Alamo-Two Layer soil

IEEE Transactions on Power Delivery, Vol. 8, No. 4, October 1993
1890
A COMPARISON AMONG EIGHT DIFFERENT TECHNIQUES TO ACHIEVE AN OPTIMUM ESTIMATION
OF ELECTRICAL GROUNDING PARAMETERS I N TWO-LAYERED EARTH
J.L. d e l Alamo, E . E . , Ph.D., I E E E Member
Dept. o f E l e c t r i c a l Engineering
U n i v e r s i t y of V a l l a d o l i d
SDain
ABSTRACT.-This paper intends t o compare t h e performance
o f e i g h t techniques, based on f i v e d i f f e r e n t methods,
used t o f i n d t h e e l e c t r i c a l grounding parameters o f a
two-layered
earth
(resistivities
and
thickness)
corresponding
to
a specific
mathematical
model.
Parameter e s t i m a t i o n i s c a r r i e d o u t - i n such a way as t o
g e t an optimum f i t t i n g between t h e s e t of r e s i s t i v i t y
values measured i n f i e l d by means o f Wenner’s Method,
and those c a l c u l a t e d from t h e mathematical model u s i n g
such parameters.
F i r s t l y , t h e paper i s devoted t o p r e s e n t i n g a
review o f t h e mathematical b a s i s o f f i v e o p t i m i z a t i o n
methods used f o r t h e implementation o f t h e d i f f e r e n t
techniques s u b j e c t t o comparison.
Secondly, t h e paper i s dedicated t o d e f i n i n g t h e
implementation o f t h e a l g o r i t h m s f o r
the eight
techniques. Three o f them a r e o f F i r s t Order Gradlent
type, one more o f Second Order (Newton Method), t h e
f i f t h one i s based on t h e Levenberg-Marquardt Method,
t h e s i x t h one on t h e Generalized I n v e r s e Method, t h e
seventh one on a Quasi-Newton Method and t h e l a s t one
on a mixed Newton-Generalized I n v e r s e method.
Algorithms a r e a p p l i e d t o s i x t e s t cases
comparison o f t h e r e s u l t s i s shown. Furthermore,
algorithms
to
improve
existing
procedures
presented.
and
new
are
V pa(xP,r , )
Gradient of t h e f u n c t i o n mathematically
J
m o d e l l i n g t h e ground w i t h t h e parameters 5 ,
evaluated a t p o i n t P o r i n s t e p p.
x - xp
A 6
F(x)
n
i
vector
3
xp
from
= O b j e c t i v e Function f o r l e a s t squares f i t t i n g
between measured values and computed values,
according t o a s p e c i f i c mathematical model.
O b j e c t i v e Function evaluated a t p o i n t P o r i n
F(xp)
step P
Superindex t o i n d i c a t e t r a n s p o s i t i o n o f a
matrix o r vector
Residual Vector w i t h m elements o f t h e
weighted d i f f e r e n c e s . Each element i s i n t h e
(r.)
form d j = (pm(rj)-pa(x,r.))/P
J
m~
= Jacobian M a t r i x w i t h p a r t i a l d e r i v a t i v e s o f
weighted d i f f e r e n c e s w i t h respect t o each
parameter. dimension v x m.
1
J . . = -- apa(X, r .)/ax .= ad ./ax.
J1
Pm
J
1
J
1
t
D
J
V F(xp)= Gradient o f t h e O b j e c t i v e Function evaluated
a t point P o r i n step p
Dimension v x 1. V F(xp)
2 Jt
D
n
U n i t M a t r i x , dimension
H
Hessian M a t r i x o f F ( x ) evaluated a t p o i n t
o r i n step p, as b r i e f n o t a t i o n o f :
LIST OF SYMBOLS
[
Upper l a y e r r e s i s t i v i t y , i n 0.m
VF(_Xp)
1.
v x v
Dimension
P
v x v
Lower l a y e r r e s i s t i v i t y , i n Q.m
R e f l e c t i o n c o e f f i c i e n t ( p 2 - p , )/(p2+P,
1
Upper l a y e r t h i c k n e s s
Vector with parameters d e f i n i n g t h e s o i l .
I n general w i t h v elements Xi
3
For two-layer v
Probe spacing (F.Wenner),
r/z
(i-1
J
E
i n meters.
V
P2’Pl
j
J
second d e r i v a t i v e s .
= Vector o f weighted d i f f e r e n c e s , w i t h j rows.
I t s elements (Generalized I n v e r s e Method) are:
.....v).
Number o f f i e l d measurements.
Measured r e s i s t i v i t y (Q.m) f o r each spacing rJ.
p,(&,r.)=
= M a t r i x o f weighted r e s i d u a l s d . and t h e i r
R
.....
= 1
m. B r i e f l y pm.
Apparent r e s i s t i v i t y (0.m).
11
E
]I2=
Gradient o f t h e squared kuclidean Norm o f
E
i n the
Vector o f weighted d i f f e r e n c e s
Generalized I n v e r s e Method.
INTRODUCTION
Computed value
u s i n g t h e mathematical model f o r each spacing
r . and v e c t o r 8
B r i e f l y pa.
J
.
93 WM 079-4 PWRD
A paper recommended and approved
by the IEEE Transmission and Distribution Committee
o f the IEEE Power Engineering Society for presentation at the IEEE/PES 1993 Winter Meeting, Columbus,
OH, January 31 - February 5, 1993. Manuscript submitted January 31, 1992; made available for printing
November 3 , 1992.
A t t h e i n i t i a l s t e p o f e l e c t r i c a l grounding design
o f a S t a t i o n , Substation, Transformer D i s t r i b u t i o n o r a
L i n e support, i t i s necessary t o o b t a i n t h e parameters
d e f i n i n g t h e s o i l where such grounding i s going t o be
done. The parameter e s t i m a t i o n must always be c a r r i e d
o u t based on a s e t o f measurements o f s o i l r e s i s t i v i t y .
The i n t e r p r e t a t i o n o f measurements c a r r i e d o u t i n
t h e f i e l d through a technique corresponding t o a
s p e c i f i c mathematical model o f t h e s o i l c o n s i s t s i n
f i n d i n g a s e t o f e l e c t r i c a l grounding parameters
d e f i n i n g i t s e l f i n such a way as t o a t t a i n a good
fitness,
preferably
an optimum f i t n e s s ,
between
measured values and those which would r e s u l t u s i n g such
parameters i n t h e mathematical model.
0885-8977/93/$03.000 1993 IEEE
1891
The m a j o r i t y of t h e times measurements w i l l
i n d i c a t e t h a t t h e s o i l , w i t h i n a l i m i t e d extension,
behaves as i f i t was composed [ S I [ 9 ] [12] [15] o f two
h o r i z o n t a l layers.
3rd.-We admit t h a t we have a s e t o f r e s i s t i v i t y
values p ( r . ) c o l l e c t e d i n f i e l d f o r d i f f e r e n t spacing
m~
rj
(j
1
.......m),
4th.-The
I n t h i s paper we w i l l
consider t h a t s o i l
measurements have been made through t h e F. Wenner
Method, and so t h e mathematical model corresponds t o
t h i s Method. Along t h i s , several i n v e s t i g a t o r s have
with
solutions
to
the
problem
of
contributed
two-layered parameter s o i l e s t i m a t i o n t o achieve t h e
b e s t p o s s i b l e f i t between t h e measured and computed
E P R I [61,
r e s i s t i v i t y values. The c o n t r i b u t i o n s from
C.J.Blatner
[7],
ANSI/IEEE
[91,
F.Dawalibi
and
[ l o ] , C.J.Blatner [121, A.P.Meliopoulos
C.J.Blatner
[13] and, r e c e n t l y , J.L.de1
Alamo [ l a ] should be
highlighted.
