1337IEEE Transactions on Power Apparatus and Systems, Vol. PAS-98,No.4 July/Aug 1979

PERFORMANCE SENSITIVITY OF A POWER LINE CARRIER CHANNEL

S. Cristina, IEEE Member M. D'Amore, IEEE Member

University of RomeRome, Italy

ABSTRACT

A new sensitivity model to compute the partial deri-vatives of the carrier channel attenuation coefficientwith regard to numerous line and earth parameters, ispresented. The propagation mathematical model of thevoltage carrier signals takes into account the discon-tinuities of a lossy overhead line: it allows a fastcomputation of the voltage overall transition matrixand, hence, of the propagation coefficient. The propagation and sensitivity models are used to carry out thecoupling connections performances of a practical nonuniform horizontal line. The computed attenuation coeffi-cients are in closer agreement with the already knownexperimental values. The sensitivity analysis shows theinfluence of some parameters on the attenuation curvesas a function of frequency. Such curves are predeter-mined also by means of comparative-type formulae, whichinclude the frequency sensitivity functions.

INTRODUCTION

In a previous paperl, a new sensitivity model of thehigh frequency modal quantities was defined for alossy three-phase uniform line above a conducting ground.The transmission system mathematical model takes into account the dependence of the line and earth parameters onthe frequency. The partial derivatives of the propaga-tion matrix, of the modal attenuation constants andphase velocities with regard to numerous line and earthparameters, were presented. The sensitivity analysis,worked out'2 for practical horizontal lines with sky-wires, has shown the influence of each parameter on modal propagation.

The results of a mode-by-mode sensitivity study aredifficult to interpret when different modes are excitedtogether in the propagation phenomenon. This takes place both for line discontinuities and for various typesof transmitter coupling connections.Discontinuities arecaused by phase conductor transpositions, network interconnections, terminal loads. The transmitter coupled tothe center and one of the outer phases, often utilizedfor it results in a low attenuation produces all threemodes on the line.

In order to compute the propagation coefficient andits parameter sensitivity for a complete power-line carrier channel, it is necessary to define new propagationand sensitivity mathematical models. The modal analysisshould be used to study the voltages and currents propagation on non uniform lossy lines by applying alreadyknown algorithms3 94 . The computation of the partial de-rivatives of the voltage and current overall transitionmatrices, poses a number of new difficulties.The evaluation of the influence of the line and earth parameterson the carrier signals attenuation, is of a-considerableinterest, however, for the optimal choice of the trans-mitter and receiver coupling connections and of trans-

mission frequency.

Purpose of the Paper

This study is set out to:

- define a matrix mathematical model for the fast computation of the voltage propagation coefficient for anyphases coupling connection of a lossy non uniform power line;

- define a sensitivity model to compute the partial de-rivatives of the voltage overall transitionmatrix andof the propagation coefficient with regard to line andearth parameters;

- apply the propagation and sensitivity models to carryout the carrier channel performances of a practicalnon uniform horizontal line with skywires, for whichthe experimental values of voltage attenuation coefficient are already known5.

1. CARRIER SIGNALS PROPAGATION MODEL

Propagation coefficientConsider a three-phase lossy line with skywires on a

conducting ground. Discontinuities such as sections in-terconnections, phase conductors transpositions, trans-mitting and receiving terminations, are present in thenon uniform transmission line.

The phase voltages at line both ends are the entriesof the following column vectors:

(la)

(lb)

T I T 2T 3T t

R iR 2R 3R t

where the upperscript point indicates complex quantityand the underscript 't' matrix transposition.

Transmitting and receiving voltages VT and VR of acomplete carrier channel,are expressed as functions ofphase voltages:

(2a)

(2b)

VT = fT(1 T' A2 T, 3 T)

VR = fR(E E2 R, 3 R)

according to the terminal connections.The voltage propagation coefficient is:

(3a)

in which:

(3b)

m = 20 log 1 = a + jb

r = VTI\ = IVT-VR Iexp(jy)The attenuation and phase coefficients are:

(4a) a = 20 log VT VR

(4b) b = 8.69y with -r < y < r

F 79 258-5 A paper recommended and approved by theIEEE Power System Communications Committee of the IEEEPower Engineering Society for presentation at the IEEEPES Winter Meeting, New York, NY, February 4-9, 1979.Manuscript submitted September 5, 1978; made availablefor printing November 14, 1978.

