TRANSMISSION LINE MODELING FOR

REAL-TIME SIMULATIONS

Maria Isabel Silva Lafaia Simões

Dissertation submitted to obtain the degree of Master in

Electrical and Computers Engineering

Committee Members

President Prof. Paulo José da Costa Branco (DEEC, IST)

Supervisor Profa Maria Teresa Nunes Padilha de Castro Correia de Barros (DEEC, IST)

Member Prof. Jean Mahseredjian (École Polytechnique de Montréal)

Member Prof. José António Marinho Brandão Faria (DEEC, IST)

November 2012

Agradecimentos

Agradeço em primeiro lugar à minha família � ao meu pai, à minha mãe, ao meu irmão e à minha

madrinha, pelo amor constante e incondicional que me dedicam.

Agradeço à professora Teresa Correia de Barros pela sua orientação e, sobretudo, pela con�ança que

deposita em mim.

Agradeço ao meu colega Pedro Cruz pela sua amizade, conselhos e palavras de apoio. Agradeço ainda

ao colega e amigo Miguel Fragoso pelo seu exemplo de dedicação e camaradagem.

A todos os que me inspiraram ao longo dos meus estudos no Instituto Superior Técnico � Obrigada!

i

Resumo

A simulação em tempo-real de sistemas de energia eléctrica é uma ferramenta importante sempre que

é necessário incluir um componente físico no sistema em estudo, em vez do seu modelo matemático.

O tempo-real é difícil de atingir em simulação digital, pois a um aumento de exactidão corresponde,

geralmente, um aumento do tempo de processamento. Torna-se assim necessário combinar arquitecturas

de processamento paralelo com a utilização de modelos e�cientes. As linhas de transmissão permitem o

processamento paralelo ao dividir uma grande rede em pequenas sub-redes independentes.

Uma representação exacta da linha exige que se considere a dependência na frequência dos seus

parâmetros, o que coloca um desa�o na de�nição de um modelo adequado. O objectivo desta dissertação

é estabelecer os procedimentos para uma aproximação dos parâmetros de propagação em modelos de

linha adequada para simulações em tempo-real.

O estudo dos modelos existentes constitui uma base para o desenvolvimento do RT_WB Line, que

é uma reformulação do modelo WB Line do EMTP-RV, em linha com o objectivo de tempo-real. Para

atingir uma exactidão superior com recursos reduzidos, consideram-se duas optimizações relativas à iden-

ti�cação dos atrasos modais e à distribuição dos pólos pelos modos.

O RT_WB Line é validado através de simulações no domínio da frequência e do tempo, considerando

soluções exactas ou o WB Line como referência da exactidão pretendida. Os testes con�rmam que o

modelo desenvolvido permitirá, no tipo de aplicações nas quais é relevante o tempo-real, reduzir tempos

de processamento, por redução do número de operações requeridas, sem prejuízo da exactidão das soluções

obtidas.

Palavras-chave: Simulação em tempo-real, transitórios electromagnéticos, parâmetros dependentes

da frequência, RT_WB Line, identi�cação optimizada dos atrasos modais, distribuição optimizada dos

pólos pelos modos.

ii

Abstract

Real-time simulation of power systems transients is an important tool when there is a need to include

physical elements in the system under study, rather than their mathematical models. However, real-time

is hard to achieve in digital simulations, where accuracy runs oppositely to processing speed. It is there-

fore necessary to combine parallel processing with e�cient numerical techniques for model computation.

Transmission lines allow parallel processing in power systems studies, by dividing large networks into

smaller independent subnetworks.

Accurate line representation requires the use of its frequency dependent parameters. This poses a

challenge on the de�nition of an adequate line model. The goal of this dissertation is to establish adequate

numerical techniques for approximating the propagation parameters for transmission line modeling, al-

lowing real-time simulations.

The study of existing line models provides the basis for the development of the RT_WB Line, which

is a reformulation of the EMTP-RV model WB Line (based on the Universal Model [3]), in-line with the

real-time simulation target. To ensure additional accuracy with reduced �tting resources, two optimiza-

tions are suggested, concerning the computation of the modal delays and the assignment of the modal

poles.

The RT_WB Line performance is validated through frequency and time domain tests, considering

the exact solutions or the WB Line as a reference of accuracy. The tests con�rm that the RT_WB Line

allows, for the applications in which the real-time is important, a reduction of the processing time, by

reducing the required computations, without prejudice to the accuracy of the solutions.

Keywords: Real-time simulations, electromagnetic transients, transmission line modeling, frequency

dependent parameters, RT_WB Line, optimized modal delay computation, optimized modal poles as-

signment.

iii

Contents

Agradecimentos i

Resumo ii

Abstract iii

List of tables vi

List of �gures vii

Symbols and abbreviations x

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective of the present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Organization of the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Fundamentals on multiphase transmission line theory � a brief review 5

2.1 Phase domain solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Modal domain solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 The propagation function of a transmission line . . . . . . . . . . . . . . . . . . . . . . . . 8

3 EMTP-RV and transmission line modeling 11

3.1 Background on line modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Numerical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Major challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.2 Recursive calculation of convolution integrals . . . . . . . . . . . . . . . . . . . . . 14

3.2.3 Rational approximation of transmission line functions . . . . . . . . . . . . . . . . 16

3.2.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.3.2 Asymptotic Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3.3 Vector Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3.4 Asymptotic Fitting versus Vector Fitting . . . . . . . . . . . . . . . . . . 19

3.3 Transmission line models provided by the EMTP-RV 2.3 . . . . . . . . . . . . . . . . . . 19

iv

3.3.1 CP Line � constant parameters line model . . . . . . . . . . . . . . . . . . . . . . 19

3.3.2 FD Line � frequency dependent line model . . . . . . . . . . . . . . . . . . . . . . 20

3.3.3 WB Line � wide-band line model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Model testing and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Frequency response � short-circuited and open-ended line . . . . . . . . . . . . . . 21

3.4.2 Line energization and single-phase short-circuit . . . . . . . . . . . . . . . . . . . . 24

3.4.3 Current induced by phase coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.4 Model e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Wide-band model for real-time simulations � RT_WB Line 31

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Optimized �tting of the propagation function . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Optimal modal delay identi�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.2 Optimal modal poles assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Computer program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4.1 Main program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.2 Propagation parameters computation . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4.3 Yc �tting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4.4 H �tting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4.5 Output generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 RT_WB Line model validation 41

5.1 Validation perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Frequency response � short-circuited and open-ended line . . . . . . . . . . . . . . . . . . 42

5.3 Line energization and single-phase short-circuit . . . . . . . . . . . . . . . . . . . . . . . . 46

5.4 Current induced by phase coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Conclusions 53

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 Completion of proposed objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.3 Proposals for further improvements in line modeling . . . . . . . . . . . . . . . . . . . . . 55

Bibliography 57

A Transmission line used in model testing 59

v

List of Tables

3.1 Analytical short-circuit frequency response and approximating errors according to the

EMTP-RV 2.3 models, in terms of the magnitude of the current at the sending end of

phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Analytical open-end frequency response and approximating errors according to the EMTP-

RV 2.3 models, in terms of the magnitude of the voltage at the receiving end of phase