I n [16]
a very compact f o r m u l a t i o n f o r t h e
a n a l y s i s o f t h e e v o l u t i o n forms o f c a l c u l a t e d apparent
r e s i s t i v i t y i n m u l t i - l a y e r s o i l s i s presented, and an
e s t i m a t i o n procedure i s c a r r i e d o u t based i n a p - r
comparison technique. However no a n a l y t i c a l s o l u t i o n t o
t h e t h e problem o f parameter e s t i m a t i o n i n such kinds
o f s o i l s i s included.
I n references [ll] and [13] e s t i m a t i o n o f s o i l
parameters i s made when measurements have been c a r r i e d
o u t f o l l o w i n g t h e d r i v e n rod method. I n [12] a
comparison o f r e s u l t s i s shown.
As i s i n d i c a t e d i n [ 7 ] [13] and 1181, t h e m a j o r i t y
o f these techniques a r e n o t very e f f e c t i v e because,
even i f t h e s o i l was a b l e t o be shaped as two-layers,
t h e a l g o r i t h m s present slow convergence o r have n o t
enough s t a b i l i t y t o converge t o t h e s o l u t i o n .
However
i t has been shown [ l a ] t h a t a second order g r a d i e n t
technique i s capable o f a r r i v i n g a t t h e intended
absolute minimum o f t h e O b j e c t i v e Function.
The research work c o n s t i t u t i n g t h e s u b j e c t o f t h i s
paper i s one o f t h e f i r s t r e s u l t s from t h e Research
P r o j e c t No. 132197 sponsored by both OCIDE, O f i c i n a de
Coordination e I n v e s t i g a c i d n de D e s a r r o l l o E l e c t r o t k c n i c o (Coordinating Bureau o f Research and E l e c t r i c a l
Engineering Development), and t h e Company
IBERDROLA,
i n Spain, t o d e f i n e t h e b e s t t o o l f o r optimal parameter
e s t i m a t i o n i n a m u l t i l a y e r s o i l from r e s i s t i v i t y
measurements through t h e Wenner Method.
METHODOLOGY
We wish t o compare t h e e f f i c i e n c y o f d i f f e r e n t
methods o f s o i l parameter e s t i m a t i o n f o r t h e optimal
f i t between a s e t o f measured r e s i s t i v i t y values i n
f i e l d and those which would r e s u l t from t h e a p p l i c a t i o n
of a mathematical model u s i n g these parameters.
For t h i s o b j e c t i v e we use a homogeneous treatment
o f t h e problem, based on t h e f o l l o w i n g hypothesis:
1st.-We admit t h a t t h e s o i l i s composed by two
horizontal
homogeneous
layers
or
strata,
being
s e m i - i n f i n i t e , with d i f f e r e n t r e s i s t i v i t y , p1 and P2 ,
and z
being t h e t h i c k n e s s o f t h e upper l a y e r . L e t
K = ( p -p )/(p2+pl)
the r e f l e c t i o n coefficient.
2 1
2nd.-For t h e computed apparent r e s i s t i v i t y , u s i n g
t h e F.Wenner Method as model, w i t h spacing r between
probe electrodes, we w i l l use [81 [ l o ] 1121:
where
A
1
=
+
l.....m
( 2 n
z / r ) 2 and B = A +
pm(r1), pm(r2)
6
vector
,.....Pm ( r m1.
defining
the
soil
parameters w i l l be used i n each treatment:
? =
[z']
or, a l t e r n a t i v e l y f o r the
comparison w i t h t h e proposed
technique i n [ 9 ] :
[?I
_X
5h.-We w i l l n o t e t h e weighted d i f f e r e n c e d .
J
(j =
l...m)
between measured and computed values f o r a
c e r t a i n spacing r j and w i t h a s e t o f parameters 5 , by
t h e expression:
J
t
'
t h e weighted d i f f e r e n c e s w i l l c o n s t i t u t e t h e
v e c t o r D, w i t h dimension m x 1.
All
6th.-We w i l l consider t h a t t h e s e t o f parameters
&
which b e s t shapes t h e s o i l i s t h a t which optimizes t h e
square o f a f u n c t i o n F(X), d e f i n e d as t h e Euclidean
,I1
D 112, o f v e c t o r D. T h i s corresponds t o t h e
Norm
m a j o r i t y o f t h e adjustment f u n c t i o n s used i n t h e
l i t e r a t u r e [61 [ l o ] as:
j
=
t....m
METHODS TO BE USED
To compare t h e procedures we w i l l
implement
e s t i m a t i o n techniques based on t h e f o l l o w i n g methods:
Steepest descent method
Levenberg-Marquardt method
Newton method
Generalized I n v e r s e method
Quasi-Newton method
STEEPEST DESCENT METHOD
Based on t h i s method,
several techniques o f
parameter e s t i m a t i o n have been presented, such as those
proposed i n EPRI Research P r o j e c t 149-Report EL 2699,
References
[61 [ l o ] ,
and ANSI/IEEE Std. 81-1983
Ref.[9]. Also presented i n t h i s paper i s an a u x i l i a r y
technique based on t h i s method t o achieve an absolute
minimum f o r t h e o b j e c t i v e f u n c t i o n o f e r r o r s between
measured and comouted values.
I t i s known t h a t t h e negative g r a d i e n t i s i n t h e
d i r e c t i o n o f descent step. So approaching t h e minimum
w i l l be c a r r i e d o u t through an i t e r a t i v e procedure such
that:
XP+'
= Xp
-
-
A
V F(Zp)
Where A i s a p o s i t i v e s c a l e f a c t o r .
The main problems o f convergence, when u s i n g a
technique o f t h e steepest descent method, a r e caused by
t h e s c a l e f a c t o r A.
To avoid i n p a r t t h e s c a l i n g problem we can s e l e c t
an u n i t v e c t o r d e f i n e d as:
VF(Xp)
/ IIF(&p)I(
and then t h e process o f approaching t h e minimum w i l l
be expressed i n t h e form [181:
Pa = P1 ( 1 + 4
n
i.e.
same
3
1892
Steepest descent method i s a poor search s t r a t e g y .
I t u s u a l l y works q u i t e w e l l [ 5 1 d u r i n g e a r l y stages o f
t h e o p t i m i z a t i o n process. However, as a s t a t i o n a r y
p o i n t i s approached, t h e method u s u a l l y i s i n e f f i c i e n t
when small orthogonal steps are taken.
F i r s t d e r i v a t i v e s o f t h e O b j e c t i v e Function are
needed f o r a l l o f t h e techniques being analyzed. A l l
t h e r e q u i r e d i n f o r m a t i o n i s gathered i n t h e Appendix.
must
It
formulation,
be
noted
that
according
t h e changes i n v e c t o r
to
from _Xp
to
P
where
U
zero,
the following
+
[ H
i s called
CT
P ]
AXp
=
minimum c o n d i t i o n
-
is
VF(X P )
t h e Levenberg-Marquardt parameter.
For t h e problem we a r e considering, and according
t o t h e f o r m u l a t i o n i n t h e Appendix:
Ift h e r e s i d u a l s were l i n e a r , then:
that
5’’
can be computed through t h e expression:
where
Lagrangian
reached:
(4)
I t must be noted t h a t , i n a c e r t a i n way [ 1 4 ] , t h e
m a t r i x o f r e s i d u a l s R and i t s second d e r i v a t i v e s have
been s u b s t i t u t e d by t h e diagonal m a t r i x CT 1.
i s a s c a l e f a c t o r s i m i l a r t o X.
