Kothari D.P., Nagrath I.J. Modern Power Systems Analysis
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A
. flTY'
-
t
I
0x.=
0
The
multiplication
of
two
vectors
x
and y
of
same
dimensions
results
in a
very
important
product
known
as
inner
or
j9g!g!
pfodqcij.e.
*tv
AD",y,Aytx
i:l
Also,
it
is
interesting
to
note
that
xTx
=
lx
12
cos
d
4
"tY'
lxllyl
,
wtere
Q
is
angle
between
vectors,
lxl
and
lyl
are
the geometric
lengths
of
vectors
x and
y,
respectively.
Two
non-zero
vectors
are
said
to
be
orthJgonral,
if
*ty=
o
(A-6)
The
unit
'tector
e
the.rest
of
the
componentb
are
zero,
i.e.
0
0
1
0
0
kth
component
Some
Fundamental
Vector
Operations
A
€k:
(A_3)
(A-4)
(A-5)
Two
vectors
x
and
y
are
known
as
equal
if,
and
only
if,
.yk=
!*for
k
=
r,2,
...,
n.
Then
we
sav
x=y
The
product
of
a
vector
by
a
scalar
is
carried
out
by
multiplying
each
component
of
the
vector
by
that
scalar,
i.e.
If
a
vector y
is
to be
added
to
or
subtracted
from
another
vector
x of
the
same
dimension,
then
each
component
of
the
resulting
vector
will
consist
of
addition
or
subtraction
of
the
corresponding
components
of
the
vectors
x
and
vr,i.e.
The fnllnurinc nrnnArfioo ^*^ ^--+:^^l^l^ r^ rL- --' ' -
Yvrrr6yrvPvr!avD4rv4yI-,uU<rUItrLUurcveL:t0rargeDra:
x*!=y+x
x+(v+'z)=(x+y)+z
ar
@zx)
-
(afiz)x
(ar+
e)x-
dF+
a2x
MATRICES
Definitions
Matrix
'
'
'\n
m x
n
(ot
m'
n)
matrix
is
an
ordered
rectangular
array
of
elements
which
may
be
real
numbers,
complex
numbers,
functions
or
operators.
The
matrix
(A-7)
is
a rectangular
array
of
mn
elements.
. .
o,t
!"notes
the
(i,
i)th
element,
i.e.
the
element
located
in
the
ith
row
and
the
7th
column.
The
matrix
A
has
m
rows
ano
n
coiumns
and
is
said
to
be
of
order
mxn.
When
m
=
rt,
i.e.
the
number
of
rows
is
equal
to
that
of
columns,
the
matrix
is
said
to
be
a
square
matrix
of
order
n.
An
m
x
1
matrix,
i.e.
a
matrix
having
only
one
column
is
called
a
column
vector.
An
I
x
n
matix,
i.e.
a
matrix
having
only
one
row
is
called
a
rore
vector.
*Sometimes
inner
product
is
also
represented
by
the
following
alterr
ative
forms
x
.
y,
(x,
y),(x,y).
.
,1
r\!r
's.t
frffi-H
Modern
Power
Svstem Analvsis
Diagronal
matrix
A diagonal
matrix
is a square
matrix
whose elements
off
the main diagonal
are
allzeros
(aij=
0for i+j).
NUII
matrix
If
all the
elements
of the square
matrix
are
zero,
the matrix is
a nuII or
zero
matrix.
(A-8)
Unit
(identity).
matrix
A unit
matrix /
is a diagonal
matrix with
all diagonal
elements
equal to unity.
If
a unit matrix
is multiplied
by
a constant
(),),
the resulting
matrix is
a diagonal
matrix
with
all diagonal
elements
equal
to
2. This matrix
is known
as a scalar
maffix.
=3x3scalarmatrix
Determinant
of a matrix
For
each square
matrix,
there exists
a
determinant
which
is
formed
by taking
the determinant
of the elements
of the
matrix.
T
det
(A)
=
tAt
=
213
2l
l-l
2l
, l-1
3l
'1,
+l-
r-
t1
I 4l*
|
1
2l
=2(8)+('6)+(-5)=5
l-1
-l
l2
(A-e)
(A-10)
ll:Lll;l
L-
^J
=4x4unitmatrix
Transpose
of a
matrix
'l.h-.e
transpose
of
matrixA
denoted
by At
is the
matrix
formed
by interchanglng
the
rows
and columns
of A.