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Number poles used by the FD Line, available on the EMTP-RV 2.3, in the approximation

of the propagation parameters of a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Number of poles used by the WB Line, available on the EMTP-RV 2.3, in the approxi-

mation of the propagation parameters of a line . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1 E�ect of using optimized modal delays � average error of approximating propagation

functions according to di�erent order applications of the RT_WB Line, in mode and

phase domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Average error of the approximation of H, according to di�erent order applications of the

RT_WB Line. Use of equal (E) or optimized (O) distribution of the modal poles . . . . . 35

5.1 Number of poles used for the approximation of the propagation parameters of a line, for

the applications of tested models � WB Line and RT_WB Line . . . . . . . . . . . . . . 41

vi

List of Figures

2.1 Multi-phase transmission line and convention used to de�ne the phase currents and voltages. 5

2.2 Illustration of the physical meaning of the propagation function � current source in parallel

with characteristic admittance connected to a short-circuited transmission line. . . . . . . 9

2.3 Illustration of the physical meaning of the propagation function � left: current unit impulse

applied to the line; right: short-circuited line response to a current unit impulse. . . . . . 9

2.4 Illustration of the physical meaning of the propagation function � alternative representa-

tion of h(t) through a time function translated to the origin, h′(t). . . . . . . . . . . . . . 10

3.1 Illustration of the method of recursive convolutions: system represented by its impulse

response h(t). The function g(t) is the response of the system to an input signal f(t). . . 14

3.2 Illustration of the method of recursive convolutions: Input signal f(t− z) as a function of

z inside the interval [−∆t; 0]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Inclusion of losses in a CP Line Model, in the form of lumped resistances. . . . . . . . . . 20

3.4 Circuits used to study the short-circuit and open-end frequency responses according to

the line models available on the EMTP-RV 2.3 . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 Short-circuit frequency response according to the EMTP-RV 2.3 line models, in terms of

the current at the sending end of phase 1 (CP Line � thin, FD Line � dotted, WB Line

� bold). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Open-end frequency response according to the EMTP-RV 2.3 line models, in terms of the

voltage at the receiving end of phase 1 (CP Line � thin, FD Line � dotted, WB Line �

bold). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7 Circuit used to study the time response to energization followed by single-phase short-

circuit, according to the line models available on the EMTP-RV 2.3. . . . . . . . . . . . . 24

3.8 Response to line energization at t = 20 ms according to the CP Line � thin, and to the

WB Line � bold, in terms of the voltage at the receiving end of phase 1. . . . . . . . . . 25

3.9 Response to a short-circuit at the receiving end of phase 3 at t = 180 ms according to CP

Line � thin, and to the WB Line � bold, in terms of the voltage at the receiving end of

phase 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.10 Circuit used to study the phenomena of phase coupling according to the line models

available on the EMTP-RV 2.3, in terms of the current induced in phase 3 by energization

of phase 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vii

3.11 Current induced by energization of phase 1 � time evolution of the current at the sending

end of phase 3 during the �rst 20 miliseconds of the transient according to the EMTP-RV

2.3 line models (CP Line � thin, FD Line � dashed, WB Line � bold). . . . . . . . . . . 27

3.12 Current induced by energization of phase 1 � time evolution of the current at the sending

end of phase 3 during the �rst second of the transient according to the EMTP-RV 2.3 line

models (CP Line � thin, FD Line � dashed, WB Line � bold). . . . . . . . . . . . . . . . 27

4.1 General structure of the program developed to compute applications of the RT_WB Line

model in-line with the real-time simulation target, with respective input and output data. 36

5.1 Circuits used to study the short-circuit and open-end frequency responses according to

the WB Line and RT_WB Line models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Short-circuit frequency response (0.1 Hz - 1 MHz) � analytical (bold), WB Line (thin)

and RT_WB Line (dashed). Current at the sending end of phase 1. . . . . . . . . . . . . 43

5.3 Relative error of the short-circuit frequency response (0.1 Hz - 1 MHz) � WB Line (bold)

and RT_WB Line (thin). Current at the sending end of phase 1. . . . . . . . . . . . . . . 43

5.4 Detailed relative error of the short-circuit frequency response (700 Hz - 10 kHz) � WB

Line (bold) and RT_WB Line (thin). Current at the sending end of phase 1. . . . . . . . 44

5.5 Open-end frequency response (100 Hz - 1 MHz) � analytical(bold), WB Line (thin) and

RT_WB Line (dashed). Voltage at the receiving end of phase 1. . . . . . . . . . . . . . . 45

5.6 Relative error of the open-end frequency response (100 Hz - 1 MHz) � WB Line (bold)

and RT_WB Line(thin). Voltage at the receiving end of phase 1. . . . . . . . . . . . . . . 45

5.7 Detailed relative error of the open-end frequency response (700 Hz - 10 kHz) � WB Line

(bold) and RT_WB Line (thin). Voltage at the receiving end of phase 1. . . . . . . . . . 46

5.8 Circuit used to study the response to line energization followed by single-phase short-

circuit, according to the WB Line and to the RT_WB Line applications. . . . . . . . . . 46

5.9 Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)

and RT_WB Line (thin). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.10 Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)

and RT_WB Line (thin). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.11 Circuit used to study the phenomena of phase coupling according to the WB Line and to

the RT_WB Line applications, in terms of the current induced in phase 3 by energization

of phase 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.12 Current at the sending end of phase 3, induced by energization of phase 1, at the �rst 20

miliseconds, according to the WB Line (bold), and to the RT_WB Line (thin). . . . . . . 48

5.13 Current at the sending end of phase 3, induced by energization of phase 1, at the �rst

second of simulation, according to the WB Line (bold), and to the RT_WB Line (thin). . 49

5.14 Current at the sending end of phase 3, induced by energization of phase 1, at the �rst

20 miliseconds, according to the WB Line (bold), and to the RT_WB Line applications

(low order � thin; high order � dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

viii

5.15 Current at the sending end of phase 3, induced by energization of phase 1, at the �rst sec-

ond of simulation, according to theWB Line (bold), and to the RT_WB Line applications

(low order � thin; high order � dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.16 Current at the sending end of phase 3, induced by energization of phase 1, at the �rst 50

second of simulation (reaching steady-state), according to the WB Line (bold), and to the

RT_WB Line applications (low order � thin; high order � dashed). . . . . . . . . . . . . . 50

A.1 Spacial con�guration of the transmission line used throughout this dissertation. . . . . . . 59

ix

Symbols and abbreviations

• When denoting system variables, upper case letters refer to frequency domain quantities, whereas

lower case letters denote time domain quantities. For example, V for frequency domain voltage

and v for time domain voltage.

• Bold letters distinguish between matrix or vector quantities and scalar quantities. For example, a

vector of scalar voltages is written as v = [v1, v2, · · · , vn]T , where T stands for transposition and

n is the vector dimension.

• A bar-hatted letter denotes a complex quantity, with real and imaginary parts. For example,

γ = α+ jβ.