One o f t h e d i f f i c u l t i e s i n t h e implementation o f
t h i s method [14] c o n s i s t s i n c o n t r o l l i n g t h e value o f
NEWTON METHOD
Considering t h a t around a p o i n t ( v e c t o r )
Xp,
parameter
the
O b j e c t i v e Function i s shaped by means o f a q u a d r a t i c
f u n c t i o n , t h e value o f t h e f u n c t i o n a t any p o i n t i n t h e
p r o x i m i t y o f t h a t p o i n t can be expressed by:
The sequence o f searching o f t h e minimum has t h e
form [ 5 ] :
AEp = - H-’ VF(Xp)
The elements o f t h e Hessian m a t r i x (and those o f
t h e Gradient v e c t o r ) are reported i n t h e Appendix.
The
descent
property
is
paramount
for
an
o p t i m i z a t i o n a l g o r i t h m , t h a t i s , each s t e p must proceed
d o w n h i l l [14]. I n Newton’s method t h i s i s p o s s i b l e , f o r
any
AX i f and o n l y i f t h e Hessian i s p o s i t i v e
definite.
Newton’s Method, i n general, does n o t converge t o
a s t a t i o n a r y p o i n t [ 5 ] when we s t a r t w i t h an a r b i t r a r y
i n i t i a l p o i n t . For t h e problem we are considering, and
according t o t h e f o r m u l a t i o n o f t h e Appendix:
(3)
Jt
J
i s always p o s i t i v e - d e f i n i t e .
The
where
p o s s i b i l i t i e s o f approaching t h e minimum are then
R , which
contains
dependent, mainly, on t h e m a t r i x
t h e r e s i d u a l s and t h e i r second d e r i v a t i v e s . Obviously,
and according t o t h e preceding e x p o s i t i o n ,
i f we
succeed by means o f an a u x i l i a r y technique i n g e t t i n g
a l l t h e elements o f
R
o f small magnitude, t h e
a l g o r i t h m w i l l s u r e l y converge t o a minimum.
T h i s i s p r e c i s e l y t h e reason f o r t h e e f f i c i e n c y o f
t h e technique proposed i n Ref. [ l a ] , i n which t h e
i n i t i a l process o f approaching i s assigned t o a F i r s t
Order Gradient Technique, where t h e r e s u l t s are, as
i n d i c a t e d , very e f f e c t i v e .
0.
=
On t h e o t h e r hand, as VF(Xp)
f o l l o w s t h a t , a t t h e minimum where VF(Xp)
then 9 must be
t h e minimum, t h e
accurate b u t t h e
To a v o i d t h i s ,
must be used.
2,
.Ut
D it
if D # @
s i n g u l a r . Consequently, as we approach
successive steps become more and more
Jacobian M a t r i x i s more d e t e r i o r a t e d .
Jackson’s s i n g u l a r decomposition [31
GENERALIZED INVERSE METHOD
T h i s technique i s presented i n [ll] and [13]
applied
to
the
parameter
estimation
using
as
mathematical model t h a t corresponding t o t h e Resistance
measurements through t h e Driven Rod Method. As a
complement i t i s i n d i c a t e d i n [13] t h a t t h i s technique
can be used when t h e measurement method i s t h e Wenner
Method. I n such case parameters pl, p 2 , z are used.
I n our case we have examined t h e p o s s i b i l i t i e s o f
t h e Generalized I n v e r s e Method o b t a i n i n g s y s t e m a t i c a l l y
t h e f i e l d measurements through t h e Wenner Method and
u s i n g as parameters pl, U, z.
The o u t l i n e o f t h e method i s as f o l l o w s :
L e t us assume t h a t we. are a b l e t o estimate a s e t
o f parameters X . (i = l...v,
i n our case v = 31, such
t h a t t h e r e s i s t i v i t y values obtained through t h e
r.
w i l l be
mathematical model f o r every spacing
J
s u f f i c i e n t l y c l o s e t o t h e measured r e s i s t i v i t y values
p m ( r . ) , so t h a t we can evaluate them from an T a y l o r
J
expansion around t h e computed values
p (X,r.)
a -
J
That i s :
The weighted d i f f e r e n c e ( e r r o r ) between measured
and computed values w i l l appear i n t h e f o l l o w i n g form
f o r every spacing r .
J
LEVENBERG-MARQUARDT METHOD
The aim o f t h e Levenberg Method i s t o search f o r ,
a t each s t e p o f approaching t h e minimum, t h e smallest
value reached by t h e preceding approximation of F(_X),
i n such a way t h a t t h e change
AX r e q u i r e d t o f i n d i t
i s r e s t r i c t e d t o a hypersphere o f u n i t r a d i u s which
c e n t e r s on XP. I n o t h e r words, a t each step we t r y t o :
minimize
such t h a t
Making
the
F(X)
AXt.
derivatives
AX
of
= 1
the
corresponding
111
j
= l...m
J
; i
i
= 1...3
and t a k i n g i n t o account t h a t t h e expressions o f t h e
elements o f
D
and
J , (see l i s t o f symbols and
Appendix), i t f o l l o w s t h a t f o r t h e v e c t o r E w i t h m
elements e . :
J
E = D + Jt AX
We e s t a b l i s h t h a t t h e best e s t i m a t i o n o f t h e
parameters o f t h e model w i l l be t h a t which makes t h e
Euclidean Norm square o f t h i s e r r o r v e c t o r minimum,
that is:
1893
minimize
11
b).-Newton Method
112
E
AXp
f o r which
= - [ Jt J
c).-Levenberg-Marquardt
a
- 11
ax
E
112 =
0
that is, for j
=
1.
. .m
= -
Axp
d).-Generalized
2 Jt D
and t h e second
2 ( Jt J
1,
R I-' .Jt D
[ Jt
J
+
n I
I-' Jt
D
I n v e r s e Method
AXp
Consulting t h e Appendix we see t h a t t h e f i r s t term
i n t h e preceding expression i s :
+
Method
f;
-
[ lIt 9
e).-Quasi-Newton Method
XP+l - Xp = A Xp = - l7-l
I-'
Jt D
( V F(Xp)
f o r m u l a t i o n necessary
methods has been presented.
All
to
-
V
F(Xp+'))
implement
such
t h e r e f o r e , t h e sequence o f minimum searching i s :
IMPLEMENTATION
AXP
- [ .Ut J I-'Jt D
(5)
Note t h a t i n t h i s case, t h e f u n c t i o n t o be
optimized i s n o t t h e Euclidean Norm square o f D, as i n
t h e preceding techniques, b u t t h e Euclidean Norm square
o f E. I n any case t h e f i t i s performed over weighted
d i f f e r e n c e s . For comparative purposes, when a p p l y i n g
t h e Generalized I n v e r s e Technique,
we w i l l a l s o
The techniques
implemented so f a r ,
i n the
parameter e s t i m a t i o n problem o f a ground shaped as
two-layers,
from measurements made u s i n g Wenner's
method have been based on: F i r s t Order Gradient [ 9 ] [ 6 ]
[ l o ] , Generalized I n v e r s e [13] and Second Order
Gradient [18]. These techniques are implemented here i n
order t o compare them w i t h t h e new proposed techniques,
and are a p p l i e d t o t h e same cases.
evaluate t h e o b j e c t i v e f u n c t i o n
11
D
112
The customary d i f f i c u l t i e s i n t h e a p p l i c a t i o n o f
this
technique
are
similar
to
those
of
the
Levenberg-Marquardt Method, as r e l a t e d t o t h e product
Jt
J
. Consequently
t h e Jackson S i n g u l a r Decomposition
[3]
should be applied.
Indeed,
if
JtJ
is
approximately s i n g u l a r , one o r more o f i t s eigenvalues
c o u l d be c l o s e t o zero. I t can be seen t h a t a small
eigenvalue w i l l cause a l a r g e change i n one o r more o f
t h e elements o f A5.
