A “ATE mean ANN-Y‘Sfi OF ELECTRIE
FOWER WW5

TM: {6" the that» a“? @h. 9.
MICHSGAN STATE UNNERSWY
Mber‘: L. was

1963-

THEsxs

LIBRARY

 

This is to certify that the

thesis entitled

A STATE MODEL ANALYSIS OF
ELECTRIC POWER SYSTEMS

presented by

Albe rt L . Duke

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Electrical Engineering

 

Date August 8, 1963

 

 

ABSTRACT

A STATE MODEL ANALYSIS OF
ELECTRIC POWER SYSTEMS

by Albert L. Duke

Many of the electro-dynamic problems of electric
power systems have been extensively investigated in the past
primarily through the extensions of steady—state concepts, con-
cepts which are applicable for linear conditions but not for the
nonlinear conditions that often exist. No formal attempts have
previously been made to analyze electric power systems in terms
of the state models used effectively in nonlinear control and other
system studies, nor to utilize the large body of theory developed

for the effective study of such system models.

To generate the system state models used in this
thesis it is convenient to express the component models in the
state form. Such models are developed in terms of two sets of
variables, consisting of l) line-to-line voltages and two line cur-
rents and 2) voltage from one line to neutral and the neutral cur-
rent. These variables are used to establish identical topological
representations for both three and four wire components. The
form of the model is then simplified by using a set of linear trans-

formations of variables to define two sets of variables which are

Abstract -2- Albert L. Duke

independent for balanced operating conditions and which can be as-
sociated with elements of a linear graph. The linear graph serves
as a basis for selecting the variables to be used in the state vector
and as a basis for formulating the state model of the system. In this
application, the graph serves the same purpose in dynamic studies that
symmetrical component sequence graphs serve in steady-state studies.
A systematic procedure is developed for modeling large
electric power systems in terms of state models of subassemblies.
Such a technique is essential to the study of large-scale systems.
A typical system is utilized to exemplify the techniques involved and
to illustrate the necessary detail.
The system solution is discussed and the stability
concepts used in the power system industry are related to the mathe-

matical concepts of stability and existence of solutions.

A STATE MODEL ANALYSIS
OF ELECTRIC POWER SYSTEMS

by

Albe rt L . Duke

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Department of Electrical Engineering

1963

ACKNOWLEDGEMENTS

The author wishes to express his appreciation to
the many pe0ple who have assisted in making this thesis
possible. It is impossible to name all who have helped but,
the thesis could not have been completed without the assis-
tance and encouragement of the author's major professor

and thesis advisor, Dr. H. E. Koenig, and the understanding

and patience of the author's family.

ii

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF FIGURES
CHAPTERS

II

III

IV

VI

VII

VIII

INTRODUCTION

Historical Review
Recent Developments

COMPONENT REPRESENTATION ............

SYNCHRONOUS MACHINE MODE L ...........

MODELS OF OTHER COMPONENTS ..........
Three-phase Transformer Bank ..........

T rans mis sion Line 5

Induction Motor

Fault C onditions

SYSTEM MODE LS

System Topology
Formulation of System Models ..............

OOOOOOOOOOOOOOOOOOOO

EXAMPLE OF A SYSTEM MODEL .............

System Clas sification

OOOOOOOOOOOOOOOOOOOO

Subgroup A - Transmission Lines ..........
Subgroup B ~ Transformer Banks ..........
Subgroup C - Synchronous Machines .........
Model of Entire System ....................

EXISTENCE AND STABILITY OF SOLUTIONS . .

SUMMARY

OOOOOOOOOOOOOOOOOOOO

ii

iv

I—‘

17

3O
31'
37
40
44

47
47
51

56
57
57
62
66
67

72

79

Figure

533
5-4
6-1
6-2

6-3
6-4

LIST OF FIGURES

Representation of Three - and Four -
Terminal Ports of Three-phase Components

A Device for Measuring Transformed
Voltage

Representation of a Synchronous
Machine as a Five -Port Component ,

Representation of a Synchronous
Machine as a Three-port Component

Application of a Tree Transformation .

Single Line Diagram of a Typical
Power System .

System Graph of the System Represented by
the Single Line Diagram of Figure 5-1 .

System Graph for Two L-Section
Transmission Lines . . .

Vector Graph of the System of Figure
5-2(a) Showing One Possible Tree

System Subgraph of Subassembly A .

Subgraph Al

Subgraph A2

Subgraph C 1

iv

Page

.15

.17

.20

.33

.47

.48

.49

.54
.58
.62
.64
.66

I INTRODUC TION

Historical Review

The development of the electric power system complex
of today has taken. place in large measure through the application
of several distinctive but interrelated disciplines to solve the
myriad problems of the industry. One of the first disciplines, the
use of phasors to represent sinusoidally varying time functions, is
perhaps the most widely known and commonly used type of analysis
inpower system engineering today. The use of phasors has served
to greatly reduce the labor involved in manipulating the transcen-
dental functions arising from the introduction of alternating cur-
rent components. This type of analysis is a very direct and ef-
fective approach to the problems involving single -phase systems
operating at steady-state conditions. The same approach is applied
to the steady-state analysis of three-phase balanced systems when
reduced to three equivalent single -phase systems and is effective
in providing answers to a large class of problems.

The steady-state analysis of the unbalanced power system
as presented by Fortescue [l] and further developed by Wagner and
Evans [2] , Clarke [3] and others [4], extends the use of phasors to
unbalanced systems. The application of this type of analysis gives a par-
ticularly effective method, which is in widespread use, for designing
protection against sustained system faults.

The more general dynamic or transient analysis problems
can be classified into two general subclasses, the short-time or

1

-2-

so-called electrical transient problem, and the longer-term
dynamic problem, sometimes called the stability problem. These
problems have until recent years been considered only in a piece-

meal fashion .

The electric transient problem is almost always approxi—
mated by linear models, the rotating machinery in the system
being ccnsidered as having constant velocities for the duration of
the problem. Such a linear analysis also neglects the phenomena
of hysteresis, saturation, and variation of resistance with frequency.
In the past, attempts have been made to include such effects by the
use of techniques similar to the "describing functions” discussed by
Truxal [5 Chapter 10] and others, e. g. , transient reactances and

subtransient reactances.

The problem of "stability of power systems" is in reality
more than one problem. The "transient stability'problem normally
considered,is a true stability problem in the Liapunov [6] sense
while the common "steady-state. stability” problem defined by Crary
[7 Section 2.5] and Kimbark [8 Chapter 1] is in reality not a stability
problem but an existence problem. Until recently, analytical or
graphical solutions to these two types of problems could only be ob-
tained by making several rather gross approximations , some of which
are: neglecting of damping factors in the dynamic equations, assuming
constant rms voltages during mechanical oscillations, and isolation
of machines by pairs. Examples of the graphical methods used

include the ”power circle diagrams" for studies of the existence

-3-

problem and the "equal area method" for the stability problem [7], [8}.
Prior to the advent of the large digital computer the analytical solu-
tions were limited almost completely to the ”swing equations" with
either point-by-point or analog solutions. Systems considered were
generally limited to one machine operating on an "infinite bus" or

to two similar machines operating together. Extensions which were
made to large systems were apparently based upon something similar to

Bellman's "principal of wishful thinking" [9 page 7].
Recent Developments

Prior to the period of the early 1950's most studies in-
volving larger systems composed of nonlinear elements were limited
of necessity to the approximations discussed previously or to analog
simulation devices such as the a. c. network analyzer. Subsequently,
with the widespread use of large digital computers and the develop-
ment of the electronic analog computer, investigations began to be
undertaken on a theoretical basis. The first attempts to use numeri-
cal methods were extensions of the existing techniques to obtain
machine solutions. Applications of electronic analog computers we re
on a somewhat more fundamental basis in that the differential
eqiati'cns of the components were considered to be solved rather than
the system performance studied by comparison to analogous com-
ponents.

The theoretical work in power systems that is necessary to
take full advantage of machine techniques has been only recently begun.

What might be called the first of this, by Lyon [10], was not actually

-4-

directed at computer solutions but at transient studies of electrical
machines. However, for the first time in a major study: the sym-
metrical component transformations were applied to instantaneous
variables rather than to phasor variables. White and Woodson [11]
along with Koenig and Blackwell [12 Chapters 11-14] extended these
ideas using additional transformations and a more detailed analysis
of several components, with a generalized modeling of the rotating
rmchinery components. Koenig and B1ackwell,along,with Gilchrist
[l3],using transformed instantaneous variables, introduced the multi-
terminal component representation and began the theoretical develop-
ment necessary to build a generalized system discipline highly suit-
able for computing-machine solutions. In introducing these concepts
to the power system field, Koenig and Blackwell considered several
types of polyphase systems such as single machines with known
terminal conditions. Both linear and nonlinear representations of
systems of two and three synchros and systems of two synchronous
machines and other similar systems were considered by them with
analytical solutions given in closed form for special conditions of
operation. Gilchrist obtained numerical solutions to the more gener-
al nonlinear mathematical models describing the dynamics of two
interconnected synchronous machines and one machine operating on

an "infinite bus ".

The system discipline, in which the component characteristics
and the system topology are considered explicitly seems to offer the
most logical means to build a discipline capable of being extended in-

definitely both in system magnitude and in SOphistication. Such a

-5-

discipline, of course, must satisfy the correspondence principle,
i. e. each of the previously mentioned disciplines must appear as
special cases of the general discipline. It should be recognized at
the outset, however, that, due to the nonlinearities of several of
the components and the large numbers of components involved,
there is no simple panacea which will supply the desired answers
when the proper buttons are pushed and the crank turned. The
large number of different disciplines, techniques and methods of

solution can and should be brought under one central discipline.
General System Analysis

This thesis is considered to be an extension of the systems
discipline as it applies to three-phase power systems. By means
of the techniques developed here a capability is provided for mathe-
matically modeling systems composed of large numbers of the type
of nonlinear components normally used in three-phase systems.
System models are established in such a form that disciplines de -
veloped in other areas can be brought to bear on the subject either
directly or indirectly. This study is not to be thought of as a means
to obtain a more accurate representation of certain components. No
such attempt has been made. Rather, the purpose has been to apply
the concepts and the advances made in the studies of systems during

the past few years to the specific problems of electric power systems.

