3) are gone. In the case of Line 8 getting cut, all three
terms in (4.3) stay nonzero. The results we get are equally good for this case. The reason
behind such accurate estimates is that the steady-state errors of the state predictions are
small, and the measurement vector is either a zero-order or a first-order function of the line
parameters of interest. Therefore, generally the first-order approximations from (4.3) have
given us very good results.
Although giving us impressive results, the possible risk of the assumption about the
one-to-one correspondence still persists. In order to ensure reliable fault isolation, we have
to develop a complementary tool to (4.3).
4.3.3
Exploring on the Nonlinear DAE Swing Model:
Complement Tool For Fault Isolation
The nonlinear observer model in (2.2) is a possible alternative in helping us get around the
risk we discussed in the previous analysis.
First we show the nonlinear observer model in (4.8) and the nonlinear DAE swing system
47
model in (4.9):
u, w, B 9 )
fom(,
+ L(--
f (x, u, w, B 9 )
x,B
9
U*, 0, B"") + Le-y
=BC
= fact (,u *, 0,
=
0
(4.8)
9
B9ct
Using the Taylor series expansion we can generate a second-order approximation of the
actual system model in terms of the observer model and its first and second derivatives in
(4.10):
fact (,U*,
~ofnom(5,U*0Bg"m) + Ofnom(X, u*, 0, Bnom)_
0, Bct)
Ox
Ofnom(X, u*, 0, B
9
"om)
B+±
1-
2
fnom(-,
+B 9
2
102 fnom( f*, 0, B " m )~2
B9
2 f
B
n
+ 2 fnom(5, u*, 0, B90 m
OB 9 0x
+
, 0, B non) g2
;x
, 9
**'
2
1x
)B xi
9
(4.10)
After that, we use (4.8), (4.9) and (4.10) to obtain (4.11). This equation shows the relation
between the steady state output error (Ey) and the change in the susceptance of Line 9.
= fnom(i, u*, 0, B 0"")
LE
Dx
2
+ 1 fnom (7,
+
+
O&fnom(', u*, 0, Bnom)B
xBF
0
U*, 0, B1 ")
Ox
2
2
+ 2 fnom(5, U*, 0, B
OB 9 a_
+ 2
)
2
norn(5, u*, 0, Bom)
OB2
B9
(4.11)
iFB
An important consideration is the distinction between the relations shown in (4.11) and
(4.3). We expect the relation in (4.11) to give richer information, but to be less sensitive to
a choice of sensors used by the observer. The claim can be substantiated by investigating
the structures of the derivatives in (4.11) and (4.3). Although the relation in (4.11) has not
been proven to guarantee the one-to-one correspondence between eY and
5
or B 9 , it can
help double check the conclusion we get from (4.3).
From the analytical point of view, the constraint in (4.11) is promising in helping us
achieve better fault isolation.
However, (4.11) in estimation tests like those we did for
(4.3) has not given us satisfactory results.
We leave the resolution of this for a future
48
investigation.
4.4
Summary
We have finished our preliminary study on the use of our network observer in fault isolation.
First we presented a simple method of multiple observers for fault isolation in the power
network.
Besides presenting the applicability of the scheme in doing fault isolation, the
multiple observer scheme helped us show that the inherited model of an observer is a
crucial factor for an observer to achieve optimal state predictions.
After that, we studied the geometric pattern of the steady-state output error vectors.
The simulations we did have confirmed for us that the exhibited pattern was not too sensitive
to small line-parameter disturbances and small power variations. The pattern also suggested
a useful way to do fault detection and diagnosis using the directions and the magnitudes of
the steady-state output error vectors resulting from different faults.
Next we tried to gain insight into the network behavior after a complete single line-out
fault occurs. We also provided examples of using Taylor series expansion in the analyses to
understand the relation between a steady-state output error vector and its corresponding
disturbance on line-parameters.
The relation we derived from the measurement model
provides us satisfactory estimates, but that from the DAE swing model gives us a poor
results. Further investigation of the underachieving second order approximation from the
relation we get from the DAE swing model is still needed.
49
Chapter 5
Summary
5.1
Brief Content Overview
Here, we provide a brief summary of the work in this thesis.
In Chapter 1, we presented our motivation for creating a power network observer and
some basic terminology of monitoring processes.