Similarly,
given t h a t t h e
variances o f t h e model parameters are i n v e r s e l y
p r o p o r t i o n a l t o t h e square r o o t o f t h e eigenvalues [ 3 ] ,
i t may happen t h a t small values o f variance w i l l y i e l d
l a r g e standard d e v i a t i o n s .
QUASI-NEWTON METHOD
This
term
refers t o the
methods
i n which t h e
minimum searching d i r e c t i o n a t every p o i n t
t h e form
- A VF(Xp),
where
A
Xp
i s of
i s a positive-definite
m a t r i x approximating t h e i n v e r s e o f t h e Hessian M a t r i x .
The
gradient
direction
is
then
deflected
by
p r e m u l t i p l y i n g i t by
A . Probably, t h e Quasi-Newton
Methods a r e t h e most e f f e c t i v e n o n l i n e a r o p t i m i z a t i o n
methods f o r general problems [14].
I n a Quasi-Newton
Method,
the d i r e c t i o n o f
movement towards t h e minimum i s e s t a b l i s h e d such t h a t :
The a l g o r i t h m used f o r t h e treatment o f t h e
problem we are c o n s i d e r i n g i n t h e a n a l y s i s o f t h e cases
proposed i n 1181, i s t h e same one proposed i n 141, and
now implemented by means o f t h e subroutine ZXMIN from
t h e IMSL l i b r a r y [81.
GENERAL CONSIDERATIONS
For each implemented technique:
-The s t a r t i n g p o i n t i s from a s e t o f f i e l d
r e s i s t i v i t y measurements, u s i n g Wenner's Method.
-The i n i t i a l values f o r each i t e r a t i v e process
are:
For
py , t h e r e s i s t i v i t y corresponding t o t h e
smaller d i s t a n c e o f t h e measurements l i s t
For
,
pi
t h e r e s i s t i v i t y corresponding t o t h e
g r e a t e r d i s t a n c e o f t h e measurements l i s t .
KO, i f a p p l i c a b l e ,
For
t h e preceding c r i t e r i a .
For zo,
a concordant value w i t h
1 m i n a l l cases.
- A l l t h e software Programs have been executed i n a
HP, Vectra RS 20/C w i t h mathematical coprocessor.
For t h e purposes o f f i n a l comparison, t h e value o f
t h e same O b j e c t i v e Function F, t h a t i s , 1) D 112, t h e
Euclidean Norm o f OF, t h a t i s )(VF(X)II, and t h e run t i m e
are evaluated.
CASES TO BE SOLVED
SUMMARY
The r e s u l t s i n r e l a t i o n t o i t e r a t i v e process
f o r m u l a t i o n i n t h e search f o r t h e Optimum i n t h e
d i f f e r e n t methods examined f o r t h e treatment o f our
problem, can be summarized as: (see l i s t o f symbols)
a).-Steepest
I n a d d i t i o n t o t h e preceding techniques, t h e
f o l l o w i n g techniques are a p p l i e d t o s i x s e l e c t e d t e s t
cases [18]: An improved F i r s t Order Gradient which
a l l o w s reaching t h e absolute minimum o f t h e o b j e c t i v e
f u n c t i o n ; a technique based on t h e Levenberg-Marquardt
method; another one based on t h e Generalized I n v e r s e
Method s i m i l a r t o [ I 3 1 b u t a p p l i e d t o d i f f e r e n t s e t o f
parameters; one more based on a Quasi-Newton Method
and; f i n a l l y , a mixed Second Order-Generalized I n v e r s e
technique which a l l o w s an improvement over t h e one
proposed i n [18] regarding run-time.
Descent method
AXp
= -
I-( Jt
D
The techniques w i l l be a p p l i e d t o s i x cases
proposed i n [18]. Case U1 corresponds t o references [61
[lo].
Most o f t h e Programs have been b u i l t on TurboC and
TurboC++ (Borland) v e r s i o n 2.0. Two o f t h e techniques
have been implemented u s i n g IMSL r o u t i n e s , i n i t i a l l y
foreseen t o be run on VAX 11 and adapted t o be compiled
i n FORTRAN M i c r o s o f t Version 4.0 and L i n k e r Version
5.01.20.
1894
CAHRT. METHODS OF [hl AND
Fig.l.-FLOW
[e]
Meas: File *.OAT
=
Niterup
/Guess a vector as s t a r t i n g p o i n t
I
I
500
COmPute
I1
Compute Gradient's Components
vn~(xP)=
Normalize it
1
VF(&P)/lVF(Xp)~
F(~P:
FOi??l THE CHANGES
-
=
VF(Xp)
- 0,
I
r
111
111
VF(x_C)
normalize i t
XVnFt:P)
searching the maximum
decrease o f
F(z)
I
CORRECT the Vector t o the
IV
NEW value
[ A I vector
Build
$-I
=
E~
+
I
V
VI
Ye;bSOLVE
VII
Fig.2.-FLOW
CHART. PROPOSED FIRST ORDER GRADIENT TECHNIQUE
Fig.4.-FLOW W R T . P R O W S E O TECHNIQUE BASED IN A
GENERALIZED INVERSE METHOD
Meas.Files *.DAT
.Niterup = 500
R a n d m steps=RST=100
Heas: File *.DAT
Niterup
+
___)
I1
C m p u t e Objective Funct. F(Xp)
c m p u t e Gradient's c a n w n e n t s
Normalize it
VnF(Xp)
I
I
k =
+
,
I
'
I1
~~
I
= 50
VF(X~)
= VF(Xp)/lVF(~')~
I
11
I1
I11
I
Generate r a n d m l y
Build [XI vector
Compute 5; = 5' -[XIhnF(xp)
I
C m p u t e D and
I
cmpute
[ Jt J I-'
Jt D
i-'
p:pt1
I
Cmpute
IV
II
"I1
,m,o
A
cl
Resu 1ts
I
t
xP1z
3' -
[JtJ]-'JtD
ill
Results
1
1895
1 s t .-IMPLEMENTATION OF THE METHOD PROPOSED I N ANSI/IEEE
Std.81-1983,
Ref. 191
T h i s i s a F i r s t Order Gradient Technique (FOGT).
It uses as parameters
pl, p2 , z, and i t has been
implemented according t o t h e Flow Chart shown i n F i g . l ,
by means o f a TurboC Program.
I n t h e Stage 11, f o r t h e c a l c u l a t i o n o f t h e
O b j e c t i v e Function and f o r
lack o f an e x p l i c i t
c r i t e r i o n i n 191, t h e r e q u i r e d terms a r e used u n t i l t h e
new term brought t o t h e s e r i e s w i l l c o n t r i b u t e with a
value l e s s than l.E-5 o f t h e sum c a r r i e d out. I n any
case, t h e maximum number o f terms o f t h e s e r i e s i s
r e s t r i c t e d t o 10,000.
I n Stage 111, although n o t i n d i c a t e d i n Ref. [ 9 ] ,
we have normalized t h e Gradient, u s i n g f o r t h i s t h e
Euclidean Norm, as otherwise i t i s impossible t o reach
a solution.
The elements o f t h e diagonal o f [XI have been
taken as z = .005 Ipl I , .005 lp21 and I z I , (see Ref.
[91). The c r i t e r i o n t o s t o p t h e i t e r a t i v e process when
IAF(X)Itl.E-3,
as i n d i c a t e d i n [91, has been v e r i f i e d
as a b s o l u t e l y i n e f f e c t i v e , s i n c e t h e process f u l f i l l s
t h i s c o n d i c t i o n almost immediately, s t i l l being f a r
from t h e minimum, as can e a s i l y be checked. A value o f
l.E-5 has been used instead.