The critical factor in the use of the systems concept is the
development of the component model. It would seem, with the many

hundreds of studies made throughout this century on the subject of

-6-

polyphase equipment, that a highly suitable model would have
been developed. This has not been the case although some of the
models developed were similar to the model developed here. The
component models developed here and to an even greater extent the
concepts involved in the development have appreciable significance.
An important property of balanced three-phase or, for that
matter, balanced n-phase components is that ‘the coefficient
matrices of the component equations, whether algebraic or time
varying, are cyclic symmetric [14]. Use of the symmetrical com-
ponent transformation applied to the instantaneous phase variables
takes advantage of the cyclic symmetric properties to diagonalize
the coefficient matrices in the component models. If the phase
connections were identical for all the components in a given elec-
trical power system then without question this transformation
would be effective for general system studies. The use of both
delta and wye connections of the phase windings in the same system
in large measure negates the advantages gained through the use of
the symmetrical components variables. The two-phase or afl com-
ponents of Clarke [3] obtained either from the measured phase vari-
ables or from the symmetrical components variables have the ad-
vantage of providing a model which is more suitable for computer
solutions in that the entries of the coefficient matrices are real
rather than complex numbers. The basic problem of the delta -wye
interconnection nevertheless remains the same as for the symmet-

rical component model.

-7-

Other variables frequently used are the so-called rotating field
or fb components of Y.H. Ku[15] which, like symmetrical components,
lead to complex coefficient differential equations. The real coefficient
counterpart of the fb components are the so-called cross-field or dq
components extensively developed by R. H. Park [16]. The use of
these components results in the removal of the effects of the machine
rotation from the equations of themodel. From an alternative point
of view, removing the rotational effects from the equations is the same
as changing the coordinate system to a rotating reference frame, there-
by simplifying the form of the model for the study of one machine and
to a lesser degree for two machines. Considerable difficulty is en-
countered, however, in the analysis of systems involving several
machines, each with a different frame of reference. The component
model developed in this thesis retains the desirable properties of these
transformations and also provides some additional properties which

are necessary for general system use.

The variables used in modeling the terminal characteristics
represent a set of measureable variables referred to as the x variables,
consisting of one line -to-neutral and two line -to-1ine voltages along with
two line currents and the neutral current. Application of a simplifying
transformation of the measured variables provides two sets of variables,
one called the scalar variables, very similar to the well known zero
sequence variables and the other, called the vector variables, some-
what similar to the dq components. One difference between the dq com-
ponents and the vector variables is that in this thesis the ”nomial

system frequency, a), rather than the individual machine rotation is

-8-

used as a common frame of reference for all machines of the
system. The component models based on the x variables are par-
ticularly useful in steady-state studies. In dynamic studies where
the only feasible means of solution is by computing machines a
more suitable form of model is required. To effectively utilize

the modern developments in system theory the component models
are placed in a derivative explicit or state model form. Theorems
developed by Wirth [17] for formulating state models are used
here. The state models of power systems as established in this
thesis also make it possible to apply the existence theorems of
Ince [18] , Murray and Miller [19] and Wirth, and bring the electric
power system within the framework of the stability and optimiza-
tion theorems presented by Liapunov, Bellman, Pontryagin and
many others. In this thesis a preliminary investigation of the ap-
plication of some of the existence and stability theorems is carried

out, using a typical electric power system as an example.

II COMPONENT REPRESENTATION

The components to be discussed here are all in common
use in three-phase systems. These components are usually distin-
guished by the presence of one or more three -phase ports having
either three or four terminals each.as indicated in Figure 2 - 1(a).
As a conventiOn, the terminals of each three—terminal port are let-
tered a, b and c and of each four -termina1 port a, b, c and n, as in-
dicated. Three-phase systems are formed by interconnecting the
corresponding terminals of two or more ports; i. e. , connecting a to

a, b to b, c to c and, for four-terminal ports, connecting n to n.

 

 

 

 

b c b

ao—— ‘F——* a

Three Four 3 n
b Terminal Terminal b x2 x x x

Port Port

xl
Three-Terminal Four-Terminal
~———-o n Port Port

 

(a) (b)

Figure 2-1. Representation of Three and Four Terminal Ports
of Three-phase Components.

The component topological representation used in this thesis con-
sistscfafliree-element terminal graph as shown in Figure 2 - 1(b)

for both the three-terminal and the four-terminal ports.

-10-

The representation for each four-terminal port is established
directly while the representation of each three-terminal port
is established by the addition of the reference terminal n and
the trivial element x1 which has the terminal equation ixl = O.
This scheme presents each port as having an identical topo-

lOgical representation. The significance of this scheme be-

comes evident in the later development.

The characteristics of the components are modeled

by associating with each port the vector 3* of variables:

 

 

 

vxl 1x1
v” d I ' 2 1
‘0': - sz an —x - 1X2 ( - )
vx3 1x3
I

 

In general, if the vectors Ex andlx are interrelated by a

second or third order coefficient matrix of the form:

 

* The two vertical lines are used throughout the thesis to
represent a vector or a matrix.

** The underscore is used to represent a vector variable or
a matrix.

-11-

 

 

a -a -a
2aZ a2 1 2 2
92- or _3 = -a2 2&2 a2 (2-2)
a2 2az
-a2 a2 2a2

 

 

the component is said to be a balanced, algebraic component.
The form of 92 and C_3 suggests that a more convenient mathe-
matical model of three-phase components of this type can be
realized by applying a symmetric transformation of variables.

This transformation is designed to diagonalize the coefficient

matrices (£2 and C_3 and is obtained by taking Y-T : I Ex
- T
= * '
and_l__T (I ) lx With
J3 _1__ _l_ __l 0 0
«[3 J3 «f3
T - 1 1 d T"1 T - 1 1
_ ' o __ _" an (_ l‘ ‘ 0 _- _'_
J2 «f2 «f2 «f2
0 L _l_ :2; _3_ _3_
6
J J6 J6 J6 J6 (2-3)

 

 

 

 

* The superscript T is used to represent the transpose

of a vector or a matrix.

-12..

When the voltage vector at a four ~terminal port

is of the form:

Cos(oot-6)
v = v «f3 Cos(u)t-9 - 150°) (2-4)

J3 Cos(oot-9 + 150°)

 

 

the terminal voltages are said to be balanced with "carrier"
frequency 0.). The transformed port vector V is then of the

form:

vt1 0
\_/’T= vt2 = {—3- V Sin(wt-6) (2-5)
«f2
f
vt3 . -Cos(o)t-6)

 

 

 

 

where it is evident that vt2 and vt3 form an orthogonal basis

for the two dimensional vector space. The presence of the zero

in the top position is used in the later development. For balanced
systems the "carrier" is removed from the system model by means

of a second transformation of variables established by taking Em =

M KT and“!In = MIT With

_13-

l O O
_I\_/I_ = 0 Sinwt -Cos out (2-6)
0 Coswt _ Sin wt

 

 

where M is, of course. orthogonal. It is to be noted that M and
I are non-singular for all t with determinants equal to unity.
For the balanced terminal voltages the port vector Em takes the

form:

vml 0
«f3
1/ = vmz =——— V C086 (2-7)
m N/‘z
vm3 -Sin6

 

 

 

 

Combining the two transformations given above:

V =MTVandI =(M)(T)IorI=TM (2-8)
--——x --m x -—

-14-

}. l l

«[2 J2 \f2
1113: 3-1-2 0 Cos (6x - 120°) Cos ((1312 + 120°)
«f3 0 -Sin (wt - 120°) -Sin (wt + 120°)

 

 

The question arises at this point, what is the effect of the trans-
formations on the terminal graphs of the variables Xx and Ex?

The oriented linear graph as used by Reed [20] and others for
two terminal elements is based upon using an oriented line segment
to represent a pair of measurements taken in a specified manner.
Koenig [12] and others extended this approach to multiterminal components
through the use of n-l elements to represent 2(n-l) measurements on
n-terminal components. This procedure can be further extended to
identify the variables defined by the above transformations with a set
of measurements. This identification is established by considering the
measuring device shown in figure 2-2, which consists of d. c. amplifiers
and idealized a. c. generators with each generator having two quadrature
fields. The functioning of this device can be seen to be exactly that of
the transformations I and M. When suitable values are taken for the
resistors shown, the potentials at the points j, k and 1 represent the

v andv respectively. When the rotor-stator

voltage variables th’ t2 t3

coupling coefficients of the a. c. generators have the proper values, the
voltages represented as v1, v2 and v3 are seen to be the variables
resulting from the application of the transformation, M, to the vector
of variables, VT. Similar devices serve to identify the transformed

currents and also the inverse relations giving the vectors of variables

in terms of the transformed variables.

-15-

 

b'W—l

 

 

 

 

 

 

 

.k 7% v2 2

(v

 

 

W

1
c ‘ 2
V3 V3

Figure 2-2. A Device for Measuring Transformed Voltages.

 

 

 

 

 

 

 

All component and system models and all analyses given in
this thesis are presented in terms of the port vectors Xm and_I_m or
their subsets and the associated graphs defined by the type of instru-
mentation shown in the figure. The form of mathematical model re-
lating these port vectors can be developed from models of the com-
ponents such as those given by Koenig and Blackwell [12 Chapter 11],
or similar forms wherein the phase windings are represented as

three- Ct four-terminal ports.

The models so obtained can certainly be presented in any
one of several forms-each involving a degree of approximation. Any
mathematical model must be balanced between the two extremes of

being so simple as to produce no useful results and of being so complex

-16~

as to be mathematically intractable. In general, the approxi-

mations used here closely parallel the assumptions made by

other investigator s .

The final form of the component model developed is

the so-called state model which has the general form

gm 313w), 3m. 5m]

(2-10)

9+1

 

:0) 903m. :(t), 110)]

 

 

 

‘3 .| ‘fl .‘I‘-, r

In the next section these state model forms are shown explicitly

for the important example of a synchronous machine.

III SYNCHRONOUS MACHINE MODEL

The procedures used in establishing a model of a
component of a three-phase power system, as well as some of
the implications of the model itself can be further clarified by
means of an example. The example of the synchronous machine
used here represents an important nonlinear component in elec-
tric power systems. The stator is considered as having three
isolated single-phase ports in one model and as a three-phase

three-or-four terminal port in a second model.

A mathematical model of the synchronous machine
with isolated phase windings may be established by considering

it to be a five-port component as illustrated in figure 3-1.

Phase Ports

MW... 13‘ W” Ib'lc [C'

 

 

 

 

Port ‘31 b, C r ?,
__, r

5% . _Field y 1' v
Port

g —° 1" , , ,

/TI7‘I'7'7'7 a' b' c' r' g

 

 

 

Figure 3-1. Representation of a Synchronous Machine as a
Five—port Component.