Note that
(Ar)r
=
A
Symmetrtc
matrix
A
square
matrix
is
symmetric,
if
it is
equal
to
its transpose,
i.e.
AT=A
Notice
that
the
matrix
A
of Eq.
(A-9)
is
a symmeffic
matrix.
Minor
The mino,
Mij
of an
n
x
n rnatrix
is
the determinant
of
(n
-
l) x
(n
-
1)
matrix
formed
by deleting
the
ith
row and
the
7th
column
of the
n
x
n matrix.
Cofactor
The
cofactor
AU
of
element
a,, of
the matrix
A is
defined
as
aU
=
Gl)'*t
M,i
Adjoint
matrix
The
adjoint
matrix
of a
square
matrix
A is
found
by
replacing
each
element
au
of matrix
A by
its cofactor
A,,
and
then
transposing.
For
example,
if
A is
given
by Eq.
(A-9),
then
l-t
l1
l2
lr
l2
l-1
1
3
l3
lz
t-
I
l-
I
aoJA=
MotJellr Powor
S
6
7
-5
-\l
_;l
)l
(A-11)
A square
matrix is called
singular, if its associated
determinant
is zero, and
non-singular,
if its associated
determinant is
non-zero.
ELEMENTARY
MATR.IX
OPERATIONS
Eguality
of matrices
Two matrices A(m
x
n) and B(m
x
n) are said to
be equal, if the only if
ai=
bij
for i
=
7,2, ..., ffi,
.i
=
1,2,
...,
fl
Then
we
write
A= B
Multiplication
of a matrix
by a scalar
A
matrix
is
multiplied by a
scalar a
if
all the mn elements
are multiplied by
a. i.e.
(A-
12)
Addition
(or
suhtraction)
of matrices
To add
(or
subtract) two matrices
of the same older
(same
number of rows, and
same number
of columns),
simply add
(or
subtract)
the corresponding elements
nf the frxrn rnefr;cce i e rrrhen frxrn rnqfri/-ec A enA Il nf lhe cqrnc o,rAcr
qrc
added,
a
new matrix
C
results
such that
C-A+ B;
whose
ryth
elernent equals
cij= aij * bij
l------_-__-ir1
r
,Example
I I
l:r
I
i-? O-r f)
-11
IA
A=i
'--lz
-rl"-Lo
3.1
then,
Appendix
A
I
,6tl
I-
15
-1-l
C=A+B=l'
I
L2
2)
Addition
and subtraction
are defined
only
for
matrices
of
the
same
order.
The fbllowing
laws
hold lbr
addition:
(+
fhe
cammatatLve
ln+y:
A +
B
-B
+
,{
:
(ii)
The
associative
law:
A +-
(B
+
C)
=
(A
+
B) +
C
Further
(A
tB)r=
Ar
+
Br
Matrix
Multiplication
The
product
of two
matrices
A
x
B
is
defined
if
A
has
the
same
number
of
columns
as the number
of
rows
in
B.
The
natrices
are
then
said
to be
conform.able.If
amatrix
A
is of
order
mx
n and
B
is
an
n
x
q
matrix,
the
product
C
=
AB will
be
an m
x
q
matrix.
The element
c,, of
the product
is
given
bv
,l
r ,
cii
=
)_rai*0*i
k:l
Thus
the
elements
cu are
obtained
by
multiplying
the
elements
of
the
ith
row
ofA
with the corresponding
elements
of theTth
column
of B
and
then
summing
these
element
productp.
For
example
where
ctt= attbt,
+
arrb^
ctz=
anbn
+
arrb2
czl=
uzlbt, +
arrb21
czz=
aztbtz
+ crrb2
If the product
AB
is defined,
the product
BA
may
or
may
not
be
defined.
Even
if
BA
is defined,
the
resulting products
of
AB
and, BA
are
not,
in general,
equal.
Thus, it
is important
to
note
that
in
general
matrix
multiplication
is
not
commutative,
i.e.
AB+BA
The
associarive
and
distributive
larr-s
hold
for
matrix
mulriplicarion
(wtren
the
'aryprrtixiate
erpernirnys
.are
def;tv.4)-
re.