2nd.-IMPLEMENTATION OF THE METHOD PROPOSED I N EPRI
-Research P r o j e c t 1491 Report EL 2699, Ref. [61
and [ l o 1
T h i s i s a F i r s t Order Gradient Technique (FOGT).
I t uses as parameters p , K , z.
I t has been implemented u s i n g t h e same Flow Chart
o f F i g . l by means o f a Program w r i t t e n i n TurboC.
I n Stage 11, f o r t h e c a l c u l a t i o n o f t h e O b j e c t i v e
Function, t h e r e q u i r e d terms a r e used u n t i l t h e new
term brought t o t h e s e r i e s w i l l c o n t r i b u t e w i t h a value
l e s s than .001 o f t h e sum c a r r i e d out.
I n Stage 111, although t h i s i s n o t i n d i c a t e d i n
Ref. [61, we have normalized t h e Gradient, u s i n g f o r
t h i s t h e Euclidean Norm, as suggested i n , as otherwise
i t i s impossible t o reach a s o l u t i o n .
I n Refs.[6] and [ l o ] no c r i t e r i o n i s s p e c i f i e d f o r
t h e s e l e c t i o n o f [1], except t h a t a "proper s e l e c t i o n "
must be made. Thus we assume t h a t t h e authors adopt f o r
t h e diagonal m a t r i x [ X I a value according t o R e f . [ 9 ] ,
t h a t i s T; = .005 IplI , .005 I P 2 1 and .005 I z I .
The c r i t e r i o n t o s t o p t h e i t e r a t i v e process when
IAF(X)I<l.E-3, as i n d i c a t e d i n
[61 1101
has been
v e r i f i e d as a b s o l u t e l y i n e f f e c t i v e , s i n c e t h e process
f u l f i l l s t h i s c o n d i t i o n almost immediately, s t i l l being
f a r from t h e minimum, as can e a s i l y be checked. A value
o f l.E-5 has been used instead.
3rd.-PROPOSAL
OF
TECHNIQUE
is
AN
IMPROVED
FIRST
ORDER
GRADIENT
An improved F i r s t Order Gradient Technique (FOGT)
presented,
u s i n g as parameters
K,
Z,
implemented f o l l o w i n g t h e f l o w diagram o f Fig.
means o f a TurboC Program.
2, by
terms of t h e s e r i e s i s r e s t r i c t e d t o 1,000.
I n Stage I V , a random p o s i t i v e number i n
i n t e r v a l ( 0 , l ) i s generated f o r t h e s c a l e f a c t o r
T h i s i s s i m i l a r t o t h e treatment t h a t used i n [ l a ]
one o f i t s c a l c u l a t i o n stages. We use as v e c t o r
scale f a c t o r s 1000 1. .02 1 and .75 1
the
1.
in
of
The Euclidean Norm o f t h e Gradient v e c t o r
v e r i f i e d t o be l e s s than l.E-3 t o make a p r i n t - o u t
the results.
is
of
4th.-SECOND
ORDER GRADIENT TECHNIQUE (SOGT)
I t i s a technique based on t h e Newton Method which
takes advantage, o f t h e behavior o f a F i r s t Order
Gradient technique f o r approaching t h e minimum, and o f
t.he q u a d r a t i c convergence o f t h e Newton Method when i t
i s i n an area c l o s e t o t h e s o l u t i o n .
The a l g o r i t h m was described i n Ref.
implemented i n TurboC. I t s Flow Chart i s
Fig.3.
5th.-TECHNIQUE
(LMT 1
BASED
ON
We use as parameters
[181 and
shown i n
LEVENBERG-MARQUARDT
METHOD
, z.
p, K
I t has been implemented by means o f t h e Routine
ZXSSQ from Ref. [81 w i t h o u t e x p l i c i t c a l c u l a t i o n o f t h e
f i r s t d e r i v a t i v e s . For c a l c u l a t e d apparent r e s i s t i v i t y ,
t h e I n t e g r a l Form [181 has been used, given t h a t t h e
program i s being used t o i n v e s t i g a t e t h e p o s s i b i l i t i e s
o f m u l t i - l a y e r modelling. The r e s u l t s do n o t d i f f e r
from those obtained u s i n g t h e i n f i n i t e s e r i e s . The
source code used t o d e f i n e t h e Function has been
implemented i n FORTRAN 77. For t h e e v a l u a t i o n o f t h e
i n t e g r a l , t h e Romberg Method has been used, being
implemented [17] by means o f QROMO, MIDINF and POLINT
Routines.
The BesSel Function values
have been
approximated
by
polynomials
using
the
Function
BESSJO(X) from Ref. 1171.
As convergence c r i t e r i o n , t h e Euclidean Norm o f
t h e Gradient o f t h e O b j e c t i v e Function i s r e q u i r e d t o
be smaller than l.E-5.
6th. -TECHNIQUE BASED ON THE INVERSE GENERALIZED METHOD
(IGT 1
We use as parameters
pl,
,
K
z.
I t has been implemented using t h e Flow Chart shown
i n Fig.4, by means o f a TurboC Program. The process i s
stopped when t h e v a r i a t i o n i n each parameter i s l e s s
than l.E-5.
The Program has been implemented i n such a way as
t o r e j e c t t h e measurements o u t s i d e o f an a d j u s t a b l e
range o f e r r o r , by d e f a u l t 15%, t o avoid s i n g u l a r i t i e s .
7th.-TECHNIQUE
BASED ON A QUASI-NEWTON METHOD (QNT)
We use as parameters
pl,
K
,
z.
I t has been implemented by means o f ZXMIN Routine
from R e f . [ 8 ] , w i t h o u t e x p l i c i t c a l c u l a t i o n o f t h e f i r s t
d e r i v a t i v e s . For c a l c u l a t e d apparent r e s i s t i v i t y t h e
I n t e g r a l Form has been used, because t h e program i s
being used t o i n v e s t i g a t e
the possibilities o f
multi-layer
modelling.
The
Hessian
Matrix
is
i n i t i a l i z e d as a u n i t m a t r i x .
I n Stage 11, t h e sum o f t h e terms f o r t h e
c a l c u l a t i o n o f F(8) i s stopped when t h e c o n t r i b u t i o n o f
8th.-TECHNIQUE
a new term t o t h e s e r i e s i s l e s s than 1.E-5 o f t h e sum
already c a r r i e d out. I n any case, t h e maximum number o f
The
exposed
considerations
concerning
the
convergence c h a r a c t e r i s t i c s o f t h e Newton Method when
BASED ON A MIXED METHOD (MMT)
1896
t h e process of approaching t h e minimum i s c a r r i e d o u t
i n t h e p r o x i m i t i e s o f t h e minimum, were already used
favorably i n [181, s t a t i n g t h e t o t a l e f f e c t i v e n e s s o f
t h e procedure.
acceptable values i n almost any cases. The lst(F0GT)
Technique achieves acceptable values, which does
not
happen with t h e 2th(FOGT).
TABLE V I I 1 . - M I X E D
I t i s expected t h a t , improving t h e i n i t i a l values,
t h e process w i l l converge w i t h g r e a t e r s e c u r i t y and
rapidity
to
the
minimum.
Among t h e
techniques
examinated, as we w i l l see i n t h e f o l l o w i n g paragraph,
t h e one which reaches t o r e s u l t s c l o s e t o t h e optimum
w i t h a h i g h r a p i d i t y i s t h e technique based on t h e I G T ,
and w i t h enough accuracy t o be u s e f u l i n a h i g h number
o f practical applications.