The form of the equations usually used in modeling the terminal
characteristics of the five-port synchronous machine without dam-

per windings is developed elsewhere [ll], [12] and is of the form

-17-

-18-

 

 

 

 

 

 

Va“) Raa+§fLaa(¢) ddeabW) ditLaCW) di't'Larm) ‘ i‘am -
d d d I d .
vbm mLabm Rbbi-HELbbW) a.Lbcun aims) :bm
Vc(t) = 'difLacw) £1”me ~Rcc+dlchcM £1?ch icm
vrlt) 'gfLarlm . dgberm) gchrm) Rrrl'ddtl‘rr lirlt)
. .
Larm) é .
Tm Iia(t)tb(t)ic(t)l g], Lbrm i,<t)+(B<¢)+J§.)¢(t>*
LCI(¢)

 

 

(3-1fa

The inductance coefficients in these equations, in general,
are functions of shaft position as indicated and may also be functions
of the currents, if the latter type of nonlinearity'~ is to be included in

the model. The coil inductances Laa’ Lbb’ L L L L (1

cc' ab’ at’ be an
er are periodic functions of shaft pesition and each can be represented

in terms of Fourier Series. If the rotor and stator are cylindrical and con—

centric, i.e. , if the slot andalient pole effects are neglected, then

T

The overdot, e. g. 5:, is used to represent the derivative with
respect to time.

-19-

these coefficients can be considered independent of (5. Under
the same conditions the coeffiCients Larw)’ Lbr(¢) and Lcrm)

are taken as:

 

 

Lar(¢) Lar Cos ¢
Lbr(¢) : Lbr Cos (91 - 9b) 1‘ (3-2)
1
a
1 LCM) LC. COS (9’ - 9.) :4;
i .' 2'
s i I 1

 

 

When stator windings are identical the third order coefficient
matrix associated with the stator ports is cyclic symmetric, i. e.

Lab : Lac = Lbc’ Laa : I"bb : Lcc and Raa = Rbb

coupling coeffiCients are equated, 1. e. , Lar : Lbr = Lcr and 6b =

= R and the
CC

- 6 c = 1200. When the salient pole effects are included, an ad -
ditional term is included in the series expansion of all inductance

coefficients except er.

Perhaps the most severe limitation of the model of equations
(3-1) is the fact that the inductance coefficients are considered in-
dependent of the currents and an attempt to include these nonlineari-
ties at this point leads to intractable mathematics. If saturation ef-
fects must be included in the model it is easier to do so at a later

stage in the development.

-20-

The coefficient B in the torque equation is usually con-
sidered as a constant although there is no difficulty in so far as
numerical solutions are concerned when the coefficient is con-

sidered to be a function of ¢.

The form shown in equations (3-1) serves as a basis for
deriving the general form of a more acceptable model. Connecting
the stator windings of the machine as a four-terminal wye reduces
the machine from a five -port component to a three -port component

as shown in figure 3-2.

Stator Port

 

 

 

 

a c n
Mechanical I I I I b
Port r c I)" l]
sh— Field ‘1 v )r
gort
g 1" a rh' é!

 

 

 

rrrfr'r'n

Figure 3-2. Representation of a Synchronous Machine as a
Three -port Component

When the characteristics of the four-terminal three-
phase port are modeled in terms of the vectors Xx and_i_x defined
in equation (2-1), the terminal equations as derived from the previous

five-p0 rt model are

 

-21-

 

 

 

 

 

 

 

in‘t’ za -2; -7: g M(¢) ixl(t)
vx2(t) -z 22 z M(¢-150°) ix2(t)
vx3(t) -z z 22 M(¢+150°) ix3(t)
..-- ............................................... I
iv (t) .l—MM) M(¢-150°) M(¢+150°) zr ir(t)
; 1' 43' f 1
Cos ¢
p . . . d . '
T s“) = 2 lel(t) 1x2“) 1x3“)I W Lar J3Cos(¢-150) Ir“) + Tm(m
\f3Cos(¢+150)
(3-3)
where Za = Rail-((11? La’ Z : Ra+§E(La-Lab)=R+:_tL
Z = R +5-1— L M(¢):\/—3—<-1- L Cos ¢ (3-4)
r r dt r ’ dt ar

T (b) = 1% (B + J g?) ZNt), P =No. of poles.

The form of the equations (3-3) as well as the values of the coef-
ficients, can be determined directly from measurements. For

example, if i and irare set equal to zero then, by applying

x2’ 1x3’
a suitable signal to the terminals represented by xl, the values of
the coefficients Ra’ La’ L, R, and Lar are determined from the

measured relationships between suitable pairs of variables.

-22-

The three-terminal machine model can also be established either
by derivation or by measurement. The relations between the co-
efficients of the three and four terminal ports, as well as the dif-
ferences due to wyeor delta internal connections, are apparent
when the models are derived from the five -port model cf the previ-
ous section. The differences in the wye and delta configurations,
as would be expected, are reflected in the values of the constant
coefficients and in a shift of the rotational reference. Another dif-
ference, usually ignored or not recognized, is the presence of an
auxilary equation of the form d]: i = Ki for the delta configuration.
This term is insignificant to the external-characteristics. The three-

terminal model can be written as

 

 

 

 

 

 

ix1(t) = 0
vx2(t) 22 2 J3 M(¢-150) ix2(t)
vx3(t) 2 22 J§M(¢+150) ix3(t) :
'1
vr(t) ~13 MW-ISO) J3 M(¢+150) zr . ir(t)
. z i

Cos(¢-150) .

Tam =§ lixzm ix3ml g5 JiLar c6.(¢+1so) irm +%(B+J-§E) M0

The delta coefficients of equations (3-5) are related to the wye coefficients
of equations (3-4) with

Z: Z,L :
Y

0
A 3 L ,¢ =¢-90 (3-5)

-23-

Note that the coefficient matrix of the above voltage-current
equations is a sub-matrix of the corresponding four-terminal

model.

The application of the transformations of variables to
both the four -terminal and three-terminal model is greatly fa-
cilitated by compacting the notation. The four terminal model
of equations (3-3) after applying the transformations 2 and M ,

is written in matrix form as

 

 

 

 

 

 

 

 

 

 

T T d

-m“) Mléxl M Mia-$.11“) Em“)

v,(t) 3d; [garmnT EMT 2,. ' 1,0)
(3-6)

T S(t) 2%) [£m(t)]T LII—l 3% liar”) ir(t) + TmUl)

where
Za -Z -Z Cos ¢
_zx= -z 22 z _1_-ar(¢) = Lar J3 Cos(¢-150°)

-2 2 22 J3 C‘os(¢+150°)

(3-7)

and dea) = T2. (B + J 345) 60)

-24-

Since the product '_l‘__Z_.x IT is diagonal and since the
column matrix obtained from the product 3 Ear”) has a zero in
the t0p position, the top equation in the four-terminal model is
independent of all remaining equations and is of the simple form
vm1(t) = Z1 im1(t). The relation between the coefficient Z1 and
the co efficients in the five-port model is Z] = Ra+ Elitu‘a + ZLatl'
The remaining equations of the four-terminal model are of the
same form as thoseobtained by application of the transformations

of the variables to the three-terminal model of equations (3-5).

Since the rotational velocity of the machine is nearly
equal to the nominal frequency of the system of which it is a part
it is desirable to use the difference function o.(t) = ¢(t) - out as
a variable in establishing the model rather than ¢(t). The result-

ing equations then can be expressed as

vii): Z111“)

gm Ii; -(a+w)g_.m<a) _I_(t)

ll“

151,16) _1_ (1:)

19mm)? L. 11.0)

 

r

 

 

 

 

 

 

 

 

vr(t) -d [Em(°’]T R ir(t)

.rvwv-

D

T30) =-§ [1(01T 3mm) ire) + Tm<d> (3-8)

 

where:

I?u
n

«L

 

'1"
ll

 

-wL

fills:

 

 

3mm = L...

ya) =

-25-

Sins.

-Coso.

 

(’6)

m2

V

 

(t)

m3

 

 

10:)

 

i

(’6)

m2

(1:)

m3

 

 

Coso.

‘Sino.

. P P d .
r (for wye machine) and Tm(o.) _ —2- Bw +E(B +J-a) <1

In realizing this result it is necessary first to expand the derivatives

indicated in Eq. (3-6).

This set of equations, (3—8), explicit in voltages

and torques, represents a very useful model of a three-phase synchro-

nous machine and is very similar to the equations used by other in-

vestigators .

 

Specifically, ifo. andd are zero, the form of the

* The prime is used here to represent the derivative with respect

to the argument of the function e. g. f'(x) = d
'5—

X

f(x) .

 

~26-

equations (3-8) reduces to that of the direct and quadrature
component equations used extensively in power system studies.

Essentially all theoretical work on power systems has been based

on this type of model.

The state model form of equations (3—8) is obtained by
solving for the derivatives. Although the only requirement for
existence of the required inverse is the non-vanishing of the de-
terminant L(LLr - LI: ), it is noted that for practical machines
the determinant is also positive. Since the only other leading

principal minors L and L2 are also positive the coefficient matrix

is positive definite.

The resulting state model can be expressed in terms of
matrix products or these multiplications can be executed and the

resulting equations expressed functionally. The detailed model in

the form of matrix products is

di-lv-Rli or i-0 ‘11—-.5—31
at 1 “11—1 l r; l l — ’ at 3' L3 5
. 2 . 2 .
12(t) LL -L Sim L SinaCoso. -LL Coso.
r m m m
9.. i (t) =———IZ LfnSim Cosa LLr-Ligjoszo. -LLmSim
dt 3 LLC
i (t) -LL Cosa -LL Sim L2
r m m

 

 

 

 

-27-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v2(t) R -00L -Lm(c1+w)Sim 12(t)
x v3(t) - 01L R Lm(o. +0.1)Coso. 13(t)
v (t) -L dSino. L ciCoso. R i (t)
r 1 n1 n1 r r
I L
2 —Sin1
d . P 1300 B P L . . .
at“ '27 T 6“) "3‘ ‘ T a - __E l 12(913‘91 1 c666] 1r“)
4J
(3-9)
where L‘2 = L L - L2
c r Hi
When these multiplications are carried out and the notation
condensed,the result is expressed as
-LLrR+Lm(o.)[Lm(o.)]TR -wLm(o.) RrLr'rla)
I(t) ‘ — “ ‘ —————" “ I(t)
_ L L‘2 L LCz "
5.1.. -
dt 7
T I
I .
. ”ll-‘mlall E -L R .l
1r“) 2 r i.(fi
L 2 1. r
1 1 C LC

 

-28..

 

 

 

 

 

 

 

1.1.1,}; - gmwgmmHT l‘m‘” I(t)
L2 L —
C
- a
1% lime)? ° 1"“)
C
LLr_I_J' -_I_.m(a) [ldmhflrr Edam V(t)
LL? L z - ‘-
C C
+ .
ugh)?" 1 . (t)
V
LC LC r

 

 

 

B B P2

did“) : 21; Ts(t) “j w'rde) + 73'— HUNT

15mm 1,.(0

(3-10)

Equations (3-9) and (3-10) can also be written in a more tractable

functional notation.