Then, in Chapter 2, we first illustrated the fundamental nonlinear models including
the DAE swing model and the measurement model; both are crucial in the working of our
network observer. Next, we went through the processes of linearizing and collapsing the
nonlinear models to achieve a state-space swing model. The ensuing state-space model of
the system enables us to follow the classical method of obtaining an observer model. Using
the LQE method, we can then compute a linear gain for the linear observer. The integration
of the gain and the nonlinear models forms our power network observer model.
Chapter 3 showed numerial studies of our network observer. Having the classical ninebus example as a test system, the investigation uses two representative events: a line drop
and a change in the power injection at a load node, to examine the performance of our
observer. The results of the observer predictions were good in terms of both fast convergence
rates and low offsets between the real state values and predicted ones. Also the result of
the simulation comparing the performance of the observer with the LQE gain to that of
the observer with the pole-placement gain suggests that the observer with the LQE gain
usually outperforms the other. This evidence helps support our decision in using the LQE
method to compute our network observer gain.
Chapter 4 suggested two fault isolation methods using our network observer.
50
Both
studies focused on fault isolation of complete line-out faults in the classical nine-bus network.
Despite being computationally expensive and unscalable, the multiple observer scheme is
an accurate and simple fault isolation method, especially for small networks.
Another
important lesson we learned from this scheme is that the similarity between inherited swingstate model of the observer and that of the real system is crucial for our observer to achieve
good state predictions. The other method we focused in Chapter 4 was the nominal observer
scheme, which uses the observer whose model is the same as the nominal model of the real
system to do fault isolation. This method uses the direction and the magnitude information
of stable e, vector patterns exhibited after failure as a criterion for fault isolation. In the
end, we provided an analysis based on the nonlinear measurement model to gain insights
into the relation between a steady-state output error vector and its corresponding fault.
The numerical results of the analysis based on the nonlinear measurement model were very
satisfactory.
We then explored on the analysis using the nonlinear DAE swing model.
Although the analysis was thought to be promising, the result of numerical studies were
doubtfully unsatisfactory.
5.2
Suggestions for Future Studies
Here we provide suggestions for future extensions of the work in this thesis.
To deploy an observer-based method, we have assumed throughout that we know the system parameters values. However, in practice, some system parameters cannot be measured;
as a result, the fault isolation using only an observer-based method may not be sufficient.
Thus, we suggest that a combination between parameter estimation and observer-based
method will be of considerable interest to make the fault isolation scheme less sensitive to
information insufficiency and (expectedly) more adaptive.
The decision criteria for determining a suitable choice of sensors and placements for a
power network observer will definitely need to be sorted out. Besides knowing the trade
off between different types of sensors, the decision rule should also take the topological and
electrical characteristic of the network into account. Having such rules, we hope to put more
trust on our observer to be less volatile to disturbances and be more accurate in giving out
estimates.
A more systematic way of determining proper values of
51
Q
and R of the real systems
is essential to make our numerical study realistic.
Although both values are considered
as design parameters in our thesis, we do not have high degrees of freedom in choosing
their values because they actually represent real "noises." Therefore, to ensure that we will
achieving similar performances that we got in simulations in the real world system, we must
develop a pragmatic way to determine the value ranges of
Q
and R in the real systems and
use those values as our design parameters.
More research on comparison between the performance of the network observer with
the LQE gain and that of the observer with the pole-placement gain should be conducted.
In this case, interesting issues are determining the scenario in which one outperforms the
other, finding by how much better one's performance is compared to the other's and checking
whether the topology of the power network does affect the performances of both observers.
In Chapter 4, we investigated the patterns of the steady-state output error vectors
resulting from complete line-out faults. It is worth further investigating the scalability of
the nominal observer scheme. Additionally, it would be interesting to expand the scope of
our study by exploring on the characteristics of single faults of other types. For examples,
we would like to know the exhibited patterns of EY vectors resulting from faults of other
types. Ultimately, we may use pattern recognitions of the
EY vectors to help us determine
the types of faults happening to the power network.
Finally, other extensions should be pursued such as fault identifications of multiple faults
of either the same type or different types. However, it is crucial that we continue developing
solid understandings of network's behaviors both before and after disturbances. Although
we realize that there is much improvement needed for the work in this thesis, we hope
that the power network observer we developed will be a valuable building block that will
help researchers better understand complex interactions of the power network and more
effectively achieve efficient fault monitoring processes of the power networks.
52
Bibliography
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