Overall Transition MatrixIn order to compute the propagation coefficient

mn, it is necessary to evaluate 4R, then, in general,thereceiving vector [ER] if VT is known. For this purpose,the phase voltages are transformed into modal voltagesand their attenuated values are recombined at each line

0018-9510/79/0700-1337$00.75 1)1979 IEEE

1338

discontinuity. However, in order to simplify the propa-gation and sensitivity models, the reflected standingwaves are not considered.

The propagation matrix of generic 'n' reflection-freeline section, is:

(5) 1iPn] = [Zn]4iT1JThe series-impedance matrix [Zn], which includes both

earth and conductor frequency dependence and skywireseffects, takes into account the diagonal matrices of theconductor resistance and internal reactances, the reac-tance matrix due to physical geometry of the conductorsrelative to the earth plane, the matrices of the resis-tances and reactances which consider the contribution ofthe earth-return path. The shunt-admittance matrix(Y.],neglecting the conductance of the air path, takes intoaccount the matrix of the shunt-susceptance only.The expressions of [1Z] and [4ni are presented inexplicit formin a previous paper1. The voltage transition matrix is:

(6) [1 I = [AlJJ r] [M] 1sTI Ti Ti Ti

where [Mn] is the voltage transformation matrix that appears in the equation:

(7a) [M1 *[P 1.1M ] =diag{A A A iTi Ti fl fln 2fl 3n

in which 2kn are the eigenvalues of the propagation ma-trix.

The modal voltages transition matrix has the follo-wing expression:

(7b) [ n] =diag{exp(-mt )L )exp(_m(2)L )exp(-(3)Ln}where m(k) =/2 is the propagation constant of the k-thmode and L. is the line section length.The modes are numbered in order of increasing modal impedances.

The voltage transition matrix of generic 'v' line discontinuity between sections 'n' and 'n+l', is:

(8a) 1bdI = 2[1i , jy*dv cTI,n+l c-

where

(8b) Cn9n+1 cn + cn+l

The voltage transition matrices of the both terminaldiscontinuities are:

(9a) [idT1=2[YcT,T] [iTi [IdR =2[ crI,R [ cn

in which

(9b) [YcnT1 [ +1cn T1 , 1R'=14[cn1+RI

where [Y]T and [iR' are respectively the transmitter andreceiving load admittance matrices.

The characteristic admittance matrix can be obtainedby means of the following equation:

(10a) [Y ]- =[I I-1 .[AM ]R[m°]_j1CTI 'n ~ Ti ni Ti

in which

(lOb) [mO° = diag{m1( m(2) m(3)}If p are the line sections and q the line disconti-

nuities, the voltage overall transition matrix is:

(11) '1 ;dR1 'spJ1 dqI *1sp- 5dq-lW 5DS2 I di]

S1 dT

Finally, the receiving phase voltage vector can becomputed:(12) [AR] [ *1E1T]

from which IR is obtained.

2. CARRIER SIGNALS SENSITIVITY MODEL

Propagation Coefficient SensitivityThe partial derivative of the propagation coefficient

with regard to generic parameter x of the transmissionsystem, is:

V(13) a-= Sim -8#6 W 1SVax x R x

The partial derivatives of the attenuation and phasecoefficients are, respectively:

aa a *(14a) .X = S = Re{§S}( ax =x x

14 b .b Im{Sm}~l, ax x x

It is thus necessary to evaluate the partialvative of the receiving voltage:

deri-

avR VR E E E(15)

V

(SRR, S 2R, S 3R)aX x R x x x

and, in general, the parametric sensitivity vector ofthe receiving phase voltages:

ER E E E(16) [S I =[ X R 2XR S

Taking into account eq. (12), the sensitivity vector(16) is obtained through the following equation:

(17) [ER1 1 [-I1 E¢4I TIx aX T

The partial derivative of the overall transition ma-trix indicated in eq.(ll) can be computedby means of thealgorithm described in Appendix.The flow-chart appearingin Fig. 1 shows that all partial derivatives derive fromthose relating to the series-impedance and shunt-admit-tance matrices. There were presented, in a previous pa-per', the expressions of the derivatives of 1Z1 and of[1Yn] with regard to the following parameters: radius r,resistivity p, permeability p, stranding factor K. ofthe subconductors in the bundle, number n and spacing sof the subconductors, equivalent height heq of the pha-se bundle, distance A between the phases, earth resistivity Pe and permeability Pe, frequency f.