 

’ i1(t) K1 11m + K2 vl(t) or 11m = 0
in) g, gut), in), am. am, mm + 316011140
53. 1(0 = 62 (in), i<t)-d<t)a<t) vm) + 3261(0) 30)
out) 63 (in). 1m. d(t)a(t)) + Kr -r (t)
o.(t) d(t) (3-11)

 

 

 

-29-

The form of the model of the three-phase synchronous ma-
chine (either delta or wye) developed here includes most of the major
nonlinearities evident in the machine. The machine is modeled in
terms of a set of variables related to the measured or x variables
by a linear transformation. These variables can be measured direct-
ly using special instruments. The detailed form of the model in
equations (3-9) shows that the equations are nonlinear in the speed
variable 6. . If this variable is held constant the model consists
of a set of linear first order differential equations and one algebraic
equation. Even when the variable (i is considered to be time varying
it is noted from equation (3-10) that the equations are linear in the

variable V(t).

IV MODELS OF OTHER COMPONENTS

The procedure used to establish the state model for
other three-phase components follows that given above for the

synchronous machine, and can be summarized by the following

three steps:

(1) Model the component in terms of the standarized
variables X (t) andlxu) to realize a set of equations

in the form

31(3),) = 329..) (4-1)
(2) Transform the variables Vx(t) andlx(t) to the more
convenient vectors Emu)" and _Im(t) to establish

the model

The vectors Km and_l_rn are partitioned into subvectors referred to

as the "scalar" variables

V(t) : vm(t)
i(t) = 1mm
and the "vector" variables
vm2(t) imam
V(t) = I(t)=
— vm3(t) - im3(t)

-30-

-31-

(3) Establish the state model form by solving
for the derivative vector.

This solution involves the inversion of a coefficient
matrix. The conditions for existence of this inverse place
restrictions on the values of the coefficients of the component
as already discussed for the case of the synchronous machine.
Convenient forms of models thus established for several com—

ponents, together with a short discussion of the limitations of

these models, are found in the following paragraphs.

Three -phase Transformer Bank

A model of a transformer, either single-phase or three
phase, is frequently required to include the effects of the magne-
tizing currents. When these effects are included, the model may
be established in a form explicit in the voltage, i. e. in terms of
the open circuit parameters. At other times it may be permis-
sable and desirable to neglect the magnetizing current. In the
latter case the model cannot be established in a form explicit in the
voltages of the windings, but must be modeled in the so-called h-
parameters. This form of model can also be used to include the
magnetizing effects by using the operator Zm or (Rm + Lm 21-15) as
illustrated in equations (4-4) rather than the form of equations(4-3).
To show the techniques involved in modeling three-phase transformers
and limitations of the models , consider first the single-phase trans—

formers. Two of the basic forms of the models for single-phase

transformers are

-32-

 

 

 

 

 

 

 

 

 

 

va(t) Ze n ia(t)
v vb
= all 7.
1a 11b
ibm ’ -n ' 0 vb(t) .-__1
(4-3)
va(t) Ze n ia(t)
v v
. ' a)! v b
16 is
vab(t) -an 1 L Vb“) "'
l 1

 

 

 

 

 

 

(4-4)

Where Ze and Zm represent the short and open-circuit parameters.

The two -winding transformer is obviously a four terminal
component and should require a three element topological rep—
resentation. However, the transformer in use is characterized
by connections to the primary terminals and to the secondary.
This use marks the transformer as a two-port isolation conponent
or when one side of each winding is grounded it is a three brminal
component. In either case the elements of the graph can be joined
by the dashed lines shown above to obtain a three-terminal repre-
sentation. Alternate forms of the transformer equations such as
the open circuit form can also be used as a basic model but this
frequently involves the problem of numerical accuracy. The former
model uses the standard open and short circuit tests to establish

the parameters, thus avoiding this problem.

-33-

In general it is convenient to normalize the transformer
turns ratio. This can be done either by using the per unit system
of measurements or by making a change of variables of the type

1 .
l _ 'l ...
vb _ nvb and 1b- In 1b

Since the transformer 'is modeled as a three-terminal
component with the turns ratio normalized, the model can be
simplified by applying a tree transformation to the above graphs

as indicated in figure 4-1.

q ‘1

Figure 4-1. Application of a Tree Transformation

The resulting models of single -phase transformers are shown in
equations(4-5) and (4-6). This form also provides better cor-
relation between the linear model and the experimental varia -

tions observed due to the saturation effects of iron-core trans-

 

formers.
. e
ve(t) Ze ie(t) <
mw
i (t) 0
m (4-5)

 

 

 

 

 

-34-

Ve(t) Ze ie(t) e

 

 

vm(t) Z i (t)

 

 

 

 

i (4-6)

The technique described for modeling the single-phase
transformer is readily extended to the three-phase transformer
bank. This is accomplished by executing the first two steps of
the procedure previously mentioned, followed by a tree transfor-
mation similiar to that used in the single-phase case. This tree
transformation is applied to the vector variables and also to the

scalar variables of each four-terminal port.

The model of the three-phase transformer bank thus

established for the transformers connected wye -wye, with grounded

neutrals and including magnetizing currents is el
d rnl
vel(t) Re+Ledtl 0 0 i (t)
I e1
-..-- ......... § ................. ----
5 d .
Vez(t) = 0 ‘; Re'l'Lea—t- -wLe 1e2(t)
i
v (t) 0 mi. R +1. .9. i (t)
e3 I e e e dt e3
l

 

 

 

 

 

 

 

-35-

 

 

 

 

 

 

 

 

 

 

d 3
t
vm1( ) Rm+ Lm-a-E: 0 0 1ml(t)
I
---- """""""" | """""" ('1 """""" "'- m3
szm 0 : Rm+ 1"de -(.0L2 1m2(t)
I
I d .
I 1 I f
(4-7)
On separating the derivative terms and partitioning as indicated
the model is
V(t) = RI(t)+L dI(t) :2
_e -—e e e 33—6
v E
V(t): RI(t)+L-(-i-I(t)
_m —-m m dt —m
v (t) = R 1 (t) L+i 1 (t) E:
e e e edt e
V...
V(t) = R i(t)+L 511 (t)
m m m mdt m (4-8)

Solving the equations for the derivatives, the final model for

the wye -wye transformer with grounded neutral is

-36-

 

 

 

 

 

 

 

 

 

 

 

l l
gem T." 3.3.“) r: gem
e e T
.E
d _
at ‘ + V32
; 1
1m“) L Bm—Im(t) rm Em“)
Re . l
ie(t) ' L— ie(t) 1:" V6“)
e e <
e
d .
dt' = + ]'m
Rm 1
m m

 

 

 

 

 

(4-9)

The models for three-phase transformers connected delta-wye
and delta-delta are similar to the models given above. The
differences are l) for the grounded wye -delta connection, the
scalar equations become ib : 0 and v = Z i (or v = Z i if
a e a a m a
the Y is ungrounded) and the matrix used to normalize the turns
ratio takes the form shown by equations (4-10) rather than the
previous form N = n _l_J and 2) for the delta-delta connection, the

scalar equation reduces to ia = 0 and 1b = ‘0 with N remaining as

N=n£l.

-37-

 

 

(4-10)

Transmis sion Lines

Any one of several lumped parameter forms can be
used as a model of a power transmission line. The specific
form depencb on-the operating voltage level and the length of
the line. One basic form from which other forms can be easily
obtained is referred to as the "L" line or "L" section. This
form considers all leakage effects, including resistive and ca-
pacitive leakage from line-to-line as well as line-to-ground,to
be lumped at one end of the line. The technique for obtaining
the state model of an ”L" section is almost identical to that of

the three-phase transformer of the previous section.

The basic form of the equations of the "L" section
model can be obtained easily and accurately in terms of the
measured variables. These equations for the grounded return

transmission line are of the form

 

-33-

 

 

 

 

 

 

vxl(t) Za -Z -Z l 0 0 1x1“)
vxz(t) -Z ZZ -Z 0 l 0 1x2(t)
vx3(t) -Z Z 22 O 0 l 1x3(t)
1' (t) -1 o 0 Y 1Y —1-Y v' (t)
x1 0 3 o 3 o xl

. l .

1 _ _ 1
1x2“) 0 1 0 3Y0 Y1 Y2 1x2“)
., _ _1_ ., s
1x3(t) 0 0 1 3Y0 Y2 Y1 1 3t)

1 x
b l c b c
x 3 x2 x3
a a
xl Y x1
14.1!

(4-11)

The model of the line in terms of the transformed
variables can be expressed either as atwo port vector model

or as the following ”three terminal" vector model.

-39-

 

 

 

 

 

 

 

 

 

yem 26 3 _Ie<t) 2
= v 22

3mm | _<_> _m ; 3mm 1

ve(t) 2o 0 ie(t) :
= Y m

1mm I i 0 Yo vm(t)

 

 

 

 

 

 

 

 

 

(4-12)
where Z : R + L d and Y = G + C d have the forms
—e -e -+:dt —4n -Jn -4n at
d d
Re+Ledt -Q)Le g'l'C'aY "(DC
Z = Y :
_e —m
wJL R. +-L. ii i we g +<:
e e e dt] 5 dt
a :

 

 

(4-13)

-40-

The state model is then

 

 

 

E.
1 ._1-
-I-e(t) “IT—elem L Ye“) Vin-
e e
v (t) -10 v (t) l I (t)
_m c—m—m C —m
d
at = + e
160:) -3}:— iem T}- vem 7‘
e e
7m
vm(t) -§ v (t) % 1mm

 

 

 

 

 

 

 

(4-14)

This model can be easily reduced to the short line form
by setting_Im and im to zero. To extend the model to the T or 1r
models, two or more "L" sections are cascaded by considering

them as components of a system, as discussed in a later section.