METHOD RESULTS:
Iterations :
8 o f INVERSE GENERALIZED TECHNIQUE (0.08 min)
1 FIRST ORDER GRADIENT and
4 SECOND ORDER GRADIENT (0.40 min)
GRADIENT Maximum p e r m i s s i b l e value
1E-005
The proposal c o n s i s t s i n u s i n g as i n i t i a l values,
f o r t h e Newton Technique presented i n [18], those
p r e v i o u s l y generated by t h e technique based on t h e I G T
as presented i n t h i s paper. To o b t a i n t h e Flow Chart,
i t i s s u f f i c i e n t t o l i n k those o f F i g ' s 4 and 3.
HESSIAN M a t r i x . Last i t e r a t i o n
RESULTS
STARTING VALUES (two-layer s t r u c t u r e )
UPPER LAYER R e s i s t i v i t y (0.m) = 30.2
BOTTOM LAYER R e s i s t i v i t y (0.m) = 7.1
DEPTH UPPER l a y e r (m) = 1
OBJECTIVE FUNCTION = 6.35825484
I n Tables I through V I , t h e values obtained by
a p p l i c a t i o n of these techniques t o t h e proposed cases
i n [181 are c o l l e c t e d . The estimated values f o r pl, p2,
z,
t h e value o f t h e O b j e c t i v e Function F, Euclidean
Norm o f t h e Gradient and Run Time a r e reported.
The e x c e l l e n t r e s u l t s obtained can be appreciated,
i n general, with t h e 4th(SOGT), 5th(LMT), 7th(QNT) and
8th(MMT) techniques.
I n Table V I 1 t h e number o f
i t e r a t i o n s r e q u i r e d f o r t h e MMT, compared t o t h e SOGT
i s shown, where i t can be appreciated t h a t t h e MMT i s
even b e t t e r than t h e SOGT from Ref. [181. T h i s i s t r u e
w i t h respect t h e run-times,
which a r e d r a s t i c a l l y
smaller than those of t h e SOGT.
I t must be noted, on t h e o t h e r hand, t h a t i n t h e
proposed MMT, t h e i t e r a t i o n s o f F i r s t Order Gradient
Technique a r e almost unnecessary. I t i s v e r i f i e d ,
i n d i r e c t l y , t h a t i n t h e process o f approaching t h e
minimum, t h e I G T i s much more e f f e c t i v e t h a t any FOGT
1, as i n t h e case
which employs t h e diagonal m a t r i x
o f t h e SOGT when i t uses a FOGT i n i t s process o f
approaching t h e minimum.
(SOGT)
Ref.[91
1 s t Order
2nd Order
(MMT)
1 s t Order
This
paper
2nd Order
I
I
14
2
4
27
5
11
9
10
8
7
9
21
1
0
1
2
0
4
5
5
5
5
5
6
I
As an example, t h e corresponding r e s u l t s , from
a p p l y i n g t h i s MMT technique, t o a s e t o f measured
values f o r i m p l a n t a t i o n o f a Substation i n Meleno,
LeganBs, MADRID (Spain), a r e l i s t e d i n Table V I I I . The
discrepancy i s expressed i n percent as: 100 (pm-pa)/pa
I n r e l a t i o n w i t h t h e value o f t h e O b j e c t i v e
f u n c t i o n F. t h e 4 t h (SOGT), 8 t h (MMT), 5 t h (LMT), 7 t h
(QNT),
and 3 r d (FOGT) Techniques are p r a c t i c a l l y
e q u i v a l e n t , where indeed t h e F i r s t Order Gradient
technique, proposed i n 3 r d p l a c e i s included. The 6 t h
(IGMT), 1 s t (FOGT) and 2nd (FOGT) techniques , i n t h i s
order have been shown t o be s l i g h t l y i n f e r i o r .
From t h e p o i n t o f view o f Euclidean Norm o f t h e
Gradient, o n l y t h e 4 t h (SOGT), 8 t h (MMT), 5th(LMT),
7th(QNT) and 3rd(FOGT) Techniques reach a s u f f i c i e n t l y
small value. The Gth(1GT) Technique does n o t g i v e
0.0299
0.0273
0.3353
0.0273
0.1041
1.9913
0.3353
1.9913
56.5607
COMPUTED VALUES (two-laver s t r u c t u r e )
29: 800
UPPER LAYER R e s i s t i v i t y - (0.m) =
BOTTOM LAYER R e s i s t i v i t y ( 0 . m ) =
5.635
DEPTH UPPER l a y e r (m)
10.238
OBJECTIVE FUNCTION (end-value)
Euclidean Norm o f t h e GRADIENT
Meas.#
Spacing(m)
1
2
3
4
5
6
7
1.500
2.000
2.500
3.000
3.500
4.000
4.500
5.000
8
9
10
11
12
13
14
15
16
17
18
5.500
6.000
8.000
10.000
13.000
16.000
20.000
25.000
32.000
40.000
p,meas.
30.200
29.700
30.000
29.000
28.900
28.800
28.800
27.600
27.400
28.200
27.000
24.000
18.900
17.500
13.000
10.000
7.800
7.100
=
0.01387167
3.44657E-009
PaComP.
29.756
29.698
29.605
29.472
29.294
29.068
28.794
28.473
28.105
27.694
25.693
23.340
19.722
16.481
13.085
10.235
8.030
6.864
d i screp. x;
1.491
0.006
1.333
-1.601
-1.344
-0.923
0.019
-3.065
-2.508
1.828
5.088
2.826
-4.167
6.185
-0.651
-2.299
-2.868
3.445
I n r e l a t i o n w i t h t h e t i m e needed t o o b t a i n t h e
r e s u l t s . i t i s obvious t h a t t h e a u i c k e s t o f a l l t h e
techniques i s t h e 6 t h (IGT), followed, i n most cases,
by t h e 8 t h (MMT).
The 4th(SOGT)
and 5 t h (LMT)
techniques present i n general a r e l a t i v e l y reduced
run-time. The 3rd(FOGT) and 7th(QNT) techniques present
g e n e r a l l y a p r o h i b i t i v e time, which i s excusable o n l y
i n t h e second because o f t h e h i g h accuracy obtained.
The
lst(F0GT)
and 2nd(FOGT)
techniques
present,
normally, s i m i l a r and r e l a t i v e l y acceptable run-times.
~~
CONCLUSIONS
The t h e o r e t i c a l b a s i s f o r t h e e s t i m a t i o n o f a
vector o f parameters, o b t a i n i n g an optimum f i t t i n g
between a s e t o f measured r e s i s t i v i t y values and t h e
corresponding s e t o f computed r e s i s t i v i t y values u s i n g
such parameters has been reviewed. Our f i r s t o b j e c t i v e
has been t o d e f i n e t h e necessary t o o l s f o r l a t e r
implementation o f e i g h t d i f f e r e n t techniques which w i l l
a l l o w t h e e s t i m a t i o n o f parameters o f a two-layer
earth.
1
Used
Used
COMPUTED VALUES
Method
p1
z
p2
F
UV
FI
1s t
------b i n >
(min)
1s t
355.773 143.122 2.880 .008361 2.23E-3
----324.600 141.454 3.176 .014998 4.18E-2
----360.379 143.751 2.821 .007950 9.04E-4
-----
2nd
3r d
4th
6th
3r d
.00162
7.32E-8
4th
372.727
145.262 2.689
.007627 1.85E-5
5th
364.683
143.634 2.828
.00811
-l-l-l
5th
2nd
144.470 2.760
368.295
0.05023
0.06
7th
8th
0.58
TABLE I 1
RESULTS i n CASE I 2
Used
z
P2
p1
10
Fi
Runtime
13.72
491.026
92.92
.0112
4.431
5.03E-8
4 th
5th
I 241.2821-I
-I
1
I
6th
I
992.25~1.950~.011052~.047131 0.08
I1-1246.836~1058.62~2.139~.010740~9.01E-93.12
7th
I
I
7th
8th
I 246.836~1058.6312.139~.010740~8.46E-8I 0.55 I
8th
T E G l G l G l ~ l ~ l ~
RESULTS i n CASE t 5
RESULTS i n CASE t 3
-----170.582 48.896 1.452 .013698 4.14E-3
0.617
-----158.920 35.323 1.829 .019508 2.98E-2
0.38
-----161.187
1.