Normalized Sensitivity Functions

The purpose of the sensitivity analysis is to eva-luate the effects on the propagation coefficient due toinfinitesimal parameter variations. To make the sensitivities of attenuation and phase coefficients comparablein respect of the different parameters, the followingnormalized functions are defined:

(18a) sa =asx a x

(18b) sb xbsx b x

Approximate Prediction FormulaeThe sensitivity functions can be utilized in compara

tive-type prediction formulae of the attenuation andphase coefficients. In fact, if values ao, bo of the refe-rence system, described by parameters xio, are known,coefficients a ,bl of a system with different parametersxi, are computed by means of the following expressions:

(19a) a =a + ji Sa (xi, xio)(10

(19b) b,= bo + Xi sxio (xi, - xio)

1339

Fig.l. Flow-chart of the sensitivity model.

which are obtained through a Taylor series development,lirted to the terms of the first order.

From eqs.(19), taking into account eqs.(18), the formulae are obtained for the computation of the effects on

the carrier channel performances:

^a(20a) Aa% - 1. s. Ax.%1L X 3.

10

(20b) Ab% - I Sb1L0 1

due to percent finite variations of parameters.Eqs. (19), (20)provide reliable results whenever the max

imum admissible variations for each parameter are known.

3. APPLICATION TO PRACTICAL COUPLING CONNECTIONS

Transmission System Parometers

The untransposed 380 kV transmission lines consistsof three sections of various lengths, having phase bun-dles with a different number of subconductors. The horizontal configuration with skywires, equal for all thethree sections, is shown in Fig. 2, where there appearalso the values of the electrical and geometrical para-meters. The earth resistivity, measureds at 5OHz,rangesfrom 60 Q'm to 860 Q*m, with an average value equal to250 9*m. The equivalent height of the phase conductors,computed by means of the following expression':

(21). h - h +0.26(h -h.)eq min max minis indicated in Fig. 2.

The bundle conductor resistance is:

(22) R = KX.(2ktW .(ppfwl)1/2

where the value of the stranding factor K8 is equal to1.55, according to the indication given in a previouspaper'.

The resistive and reactive terms,due to the earth-returnpath, are computed by means of the general Carson's formulae!

The line impedance is variable owing to the standingwaves: it has an average values equal to 300Q and 500 Qfor phase-to-ground and line-to-line connections,respectively.

Carrier Channel AttenuationAttenuation was measured5 by closing the transmis-

sion circuit at both ends on the line impedance;the un-

used phases were grounded through 400 Q resistances.Consider the coupling arrangements indicated in Fig.3:con-nections 'a' and 'b' generate the first (interphase)andsecond (bifilar) mode respectively, connections 'c' and'd' are of the 1-phase conductor type, connection 'e'isof the 2-phase conductors type. Since the line is sym-metrical, it makes no difference to supply either outer

phase, and to connect the center phase with one of thetwo outer phases. The experimental valuess of the car-rier channels attenuation coefficients are indicated inFigg. 4,5 for a frequency between 50kHz and 400kHz:cou-plings 'a', 'c', 'e' produce low attenuations, for con-

nection 'd' a maximum attenuation occurs for f-275kHz.The voltage overall transition matrix has been com-

puted by equation derived from (11):

(23)[ l [ dRli4fS3 I [1 id2][ S2 I +tdi I [1 isl I -[ dTlwhere there appear the transition matrices of the threesections, of the two discontinuities due to the sectionsjunctions, of the two discontinuities due to the termi-nal line impedance. Eigenvalues Xkn of the propagationmatrices and transformation normalized matrices[ i-il]werecomputed by using expressions7valid for a line configu-ration symmetrical about the central conductor.