Induction Motor

The three-phase induction motor model is obtained by
considering the machine construction to differ from that of the
synchronous machine only in the number of phases of the rotor
and in the voltages applied to the rotor. The model for a wound
rotor. machine with three-phase windings on the rotor as well as

the stator is'determined by using the procedure already presented

-41..

for the synchronous machine, with the stator and rotor each
considered as three-phase ports. The model of a squirrel-
cage induction motor is obtained by a slight variation of this
procedure. The stator is considered as any other three-phase
port and the rotor as an n-phase port. The differential -
equations of the machine model are obtained in terms of the
stator x variables and the rotor phase variables. Since the
rotor is to be short-circuited in use and not interconnected
with other components, different transformations are used to
simplify the differential equations. These transformations
are essentially those used by Koenig and Blackwell [12chapter
12 and 13] for the n-phase rotor and are not considered fur-
ther in this thesis except to note that an additional normalizing
transformation is necessary in the squirrel-cage induction
motor in order that the coefficients can be determined experi-
mentally. The implications of the form of the model obtained
are similar to those of the synchronous machine. It may be
noted from the equations that, just as in the synchronous ma-
chine model, only the fundamental frequency effects are con-
sidered, thus neglecting the slot effects and the so-called "deep-
bar effects”. A more complete model for showing starting
performance might use two sets of rotor bars, particularly if

the deep-bar or double-cage type rotor is involved.

The experimental determination of the parameters of
the model can be attained through steady-state Operating tests.
The standard no-load or synchronous test and blocked rotor

test are sufficient.

-42-

The resulting induction motor model is

Induction Motor

Schematic Diagram

 

 

State Model Terminal Equations for four-wire wye-wye

C

 

 

l
' 32(“LrLrsHJ' LrsBrE)

connection
R
d . 1 l .
at 11“) = 1—: V1“) "1-- 11“)
l l
I (t) 1 (L R-dL 2 E)
—-s I? r-- sr ——
c
d —
a _
1 (t) 1 (L RE -<iLL U) 1 (LR -6.LZ U)
—r ‘1? sr—— sr— :2 —r rs—
i i C C

 

 

 

T e rminal Graph

 

 

14 t)

 

_Ir(t)

 

-43-

Lr
:2 XS“)
0

 

 

 

 

 

 

 

 

+ (4-15)
Lsr
‘2‘ Yr“)
L
c
. l B P T B .
33.6“) -r T s“) '3‘ w "2? Lsr [-I-s(t)] is“) '3 “6“)
where:
R -wL R 0L 0 -l
r r
R: : E:
.. _r _.
wL R -dL R l 0
I r r
1 I I
2=2=R+L_S1..=R+(L+2L)d
l S 1 ldt A A B 3'
d d
2 _ R+L-a?_RA+(LA-LAB) a?
2 _ 2 _ L _ L _ L fix];
LC _ LI"r - I"sr Lr _—; ’ I"sr 7-7- ’ I"sr-m—"l- TLAI
L L L
r r r
L + L 2 Z
R =— R = — R B=— B J = — J
r L: r L: l P s P 5 (4-16)

-44..

Fault Conditions

The various types of fault conditions, in fact any type

of unbalanced fault or load, can be considered as simply another

component with its own properties and characteristics. The termi-

nal equations of the various conditions are conveniently written in

terms of the x variables, and the direct application of the trans-

formations gives the terminal equations in terms of the mvariables.

Typical fault conditions follow, where the lines between terminals

indicate the fault connections.

 

 

 

 

 

 

Connections Terminal Graphs Terminal Equations
Three-phase i(t) 0
C r ‘l =
_Y v
‘ _I. r V i W) o
n .—
a ‘ ” (4-17)

Three -phase to ground

|<1

h—c

 

 

V(t) 0

)fit) 0

 

 

 

 

(4-18)

90"

a

(t)

Vml

im2(t)

(t)

m3

 

-45-

Single -Line to ground

 

 

c
I
V
_ V
/n —I-
0 -\/2 Cos wt «[2 Sin tot
= «[2 Cos wt 0 0
-J2 Sin wt 0 0
.. a
r 1

 

 

Dauble-Line to ground

I <1

lI-I

 

 

infit)

(t)

m3

 

(4-19)

 

 

 

im1(t)

vm2(tl

vm3(t)

 

 

~72

 

" all—3- (Sinwt - Coswt)

-46-

-1- (Sinwt + Coswt) 0

Line - to Line

 

b
a o
l 0 0
0 S inwt 0
0 0 Sinwt

 

 

 

 

O [in/381nm + Coswt)] [i(fiCoswt-Sinwtfl
\(2

|<

 

0
0 0
0 Coswt

 

 

 

-Coswt

0

V
m

i
m

m3

 

1(0

20)

(t)

 

(4-20)

lelt)

(t)

m3

 

(4-21)

It is to be noted from equations (4-19), (4-20) and (4-21) that the "carrier"

frequency, (1), which was removed from the coefficient matrices under

balanced conditions is not removed under unbalanced conditions.

 

V SYSTEM MODELS

System Topology

The compilation of the system graph for any given system
follows directly from the component graphs and the interconnection
pattern. With addition of trivial x1 elements as necessary, the syst-
tem graph consists of three identical subgraphs. In the sequel it is
convenient to refer to the graph in two separate subgraphs, l) the
"vector subgraph", which is composed of the subset of all x2 and x3
elements, and 2) the "scalar subgraph", which is composed of the
subset of all xl elements. Frequently the scalar subgraph is simpli-
fied by not showing the trivial xl elements explicitly. If the system
consists only of balanced components then the component equations
of the x1 subset are linear in the x1 variables and independent of
the x2 and x3 variables. These conditions imply that the system
equations consist of two independent sets, with the set associated
with x1 being a linear, homogeneous, autonomous set of equations.
Typical subgraphs for the power system represented by figure 5-1
are given in figure 5-2 (a) and (b).

A 214
m 51..

A E

 

 

Figure 5-1. Single Line Diagram of a Typical Power System.
-47-

-48-

 

 

 

 

 

m "° J J H
e f —el k —e2 h —e d
e b ' T 7
—e H
a E —-m —m —m P
v "
P. a --e
_e a
4 C
,L g G
a —e
9m )7 YE
‘1 (b
a a
(a) Vector Subgraph
a e , f

  

 

k
7 ] —<
a

   

 

(b) Scalar Sub gr aph

Figure 5-2. System Graph of the System Represented by the Single
Line Diagram of Figure 5-1.

The vector subgraph, (5-2a), represents orthogonal com-
ponents of the line-to-line voltages and line currents. The scalar
subgraph, (5-2b). is. similar to the zero phase sequence and repre-

sents the neutral voltages and ground currents.

The system model can be established from the component
models and the system graph in a number of ways. It may frequently

be desirable to combine some of the components into subassemblies.

-49-

This is illustrated by the simple example of two identical
transmission line "L" sections combined into one T section as
follows.

The "L" section models for sections 1 and 2 are given

by equation set(5-l)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 _ 1 R 1 1 v bk Eel; —I-ek Ck
—ek L —ek —ek L —ek I
ek ek V
-—mk
d
'3? = + V
I
1. 1 1 1 "mk
ka ‘ 6‘ gmk ka 6" —mk
'3 k k a
I
R v 1
8k . l b 6k 6k
i i —— v , c
ek ek ek Lek ek i -
d mk
d—t = + )1
mG
g l .
vmk ‘ - l mG e G- 1rnk
C 1 k
I 1 k 1 1 1
' (5-1)
for k = l, 2
The graph of the system is given in figure 5-3,
b b b b
c1 1 2 ('22 c1 g 1 2 1 c2
I l r ‘
| ‘ | e1 / e2 ]
I I l I
l ' ' l \ rn m2 1,
y I
Y ‘ Y r31 1 1
at 'a a a ‘a

Figure 5-3. System Graph for TwoL section Transmission Lines.

-50-

The graph gives the equationsl‘vm = v = v

ml m2

V I :-I I

1 = -m -1 and-’\Zm=—ml:-m2’—m —m1 -—m2'

m 1 m3
When these equations are substituted into the component e-
quations and the results are simplified the model of the'T”

subassembly is then

 

 

 

 

 

 

 

 

 

 

 

 

l l
-Iel "13‘ Be —I-el f Xei
e e
d l l
712' —Ie2 ' '1‘: —e —Ie2 + L— XeZ
e e
_1. v l 1
1 —m C —m —m C —m
1 i
i -£{—§ i —l- v
e1 L el L el
e e
R
d 1 2 ——e 1 + _l. V
3? e2 L e2 L e2
e e
gin l
V --—- V _ e
m c m c m
m
(5-2)
With Re = Rel : ReZ and Gm = G 1 + sz

 

The argument (t) is not shown explicitly in the remainder of the
thesis for the variables v, i, -r , of and 0. except as necessaryfor
clarity. -

-51-

Formulation of System Models

A general method for obtaining the system model from

the component model and the system graph is given by Wirth [17].

The five basic steps as given are listed for reference.

1)

Z)

3)

4)

5)

Select an appropriate forest of the system graph.

Write the fundamental circuit and cut-set equations
in a form explicit in the chord across and branch

through variables .

Substitute the fundamental circuit and cut-set

equations into the component equations.

By elementary operations, eliminate the coef-

ficient matrix multiplying the derivative vector.

Solve the algebraic equations for the variables not _
appearing in the derivative vector and substitute this

result into the differential equations.

There are, of course, many forests that can be selected as

indicated under 1). Selecting a forest is equivalent to selecting

the state variables. In selecting a forest the general require-

ments as given by Wirth are observed, namely.

1)

Z)

All the specified across drivers are made branches :

As many as possible of the elements having equations
explicit in the derivative of the across variable are

branches:

-52-

3) As many as possible of the elements having equations

explicit in the derivative of the through variables are

chords;

4) All the through drive rs are chords.

This requires that all specified voltage and rotational velocity
drivers must be branches, all specified current and torque drivers
must be chords. The elements in the terminal graphs used to model

the fault conditions must be classified as part of the branch or chord

system as follows.

1. Three-phase to ground fault

- vector and scalar elements in the tree

2. Three-phase ungrounded fault

- vector element in the tree and scalar element in

the chord set.
3. Single -line to ground fault
- vector element in the chord set and scalar element

in the tree

4. Double-line to ground fault

- vector element in the tree and scalar element in

the chord set

5. Line to line ungrounded fault

- scalar element in the chord set

-53-

Similar classifications apply, of course, to other unbalanced
conditions. In most cases all of the elements with equations
explicit in the derivatives of the across variables can be placed
in a tree, however, more frequently than not all elements with
equations explicit in the derivatives of the through variables can-
not be placed in a chord set. In the partial graph of figure 5-2
the element Jm is placed in a tree bit all other elements shown
are explicit in current derivatives and hence should be in the
chord set. No tree for which this is possible exists. A partial
remedy can be found if the lines K and L can be represented by

11' sections, thereby establishing elements on each line for which
the equations are explicit in the derivatives of the voltages. This
reduces the number of branches which are explicit in the derivatives
of currents. If this procedure is used indiscriminately, however,
computer solutions may not exist due to the size of the numbers
involved. One possible tree for the system of figure 5-2(a) is
shown below, by the heavy lines. Line J is modeled as a "T"line
and K and L as short lines. If an attempt is made to implement
the remaining steps in the general formulation procedure of
Wirth[l7] it becomes immediately obvious that large blocks of
quations must be manipulated analytically. (Approximately fifty
differential equations would be involved for the system used as an
example here). The difficulty of managing such large systems of
equations is avoided by the application of a method which involves
a repetitive series of operations applied to a smaller number of

equations. This is an extremely desirable and perhaps necessary

-54-

point in the consideration of large power systems and makes it

possible to consider simultaneous formulation of different parts

of the same system.