• System variables:

R � longitudinal/series resistance

L � longitudinal/series inductance

G � transversal/shunt conductance

C � transversal/shunt capacitance

Z � longitudinal impedance function

Y � transversal admittance function

Γ � propagation factor

α � attenuation factor

β � phase shift factor

τ � propagation delay

H � propagation function

Yc � characteristic admittance

Zc = Y −1c � characteristic impedance

i � phase current

I � phase current phasor

v � phase voltage

V � phase voltage phasor

Ti � current transformation matrix

Tv � voltage transformation matrix

x

• Parameters:

t � time

∆t � time step

ω � angular frequency

s � complex frequency

j �√−1

d � length of line

x � distance from the sending end of the line

• Operators:

F � Fourier transformation

F−1 � inverse Fourier transformation

Re � real part of complex quantity

Im � imaginary part of complex quantity

∗ � convolution

• Subscripts:

k � sending end of a line

m � receiving end of a line

s � source quantity

fit � approximating function

1, 2, · · · , n � line phase

short � short-circuit condition

open � open-end condition

• Superscripts:

k (superscript) � relative to the kth line mode

m (superscript) � modal domain

• Abbreviations:

DC � direct current

RMS � root-mean-square value

CP Line � constant parameters line model

FD Line � frequency dependent line model

WB Line � wide band line model

EMTP-RV � Electromagnetic Transients Program � Revised Version

xi

Chapter 1

Introduction

1.1 Overview

Although power systems are in steady-state most of the time, they must be able to withstand the worst

possible stresses to which they may be subjected, which usually occur during transient conditions of the

power system. Therefore, the size and cost of the equipment in a power system is largely determined by

transient conditions, rather than by its steady-state behavior.

It is of the utmost importance to accurately predict the behavior of the system. For instance, the

e�ectiveness of protective strategies in moderating transient conditions is only properly assessed based on

accurate data. Also, specially for high voltage power systems, any tolerance on equipment speci�cations

may represent a considerable increase of costs with no guarantee of optimum operation.

Two ways of studying transients in a power system are:

• Analogical simulation: the power system is represented by a transient network analyzer (TNA's),

which is a physical down-scaled reproduction of the power system components;

• Digital computer simulation: the power system components are represented through mathematical

models implemented computationally.

Transient network analyzers require physical facilities (space and equipment) and trained personnel.

The simulation of large networks using this method is �nancially very demanding. Furthermore, TNA's

have limited ability to represent real physical systems, namely, the distributed and frequency dependent

character of any component parameters.

On the other hand, the digital simulation has low requirements on space and equipment and, there-

fore, involves lower costs. It is more �exible than TNA's since any new component may be simulated

with reduced or null additional costs, provided an adequate model is known. Finally, the development

1

of computer processing capacity allows very rigorous simulations, with few simpli�cations.

Anytime there is a need to include physical elements in the system under study, rather than their

mathematical models, the simulation must be performed in real-time, that is, the quantities of interest

must have their values predicted correctly and within a prescribed period of time. This is usually the

case when studying the interaction between a power system and protection/relaying equipment, con-

trol/command systems and power electronic devices.

The real-time is intrinsic to TNA's, but it is not so easy to achieve in digital simulations, where

accuracy runs oppositely to processing speed. The design of a real-time digital simulator has mainly two

areas of development:

• Processor architecture: parallel processing allows to distribute the e�ort by several processing

units, working simultaneously. To do so, it is necessary to identify the operations that may be

taken independently, and not in a sequential manner;

• Implementation algorithms: e�ciency of numerical model computation and optimization of the

couple accuracy/complexity of the model.

Transmission lines play an important role on the de�nition of parallel processing levels within a power

system network: every time the propagation time of a line is su�ciently larger than the simulation time

step, the subnetworks connected through that line may be considered independently.

An accurate representation of a transmission line (for example, accurately representing distortion)

requires a �ne representation of the distributed and frequency dependent character of its parameters.

This poses a challenge on the de�nition of an adequate transmission line model.

1.2 Objective of the present work

The main goal of this dissertation is to establish adequate numerical techniques for approximating the

propagation parameters for transmission line modeling, allowing real-time simulations. This requires an

e�cient use of reduced modeling resources, namely, the introduction of optimization procedures that

ensure additional accuracy.

To accomplish this task, it is �rst necessary to take insight into the "state of the art" of line modeling,

speci�cally, the main challenges and its evolution. The study of the characteristics and performance of

the most used line models allows to de�ned the basic formulation to construct an accurate and e�cient

model.

After de�ning the structure of the model and introducing the adequate numerical techniques, it is

necessary to create a program that computes the developed model applications, using a pre-de�ned order

2

for the approximating line functions. The validation process consists of a set of tests in frequency and time

domain conditions and must use an application of the developed model which order of approximations

is adequate for real-time performances. A line model provided by the EMTP-RV 2.3 is included in the

test and taken as a reference of accuracy.

1.3 Organization of the text

The work presented in this dissertation is divided into 6 chapters and 1 appendix, summarized as follows:

Chapter 1: Introduction

The introductory chapter gives an overview of the problem of transmission line modeling for real-

time simulations. It also presents the objective of this work, which regards the establishment of adequate

numerical techniques for approximating the propagation parameters for transmission line modeling, al-

lowing real-time simulations. The chapter ends with a summary of the organization of the text in this

dissertation.

Chapter 2: Fundamentals on multiphase transmission line theory � a brief review

This chapter presents a brief review of the theory necessary to understand the construction of a

mathematical model to represent a multiphase transmission line in transient studies. The equations

representing the line behavior in transient conditions are �rst formulated in phase domain. The modal

domain is then introduced as an alternative for studying multiphase lines. The chapter concludes by

providing insight to the meaning of the propagation function for the simple case of a single-phase line.

Chapter 3: EMTP-RV and transmission tine modeling

EMTP-RV stands for Electromagnetic Transients Program. It is a widely used software, useful to

study transmission systems. The line models available in the EMTP-RV 2.3 provide a summary of the

evolution of line modeling.

The chapter starts by presenting the program and the main aspects that characterize line models,

namely, which functions are used to characterize the line, how the frequency dependence of line param-

eters is taken into account and whether the solution to line equations is computed in phase or modal

domain.

Some insight is also given to some of the most important techniques for model e�ciency. The �rst,

concerns the time domain equations that describe the behavior of the line in transient conditions, which

have to be computed at every simulation step and contain convolution integrals. A recursive calculation

[1] of those integrals is an alternative that tackles the high memory and processing time required for a

numerical evaluation.

3

The second technique concerns the representation of the line functions. Instead of using the actual

values of those functions for a large number of frequency samples, they can be approximated by analyt-

ical expressions in the form of rational functions of frequency. This representation is not only a basic

requirement for the use of recursive convolutions. It also allows a very e�cient representation of the

line functions and a direct analytical transformation to time domain. Two techniques used for ratio-

nal approximation of line functions are: Asymptotic Fitting [2], based on the magnitude of the original

function, and Vector Fitting [4, 5, 6], which allows to �t a set of functions using the same basic terms.

After that, the line models available on the EMTP-RV 2.3 are described, tested and compared. These

line models give a good insight to the evolution of line modeling, starting from the most basic constant

parameters model, called CP Line, to the most accurate WB Line, which takes the modal information

into account to �t the phase domain line functions.

Chapter 4: Wide-band model for real-time simulations � RT_WB Line

The objective of this dissertation is to establish adequate numerical techniques for approximating

the propagation parameters for transmission line modeling, allowing real-time simulations. This chapter

presents the RT_WB Line, which is a reformulation of the EMTP-RV model WB Line. In order to

ensure additional accuracy, it is necessary to introduce some optimization procedures, each of which is

presented and illustrated by a numerical example.

Chapter 5: RT_WB Line model validation

This chapter presents a set of tests that validate the developed transmission line model, RT_WB

Line. This is done through simulations in the EMTP-RV 2.3 environment. The tests analyze the

frequency and time domain behavior of a transmission line, according to an application of the RT_WB

Line, which uses an order for the approximating line functions in-line with the examples in literature

regarding real-time transmission line modeling. The WB Line, generated by the EMTP-RV, is taken as

a reference of accuracy.