Table I shows the modal attenuation constants andphase coefficients of the three line sections and theoverall transition matrix for five frequencies. The at-tenuation values computed for each of the consideredcoupling arrangements are shown in Figg. 4,5.

Sensitivity AnaZysisThe frequency sensitivity functions of the modal at-

tenuation constants and of the overall transitionmatrixhave been computed: the results obtained for five fre-quencies are indicated in Table II. The partial deriva-

Fig.2. Base-case configuration of 380 kVuntrasposed transmission line5 withthree sections.

section numberE ..-TI

1 2 3

9Am -L 67.781 2.817 1.074

72 mh127m dcm. 3.15 3.15 3.15

°n 3 2 2

h.14.75m I Scm 40 40 40

h 1445m -QQm 3.21.10 3.21-10 3.211

dw cm 1.25 1.25 1.93

ACSR bundle conduc

0////ZZ/f/<o1 Q(250 m

d 00 0~

r I A

1340

lb IC

+ ET r ET-: : Hi+*o-------*+ ET o E ' #_2ETO ,f * -O-# +ET:_

+ E o04 A T0-_ J,,.o....

+ET: :-#-f~AA +ET

Fig. 3. Coup zing arrangements.

tives of the attenuation and their normalized expres-sions with regard to the frequency, assume the valuesshown in Table III (base-case parameters) for each ofthe five carrier channels. The high values of Sf , ob-tained for connections ','d', indicate the con-siderable influence of frequency on the attenuation coefficient. The partial derivatives, as frequency varies,are almost constant for couplings 'a','b','c',while theyvary for coupling 'e'; for connection 'd', there appearsbetween 200kHz and 300kHz a minimum value of S a, whichis preceded by a positive value and is followed by a negative value, thus indicating the presence of maximumattenuation. Of the parameters that describe the trans-mission system, stranding factor Ks, equivalente heightheq Of the phase bundles, and earth resistivity Pe, present evaluation difficulties1 ,2 5.

The said parameters were assigned the following varia-tion ranges:(24) 1.4<K <1.8, 13m<{h 1<16m, 500m<pe_1000s- - eq out.ph<: e-~-00and the attenuation curves, shown in Figg.6,7, of eachcarrier channel, were computed.

The intent is to analytically assess the influence ofeach of the three parameters on the attenuation curvesas a function of frequency. For this purpose, absoluteSand normalized g sensitivities have been computedfor different values of parameters Ks,heqoPe. From the

analysis of the results indicated in Table III,the fol-lowing considerations emerge. Attenuation curves show amore pronounced upward trend for couplings 'a' and 'c',while they hardly change in shape for couplings 'b','d',let, as Ks increases;the percentage variations of theattenuation coefficient, are higher forconnect ions 'a'and 'c'.

The equivalent height hardly influences connectionsfa' and 'c'; the attenuation curve points more upwardfor connection 'b', with high percentage variations,andit also changes in form for connection 'e',with low percentage variations as heq decreases. The influences ofheq on channel 'd' is considerable: if heq decreases,the frequency for which the maximum attenuation valueis attained also decreases.

The earth resistivity hardly influences connectionsla', 'c'. Sensitivity function Sa decreases for connection 'b', with low percentage variations , as Pe in-creases. Parameter Pe has a remarkable influence on coupling 'd', while it hardlymodifies theattenuation curvefor coupling 'e': for both connections 'd' and 'e', thefrequency for which the maximum attenuation is attainedincreases, as Pe decreases.

Approximate PredeterminationThe frequency sensitivity function is utilized in the

comparative formula:

(25) a= a af 0 (f1 - fo)

derived from (19a) for variation of one single parame-ter.

Eq. (25) is applied in repeated steps, with a fre-quency percentage variation which is constant and equalto 10%: in the first computational step, ao is given aknown experimental value5. The attenuation curves, ob-tained by means of this semi-empirical method,are shownin Figg. 4,5 for each of the five carrier channels.