Asscrica'rv
&rrt':
4{B
[
=
A(BC)
=
-tBC
Distibutive
law:.
A(B +
C)
-
AB +
AC
(A-13)
1""
arzllbn
b,rl_
[''
r
cnl
Lau
azz)Lbn
brrJ
Lr^
czrl
tffiWl
Modern
Po
I
[1
-1
3]
Lo
z rl
Find AB
and BA.
A and B are conformable
(A
has two
columns
and
B
has two
rows),
thus we
have
l-1
-1
3l
| | r--1
0t
48=12
4
9lt BA=l
,
,'
'"-L;
;;l
"^-l-41)
A matrix
remains unaffected,
if a null matrix,
defined by
Eq.
(A-8)
is
added
to it, i.e.
A+0=A
If a null
matrix is multiplied
to another matrix
A, the result
is
a
null
matrix
A0= 0A
=
0
Also
A-A=0
Note that equation
AB
=
0
does not mean that
either A or
B necessarily
has
to
lre
a
null matrix, e.g.
[r
,l[ 3
ol=fo
0l
L0
0J
L-l
0l
Lo
0J
Multiplication
of
any matrix
by a unit matrix
results
in the original
maffix,
l.e.
AI-IA=A
The transpose
of the
product
of two
matrices is
the
product
of
their
transposes in
reverse order,
i.e.
6Dr
=
BrAr
The
concept of matrix
multiplication
assists
in
the
solution
of simultaneous
linear
algebraic
equations.
Consider
such
a Set of equations
atft + anxz+
... + abJn
=
ct
aLlxt + azzxz+
...
+ azrtr=
c2
:
Q^lXl
* C^*z+
... + A-rJn
=
C^
I,j].qffil*l
l.,i95I3n
or
n
D
"*t=
cii i
=
1,2, ..., m
i:l
Using the rules of matrix multiplication
defined
above.
Eqs
(A-14)
canle
written in
the compact
notation
as
Ax=c
(A-1s)
where
It is clear
that
the
vector:mntrix Eq.
(A-15)
is
a useful
shorthand
representation
of
the set of linear
algebraic equations
(A-14).
Division
does not exist as such in matrix
algebra. However,
if
A is
a square
non-singular
matrix,
its inverse
(A-l)
is
defined
by the relation
A_IA_AA_I_I
(A-16)
The conventional method for obtaining
an inverse
is to use
the
following
relation
A-;
_
adj A
det A
It is easy
to
prove
that
the
inverse
is
unique
The
following are the important
properties
charactenzing
the inverse:
(AB)-r
=
3-t4-l
(A-tf
=
(Ar)-r
14-r1-t
=
4
(A-18)
Matrix
Inversion
(A-17)
IfA
(A-14)
is
given
by
Eq.
(A-9),
then from Eqs.
(A-10),
(A-1I),
(A-17),
we
get
ffd-ffi.:|
Modern
power
System
Rnatysis
A-1
-+:if
:
i
_;l:f:i;
i,:_il
det
A
t
L-r
-s
5j
L
-;
-;
l.J
SCALAR
AI\rD
VE
CTOFFUNCTI
O]NIS
A
scalar
function
of
n
scalar
variables
is
defined
as
y
!
f\r,
x2,
...,
xn)
It
can
be
written
as
a
scalar
function
of
a
vector
variable
x,
i.e.
y
-
f(x)
where
x
is
an
n-dimension
vector,
(A-1e)
(A-20)
In general,
a scalar
function
could
be
a function
of
sever
y-f(x,u,p)
where
x,
u
and p
are
vectors
of
various
dimensions.
A
vector
function
is
defined
as
y-f(x,u,p)
DERTVATNTES
OF
SCALAR
AND
\ZECTOR
FUNCTIONS
(A-23)
A derivafit'e
of
a
scalar
funcritrn
1A-lt))
s,ith
re-sper.r
tr)
u
\.eL-R)r
r.ariable
-r
is
defind
as
al
vector
variables,
e.g.
(A-2r)
In general,
a
vector
function
is
a
function
of
several
vector
variables,
e.g.