Chapter 6: Conclusions

This chapter presents the �nal considerations on the performance of the RT_WB Line and on the ful-

�llment of the proposed objectives. A set of ideas are presented for further improvements in transmission

line modeling for real-time simulations.

Appendix: Transmission line used in model testing

The appendix presents a description of the spacial and electromagnetic characteristics of the trans-

mission line used throughout this work for model testing.

4

Chapter 2

Fundamentals on multiphase

transmission line theory � a brief

review

2.1 Phase domain solution

Consider an n-phase transmission line of length d, as illustrated in �gure 2.1. As it is well known,

penetration of the electromagnetic �eld in unperfect conductors introduces the frequency dependence of

the longitudinal transmission line parameters.

Figure 2.1: Multi-phase transmission line and convention used to de�ne the phase currents and voltages.

Therefore, the multiphase transmission line is characterized by its longitudinal impedance matrix

Z = R(ω) + jωL(ω) and transversal admittance matrix Y = G + jωC and described in frequency

5

domain by a couple of matrix di�erential functions:

d2

dx2V(ω,x) = Z(ω)Y(ω)V(ω,x) (2.1)

d2

dx2I(ω,x) = Y(ω)Z(ω)I(ω,x) (2.2)

where V and I are vectors containing the phase voltages and currents of the line. It is possible to deduce

a solution to equations (2.1) and (2.2) which relates V and I at the two line terminals as:(YcVk − Ik

)= H

(YcVm − Im

)(2.3)(

YcVm + Im

)= H

(YcVk + Ik

)(2.4)

The auxiliary line functions introduced are:

• the characteristic admittance matrix

Yc(ω) = Z(ω)−1√

Z(ω)Y(ω) (2.5)

• the propagation matrix (matrix exponential)

H(ω) = e−Γ(ω)d (2.6)

• the matrix of the propagation factors

Γ(ω) =√

Y(ω)Z(ω) (2.7)

The transformation of the line equations (2.3) and (2.4) to the time domain must take into account

the frequency dependence of all the line functions, resulting:

(yc(t) ∗ vk(t)− ik(t)) = h(t) ∗ (yc(t) ∗ vm(t)− im(t)) (2.8)

(yc(t) ∗ vm(t) + im(t)) = h(t) ∗ (yc(t) ∗ vk(t) + ik(t)) (2.9)

Due to coupling between the n line phases, the corresponding voltages and currents are interdepen-

dent. Therefore, line matrices Z and Y, and consequently Yc and H, are non-diagonal matrices, and

the total number of convolutions needed to compute the equations (2.8) and (2.9) is proportional to n2.

2.2 Modal domain solution

The study of a transmission line in terms of the voltages and currents of the n phases is complicated

by coupling phenomena. However, for ordinary multi-conductor transmission line con�gurations, there

are n independent propagation modes so, alternatively, the line may be studied in terms of the electric

quantities associated to its modes.

The conversion between phase and mode quantities is performed in frequency domain using a voltage

transformation matrix and a current transformation matrix as:

V = TvVm (2.10)

I = Ti Im (2.11)

6

Substituting these relations into the line di�erential equations (2.1) and (2.2) results:

d2

dx2Vm(ω,x) = Λ(ω)Vm(ω,x) (2.12)

d2

dx2Im(ω,x) = Λ(ω)Im(ω,x) (2.13)

where m stands for modal domain and Λ(ω) =(T−1

v ZYTv

)=(T−1

i YZTi

)is a diagonal matrix.

Therefore, Tv and Ti are the matrices that diagonalize the products ZY and YZ, respectively. This

means the columns of Tv and Ti are equal to the eigenvectors of the corresponding products.

Generally, ZY and YZ are di�erent and frequency dependent, and so will be Tv and Ti1. However,

it is possible to relate them to each other through:

Ti =(Tt

v

)−1(2.14)

where t stands for transposition. It is therefore su�cient to compute only one of them.

To write the line equations in modal domain, it is still necessary to convert the line functions Z and

Y to modal equivalents through:

Zm = T−1v Z Ti (2.15)

Ym = T−1i Y Tv (2.16)

where Zm and Ym are diagonal matrices. These are used to compute the auxiliary functions:

• the matrix of modal characteristic admittances:

Ymc = (Zm)−1

√ZmYm (2.17)

• the matrix of modal propagation functions:

Hm = e−√

YmZmd (2.18)

The solution in frequency domain to equations (2.12) and (2.13) is then similar to the phase equations

(2.3) and (2.4), as long as all quantities are considered in modal domain. The time domain solution is

then simply:

(ymc (t) ∗ vm

k (t)− imk (t)) = hm(t) ∗ (ymc (t) ∗ vm

m(t)− imm(t)) (2.19)

(ymc (t) ∗ vm

m(t) + imm(t)) = hm(t) ∗ (ymc (t) ∗ vm

k (t) + imk (t)) (2.20)

where ymc (t) and hm(t) are the inverse Fourier transformation of the matrices Ym

c (ω) and Hm(ω).

vm(t) and im(t) are vectors the modal voltages and currents in time.

1An eigenvector can be arbitrarily scaled, thus Tv and Ti are not uniquely de�ned. The ambiguity in their calculation

can be removed by normalizing the matrices columns to vectors of unitary euclidean length, that is, by requiring Tv1T ∗v1 +

Tv2T ∗v2 + ... = 1 with T ∗vi =conjugate complex of the i-th column of Tv. However, there is still ambiguity in the sense that

each column can be multiplied with a rotation constant ejα and still have unitary vector length.

7

Due to the independence of the line modes, the characteristic admittance and propagation matrices

become diagonal in modal domain. So, each of equations (2.19) and (2.20) in fact represents n inde-

pendent scalar equations, and each convolution represents only n scalar convolutions. In phase domain

equations (2.8) and (2.9), they represented n2 scalar convolutions.

This means that each mode can be studied as a single-phase line. However, the advantage is not as

good as may seem, since an additional set of convolutions must be taken in order to convert the modal

voltages and currents to the natural domain of phases through:

v(t) = tv(t) ∗ vm(t) (2.21)

i(t) = ti(t) ∗ im(t) (2.22)

Notice that, since the transformation matrices are frequency dependent, the last step implies calcu-

lating the inverse Fourier of Tv(ω) and Ti(ω), thus increasing the complexity of analysis.

2.3 The propagation function of a transmission line

It is worthwhile to analyze the meaning of the propagation function of a transmission line, which is easier

to understand for the case of a single-phase line. For this simple case, the propagation function is scalar

and de�ned as:

H = e−γ(ω)d = e−α(ω)d . e−jβ(ω)d (2.23)

with γ = α + jωβ, H contains an attenuation factor e−αd as well as a phase shift factor e−jβd, both

functions of frequency.

To go deeper into the meaning of H, consider a current source Is in parallel with an admittance

equal to the characteristic admittance of the line (to avoid re�ections), connected to the sending end, k,

of a line having the receiving end, m, short-circuited, as illustrated in �gure 2.2. In that case, we have

YcVk + Ik = Is and Vm = 0. From equation (2.4):

Im = HIs (2.24)

That is, the propagation function H is the ratio (receiving end current)/(source current) of a short-

circuited line fed through a matching admittance Yc to avoid re�ections at the sending end k.