CONCLUSION

The propagation coefficient of voltagegnals for phaqes coupling arrangements of

carrier si-lossy non uni

20 60 100 140 180 220 260 300 340 380

frequency in kHz

I

._

4,S3c

frequency in kHz

Fig. 4. Carrier channel attenuation of 380 kV non uniform untransposed Zineo measured5 - computed by 'exact' method --- computed by 'approximate'method

10C

0

._c

4'S

Ia

0

0 - --rc__ _ -

0 -XI

--- --'C

° _- ---

5 0-

._ 1-.- ,-

1341

20 60 100 140 180 220 260

frequency in kHz

40

30 / \\

/

20 /

0 ,4 /~~~~~

10-: (r

0 I aI

Fig.5. Carrier channel attenuation of 380kVnon uniformuntransposed Zine

o measured'- computed by 'exact' method --- computedby 'approximate' method.

form power lines is carried out by applying an accurate

mathematical model that is very suitable for computerimplementation. The proposed matricial method allows a

fast computation of the voltage overall transitionmatrixwhich includes line discontinuities.

The attenuation values obtained in the application to

a practical non uniform line, are quite in agreementwith the measured values: the standing waves, not con-

sidered in the propagation model,andtheuncertainevaluation of some parameters of the transmission system, maybe regarded as causes of error.

An algorithm is presented for the computation of thepartial derivatives of the propagation matrices, modalquantities, transition matrices, phase voltages and propagation coefficient, with regard to numerous transmis-sion system parameters.

The sensitivity model has been applied in order to

1 Q=5O lam2 Q= 1000 Qm3 hlq13 m

4 heq=16 m

5 ks =1.46 k, =1.8

6 54 321

5 -

6 5 4 32 1

O _ _ _ _ _ _ _ _ _ _ ___

20 60 100 140 180 220 260 300 340 380

frequency in kHz

c

S

C0

.4'a

20 60 100 140 180 220 260 300 340 380frequency in kHz

Fig. 6. Carrier channel attenuation computed fordifferent values of parameters K s,h eq,pe

evaluate the influence of frequency on the transmissioncharacteristics of five practical carrier channels. Thesensitivity analysis has shown the dependence of the at-

tenuation coefficient as a function of frequency, on thestranding factor, the bundle equivalent height and theearth resistivity, by using a new analytical tool.

The attenuation coefficients, computed by a compara-tive formula which includes the frequency sensitivityfunction, are in closer agreement with the experimentalvalues: such a semi-empirical method allows to reduce theamount of inaccuracies resulting from the causes of er-ror.

The computation of attenuation sensitivities with re-

gard to other system parameters will be carried in thefuture development of this study: the comparative formu-lae would allow to predict the attenuation curves of theplanned carrier channels, by using already known experi-

frequency in kHz

Fig.7. Carrier channel attenuation computed for differentvalues of parameters Ks,heqtPe.

.£la

£0

20C.a

.

la

c

c.._

300 340 380

1342

Table I. Modal attenuation constants a(k), in Np'm, phase coefficients a in rad*m , of the three line sec-tions, and voltaae overall transition matrix,

a([t1) a(2) a(3) () ;(2) p(3) |_[_0 __[_ __I

kHz | i4 i6 |i6.0 10 162 .162 6162 [¢]

1 0.978 2.695 78.67 0.1049 0.1060. 0.1196 [0.514 -0.298 -0.2151 -0.208 -0.055 0.2671

50 2 1.260 2.749 73.99 0.1050 0.1059 0.1187 -0.343 0.677 -0.343 -0.061 0.120 -0.061

3 1.257 2.739 68.84 0.1050 0.1059 0.1180 [ 0.215 -0.298 0.514j L0.267 -0.055 -0.208]

1 1.422 5.913 135.63 0.2098 0.2118 0.2328 [0.328 -0.270 -0.0561 [-0.228 -0.117 0.3471

100 2 1.815 5.768 127.33 0.2099 0.2116 0.2313 -0.310 0.614 -0.310 -0.131 0.256 -0.131

3 1.809 5.733 119.21 0.2099 0.2116 0.2304 [-0.056 -0.270 0.328] LO.347 -0.117 -0.228]

1 2.101 12.782 228.18 0.4196 0.4230 0.4547 [0.094 -0.181 0.0891 -0.102 -0.215 0.3221

200 2 2.646 12.112 213.86 0.4196 0.4227 0.4523 -0.208 0.412 -0.208 -0.244 0.477 -0.244