-af
0r,
af
0*,
oJ
orn
-
At
0x
(A-24)
It
may
be
noted
that
ihe
derivative
of
a
scaiar
function
with
respect
to
a vector
of dimension
n is
a vector
of
the
same
dimension.
The
derivative
of
a vector
function (A-22)
with
respect
to
a vector
variable
r
is
defined
as
ofa
0x
0*,
}fz
0*,
af-
0*,
0*,
0fz
0*,
af;
0*,
(A-zs)
o*n
?fz
0rn
..
af;
o*n
Consider
now
a
scalar
function
defined
as
t
-
ff@,
u,
p)
=
21fi(x,
u,
p)
+
Lzfz(*,
u,
p)
Let
us
fincf
j{
.
According
ro
Eq. (A-24),
a^
(A-26)
(A-27)
+
...
+
2*f*(4,
u,
)
6-2g)
we
can
write
f
(x,
u,
p)
(A-2e)
(A-24),
we
can
write
Let
us
now
find
LJm\rrUrP)J
ds^i.
-.
According
to
Eq.
ox
la"l
Lar, J
fijfilftf.f Modern
Power'system
Anatysts
I
}ft
0f,
af^
AppnNDrx
B
We can
represent,
as we saw in
Chapter
5, a three-phase
transmission
line*
by
a
circuit with two
input terminals
(sending-end,
where power
enters) and
two
output
terminals
(receiving-end,
where
power
exits).
This
two-terminal
pair
circuit is
passive (since
it
does not contain
any
electric
energy
sources),
linear.
(impedances
of its elements are
independent
of the
amount
of current
flowing
through them),
and bilateral
(impedances
being
independent
of
direction
of
current flowing).
It
can
be shown
that
such a two-terminal pair
network
can
be
represented
by an equivalent
T- or
zr-network.
Consider the
unsymmetrical
T-network
of
Fig.
B-1, which
is
equivalenr
to
the
general
two-terminal
pair
network.
0*,
0*,
Ofi
}fz
0r,
0r,
0*,
af^
0*,
)r
^2
0*n 0r,
o*n
REFERE
N
CES
l,
Shiplcy,
R,8,, Intoduction
to
Matric:r:s
and
r976.
2. Hadley,
G., Linear
Algebra,
Addison-Wesley
3.
Bellman,
R.,
Introduction
to Matrix
Analysis,
1960.
Power
Sy,rtems,
Wiley,
New
Pub.
Co. Inc., Reading,
Mass.,
McGraw-Hill
Book
Co.. New
VJI UJ2
(A-30)
York,
1961.
York,
Flg. B-1
For
Fig. B-1,
1s-
or
Is=
*:
Unsymmetrical
T-circuit
equivalent to
a
general
nair network
r-"
"---'-"_
the
following circuit
equations
can
be
written
I^+ Y(Vo+ I^Zr)
YV*+(l+
YZr)I*
V*+
I^Zr+ IrZ,
V^ + I^2, + ZrWn+ I^2,, + I^YZ.Z,
two-terminal
(B-1)
.+
iqY
*A
transformer
is similarly
terminals.
s
lp
Vg
Z1
22
Y
Vp
represented
by a circuit
with
two input
and two
ouput
tlCtf
I
Modern
porelsyslern
inal,sis
or
V5
=
(1
+
YZr)
V* +
(2,
+
Z"
+
yZrZr)Io
Equations
(B-1)
and
(B-2)
can
be
simplified
by
letting
A-l
+
YZ,
B=
2.,+
Zr+
yZrZ,
C=Y
D=l+yZ,
AD-BC=1
(B-s)
Fig.
B-2
:#:::representation
of a
two-rerminar
pair
network
using
ABCD
CONSTANTS
FOR
VARIOUS
SIMPLE
NETWORKS
We
have
already
obtained
the
ABCD
constants.of
an unsymmetrical
T-network.
The
ABCD
constants
of
unsymmetrical
a-network
shown
in
Fig.
B-3
may
be
obtained
in
a
similar
manner
and
are given
below:
-
A=l+
YrZ
B
=
Z,
C=Yr+Yr+ZYryz
D=I+
YrZ
(B-2)
(B-3)
using
these,
Eqs.
(B-1)
and
(B-2)
can
be
written
in
matrix
form
as
lyrl lA
Bllv*1
Lr,.J
=
Lt
olj^l
,t-0,
This
equation
is
the
same
as
Eq.