If Is = 1 at all frequencies, then its time domain transformation is a unit impulse is(t) = δ(t) (in-

�nitely high spike which is in�nitely narrow with an area of 1). Setting Is = 1 in equation (2.24) shows

that H(ω) transformed to time domain must be the impulse that arrives at the receiving end m if the

source is a unit impulse. According to (2.23), this response to the unit impulse will be attenuated (no

longer in�nitely high) and distorted (no longer in�nitely narrow) as illustrated in �gure 2.3 for a typical

single-phase line.

8

If is(t) is an arbitrary function of time, equation (2.24) transforms to the time domain as:

im = h(t) ∗ is(t) =

∫ +∞

τmin

h(τ)is(t− τ)dτ (2.25)

The convolution integral starts in τmin since h(t) = 0 for t < τmin, as illustrated in �gure 2.3. This

expression shows that im(t) is constructed as the sum of the samples of is(t) taken τ units of time ago

and weighted according to the value of h(τ).

Figure 2.4 shows that h(t) can also be expressed as a similar function translated in time to the origin.

In that case:

h(t) = h′(t− τmin) (2.26)

which transforms to frequency domain as

H(ω) = H ′(ω)e−jωτmin (2.27)

that is, a time delay in the time domain becomes a phase shift in the frequency domain.

Figure 2.2: Illustration of the physical meaning of the propagation function � current source in parallel

with characteristic admittance connected to a short-circuited transmission line.

Figure 2.3: Illustration of the physical meaning of the propagation function � left: current unit impulse

applied to the line; right: short-circuited line response to a current unit impulse.

9

Figure 2.4: Illustration of the physical meaning of the propagation function � alternative representation

of h(t) through a time function translated to the origin, h′(t).

10

Chapter 3

EMTP-RV and transmission line

modeling

3.1 Background on line modeling

The EMTP-RV 2.3 is a specialized software for the simulation of electromagnetic, electromechanical

and control systems transients in multiphase power systems. The software is used worldwide by many

utilities, companies and consultants. Its main applications include projects, design and engineering or

the solution of problems and unexpected failures.

Speci�cally, the program is useful to study transmission systems, including insulation coordination

and switching design. The transmission line models available in the EMTP-RV 2.3 provide a summary

of the evolution of line modeling, from the simplest constant parameters model, to the more complex

frequency dependent models, which approximate the line functions by analytical expressions in the form

of rational functions of frequency.

Generally, transmission line models represent the line as a multi-port system, that is, the study of the

line behavior is described in terms of the currents and voltages at the two line terminals. Nevertheless,

there are several aspects that distinguish the line models, namely:

• Characterization of the line: a transmission line is characterized by two functions, based on

the parameters per unit length of the line R, L, G and C. The �rst alternative is using the

characteristic admittance Yc and the propagation function H. In this case, the line is analyzed in

terms of re�ected and incident current waves (YcV ± I) at the two terminals k and m. The line

equations in frequency domain are:

(YcVk − Ik) = H(YcVm − Im) (3.1)

(YcVm + Im) = H(YcVk + Ik) (3.2)

11

The other alternative is using the characteristic impedance Zc = Y−1c and the propagation function

H. In this case, the line is analyzed in terms of re�ected and incident voltage waves (V ± ZcI)

at the two terminals k and m. The line equations in frequency domain are:

(Vk − ZcIk) = H(Vm − ZcIm) (3.3)

(Vm + ZcIm) = H(Vk + ZcIk) (3.4)

• Accounting for frequency dependence of line functions: for the case of a constant param-

eters model, the approximating line functions are constant in frequency. Therefore, the transfor-

mation of those functions to time domain is immediate and the line equations have no convolution

integrals. On the other hand, frequency dependent models approximate, within a frequency range

of interest, each line function by a sum of rational terms, which transforms to time domain as a

sum of exponential terms. The time domain equations contain convolution integrals which may be

computed recursively [1], as described in section 3.2.2. Frequency dependent models use di�erent

techniques to compute rational approximations of the line functions. Section 3.2.3 describes two

examples of these techniques: Asymptotic Fitting and Vector Fitting.

• Solution domain: a line model can be computed in phase domain, through a set of coupled

equations, or in modal domain, using independent equations for each mode. All EMTP-RV 2.3

line models make use of some information from the modes and all use a constant real transformation

matrix. The optimal frequency at which this matrix is evaluated may be automatically computed

by the EMTP-RV 2.3, or speci�ed by the user1.

Given the use of an approximating transformation matrix, the models in modal domain are based

on approximated modes, which represents a source of inaccuracy in relation to the phase domain

models. Furthermore, due to the interaction of the line with the outside system (which is modeled

in phase domain) it is necessary to convert the computed modal variables to phase domain at each

simulation step, increasing the model processing time.

These are the basic characteristics that distinguish the several EMTP-RV 2.3 models. Other line

models may have di�erent characteristics. For example, �tting the line functions in z-domain, instead of

s-domain [11], or considering lumped instead of distributed parameters.

In order to characterize the model e�ciency, it is also necessary to analyze how it is implemented.

From the chapter on line theory, it was clear that the computation of the line variables implies (1)

computing the inverse Fourier transformation of the line propagation parameters and (2) computing the

line equations which contain convolution integrals. Thus, the accuracy and e�ciency of the model is

1The optimum frequency determination procedure selects an optimum value of frequency for the range of switching

transients. This value is based on asymptotic conditions for the particular line under consideration. Typical values range

from 500 Hz to 5 kHz with a average around 1 kHz. The selection of an optimum value is based on the constancy of the

transformation matrix within the typical frequency range for switching transients. For studies involving other frequency

ranges (lightning, for example) the frequency should be supplied by the user.

12

greatly in�uenced by the way the line functions are represented and the techniques for computing the

convolution integrals.

Section 3.2, in this chapter, introduces two numerical techniques which are crucial to line models

e�ciency. The �rst technique involves a recursive computation of the convolution integrals in line

equations. The second consists of representing the line propagation parameters through approximating

analytical expressions in the form of rational functions of frequency. The advantages of both techniques

are clari�ed. After that, section 3.3 presents the line models available in EMTP-RV 2.3. These models

are then tested and compared in section 3.4.

3.2 Numerical techniques

3.2.1 Major challenges

The simulation of a transmission line implies the computation for each time step of the matrix equations

(2.8) and (2.9), or (2.19) and (2.20) for a modal domain analysis2. This creates two major challenges:

• The calculation of the inverse Fourier transformation of the line functions H and Yc, known in

frequency domain. This represents a preprocessing routine;

• The calculation at each time step of the convolution integrals in line equations (2.8) and (2.9).

The most direct approach is to execute these steps using the exact line functions H and Yc evaluated

at each frequency sample. The computation of the Fourier transformation of these functions results into

an equal number of time samples. The convolution integrals computed at each time step must then con-

sider the complete range of samples. This procedure is not only highly demanding in terms of memory

and processing time, but also vulnerable to integration errors.