3 2.634 12.031 201.84 0.4196 0.4226 0.4511 L .o89 -0.181 0.094] [0.322 -0.215 -0.|102

1 2.670 19.654 305.33 0.6293 0.6339 0.6737 -0.019 -0.060 0.0781 [0.014 -0.264 0.255

300 2 3.326 18.412 285.92 0.6294 0.6335 0.6708 -0.070 0.141 -0.070 -0.299 0.587 -0.299

3 3.310 18.291 271.08 0.6294 0.6335 0.6693 L 0.078 -0.060 -0.019] L0.255 -0.264 0.014J

1 3.174 26.337 372.95 0.8390 0.8447 0.8914 [-0.084 0.067 0.0141 [0.064 -0.252 0.1921400 2 3.920 24.518 349.05 0.8391 0.8442 0.8878 0.075 -0.142 0.075 -0.287 0.564 -0.287

3 3.899 24.362 331.90 0.8391 0.8441 0.8862 [L0.014 0.067 -0.0841 [L0.192 -0.252 0.064]

_1 -1 a (k)Table II. Sensitivity function Sj', in Np.m *kHz ,normalized sensitivities Sf and partial derivative of the

voltage overaZZ transition matrix with regard to the frequency.

[ 3aSa(1) ja(1) sa(2) ja(2) sa(3) ga(3) f lip] IS,I + IS;]f

= P c f f f

kHz | *o cj7 .i6

1 0.1040 0.532 0.6055 1.123 0.1261 0.802 [-0.375 -0.048 0.430 -0.749 0.453 0.295

50 2 0.1314 0.522 0.5756 1.047 0.1182 0.799 1-0.052 0.103 -0.052 0.521 -1.028 0.521

3 0.1306 0.519 0.5708 1.042 0.1108 0.805 [0.430 -0.048 -0.375] [0.295 0.453 -0.749]

1 0.0781 0.549 0.6697 1.133 0.1043 0.769 [-0.415 -0.112 0.5311 -0.434 0.426 0.0041100 2 0.0969 0.534 0.6231 1.080 0.0976 0.767 1-0.125 0.241 -0.1251 0.489 -0.966 0.489

3 0.0962 0.532 0.6181 1.078 0.0927 0.778 L 0.531 -0.112 -0.415J [0.004 0.426 -0.434J

1 0.0609 0.580 0.6927 1.084 0.0834 0.731 -0.186 -0.225 0.4171 -0.086 0.350 -0.268

200 2 0.0735 0.555 0.6366 1.051 0.0779 0.728 1-0.253 0.494 -0.2531 0.401 -0.793 0.401

3 0.0730 0.554 0.6324 1.051 0.0747 0.740 LO.417 -0.225 -0.186] L:0.268 0.350 -0.086]

1 0.0538 0.604 0.6791 1.037 0.0717 0.705 0.030 -0.307 0.2841 0.017 0.245 -0.2641300 2 0.0635 0.573 0.6213 1.012 0.0670 0.703 j-0.347 0.678 -0.347 0.281 -0.557 0.281

3 0.0632 0.573 0.6176 1.013 0.0644 0.713 [L0.284 -0.307 0.030J [-0.264 0.245 0.017]

1 0.0492 0.620 0.6562 0.997 0.0640 0.686 [0.143 -0.348 0.2131- 0.047 0.123 -0.170-

400 2 0.0573 0.585 0.5990 0.977 0.0597 0.684 1-0.394 0.772 -0.3941 0.143 -0.285 0.143

3 0.0569 0.584 0.5955 0.978 0.0575 0.693 L0.213 -0.348 0.143] [-0.170 0.123 0.047

1343

Sa -

Table III. Sensitivity function Sf, in dB-kHz , and normalized sensitivity Sf of the carrier channel attenuationfor different values of parcaneters Pe,heqKs.

base-case param. Qi=50 Om Q?1000 52m h =13m h*'16m |KI=1.4 K =1C.

o q~ ~ ~~~~~a a a a aaf f

kHz 0 110 ||6' |10@ .|16, |16

a 0.0657 1.642 0.0695 1.418 0.0642 1.736 0.0655 1.706 0.0660 1.497 0.0597 1.810 -0.0754 0.641

b 0.3763 1.555 0.4599 1.251 0.2405 1.813 0.4183 1.576 0.3426 1.523 0.3708 1.596 0.3849 1.492