(5.1)
and
is valid
for
any
linear,
passive
and
bilateral
two-terminal
pair
network.
The
constants
A, B,
C
and
D
are
called
the
generalized
circuit
constants
or
the
ABCD
constants
of the
network,
and
they
can
be
calculated
for
any
such
two-terminal
pair
network.
It
may
be
noted
that
ABCD
constants
of
a
two-terminal
pair
network
are
complex
numbers
in
general,
and
always
satisfy
the
following
relationship
Also,
for
any
symmetrical
network
the
constants
A
and,
D
are
equal.
From
Eq.
(B-4)
it is
clear
that
A
and
D
are
dimensionless,
while
B
has
the
dimensions
of
impedance
(ohms)
and
c has
the
dimensions
of admittance
(mhos).
The
ABCD
constants
are
extensively
used
in
power
system
analysis.
A
general
two-terminal
pair
network
is
often
represented
as
in
Fig.
B_2.
(B-6)
Fig. B-3
Unsymmetrical
zr-circuit
A series
impedance
often
represents
short
transmission
lines
and
transform-
ers. The
ABCD
constants
for
such
a
circuit
(as
shown
in
Fig.
B-4)
can
immediately
be determined
by inspection
of
Eqs.
(B-1)
and
(B-2),
as follows:
A=
1
B=Z
C= 0
D=l
Fig.
B-4
Series
impedance
Another
simple
circuit
of Fig. B-5
consisting
of simple
shunt
admittance
can
be
shown to
possess
the
following
ABCD
constants
A=
7
B=
0
C=Y
D=I
/El_a\
\s-u,,
Fig.
B-5
Shunt
admittance
r
It may
be noted
that
whenever
ABCD
constants
are
computed,
it
should
be
checked
that the
relation
AD-BC
=
1
is
satisfied.
For
examole,
using
Eq.
(B-8)
we
get
AD-BC=1x1-0xY=1
(B-7)
:fflfuii'l
Modern
power:
System
nnatysis
II ABCD
constants
of a
circuit are
given,
its equivalent
I-
or
a-circuit
can
be
determined
by
solving
Eq.
(B-3)
or
(8-6)
respectively,
for
the values
of
series
and
shunt branches.
For
the
equivalent
n-circuit
of Fig. B-3,
we
liuu.
(B-e)
,r=#
ABCD
CONSTANTS
OF
NETWORKS
IN
SERIES AND
FANEr,r,Ur,
Whenever
a
power
system consists
of
series and parallel
combinations
of
networks,
whose
ABCD
constants
are
known,
the
overall ABCD
constants
for
the
system
may
be determined
to analyze
the
overall clperation
of the
system.
Fig. 8-6
Networks
in
series
Consider
the two
networks
in series,
as
shown
in Fig. 8-6.
This combination
can
be reduced
to
a single
equivalent
network
as follows:
For
the
first
network.
we hal'e
t,
,,
Append8B
l,;621+
I
tr=
(ArBr+
ArBr)/(8,
+ Br)
B=
BrBzl(Bt+
Br)
C=
(Cr
*
Cr) +
(Ar
-
Ar)
(Dz-
D)l(Br
*
Ur)
(B-13)
Fig. B-7
Networks
in
parallel
Measurement
of
ABCD
Gonstants
The
generuIized
circuit constants
rnay
be computed
for
a transmission
line
which
is
being designed from
a knowledge
of the
system
impedance/admittance
parameters
using expressions
such
as those
cleveloped
above.
If\the
line is
already
built, the
generalized
circuit
constants
can
be
measured
by
making a
few
ordinary tests
on the
line.
Using Eq.
(B-4),
these
constants
can
easily be
shown to be ratios
of
either voltage
or current
at the
sending-end
to
voltage
or
current
ai the receiving-end
of the network
with
the receiving-end
open
or
short-
circuited. When
the network
is a
transformer, generator,
or
circuit
having
lumped
parameters,
voltage
and
current
measurements
at
both
ends
of the line
can be made,
and the
phase
angles
between
the
sending
and
receiving-end
quantities
can be
found out.
Thus
rhe ABCD
constants
can
be
determineC.