Alternatively, currently used line models approximate the elements of matrices H and Yc with

analytical expressions in the form of rational functions of frequency. The advantages of this approach

are:

• Direct calculation of h(t) and yc(t): the inverse Fourier transformation of a rational function of

frequency has a well known analytical form;

• Memory saving: instead of saving a high number of samples of H and Yc, it is only necessary to

keep the parameters of their approximating functions;

2For a modal domain approach, it is necessary to convert at each time step between phase and modal quantities through

equations vphase(t) = tv(t) ∗ vmodal(t) and iphase(t) = ti(t) ∗ imodal(t), where tv(t) and ti(t) are the inverse Fourier

transformation of the voltage and current transformation matrices. Many line models, however, consider real constant

transformation matrices, turning this equations into simple matrix products with reduced impact on the model e�ciency.

Thus, this step will be omitted along this chapter.

13

• Possibility of computing the convolution integrals recursively [1], greatly reducing the processing

time for each time step.

The use of this strategy implies an additional e�ort to compute the functions which approximate the

elements of H and Yc. This aditional e�ort must be included in the preprocessing routine and it does not

interfere with the requirements of real-time simulations. However, the complexity of the approximating

functions will have a direct impact on the model e�ciency. It is therefore mandatory to obtain adequate

approximations: an optimized reduced order model.

Section 3.2.2 illustrates the technique of recursive convolutions [1]. A general view of the techniques

for rational approximation of line functions is given in section 3.2.3.

3.2.2 Recursive calculation of convolution integrals

The digital simulation of a transmission line implies the calculation, at each time step, of a set of equa-

tions involving convolution integrals, in terms of the time domain counterparts of the line functions and

the voltages and currents at the two line terminals.

A numerical solution is prohibitively time and memory consuming. A much more e�cient approach is

the recursive solution of the convolutions integrals [1]. This technique consists of dividing the convolution

integral in two smaller integrals: one computed from the beginning of the simulation until the previous

time step; the second computed over the present time step.

The use of this technique reduces considerably the processing time for each simulation step, since it

is only necessary to compute the second part of the integral (the �rst comes from the previous iteration).

It also reduces the memory requirements since it is only necessary to keep track of a few past time

steps, to account for the delay of propagation across the line. Nevertheless, the application of recursive

convolutions, for the purpose of transmission line modeling, requires the line functions h(t) and yc(t) to

be represented as a sum of exponentials.

To illustrate the basics of recursive convolution, consider �gure 3.1, where a given system is rep-

resented by its impulse response h(t). g(t) is the system response to an input signal f(t), computed

through a convolution integral:

Figure 3.1: Illustration of the method of recursive convolutions: system represented by its impulse

response h(t). The function g(t) is the response of the system to an input signal f(t).

14

g(t) = h(t) ∗ f(t) =

∫ +∞

−∞h(τ)f(t− τ)dτ =

∫ +∞

τ=0

Ae−bτf(t− τ)dτ (3.5)

where h(t) = Ae−bt for t > 0. At an instant t+ ∆t, one may write:

g(t+ ∆t) =

∫ ∞τ=0

Ae−bτf(t+ ∆t− τ)dτ (3.6)

Now consider the change of variables:

z = τ −∆t dz = dτ

The application of this change of variables in (3.6) leads to:

g(t+ ∆t) = e−b∆t[∫ 0

z=−∆t

Ae−bzf(t− z)dz +

∫ ∞z=0

Ae−bzf(t− z)dz]

(3.7)

= e−b∆t[∫ 0

z=−∆t

Ae−bzf(t− z)dz + g(t)

](3.8)

The �rst integral in (3.8) may be developed if the function f(t−z) is approximated inside the interval

z ∈ [−∆t; 0] by a polynomial function. Consider, for simplicity, a �rst order approximation. According

to �gure 3.2:

Figure 3.2: Illustration of the method of recursive convolutions: Input signal f(t− z) as a function of z

inside the interval [−∆t; 0].

f(t− z) ≈ f(t)− f(t+ ∆t)

∆tz + f(t) = k1z + k2 , for z ∈ [−∆t; 0] (3.9)

where k1 = f(t)−f(t+∆t)∆t and k2 = f(t) are determined at each time step. This allows to solve the �rst

integral in (3.8) by parts, resulting into:

e−b∆t∫ 0

z=−∆t

Ae−bzf(t− z)dz = α f(t) + β f(t+ ∆t) (3.10)

where α and β are constants given by:

α =A

b

(−e−b∆t +

1

b∆t(1− e−b∆t)

)(3.11)

β =A

b

(1− 1

b∆t(1− e−b∆t)

)(3.12)

15

Equation (3.9) may now be rewritten in a recursive manner as:

g(t+ ∆t) = α f(t) + β f(t+ ∆t) + γ g(t) (3.13)

with α and β given by (3.11) and (3.12) and γ = e−b∆t. Notice that calculation of (3.13) implies only

the calculation of three products involving scalar quantities (the coe�cients are constant throughout

the simulation). In the general case of an impulse response approximated by a sum of exponentials

h(t) =∑iAie

−bit, thus resulting:

g(t) =∑i

gi(t) (3.14)

where gi(t) = hi(t) ∗ f(t). This method has proven to be very accurate and stable, as explained in [1].

Notice that the number of exponentials used to approximate the impulse response h(t) increases the

accuracy of the method, and also its processing time.

3.2.3 Rational approximation of transmission line functions

3.2.3.1 Background

As mentioned before, transmission line models characterize the line through the propagation function

and the characteristic admittance (or, alternatively, the characteristic impedance). Generally, instead of

using the "exact" value of their samples, frequency dependent models approximate those functions with

analytical expressions in the form of rational functions of frequency. Consider a general function of the

complex frequency s = jω, G(s), which could approximate any of the line functions:

G(s) = c0

∏Mm=1(s− zm)∏Nn=1(s− pn)

=

N∑n=1

cns− pn

(3.15)

where M and N ≥ M are the number of zeros (zm) and poles (pn) used in the approximation of G(s).

c0 is a constant and cn is the residue of G(s) corresponding to pole pn, which is:

cn = Res[G(s)]pn = lims→pn

dk−1

dsk−1[(s− pn)G(s)] (3.16)

where k if the multiplicity of pole pn. This way of representing the line functions presents several

advantages:

• Direct analytical inverse Fourier transformation of the line functions: the time-domain counterpart

of (3.15) is given by:

g(t) = F−1 [G(ω)] =

N∑n=1

cnepnt (3.17)

• Memory saving: instead of keeping a high number of time samples, only the parameters of the

approximating line functions are needed, which identify both their frequency and time-domain

counterparts.

• Processing time saving: thanks to the possibility of using recursive convolutions to compute the

line variables at each time step of a simulation.

16

The computation of those approximating functions represents a preprocessing routine to a frequency

dependent model. The accuracy and e�ciency of the model is directly related to the complexity of the

approximating functions. Higher order approximations provide more accurate results, but increase the

processing time associated to the model.

Two techniques have been widely used in line modeling for the rational approximation of frequency

responses: Asymptotic Fitting [2] and Vector Fitting [4, 5, 6].

3.2.3.2 Asymptotic Fitting

The Asymptotic Fitting technique has been introduced in line modeling by J. Marti [2] and it is based

on the approximation of the magnitude of the original function.

From zero up to a maximum frequency, at which the original function approaches zero or becomes

constant, the original function is compared to the approximating function. Poles and zeros are assigned to

the �tting function as needed, that is, when the di�erence between the two functions is above a maximum

accepted error. Thus, the order of the approximation is not established a priori, but determined by the

approximating routine.

For a stable model, all poles lie on the left side of the complex plane. The Asymptotic Fitting uses

only real poles and zeros to avoid ripples or local peaks in the approximating function.