 

 

 

m E e. f Jel k JeZ h Iie d
e *7 :: - 4
e
b e -m
a E m Hm D
P a a L
-'e a
a —e
c vd
./ 8 4 ‘
V ' .
.a 1.8 , ‘—

. V
a a c

 

Figure 5-4. Vector Graph of the System of Figure 5-2(a)
Showing (he Rassible Tree.

The technique used involves a separation of the system
graph into convenient subgraphs with tie points or junction points
retained as desired for present or future connection to other parts
of the system. This procedure is similar to a technique described
by Kron [21], [22] as "tearing". This is not to infer that the c --
equations are to be solved in small groups. Such is, in general,
not possible.

The technique can also be viewed as a procedure in

which the equations are manipulated to obtain two groups, one of

-55..

which is necessary for further use in the formulation process and
is retained for additional manipulations. The other group of
erpations is not necessary for further formulation but is necessary

for the solution phase and, consequently, must be retained.

VI EXAMPLE OF A SYSTEM MODEL

The concepts involved in the formulation process
can, perhaps, best be presented by using the above system
as an example. In order to provide as much simplicity as
possible and still develop the desired concepts, certain condi-

tions of operation of the system are considered.

1) The synchronous machines are considered

to operate with constant field voltage and constant torque
drivers. (The extension to different conditions, such as
controlled torque or constant horsepower or other types
of drivers, results in no more than the addition of other

similar elements and equations).

2) No additional loads and no additional tie
points are to be included. The extension to include ad-
ditional tie points for future expansion is carried out by
considering such tie points just as any other tie point
until the equations are stored. An additional equation
setting the appropriate currents to zero is simultaneously
stored until such time as the future expansion is imple-

mented.

-56-

-57-
System Classification
Since the system involved has no unbalanced components
the scalar set of variable need not be considered. The system
represented by the vector -graph can be conveniently classified in-

to three general subsets as follows :

A Transmission lines J, K and L
B Transformer Banks E, F, G and H
C Synchronous Machines A, B, C and D

Although the selection of the subgroups is to some extent
arbitrary there are frequently some rather obvious consid-
erations. For example, the elements representing nonlinear
equations such as the synchronous machine are relegated to
the last group in order to avoid the necessity of manipulating
nonlinear equations at the early stages. Also, subgroups
should be chosen, if possible, so that the tie point elements
do not form a circuit or a cut-set. An example of such a cut-
set appears in subgroup A when the leakage current of trans-
mission line J is neglected, i.e. , when element Jm is removed.
When such is the case a transformer with its magnetizing
current is included in this subassembly. The sleection of such
assemblies is not necessary but if does simplify, somewhat,
the formulation procedures.
Subgroup A - Transmission Lines

To continue the present example, consider the graph

of the subassembly A as shown in figure 6-1.

 

 

 

 

 

11
a

Figure 6-1 . System Subgraph of Subassembly

The elements 1, 2 and 3 are tie point elements of the
subgraph with the remainder of the system and for formula-
tion purposes are considered to be drivers. For convenience
it is desirable that elements 1, 2 and 3 be considered as cur-
a cutset, however,

rent drivers. If the three elements form

one of the three must be considered as a voltage driver.

The general procedure given by Wirth is followed
in modeling the system of subassembly A. The component

equations are

1 1
-I-j1 r; —eR —,1 i; 191

. d l l
Lme J ET: —Ij2 —Le —e —I-j2 —Le -j2
—m C —m —m C —m

 

 

 

 

 

 

(6-1)

-59-

. d 1 1

mm“ alk"f§k—I-k+_IT—k
k k

. d _ 1 1

LmeL df—I-L ‘ ' TL 51- IL + LL XL

A tree is selected for the subgraph A with element
Jm as a branch and with elements 1, 2, and 3 as chords.

Elements J1, J and K are selected to complete thetree as

2
shown by the heavy lines. The circuit and cutset equations

can be written explicit in chord voltages and branch currents

re spectively , as follows

 

 

 

 

 

 

 

 

 

 

 

31 1 0 0 1 —jl —31 1 1 0 -1
3'2 1 0 -1 1 3532, 132 o 0 1 1
113 e 0 1 0 1 3K 1K s- 0 -1 0 1
LL -1 1 1 0 _m” gm, 1 1 1 o
1 I 1 1 j
' 1 1 i 1 1

(6-

These topological equations are substituted into the terminal
equations and by row operations the equations are reduced to

the set

2)

 

-60-

 

 

 

 

1 1
—11 'L—Be('—I '—12 13L) L— 5K(‘—12’—IL)
e K
32 “LL-Bxgz +12) +1.1 5L —IL
K L
d 1 1
a? 13 = '1'.— §e(‘—3 +-I-L) 'fBZ-I-z
e L
I --1— R I
—L LL —L—L
‘v ._
—m m —m
1 +_1_
L; —jl LK —K
iv -_1.. —v +v .V)
LK—K LL —j1 —j2 —K
+ _1_ v +-i (-V +V +V) (6-3)
L —j2 L —jl —j2 —K
e L
—1- (-v + V + V)
L —jl —j2 —K
L
l (1 + 1 + 1 )
c 1 —2 3

 

 

At this point the importance of the previous restric-
tions on the selection of subassembly A becomes obvious. Since
elements 1, 2, and 3 are chords but could also have been branches
the top three equations of the set of circuit equations (not used as

yet) can be solved for the three voltages yjl’ ij, and XK in terms

ofV

—1'—2'—3

these variables from the system equations.
of equations obtained can be partitioned into the following
two groups of equations which shall be referred to as A(l),

A(Z), A(3), and E(l)

A(l)

A(2)

A13)

13(1)

13(2)

 

 

 

*—

+

t—g
xlld
t*
r ]k: 7F1h: 7f1h:

51th cal—‘1":
I

[O

 

 

 

 

~61-

and Em and the results used to eliminate

and E(2).

52 [.515
I"e I"K
6 5‘5
2 2
0
I?
- ..._]-
LL
U U
[.1 + :_
LL L
I
L131. -L

The system

rigs
lo

(D

 

 

 

(Fl Ic:

H

 

 

 

 

t"

 

 

 

(6-5)

 

-62-

The last two equations E(l) and E(2) can be considered to be
internal or auxiliary equations not needed in the further formu-
lation but necessary in the complete solution of the problem .
Equations A(l), A(2) and A(3) are retained with the corresponding

graph elements 1, 2 and 3 as the model of the subsystem A.

Subgroup B - Transfomzner Banks

The subassembly B can be combined with the subsystem
A as a complete mitoringroups of one or two transformers, i. e. .
each tie point considered separately. Following the latter procedure
and starting with element 1, the subgraph A1 of figure 6-2, is obtained.
with the tree Ee’ Fe and Fm selected by the same procedures as
before. The terminal equations of the components of this graph can

be written as follows

 

using a more condensed
form of functional notation
for the equation A(l). The

equations A(2) and A(3) are

 

held in "temporary storage"

Figure 6-2. Subgraph A

as they are not needed at

l
thispoint.
d
Element 1 all 3 _f_ (ll-1:129 3L, X19 K2: Km)
Transformer
E
d _ l l
atlEm ‘ "“L BEm -—I-Em +"""L XEm
Em Em
d l l
ET—IEe ' ' L‘“E BEe —IEe + _L XEe
e Ee

-63-

Transformer
F

d l l

dt-I-Fm ‘ "‘"L BFm —IFm +‘L'— XFm
Fm F

d l l

-- I : .. __ + __ V

dt —Fe LFe — e —Fe LFe —Fe

(6-6)

The topological equations are obtained from the graph as before
and substituted into the component equations and the resulting

quations reduced to the two groups of equations

 

 

 

 

 

 

 

 

 

 

l l l
3‘“ l4 raise-i4 * 13.14 'rs. 1’1
d
a? = +
, l l l
3‘” .1. tr. 5.6.1. “ 17.335 '13:. 1’1
I
(6-71
12(3) I '1 I + 1 v
—Em L-Em— m Em LEm—l
d
15(4) 3? —I1 = filly-12' 1’1' 32' -IL’ 3m)
_ 1 1 1 -1 1 1 -1
3(5) -X1-[r'r'r—'r—'r'L—1 £4“)

-64-

where
R R
l l l l —e -Fm

f(t):—-—V+— +_v+——v +(—+——)1
—4 LK—2 Le—m LEe—4 LFe—S Le LFm -l

+ (%'%)12+(%E'En+%lnns "T.’+%)I

e K ' Em Fm " m K e ‘

- gage.+§§igl) 1 .- (' e .+ :Llp ) 1

1".— L —4 L— L —5 (6-9)

Ee Fm Fe Fm

The equations E(3), E(4) and E(5) can be stored since they are
considered as auxiliary equations. Equations B(l) and B(2) with
corresponding graph elements 4 and 5 along with equations A(2)
and A(3) with the corresponding elements 2 and 3 and, of course,
the stored auxiliary equations E(l) --- E(6) constitute a state
model of the combined subsystem containing transmission lines

J, K, and L and transformers E and F.

When the above procedure is applied to the remaining
parts of subassembly B, represented in figure 6-3, the
resulting equations are the auxiliary equations E(6), E(7), E(8)
and E(9) of set (6-10) and the equations B(3) and B(4) of set

(6-12) which are retained for ports 6 and 7.

loo

Figure 6-3 Subgraph A2

 

 

 

 

. -1 -1 1 1 -l
E(6) 0 V+[ ——.1:—+-—+—-] f
l — — rG—m Ge LL I"K _5
—RK 1 1 1 1 5K —
E(7) I -— I -——-v +(——-——) +—V+(——-§l')l
d —2 LK —2 LK—l LK LL —2 LL—3 LK LL—L
"d? =
1 l l l '1
0 _\_f + [— + — + —— + —] f
E(8) _ 3 Le LL LHm LHe —6
—e l l l — E[e 1
E(9) l3 ' ire—1.3 ”'qu +‘L—L'Lj 33+(L—L'LLEL 'r; 1"...
(6-10)
where
f5 ' LL v1+{— 3 "LITI— 6+(§’<'§9‘m” 4% icml-e'é”— —I-2
" K" L" Ge- K one" Ge Gm LL IE
f6 = "Li "va "ISL 7+r'gflie‘5iqf' ’17'(?’T7‘n113'%‘%16
— c—m Ll: H: e m Hm- L e-
(6-11)
R
B(3) 36 T—Ge —16 + ”II-1 1’6 ‘ —L1 _2
d Ge Ge Ge
a-t- 2 R (6‘12)
"He 1 l
3(4) I "17’ I + V "L'—
-7 He —7 I-H-e _7 He'—3
1 1

 

 

 

 

It is important to the later discussion to note that the relations

represented by f4, _f5 and_f6 in equations (6-9) and (6-11) are linear in

the variables XI’ 1’2 and X3 .