It is
possible,
also, to measure
the rnagnitudes
of
the required
voltages
and
currents
sinrultaneously at
both
ends of
a transmission
line,
but
there
is no
simple
method to find
the difference
in
phase
angle
between
the
quantities
at the
two
ends of the line.
Phase difference
is necessary
because
the
ABCD
constants
are complex.
By measuring
two
impedances
at each
end
of
a transmission
line,
however', the
generalized
circuit
constants
can be
computed.
The
following impedances
are to
be
measured:
7
-
oo-'l.i-- o-.l i*^o'l^-^^ ..'i+L -^^^:.,:-- ^-l ^-^- ^:-^- . :-- r
"so
-
rvrrulrS-vr:\r
uuPv\r(rrrlvs
wrlrl
r9L9rvlttE-trlru
uP(,il-ull'uulteo
Zss
=
sending-end
impedance
with
receiving-end
short
circuited
zno
=
receiving-end
impedance
with
sending-end
cpen-circuiteci
Zns
=
receiving-end
impedance
with sending-end
short-circuited
The impedances
measured from
the
sending:end
can
be
determined
in terms
of the ABCD
constants
as follows:
[u,l
_
|
o,
",
'l
[u,l
Lr,
J
-
1.,
o,
)lt, )
For
the
second
network,
we can
write
I
y*
I I
A,
Brll
vo]
I t=t || |
lrr) l_c,
DrJlr*l
From
Eqs.
(B-10)
and
(B-11),
we can
write
[u'l
_lo'
u'11o,
u'l[u^]
L+ I
-
1.,
n,ll.c,
or)j^
l
(B-
10)
(B-11)
f
A,A,
+
BtC.,
AtBz
+
qD2l
[y* I
=l
lcrAz
+
D1C2
CrB'
-t
DrD,
Jlt
o l
If two
networks
are
connected
in
parallel
as shown
in
Fig. B-7,
the ABCD
constants
of
the combined
network
ca-n be
found
out similarly with
some
simple
manipulations
of matrix
algebra.
The
results
are
presented
below:
From Eq.
@- ),
with 1R
=
0,
Zso=
Vslls= AlC
and with VR
=
0,
Vn=
DVs + BIt
In= CVs
+ AIt
From
Eq.
(8-16),
with Is
=
0,
Zno= VR|IR=
D/C
and when
Vs
=
0,
Zns= VRllR=
BIA
When the
impedances are
measured from
the receiving-end,
the direction
of
current
flow is
reversed
and hence the signs of
all current
terms in Eq.
(5.25).
We can
therefore
rewrite this equation
as
AppENDIX
C
We
know
that
the
nodal
maffix
Yu.r,
and
its
associated
Jacobian
are very
sparse,
whereas
their
inverse
matrices
are
full.
For
large
power systems
the
'sparsity
of
these
matrices
may
be
as
high
as 98Vo
and
must
be
exploited.
Apart
fiom
reducing
storage
and
time
of
computation,
sparsity
utilization
limits
the
round-off
computational
errors.
In
fact,
straight-foru'ard
application
cf
the
iterative
procedure
tor
system
studies
like
load
flow
is not
possible for large
systems
unless
the
sparsity
of
the
Jacobian
is dealt
with
effectivet).
GAUSS
ELIMINATION
One
of
the
recent
techniques
of
solving
a
set
of
linear
algebraic
equations,
called
triangular
factorizatiott,
replaces
the
use
of matrix
inverse
which
is
highly
inefficient
for
large
sparse
systems.
In
triangularization
the elements
of
.u.h
,o*
below
the
main
diagonal
are
made
zero
and
the
diagonal
element
of
each
row
is
normalized
as soon
as
the
processing
of
that
row
is completed-
It
,
is
possible to
proceed
columnwise
but
it
is computationally
inefficient
and
is
theref,re
not
used.
After
triangularization
the
solution
is easily
obtained
by
hack
sub.stitution.
The
technique
is
illustrated
in
the
example
below'
Consider
the
linear
vector-maffix
equation
(B-
14)
(B-16)
(B-
17)
(B-
18)
Solving
Eqs.
(B-14), (B-15), (B-17)
and
(B-18)
we
can obtain the
values
of
the
ABCD
constants
in terms of the
measured impedances
as follows.
AD-BC
C_
p=
[using
Eq.