It is important to refer that this technique considers that the characteristic admittance (or impedance)

and the propagation function (after extracting the minimum propagation delay3) are approximately

minimum phase shift functions. For this class of functions, the phase of the original function matches

the phase of the corresponding approximating function. For the propagation function, the minimum

propagation delay is usually computed by comparing the phases of approximating and original functions.

The condition of minimum-phase-shift function is achieved by setting all zeros of the approximating

function on the left half of the complex plane.

3.2.3.3 Vector Fitting

The Vector Fitting technique [4, 5, 6] was introduced in line modeling by Gustavsen and Semlyen [14, 15]

and the program that implements this technique is available on the internet [7].

Vector Fitting technique consists of approximating a frequency response (magnitude and phase) in

an iterative manner using a prescribed set of starting poles. To illustrate the method, consider the

approximating function:

f(s) ≈N∑n=1

cns− an

+ d+ se (3.18)

The residues cn and poles an are real or complex quantities. d and e are real quantities allowing

di�erent degrees of accuracy. The objective of the method is to �t these parameters in order to obtain a

3This method has been used in line models computed in mode domain. Thus, each mode has its own propagation delay

and it is possible to represent the modal propagation function as H(ω) = H′(ω)e−jωτ , where τ is the minimum propagation

delay of the corresponding mode and H′(ω) is a minimum-phase-shift function.

17

least squares approximation of f(s) over a given frequency interval. This is a non-linear problem since

the parameters an appear on the denominator. Vector Fitting solves it as a linear problem in two stages,

both with known poles.

The �rst stage covers the poles identi�cation. Consider a set of initial poles an and multiply f(s) by

an unknown function σ(s). An additional equation is introduced for a rational approximation of σ(s),

resulting into a higher order description:

σ(s)f(s) ≈N∑n=1

cns− an

+ d+ se (3.19)

σ(s) ≈N∑n=1

cns− an

+ 1 (3.20)

Multiplying the second row of (3.20) by f(s) yields to:(N∑n=1

cns− an

+ d+ se

)≈

(N∑n=1

cns− an

+ 1

)f(s) (3.21)

=⇒ (σf)fit(s) ≈ σfit(s)f(s) (3.22)

Equation (3.21) shows a linear dependence as regards cn, cn, d and e, and it is solved as a linear

least squares problem. A rational approximation for f(s) can now be obtained. This becomes evident

by writing:

(σf)fit(s) = e

∏N+1n=1 (s− zn)∏Nn=1(s− an)

σfit(s) =

∏Nn=1(s− zn)∏Nn=1(s− an)

(3.23)

From (3.22):

f(s) ≈ (σf)fit(s)

σfit(s)= e

∏N+1n=1 (s− zn)∏Nn=1(s− zn)

(3.24)

That is, the poles of f(s) are an approximation of the zeros of σ(s). Therefore, by computing the

zeros of σ(s), one gets a good set of poles to �t f(s).

The second stage involves computing the residues of f(s). Using the zeros of σ(s) as the new poles,

equation (3.18) becomes linear in terms of cn, d and e, and is solved as a least squares problem.

In order to achieve a good approximation of f(s), it is necessary to repeat these two stages iteratively,

using the computed poles as new starting poles until an acceptable overall error is achieved. Vector Fit-

ting as been improved [5, 6] in order to accelerate convergence of the method.

All the poles are forced to be stable by inverting the sign of their real part when needed. For a fast

convergence, the initial poles should be well distributed over the frequency range of interest.

Instead of approximating a single function, Vector Fitting may be applied to an array of functions,

using the same set of poles for all. This can be very useful in line modeling, for example, for a column-wise

approximation of the characteristic admittance matrix.

18

3.2.3.4 Asymptotic Fitting versus Vector Fitting

For real-time simulations, it is important to limit the order of the line model. With Vector Fitting this is

ensured a priori, by de�ning an initial set of poles for the approximating functions. This can't be done

with Asymptotic Fitting, where the number of poles is de�ned by the approximating routine.

For the same order of approximation, Vector Fitting generally gives more accurate results. First,

because it �ts the real and imaginary part of the original function, and not just its magnitude function

like Asymptotic Fitting4. Second, Vector Fitting is not constraint to real poles, thus making it a more

�exible technique than Asymptotic Fitting.

Finally, by Vector Fitting it is possible to �t a set of functions with the same poles. This can be par-

ticularly useful, for example, for a low order �tting of the characteristic admittance of a transmission line.

Therefore, Vector Fitting is generally the most indicated technique to use in transmission line model-

ing for the possibility of prede�ning the order of the model and to achieve a more accurate representation

with lower order approximations.

3.3 Transmission line models provided by the EMTP-RV 2.3

3.3.1 CP Line � constant parameters line model

This line model is based on the work by Dommel [10].

The CP Line model is based on modal analysis and each of the n line modes is characterized in terms

of the corresponding characteristic admittance Yc and propagation function H, which are determined

through a real constant transformation matrix.

The model approximates the line as an ideal lossless line, that is, with R = 0 and G = 0. The

inductance L is considered constant and evaluated at the same frequency used for the transformation

matrix. Therefore, each modal function simply becomes:

Yc =

√C

L(3.25)

H = e−jωτ (3.26)

where C, L and τ = d√LC are the capacitance, inductance and propagation delay of the corresponding

mode. Each mode is studied as a single-phase line by using equations:√C

Lvk(t)− ik(t) =

√C

Lvm(t− τ)− im(t− τ) (3.27)√

C

Lvm(t) + im(t) =

√C

Lvk(t− τ) + ik(t− τ) (3.28)

4The technique of Asymptotic Fitting is based on the assumption that the original function is a minimum phase shift

function, which is generally just an approximation.

19

where all voltages, currents and parameters correspond to a speci�c mode. The electric quantities com-

puted from these equations must then be converted into phase domain at each simulation step, by using

the constant real transformation matrix.

This model can include the e�ect of losses in the form of a constant resistance R. To do so, the ideal

lossless line with distributed parameters is divided in two segments of half the length (that is, half the

propagation delay). The resistance R is then inserted in the form of a lumped parameter in discrete

positions: R/4 at the terminals and R/2 between the two segments. This is illustrated in �gure 3.3.

Figure 3.3: Inclusion of losses in a CP Line Model, in the form of lumped resistances.

3.3.2 FD Line � frequency dependent line model

The FD Line model is based on the work by J. Marti in [2].

This model is based on modal analysis and characterizes each of the n line modes through the cor-

responding characteristic impedance Zc and propagation function H. These functions are computed

from the line parameters in phase domain, by using a real constant transformation matrix. The line

parameters R and L are considered frequency dependent. A non-zero shunt conductance is included on

the admittance matrix Y of the line (default value 0.2× 10−9 S/km).

For each line mode, the characteristic impedance and propagation function are approximated by

Asymptotic Fitting [2] in the s-domain, as:

Zc(s) ≈ k0 +

Nz∑x=1

kxs− px

(3.29)

H(s) ≈

(Nh∑y=1

kys− py

)e−sτmin (3.30)

where Nz and Nh are the number of poles used to approximate the corresponding modal functions and

τmin is the minimum propagation delay of the mode.

3.3.3 WB Line � wide-band line model

This model is based on the Universal Line Model, presented in a work by Gustavsen et al. [3].