34.802
160.774
~
167.252
34.068
l
45.306
RV
F[
(m)
lI ~
1.570 .012834 .lo8191
p2
(0.m)
F
(min)
----123.941
5 6 . 8 5 1 9 4 . 4 8 9 ~ 1 . 5 3 6 ~ . 0 2 4 2 0 2 ~ 4 . 2 5 E - 3 0.33
2nd
132.934 1055.24 2.821 .022387 1.27E-1
3r d
125.474 1 0 9 2 . 5 j 2.71~.0207Oj8.76E-4
4th
122.319 1035.78 2.465
5th
5th
125.527 1092.831 2.712
6th
6th
117.952
7th
125.525
8th
125.526
3 rd
4th
7th
8th
l~lGlGl~lGl;l
57.279
96.489
1.626 .024027 7.33E-4
l*-l%GlLiElzl
57.344
L
96.714 1.651 .024018 4.04E-9
I
I
I
I
I
J
0.45
FI Runtime
(m)
1st
2nd
0.04
CORPUTED VALUES
(min)
1 1
~
0.97
RESULTS I n CASE #6
Method
Runtime
1.848 .010782 7.27E-6
160.7761 34.074(1.8481.010383(1.52E-8
Used
F
1
6.98
164.35
39.88 /1.6981.0107
11.52E-7
1.35
------
TABLE V I
COMPUTED VALUES
1.828 .010897 7.01E-4
57.139 93.243 1.517 .024314 4.OE-3
0.36
------
1st
3.31
Method
5th
1
.011207 8.22E-4
Used
F
3r d
I
93.395 4.399
494.883
4th
p2
(0.m)
490.688
8th
2nd
p1
(0.m)
1.63
494.883
3 rd
Used
4.13
.055909 1.23E-1
7th
2nd
Method
-1-1-1-1-1-
I
100.403 4.081
468.897
1s t
TABLE I11
92.27614.551(.01385316.18E-3
463.851
6th
1s t
6th
470.0181
TABLE V
COMPUTED VALUES
Method
I
CMPUTED VALUES
Method
Runtime
(0.m)
(0.m)
(m)
-----
1897
RESULTS i n CASE I 4
TABLE I V
994.3612.534) .02130314.53E-3
-----
I
,0206
4.60E-6
----.020702 Il.06E-4
I
4.47
I I
1 1
2.03
4.70
2.25
1.30
l
1898
As i s t y p i c a l i n t h e Steepest Descent Method i n
t h e d i r e c t i o n o f t h e Negative Gradient, t h e r e i s an
important s c a l e f a c t o r t o be taken i n t o account. An
improper s e l e c t i o n o f [ a i may cause a precarious
convergence o f t h e process.
The r e s u l t s o f t h e s i x cases examined, and many
o t h e r s which cannot be presented here because o f l a c k
o f space, show a s a t i s f a c t o r y behavior o f t h e Technique
from Ref. 191 w i t h t h e introduced m o d i f i c a t i o n s . A
s t i l l b e t t e r behavior i s presented by t h e FOGT proposed
i n t h i s paper which p r o f i t s , as i n d i c a t e d , from p a r t o f
t h e philosophy shown i n 1181, t o generate t h e diagonal
matrix [ X I .
ACKNOWLEDGEMENT
As
indicated,
the
results
and
conclusions
presented i n t h i s paper a r e a p a r t of a Research
P r o j e c t which attempts t o d e f i n e t h e b e s t t o o l f o r
e s t i m a t i o n o f s o i l parameters i n a m u l t i - l a y e r e a r t h .
The Author wishes t o express h i s g r a t i t u d e b o t h t o
OCIDE and IBERDROLA f o r sponsoring t h i s P r o j e c t . He
would a l s o l i k e t o p o i n t o u t t h e support received from
t h e Engineers o f t h e D I D I S Department o f IBERDROLA i n
t h e development o f t h i s P r o j e c t .
REFERENCES
The FOGT proposed i s , i n a l l t h e cases examined,
b e t t e r t h a n those from Ref. [61 [91 [ l o ] , w i t h repect
t o a b e t t e r f i t n e s s o f values. However, t h e execution
t i m e i s , i n a l l cases, higher.
not
#4).
[91.
are,
The 2nd Technique (FOGT) o f t h e Ref’s [ 6 ] [ l o ] i s
v e r y e f f e c t i v e i n some cases ( f o r example #1 and
I n every case i t i s i n f e r i o r t o t h a t from r e f .
The values o f t h e components o f f i n a l Gradient
t h e m a j o r i t y o f t h e times, f a r from zero.
The 1 s t Technique (FOGT) proposed i n Ref.[9], w i t h
t h e m o d i f i c a t i o n s introduced i n t h i s paper, leads t o
s a t i s f a c t o r y values and can be used i n t h e parameter
e s t i m a t i o n o f a two-layer ground.
I n any case, none o f t h e FOGT analyzed presents a
s u f f i c i e n t r e l i a b l e behavior i n t h e searching f o r t h e
optimum adjustment under any circumstance, f o r i t t o be
taken as a b a s i s f o r t h e optimum parameter e s t i m a t i o n
i n multi-layer earth.
The 6 t h Technique (IGT) r e s u l t presents components
o f t h e g r a d i e n t t o o h i g h t o perform a s u f f i c i e n t
approximation t o t h e optimum. Nevertheless i t can be
u s e f u l i n p r a c t i c e due t o t h e r a p i d i t y i n o b t a i n i n g
r e s u l t s . I n any case, i t can be considered, as has be
done i n t h e MMT, as a very useful a u x i l i a r y t o o l f o r
approaching t h e minimum f o r a l a t e r use i n a more
e f f e c t i v e Technique.
Compared t o t h e r e s t o f Techniques, t h e 8th(MMT)
proposed presents an e x c e l l e n t behavior, a c h i e v i n g i n
most cases t h e smallest values o f F(X), uVF(X)II, i n t h e
same way as t h e 4thCSOGT). Furthermore t h e lowest
execution times a r e achieved. S i m i l a r c h a r a c t e r i s t i c s
are provided by t h e Sth(LMT1, though w i t h execution
t i m e always s l i g h t l y higher. The time f o r t h e 7th(QNM)
i s a l s o very high, b u t i t s accuracy i s o f t h e same
order, even sometwhat b e t t e r than t h a t from 8th(MMT),
Ith(S0GT) and 5th(LMT) Techniques.
From t h e preceding e x p o s i t i o n i t i s concluded t h a t
t h e recommendations f o r e s t i m a t i n g t h e parameters o f a
two-layer
earth,
and
in
decreasing
order
of
e f f e c t i v e n e s s , g i v i n g weight t o accuracy ( t h e most
important) and computer run time, are: Mixed Technique
Second Order-Generalized I n v e r s e (MMT), Technique based
on t h e Levenberg-Marquardt (LMT), Technique based on
Newton Method (SOGT), Technique based on a Quasi-Newton
Method (QNT), proposed
3 r d Technique o f F i r s t Order
Gradient technique
(FOGT),
F i r s t Order
Gradient
[ l o ] and f i n a l l y t h e I n v e r s e
Technique from [ 6 ]
Generalized Technique (IGT).