50 c 0.0476 0.073 0.0210 0.044 0.0765 0.120 0.0642 0.099 0.0415 0.064 0.0395 0.062 0.0603 0.090d 0.4293 0.419 0.4628 0.440 0.4436 0.462 0.5882 0.540 0.3820 0.389 0.4453 0.438 0.4617 0.444

e 0.2828 1.858 0.2199 1.540 0.3104 2.092 0.3645 1.823 0.2291 1.881 0.2768 1.961 0.2931 1.718

a 0.0490 1.489 0.0544 1;520 0.0465 1.477 0.0501 1.518 0.0490 1.467 0.0448 1.812 0.0559 1.204

b 0.4151 1.295 0.4882 1.211 0.2893 1.450 0.4770 1.329 0.3650 0.954 0.4106 1.311 0.4221 1.269

100 c 0.0524 0.148 0.0571 0.155 0.0490 0.141 0.0549 0.154 0.0519 0.146 0.0483 0.140 0.0594 0.161

d 0.6663 0.836 0.4367 0.579 0.6970 0.920 0.9586 1.026 0.4945 0.695 0.6594 0.835 0.6709 0.829

e 0.2889 1.293 0.1798 1.051 0.3992 1.552 0.3509 1.218 0.2333 1.307 0.2841 1.320 0.2959 1.246

a 0.0384 1.016 0.0285 0.599 0.0351 0.984 0.0406 1.049 0.0369 0.981 0.0355 1.109 0.0434 0.823

b 0.4291 1.150 0.4186 1.034 0.3447 1.327 0.5098 1.189 0.3653 1.11G 0.4260 1.159 0.4339 1.135

200 c 0.0409 0.204 0.0303 0.141 0.0365 0.188 0.0433 0.214 0.0386 0.194 0.0377 0.194 0.0455 0.216

d 1.3241 1.541 0.3920 0.655 0.9265 1.068 2.1271 1.584 0.7175 1.098 1.3150 1.541 1.3320 1.533e 0.0972 0.455 0.0940 0.579 0.0015 0.006 -0.0160 -0.001 0.1369 0.738 0.0939 0.452 0.1020 0.457|

a 0.0336 0.905 0.0311 0.741 0.0300 0.867 0.0369 0.954 0.0323 0.882 0.0310 0.958 0.0376 0.834

b 0.4213 1.078 0.4436 1.025 0.3748 1.277 0.5090 1.116 0.3520 1.038 0.4182 1.083 0.4252 1.067

300 c 0.0351 0.241 0.0322 0.209 0.0308 0.219 0.0386 0.260 0.0334 0.231 0.0324 0.230 0.0389 0.253

d -0.9712 -0.892 0.3877 0.731 -0.4747 -0.960 -0.5964 -0.979 1.3697 1.811 -2.2740 -2.092 -2.1610 -1.978

e -0.0206 -0.138 0.0593 0.444 -0.1314 -1.000 -0.0465 -0.348 0.0350 0.233 -0.0232 -0.160 -0.0170 -0.108

a 0.0311 0.869 0.0296 0.758 0.0272 0.822 0.0352 0.929 0.0289 0.820 0.0295 0.934 0.0351 0.823

h 0.4068 1.026 0.4155 1.008 0.3919 1.240 0.4978 1.015 0.3366 1.055 0.4045 1.031 0.4142 1.020

400 c 0.0322 0.274 0.0303 0.246 0.0278 0.246 0.0364 0.302 0.0298 0.260 0.0307 0.271 0.0361 0.290

d |-0.4385 -0.899 0.3200 0.656 -0.0021 -0.007 -0.0587 -0.150 -1.6290 -1.508 -0.4410 -0.911 -0.4320 -0.875

e -0.0164 -0.153 0.0373 0.333 -0.0044 -0.054 -0.0122 -0.180 -0.0025 -0.022 -0.0178 -0.174 -0.0124 -0.109

mental values.In the final analysis, the presented sensitivity mod

el should be regarded as an introduction to a carrierchannel optimum design.

ACNOWLEDGEMENT

The authors would like tothank ENEL for the financialsupport of the research programme.