(B-5)]
1
AC
IC1
AC,
A A2
AC
o=(ffi)'''
By
substituting this
value of A
in Eqs.
(B-14),
value of C so
obtained
in Eq.
(B-17),
we
get
(
z\
1l/2
r>-.7 ,
-'
I
D=LRSlz*o-znr)
(Zss(Zno
-
Zo))'''
Z^o
(B-
1e)
(B-18)
and substituting
the
(B-20)
(B-21)
(B-22)
(zso(z^o
-
zo))t''
REFERE
N CE
The Transmi,ssion
and
Distribution of
Electrical Energy,
New
Delhi. 1970.
l'
L3
1'l
['t I
,l1,,I
=
t;l
l. Cotton,
H. and H, Barber,
3rd
edn.,
B.I. Publishers,
Procedure
1. Divide row 1 by the self-element
of
the row, in this case2.
2. Eliminate the element
(2,
1) by
multiplying the modified
row 1, by
element
(2,
I) and subtract
it from
row 2.
3' Divide the modified row
2 by its self-element
(f); and stop.
Following this
procedure,
we
get
the upper
triangular equation
as
[r
+
l[''-l t+l
l
z
il t=to_+l
lo
?:rll_
l-l
,
r
L t )Lxzl
Lt J
Upon back substituting,
that is first
solving for x2 and
then for 11,
we
get
o-+
1
x2=
--7:--
j
2
-
-
I
_
I
Y
-
I
_
11_3\-
5
Check
br
+ xz= 2(*)-tr
=
t
3x,
+
5rz- 3(+)
-
s(f)
=
o
Thus,
we have demonstrated
the use of the basic Gauss
elimination
and
back
substitution
procedure
for a simple
system, but the same
procedure
applies
to
any
general
system
of linear algebraic equations,
i.e.
Ax=b
(c-1)
An added
advantage of row
processing
(elimination
of
row elements below
the main
diagonal
and
normalization of the
self-element) is that
it is easily
amenable
to the
use of low storage
compact
storage
schemes-avoiding storage
of zero elements.
GAUSS ELIMINATION
USING
TABLE
OF
FACTORS
Where
repeated
solution of
vector-matrix Pq.
(C-1j
with cons tant Abut
varying
values
of
vector D is required,
it is computationally
advantageous to
split the
matrix A into triangular
f'actor
(termed
as
'Table
of
factors' or
'LU
decomposition') using the Gauss elimination
technique. If the
matrix A is
sparse,
so
is the table of factors
which can be compactly stored
thereby
not only
reducing
core storage
requirements, but
also
the
computational
effort.
Gauss
elimination using
the
table of
factors is illustrated in the
following
example.
Consider
the
following
system
of
linear equations:
(Z)xt
+
Q)x2
+
(3)x,
=
$
(2)xr+(3)xz+(4)xt=)
(3)xr
+(4)xr+(7)4=14
(c-2)
For computer
solution,
maximum
efficiency
is attained when elimination is
carried out
by
rows
rather than
the more
tamiliar
column order. The successive
reduced
sets
of
equations
are as
follows:
(l)xr
+
(t)*z+ (*)rt
=
*
(Z)xt+
(3)x2
+
(4)xt
=
)
(3)xr
+
(4)xz
+
(7)x3
=
14
(1).r,
+
(ilrr+ (|)xt
-z
(2)xz
+
(1)x,
=
l
(3)xr+(4)x2+(7)4=14
(1)x,+
(tr)rr+
(*)",
-3
(1)x2+
(*)r,
-+
(3)xr+(4)x2+Q)xt=14
(l)xr
+
(t)rr+
(J)x,
=
3
(l)x2+
(*)"r'
=
+,
i
(t),r+(*)r,-s
(1)x,
+
(t)rr+ (*)",
=
J
(t)xr+
(t)*t
-
+
(*)rt
=
+
(1)x,
+
(tr)*r+
(*)r,
(1)x,
+
(I)*t
(c-8)
x3=
These
steps
are referred
to as
'elimination'
operations.
The
solution
r may
be
immediately
determined
by
'back
substitution' operation
using
Eq.
(C-8).
(c-3)
(c-4)
(c-s)
(c-6)
(c_'7\
_J
3
2
I