~66-

Subgroup C - Synchronous Machines

At this stage of the formulation process the subgroups
A and B have been combined to form the model consisting of the
four elements 4, 5, 6 and 7 and the associated equations B(l),
B(2), B(3) and B(4) and also the stored equations E(l) --- E(9).
The next stage of the process is the combination of this model
with the model of the remaining subgroup C. The process is al-
most a repetition of the previous steps and hence is only discussed

briefly.

The machines of subgroup C can also be considered
oneat a time. The equations of the first, machine A, are written

.1: a condensed functional form and referenced as follows

d . .
C(l) 33A — gal (—IA’ la’ aa’ aa’ Va) + ialm) l[A
E(6) 31 = g (1 1 d a v) + 1 (0.)V (6-13)
dt a a2—A’ a’ a’ a’ a —a2 —A
E(7) ic1 : g (i I o. o.) + K T
dt a a3 a’—A’ a’ a a
d
E(8) Eda : ca

The last three equations E(6), E(7) and E(8) can be stored immedi-
ately since they are considered as auxiliary equations for the operating

conditions of constant field voltage and constant torque drivers. If

4
other operating conditions were to be con-
sidered then other component subgroups
could be used to extend the system to in- Figure 6-4.

clude the field and mechanical systems. Subgraph Cl

-67-

The first equation C(1) with the element A is considered to be
the state model representation of Machine A. Combining this
equation with Equation B(l) according to the rather trivial sub-

graph C1 gives the two equations:

 

 

d .
l —- = ,' , , ,
E( 0) dt—I-4 Eal(—I4 1a C1a C1a Va) +-£a (“3'4
E(ll) O:V-[f (u)+ y—flr‘l 1-1(11 6 11,111]
—A —a1 LEe ‘ EeEEe—4 —a1-'-4' a’ a' a a’

(6-14)

These two equations can be stored and the process re-
peated for the other three machines to obtain the equations which,
along with Equation E(l) - E(ll), must be solved to determine the
system operation. It is possible, although it may not be particu-
larly advantageous, to reduce the number of equations in the set
slightly by further analytical substitutions. An example of this is
seen in equations E(9) and E(lO) where equation E(lO) could be used
to eliminate XA in equation E(9). It is noted here that all of the al-
gebraic equations are linear in the variables for which solutions are

necessary although some of the equations, e. g. , E(5), E(6) and E(8), in—

volve simultaneous solutions.
Model of Entire System

The resulting mathematical model of the electric power system
represented by the vector graph of figure 5-4 consists of the following
algebraic and first order differential equations sets, where the various

vector and scalar functions g and f were defined previously.

 

1:: is .L.< ..

|<.'

65:

if:

 

 

 

 

l l l l 1 1-1
+ [ - - - -— _]- f (V ,Vm ,_V4 ’V)I I 3 ’ ’l’)
LFm F$1138 f—EmLe L _4 -2 51 Z-IErri—245
l l 1 l l
+[——-—-—-—_] 5(Vl ) I’laI)
LGm Ge LL LK —3 6’ —2 —6 —2
l l l l -l
+ [——+——+—+—] i(V ,V,V,I,I,I)
Le LL LHm LH —6 —m —2 —7 —7 —3 —6
‘ [f (0.1+ 9T1 ——1 I (-I ' d o. )]
—al LE L IjiEJe—4-gal —4’1a' a’ a’va
Ee
U
— -l l BFe . .
' [fb1(a) “LT-"1 [T351 'T—Is '5b1(’—15'1b'°b'“b"’b)]
Fe Fe Fe
U &3
- [f (0.)+——] 1[-— eI+g (-I,i,ci,c1,v)]
—cl LGe LGe—Z LGe—6 —cl —6 c c c c
U RH
l e
-[f(o,)+___]l[-—— I7+g(-,I ,d.0V.)]
—dl LGe LHe_3- LHe —d1 —7’ id (1 d d
(6-15)
‘f-‘LRL —IL + '1'.— eV‘ —3 —2)
l G V +l(I+I H)
c —m—m c — —2 —3
l 1
- __ I +— V
LEm BEm—Em LEm—l
R RK R
—e —e l l l
—I +[:— -—]I +[—+——] V -—-—
Le— -l LK Le —2 Le LK —l LK—Z
R
[’9+'§_K] 1 -_1_
L L -—6 L —m
K e
:K I -_l_v +[_1-. Hi] +_1- +r_K-;_R£] 1
LK—Z LK—l LK LL —2 LL—3 LK LL —L
"i313 +11— 2 + [f1 ‘LL1V3 + [TE '%]—IL -fl_—m
e_ L— L e — L e e
ga1(14, ia’o'a’ a ,va) + f 1((1a) V4

 

 

 

 

-69-

fibigs'lb'ab'ab'vb) +—fb1(ab) 1’5

gclg-6’1c’ac’ac’vc) +—fcl(o'c) lf6
fidlg'l’ld’ad’ad’vd) +—fd1(°a) X7
ga2g4’ia’aa’aa’va) +—fa2(a) X4

gbZQS'lb’o’b’ab’vb) +3112“) 1’5

gc2g6'ic’dc’ac’vc) +£c2(°') lr6
gdzg‘l’id'dd’ad’vd) +—fd2(°') 1'7

ga3g4’1a’o’a’aa) + K T

gbBQS’lb’ab’o’b) + K T

gc3(—I6’1c’ac'ac) + Kc T

C
gd3g-7’id’dd’ad) + Kde
d

a.
Cib
d

C
dd

 

(6-16)

-70-

The procedure used to formulate the system model of
the example can be summed up by the following steps.
1) The system is divided into convenient subassemblies
with at least one element included in the first sub-
subassembly, if possible, so that the tie point

elements do not form a cut-set.

2) A state model of the subassembly is obtained
consisting of (a) a graph element for each tie
point, representing a pair of vector variables,
(b) a vector differential equation or terminal
equation relating the graph variables,and (c)
a vector of auxiliary differential equations to
be stored until the complete system model is

solved.

3) The state model of the subassembly is combined
with those of other subassemblies until all
components have been incorporated into the

system.

The technique of formulation listed here is useful
for many power systems problems, although in some problems
it may not be possible to fully execute step two. For example,
systems with tie points forming a cut-set involve an additional
vector equation. In the example used here, if transmission line

J were modeled as a short line then the first subassembly

-71-

would produce a model of the form of equations (6-17) where

the matrix A is singular.

I) : .1} Z;

(6-17)

he.
I

30!; 3‘1)

Both equations are then considered to be a part of the state model

of the subas sembly.

VII EXISTENCE AND STABILITY OF SOLUTIONS

The final form of the state equations of the system, as
given by equations (6-15) and (6-16) for example, can be repre-

sented in vector form as

(11 o = fl (33(t1._>_c,(t1.§1(t1 1

(7-11
(2) 31m = 52(39(t1.§2<t1. 1522(0)

A numerical solution of these equations for a given set of initial

conditions on X1 is realized by the following steps.

1) The initial value of the vector, X2, is determined from
a solution of the algebraic equation given in equation (I) for X2(0),

given X1(0) and E1(0).

2) The resulting vector, XZ(0), is substituted into equation
(2) and equation (2) is then integrated by some numerical
technique such as the Runge-Kutta method [23 chapters 8, 9]

over an increment of time, h, to obtain the vector,Xl(h).

3) The vector,X1(h),becomes the new initial conditions
vector and steps 1) and 2) are carried out recursively
to realize the solution.

This procedure requires, of course, that the system of

algebraic and the system of differential equations each in itself has

a solution.

-73-

-73-

The existence of a solution to the system of algebraic
equations is considered first. In the example developed in the
previous section the algebraic equations are linear. This linear
form is a direct consequence of the form of the component medels,
their interconnection pattern,.and the technique for formulation
of the system model. The mathematical models of the synchron-
ous machines are the only nonlinear forms and these are non-
linear in the currents but not the voltages. These components
were placed in the last subgroup of components to be incorporated

in the system, thus utilizing the linear properties of the components.

Although the conditions have not been shown, in general,
under which the resulting algebraic equations are linear in the
variables represented in equations (1), the example used here is
typical of a large class of power systems models.

Since the system algebraic equations are linear in the

elements of 342, equations (1) can be expressed in the form

This equations then has a solution if_B-1 exists. In the example of
the previous section the required inverse is realized by inverting
one sixth-order and four second-order matrices, the second order
matrices having time varying entries. The existence of these in-

verses is difficult to determine except bynumerical methods.

-74-

In the event that the algebraic equations are nonlinear, existence
of the solution is determined by application of Theorem 4. 3 given
by Wirth [17 chapter 4]. Application of this theorem to the ex-

ample of the previous section is given subsequently.

Existence and uniqueness criteria for a set of differential
equations of the form represented in (7-1) are given by Murray
and Miller [19 page 42]. The necessary and sufficient conditions
for existence are that £2 must be defined and jointly continuous.
Uniqueness of the solution is assured if £2 satisfies a Lipschitz
condition. Again,these are difficult to apply. More useful condi-
tions for the existence and uniqueness of a solution to the system
models considered here are given by Wirth. In his Theorem 4. 3
the hypotheses are given in terms of a set of conditions imposed
on the components and the system topology. When these hypotheses
are satisfied a solution to the set of differential and also the set
of algebraic equations is assured. Application of this theorem
to the example here illustrates the factors involved. The hy-

potheses concerning the components require the following.

1) When the component differential equations are ex-
pressed as in equation (7-3), _F_‘(Z_) must satisfy a

Lips chitz condition.

as E%£LY0
d - = 13(2) (7-3)

32*) E (351 3‘2 31 32 ‘)

 

 

 

 

-75-

2) The matrix [gé-g- %% ] must be defined and
1 _

strictly positive definite with entries which are

bounded and satisfy a Lipschitz condition.

3) The driving functions must be such that their

derivatives exist and satisfy a Lipschitz condition.