The t h r e e f i r s t c l a s s i f i e d present c h a r a c t e r i s t i c s
o f accuracy and run-time t h a t convert them i n t o
powerful a n a l y s i s t o o l s f o r t h e s o l u t i o n t o t h e problem
o f parameter e s t i m a t i o n i n two-layer e a r t h . The I G T
must be used as a u x i l i a r y f o r t h e generation o f i n i t i a l
values.
E.D.,
Earth
conduction
effects i n
ill.-Sunde
Transmission Systems. Dover P u b l i c a t i o n s . (1968)
[ 2 l . - F l e t c h e r R., A M o d i f i e d Marquardt subroutine f o r
non-1 i n e a r l e a s t squares.
Harwell,
Berkshire,
England. Atomic Energy Research Establishment.
Report No. AERE-R.6799 (1971)
D.D.,
Interpretation
of
inaccurate,
[3l.-Jackson
i n s u f f i c i e n t and i n c o n s i s t e n t data. Geophys J.R.
A s t r . SOC. 28, pp 97-109 (1972)
[41 .-Fletcher R., FORTRAN subroutines f o r m i n i m i z a t i o n
by quasi-Newton
methods.
Harwell,
Berkshire,
England. Atomic Energy Research Establishment.
Report No. AERE-R.7125 (1972)
[5].-Bazaraa M.S., Shetty C.M., Non L i n e a r Programming.
Theory and Algorithms. Ed. John Wiley L Sons.
(1979)
[6].-Electric
Power Research I n s t i t u t e , Transmission
L i n e Grounding. Vol 1. Research P r o j e c t 1494-1.
Report EL-2699.
P r i n c i p a l Author
F.Dawalibi.
(1982)
[71.-Blatner C.J., Study
of
d r i v e n ground rods and
f o u r p o i n t s o i l r e s i s t i v i t y t e s t . I E E E PAS. Vol
101, Aug (1982), pp 2837-2850
[8].-IMSL Routines, 9 t h e d i t i o n (FORTRAN).
Houston, Texas. (1982)
IMSL I n c .
[9].-IEEE Std. 81., I E E E Guide
f o r Measuring E a r t h
Resistivity,
Ground Impedance and E a r t h Surface
P o t e n t i a l s o f a Ground System., (1983)
[lO].-Dawalibi
F., B l a t t n e r C.J.,
Earth r e s i s t i v i t y
measurement i n t e r p r e t a t i o n techniques. I E E E PAS.
Feb (19841, pp 374-382
[Ill.-Meliopoulos
A.F.,
e t al.,
Estimation o f
Parameters from
D r i v e n Rod Measurements.
PAS, No 9, (19841, pp 2579-2585
Soil
IEEE
[12].-Blatner C.J.,
Analysis o f s o i l r e s i s t i v i t y t e s t
methods i n two-layer e a r t h . I E E E Transactions on
PAS-104, No 12, (19851, pp 3603-3608
[13].-Meliopoulos A.P., Papalexopoulos A.D.,
Interpret a t i o n o f r e s i s t i v i t y measurements.
Experience
w i t h t h e model SOIMP. I E E E Transactions on Power
D e l i v e r y , VOL PWRD 1, Oct (1986), pp 142-151
[14].-Cuthbert
T.R.,
Optimization
using
computers. Ed. John Wiley L Sons (1987)
[15].-Meliopoulos
Transients.
(1988).
personal
A.P.S.,
Power System Grounding and
Ed. Marcel Dekker I n c . New York
1899
1161.-Takahashi T.,
Kawase T., Analysis o f Apparent
R e s i s t i v i t y i n a Multi-Layer E a r t h S t r u c t u r e . I E E E
Transactions on Power D e l i v e r y . Vol. 5, No. 2, pp
604-610 (1990)
[17].-Press
W.H.,
Flannery B.P.,
Teukolsky
S.A.,
Vetterling
W.T.,
Numerical
Recipes
(FORTRAN
Version). Ed. Cambridge U n i v e r s i t y Press. (1990)
A
Second Order Gradient
[18].-Alamo
J.L.
del,
Technique f o r an Improved E s t i m a t i o n o f S o i l
Parameters i n a Two-layer Earth. I E E E Transactions
on Power D e l i v e r y . Vol. PWRD 6, No. 3, (19911, pp
1166-1 170
Expressions f o r
aPa/aXi
can be found i n [ 6 ] and [ 9 ]
COMPONENTS OF THE HESSIAN MATRIX
The Hessian M a t r i x components a r e r e s p e c t i v e l y :
n = I...”, i = 1...3)
(For any case j = l.......m,
= 2 1 ( Aad] 2 + 2 1 - Ja24
d ,
axi2
ax
j
axi2
J
APPENDIX
GRADIENT COMPONENTS
Remembering t h a t :
F(X) represents t h e O b j e c t i v e Function F(p,,K,z),
/I
F ( P ~ , P ~ , Z ) .I n any case F ( X )
where t h e elements o f
differences d .
between
J
D
D
are
measured
or
t
112
D .D,
t h e weighted
and computed
values.
pa
represents
the
pa(~,,K,z,rj)
or
mathematical
model
either
P , ( P ~ , P ~ , Z , ~ ~ ) .I n
general
3.
i s t h e number o f parameters
For a l l cases
j = l.....m,
n
-=21;---dj=-21aF
ad.
1
m
’
axi
=
I n our case
l...*m,
v
3
each one o f t h e parameters, then:
Ji
ad.
axi
1
aP
P,
axi
- -a
2
.-!
1Jji Jjk
+ 2
4- G:,
a2d,
dJ.
o r , i n m a t r i x form:
H z 2 ( J t J + R )
i f k
where
R
i s a 3x3 square m a t r i x
containing
weighted d i f f e r e n c e s and t h e i r second d e r i v a t i v e s .
1axi2
a2d
J
j
d.
J
and
Rik
i f k
=
1%
j
the
d,
axiaxk
BIOGRAPHY
j
=
-
More d e t a i l e d expression f o r these can be found i n [18]
d.
If
9
represents t h e Jacobian M a t r i x o f p a r t i a l
d e r i v a t i v e s o f t h e weighted d i f f e r e n c e s d j r e l a t e d t o
J..
axi
j
axiaxk
l1
aPa
axi
a2F
--
R.. =
i = 1. ..3
ad.
Jgi+21J2dj
2-
where
pa(?, r j ) .
v
a2 F
- - 2 1
ax
r . i s the
m~
J
probe spacing a t which t h e measurements have been
made, j = l . . . . m
i s a b r i e f notation o f p ( r . ) ,
p,
i f k
and b e a r i n g i n mind t h e expression o f t h e elements
Jji,
it follows that:
which i m p l i e s VF(X) = 2 .Ut
The elements, i n a more e x p l i c i t form are:
D
J.L.de1
Alamo was born i n Albacete
(Spain),
on August
18,
1943.
He
received an E l e c t r i c a l Engineering
degree
from V a l l a d o l i d
University
(19631, a B.Sc. degree i n Phys. Sci.
from S e v i l l a U n i v e r s i t y (19751, M.Sc.
degree i n E l e c t r o n i c s from S e v i l l a
U n i v e r s i t y (1978), and Ph.D. degree
from V a l l a d o l i d U n i v e r s i t y (1984).
From 1964 t o 1981 he worked f o r several Power System
Companies i n S t a t i o n s , Substations and High Voltage
Ingenieros
Lines Design. I n 1981 he j o i n e d t h e E.T.S.
I n d u s t r i a l e s , U n i v e r s i t y o f V a l l a d o l i d , and i s now F u l l
Professor i n i t s E.E. Dept.