APPENDIX

The partial derivative of the voltage overall tran-sition matrix, is:

Dx [¢' ax 1 dRl] *['sp] * dq * sp-l] *d -1]Ia.(26) ..*[$s2 1 .141dI .14SI 1 .14dT1+1 IdR] -4 ispI

82dq di s1$i dT1S 1. idR L axs

The partial derivative of the n-th line section tran--sition matrix, is:

a F1(27) ax s[L] n ]n[n Lax *[ eM +

where:~ *1 %1 L n1(28) a- 1401 = -L - M0 10

(1)m(2) (3)

(29) ax [mn]j - diag{S6n Sn S n Ix x x

(30) ax- [ AnJ -'-- [ In -1 *; [i In *[iMnI

(31) aI[i I = [-I [Mn[ I a[[A Iax ri ax 2fl aX 3

The partial derivative of eigenvector [Mkn], present

in (31), is computed in such a way as to satisfy the following equation, in which it appears as an unknown quan-tity:

[k'n [ I] -[ pnl Sax [kMln] [¢ [PrXl(32a)

+ax [I] -IMknI

where [I] is the unit matrix.If the line configuration is horizontal or symmetrical

in respect of the central conductor, the propagation ma-

trix and the normalized transformation matrix, are:

11 12 13

(32b) [ PI= p p p , [Ml= M21 22 21 21

P P 1i 1L_ 13 12 11__,

1 1

M23

-1 1

where:

(34a)--4 YeI 4,Zn][I MnII1 m *[.M1mI + z I -1

. ft In] .[In*[io-IA +1 2n * LM',7Lm11- rel c 1 l 11 1 -t

FI0[Mn+ L i In1 n) Tx

in which the derivatives of the inverse matrices showexpressions similar to (30), and moreover:

(34b) [Y J=I[i ]+ Iax cn,n+l ax cn ax cn+l

The partial derivative of the receiving discontinui-ty transition matrix is similar to (33), while, for thetransmitting end

ax dT cn,T T

where

(36) aax [cn,T1 =ax Lcniwith7

(32c) M1 =2? (AP ) , M =2P - )21 21 1 22 23 21 3 22

In this case, the partial derivative is easily obtain

ed:

0 0

(32d) a [iM- ax 21ax 21

0 0

0

a

aX 2

in which:

a 0 a .I a. 1 a a(32e) ax 2i 2i 2laP21 - i 22 ax ax22 )

i=l, 3

The partial derivativetransition matrix, is:

(33)

of the v-th line discontinuity

-~;[x 0dv' =2{ x yn,n+l *iYCI

+

+ lY< n, n+l] *[3 [Y]}

REFERENCES

(1) S.Cristina, M.D'Amore, "Propagation On PolyphaseLossy Power Lines: A New Parameter Sensitivity Model", Paper F78 086-1 presented at IEEE PES WinterMeeting, New York, January 29, February 3, 1978.

(2) S.Cristina, M.D'Amore, "Sensitivity Analysis of Modal Quantities For Horizontal Transmission Lines",presented at IEEE Canadian Conference on Communications and Power, Montreal, October 18-20, 1978.

(3) L.O.Barthold, "Radio-Frequency Propagation on Po-lyphase Lines", IEEE Trans. on PAS, July 1964, pp.

665-670.

'4) M.C.Perz, "A Method of Analysis of Power Line Car-rier Problems on Three-Phase Lines", IEEE Trans. onPAS, July 1964, pp. 686-691.

(5) M.Aiminio, C.Starace, "Caratteristiche di trasmis-sione in alta frequenza di una linea a 380 kV", LaElettrotecnica, vol. LV, n.10, October 1968,pp.717-733.

(6) G.V.Mikutski, L.S.Perelman, V.V.Sidelnikov and Yu.P.Shkarin, "Recherches sur les caracteristiques en

haute frequence de lignes a 750 kV a fils de terreisoles", CIGRE, Report No. 35-03, 1970.

(7) M.C.Perz, "Propagation Analysis of HF Currents andVoltages on Lossy Power Lines", IEEE Trans.on PAS,vol. 9-2 pp. 2032-2042, November/December, 1973.

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