All component models developed in this thesis satisfy
the first condition. The matrix specified in the second condition
is defined with bounded entries which satisfy a Lipschitz condition
for all components. The parameters of the component must be
such that the matrix is strictly positive definite. The third
condition is a restriction on the driving functions and is, perhaps,
too strong for a, few conditions of interest such as step functions.

The hypothesis stating the restriction on the topology of the
system requires that:

4) The across drivers can be placed in a tree and

the through drivers in a chord set.

(Additional hypotheses are applicable to any algebraic component

models that are of the perfect coupler or gyrator type).

The nonlinearities in the system model discussed above
are introduced by the synchronous machines. If the speed of each
synchronous machine is held constant then the system model is
linear, and written as

3“; 22(1) = .13 3(1) + 13m 17-41

-76-

Where the matrix K is a constant matrix. The closed form

solution in this case is

31(1) = ePt 5(0) + e§T§(1-)d-r (7-5)

The matrix eBt can be evaluated analytically or numerically

as the circumstances dictate

Stability problems as encountered in electric power
studies fall into one of two categories: The so-called steady-
state stability and the transient stability. Steady-state stability
studies are concerned with the problem of establishing the range
in loads or other steady-state operating conditions under which
all synchronous machines in the system remain in synchronism.
This implies that the rotational variables (ii of all the synchonous
machines are zero. The system is said to be unstable in the
sense used here, if for a given set of Operating conditions some
of the machines of the system are operating at other than synchro-

nous speed, i. e.. for at least one machine 6 ,é 0.

Transient or dynamic stability studies are concerned, first of
all, with the question as to whether or not all machines will remain
in synchronism when subjected to sudden loads and. secondly, as to
how large these pertubations can be and how long any resulting os-
cillations are sustained.
These problems are all related to the mathematical concept
of stability in he Liapunov sense, a definition of which is given by

LaSalle and Lefschetz [6 chapter 2] and others [24].

_77-

The state model for power systems studies has been

shown to be of the form

.Y £40.71}: 9., 2)

l1—1

E5(_Y:la is g)
a? = (7'6)
E601! _11 iii)

In-

ID
|9

 

 

 

 

In terms of this model, the concept of steady-state stability in
power systems implies that the eqUilibrium point has the co-
ordinates: _Y=£(V_l=_l_(2, i=5: 2 , andg 2K4, where 13 ,

i = l --- 4 is a constant. If the machines are not in synchronism
then the coordinates of the equilibrium point are X = 1511 , _I_ = _Kz' ,
9.; = E :5 Qandg = _I_{4't +30, where again_' is a constant. The
equilibrium points may or may not be stable in the Liapunov
sense. This implies that the steady-state stability problem is
not a stability problem according to Liapunov but rather a question
of the existence of an equilibrium point satisfying certain restric-
tions. The system model thus serves as a formal and practical
basis for investigating steady-state stability problems and does
not require the solution to the differential equations.

The transient stability problem as encountered in power
systems studies is precisely the Liapunov stability problem. The
problem of establishing the size of a sudden load or other pertu-
bation that can be tolerated by the system without causing loss of
synchronism is referred to as the extent of stability in the Liapunov

sense.

-73-

Many of the problems involving dynamic stability can
be answered without realizing a solution to the system model, if
a Liapunov function can be developed. Unfortunately, the deter-
mination of such a function appears formidable and there are,

as yet, no systematic procedures for doing so.

VIII SUMMARY

The problems involved in electric power systems have
been investigated extensively over a period of several years.
This thesis mertions the more prominent disciplines which have
developed in answer to specific problems. The use of these
disciplines has resulted in solutions of a limited nature in that
only special systems or restricted regions of operation have
been considered. Only a few investigators have considered a
generalized approach to the system involved. This thesis has
continued the investigation of a generalized approach to the study

of dynamic electric power systems.

Appropriate models of components have been developed
which offer distinct advantages over other component models.
These models of three-phase components are given in terms of
two sets of variables. One set of variables is composed of two
line -to-line voltages and two line currents. The other set is
composed of the voltage from one line to neutral and the neutral
current. These variables are used to establish identical topo-
10gical representations of components for both three- and four-
wire systems. The form of the component model has been simpli-
fied by the application of a set of linear transformations of vari-
ables. These sets of variables,referred to as the vector and

scalar variable 5, are

-79-

-80-

also established by measurements using special instruments,
thus defining elements of a linear graph to identify the variables.
The final form of the component model,referred to as the state
model, is formed by solving for the derivative vector in the dif-
ferential equations.

The synchronous machine model has been developed
in some detail as a typical example of the component modeling
procedures involved and also as an important component dis -
playing extensive nonlinear effects. Other component models
have been developed in order to be able to formulate models of
typical systems and to gain appreciation of the factors involved
in modeling components as well as an appreciation of the char-
acteristics of the components themselves. ”The components
selected for this purpose are components in common use in all
power systems. Some of the components such as transmission
lines and transformers involve two interrelated electrical ports,
each to be connected into the power system. Others, such as
the squirrel cage induction motor, have one port in the system
with another port interrelated but isolated electrically from the
system. Some fault conditions have been developed as repre-
sentative of components with algebraic models. Perhaps the

principle characteristic of this type of component is that the
presence of such components in a system creates a dependence

between the scalar variables and the vector variables.

-81-

The state model of the complete system has been
established from the component models arid the interconnection
pattern of the system as represented by the system graph. A
systematized formulation procedure was developed to be applied
to many power systems composed of several components. This
procedure, based heavily on the application of theorems devel-
oped by Wirth [1?], involves the separation of the system into
subassemblies consisting of some convenient subgroup of com-
ponents. The first phase of the formulation procedure involves
establishing a state model of each subassembly consisting of 1)
tie point elements to be used in the system graph, 2) a set of
terminal equations relating the variables represented by the
tie point elements and 3) a set of auxiliary algebraic and dif-
ferential equations relating other variables in the system to
the state variables. These auxiliary equations may be present
either due to the functional dependence of certain variables or
to the desirability of using numerical techniques for large systems
of algebraic equations. In the second phase of the formulation
procedure the auxiliary equations of the subassemblies are
stored, the subassemblies are grouped together into large sub-
assemblies and a state model of the larger subassembly is
established similarly. This type of concatenation is continued

until all components have been assimilated into the system.

A typical system problem has been carried out as an

example of the use of the state model form of the components

and of the technique of formulation of large systems.

-82-

Considerable detail has been presented in this example in
order that some appreciation of the applicability of the procedure
can be obtained.

One of the major contributions of this thesis is the
development of a procedure to formulate large dynamic system
models. A systematic algorithm has been developed for gener-
ating a system graph from the component models of the system.
This graph represents the dynamic variables of the system in a
manner similar to themethod used to represent the steady-state
symmetrical components variables by means of the sequence
graphs.

The solution to the model of the example has been
discussed. Some of the existence and stability properties of this
model were examined with inferences extended to a large class of
power systems problems. Identifications have been made between
power systems stability concepts as used in the industry and con-
cepts of the existence of solutions and stability of systems repre-
sented by differential equations. The direct application of the
properties of the system model to problems of stability has been

discussed briefly.

10.

ll.

12.

13.

LIST OF REFERENCES

Fortescue, C. L. , "Method of Symmetrical Coordinates
Applied to the Solution of Polyphase Networks", Trans-
actions, A.I.E.E. , Vol. 37, pt. II, l9l8,pp 1027-1140.

Wagner, C.F. and Evans, R.D. , "Symmetrical Com-
ponents", McGraw-Hill, New York, 1933.

Clarke, E. , "Circuit Analysis of A-C Power Systems"
Vol. 2 , Wiley, New York, 1950.

Transmission and Distribution Reference Book,
A.C. Monteith, Westinghouse Electric Corporation, 1950.

Truxal, J.G. , "Automatic Feedback Control System
Synthesis", McGraw-Hill, New York, 1955.

La Salle, J. and Lefschetz, S. , "Stability by Liapunov's
Direct Method with Applications", Vol. 4, Academic
Press, New York, 1961.

Crary, S.B. , ”Power System Stability", Vol. 1, Wiley,
New York, 1945.

Kimbark, E.W. , "Power System Stability", Vol. 1,
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Bellman, R. , ”Dynamic Programming", Princeton
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Lyon, W. V. , "Transient Analysis of Alternating-
Current Machinery", Technology Press, Cambridge, Wiley,
New York, 1954.

White, D.C. and Woodson, H.H. , ”Electromechanical
Energy Conversion", Wiley, New York, 1959.

Koenig, H.E. and Blackwell, W.A. , "Electromechanical
System Theory", McGraw-Hill, New York, 1961.

Gilchrist, R. W. , "A Digital Computer Solution of Synchronous

Machine Differential Equations in Some Power System Stability
Problems", Thesis, Michigan State University, 1960.

-83-

14.

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l6.

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22.

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-84-

Koenig, H.E. and Blackwell, W.A. , "On the General
Properties of Terminal Equations for Polyphase
Machines”, Matrix and Tensor Quarterly, Vol. 9,
no. 1, pp 2-12, September, 1958,pp 2-12.

Ku, Y.H. , "Transient Analysis of Rotating Machines
and Stationary Networks by Means of Rotating Ref-
erence Frames", Trans. A.I.E.E. , Vol.70, pt. I,
1951, pp 943- 957.

Park, R.H. , "Two -Reaction Theory of Synchronous
Machines", Trans. A.I.E.E. , Part I, Vol. 48, 1929,
pp 716-727.

Wirth, J. L. , ”Time Domain Models of Physical Systems
and Existence of Solutions", Technical Report No. 1,
NSF G-20949, National Science Foundation, Division of
Engineering Research, Michigan State University, 1962.

Ince, E.L. , "Ordinary Differential Equations", Dover
Publications, Inc. , New York, 1956.

Murray , F.J. and Miller, K.S. , "Existence Theorems
for Ordinary Differential Equations", New York University
Press, New York, 1954.

Reed, M. B. , "Foundation for Electric Network Theory",
Prentice Hall, New Jersey, 1961.

Kron, G. , "Tearing and Interconnecting as a Form of
Transformation", Quarterly of Applied Mathematics,
Vol. XIII, No. 2, July, 1955. pp 147-159.

Branin, F.H. , Jr. , ”Kron's Method of Tearing and its
AppliCations", Proceedings of the Second Midwest
Symposium on Circuit Theory, Michigan State University,
1956. PP (2‘1)'(2’79)-

Kunz, K.S. , ”Numerical Analysis", McGraw-Hill,
New York, 1957.

Hahn, W. , ”Theory and Application of Liapunov's Direct
Method", Prentice-Hall, Englewood Cliffs, New Jersey,
1963.

R0

‘1

0:.) 111.1: 01111