The WB Line model describes the line in phase domain through matrices Yc and H. The line pa-

rameters R and L are considered as frequency dependent. A non-zero shunt conductance is included on

20

the admittance matrix Y of the line (default value 0.2× 10−9 S/km).

The admittance matrix Yc(ω) is �tted column-by-column by using Vector Fitting [4]. The elements

of the propagation matrix H(ω) are all approximated with the same poles and delays, de�ned by the

approximated modes. These approximations are described in equations (3.31) and (3.32), where:

Ycij (ω) ≈ k0 +

Nj∑x=1

kxjω − px

(3.31)

Hij(ω) ≈n∑k=1

(Nk∑m=1

cmkijjω − pmk

)e−jωτk (3.32)

where:

• n is the number of line modes,

• Nj is the number of poles used to �t the elements of the jth column of Yc,

• Nk is the number of poles used to �t the kth modal propagation function and

• τk is the minimum propagation delay associated to the kth mode.

The poles of Yc are generally real, whereas those of H may be real or complex. The modal poles and

delays that approximate H are obtained by applying Vector Fitting to each modal propagation function.

The residues cmkij are computed from a set of samples of H, by solving a linear least squares problem.

3.4 Model testing and comparison

This section presents a set of tests that analyze and compare the line models provided by the EMTP-RV

2.3, in terms of e�ciency and accuracy. The tests contemplate frequency and time domain simulations

including the line described in appendix, represented by the models. For this line, the optimal frequency

computed by the EMTP-RV to evaluate the constant real transformation matrix and line parameters is

1.0956 kHz, for all line models.

3.4.1 Frequency response � short-circuited and open-ended line

This test is specially adequate to infer the accuracy of the line models: given the simplicity of the

boundary conditions, it is possible to derive the analytical expression of the frequency response of the

line, and use it as a reference to analyze model accuracy. The two cases - short-circuited and open-ended

line, are illustrated in �gure 3.4, where the source feeding the line is a three-phase ideal symmetrical

source of 1V�RMS value.

21

Figure 3.4: Circuits used to study the short-circuit and open-end frequency responses according to the

line models available on the EMTP-RV 2.3

The analytical formulas of the frequency response of a line in short-circuit and open-end are:

Ik_short = Yc

(I− H2

)−1 (I + H2

)Vs (3.33)

Vm_open =(I− H2

)−1H 2Vs (3.34)

where I is the identity matrix. Ik_short is an array with the phase currents at the sending end of the

short-circuited line. Vm_open represents the phase voltages at the receiving end of the line in open-end.

Based on equations (3.33) and (3.34) and on line data, it is possible to compute the exact value

of these frequency responses. Tables 3.1 and 3.2 show the exact values of the current/voltage and the

deviation of the values computed according to the three line models. For simplicity, only the frequencies

0.1 Hz, 50 Hz, 1 kHz and 100 kHz are presented.

Table 3.1: Analytical short-circuit frequency response and approximating errors according to the

EMTP-RV 2.3 models, in terms of the magnitude of the current at the sending end of phase 1

Model 0.1 Hz 50 Hz 1 kHz 100 kHz

Analytical 0.059407 0.022864 0.0017535 0.0048819

CP Line −38.478% −11.304% +3.0265% −28.848%

FD Line +16.423% +0.5536% +3.1092% −0.3263%

WB Line +11.668% −0.6456% +3.1400% +0.0139%

Observing table 3.1, which concerns short-circuit results, it is evident that the CP Line is generally

the least accurate model. An exception is veri�ed at 1 kHz, where the model provides the best approx-

imation of the line response. Notice that this frequency is very close to that chosen by EMTP-RV to

compute the transformation matrix for all models and to process the propagation parameters of the CP

Line.

As regards the open-end results in table 3.2, the errors of the frequency response according to all

22

models are smaller than the ones concerning the short-circuit scan. The CP Line is again the least

accurate model, with the exception of very low frequencies.

Table 3.2: Analytical open-end frequency response and approximating errors according to the

EMTP-RV 2.3 models, in terms of the magnitude of the voltage at the receiving end of phase 1

Model 0.1 Hz 50 Hz 1 kHz 100 kHz

Analytical 1 1.006 1.8731 0.96315

CP Line 0% +0.0696% −0.8494% +35.393%

FD Line −0.0002% −0.0020% −0.7763% −2.0098%

WB Line −0.0001% +0.0129% −0.9332% −2.1639%

Concerning the FD Line and the WB Line, both give very similar and acceptable results. Actually,

the FD Line is more accurate for many frequency points, both for short-circuit and for open-end condi-

tions. However, in these cases, the di�erence between the two models is almost negligible.

A comparison of the EMTP-RV models performance is found in �gures 3.5 and 3.6. For the short-

circuit response, the plots represent the sending end current of phase 1 according to the three models,

whereas the open-end response is analyzed regarding the phase 1 receiving end voltage.

These plots show the great di�erence between the results generated with the CP Line in relation to

the other two models. Regarding the short-circuit scan, there is a great di�erence of results both for

low and high frequencies. For the open-end scan, the higher frequencies are more critical. Concerning

the FD Line and the WB Line, the results seem coincident for most of the frequencies, except for the

short-circuit scan, where the results of the two models diverge for low frequencies.

Figure 3.5: Short-circuit frequency response according to the EMTP-RV 2.3 line models, in terms of

the current at the sending end of phase 1 (CP Line � thin, FD Line � dotted, WB Line � bold).

23

Figure 3.6: Open-end frequency response according to the EMTP-RV 2.3 line models, in terms of the

voltage at the receiving end of phase 1 (CP Line � thin, FD Line � dotted, WB Line � bold).

3.4.2 Line energization and single-phase short-circuit

This test shows how the models represent the behavior of the line under two common transient conditions:

line energization and single-phase short-circuit. Figure 3.7 illustrates the circuit used, where a three-

phase line is connected through ideal switches to a three-phase ideal symmetrical source of 1V peak

voltage and 50 Hz. The three phases are connected to the source simultaneously at t = 20 milliseconds.

After the transient of line energization, a short-circuit occurs at t = 180 milliseconds in phase 3 of the

line. The voltage at the receiving end of phase 1, vm1(t), is observed.

Figure 3.7: Circuit used to study the time response to energization followed by single-phase

short-circuit, according to the line models available on the EMTP-RV 2.3.

The results of this test are plotted in �gures 3.8 and 3.9, which represent only the CP Line and the

WB Line approximations. This is done for simplicity, given the FD Line generates practically the same

results as the WB Line. As regards the CP Line, the results are noticeably di�erent for both transient

24

conditions. Particularly, this model doesn't manage to represent the distortion introduced by the line

in the energization transient, as perceived by the square waves in �gure 3.8. Furthermore, the CP Line

also shows a stronger attenuation. On the other hand, the WB Line shows more realistic results, with

a smoother wave, representing the distortion phenomena.

Concerning the transient induced by the short-circuit on phase 3, illustrated in �gure 3.9, the di�er-

ence between the results of the two models basically resumes to a delay and weaker attenuation in the

response generated by the CP Line.

Figure 3.8: Response to line energization at t = 20 ms according to the CP Line � thin, and to the WB

Line � bold, in terms of the voltage at the receiving end of phase 1.

Figure 3.9: Response to a short-circuit at the receiving end of phase 3 at t = 180 ms according to CP

Line � thin, and to the WB Line � bold, in terms of the voltage at the receiving end of phase 1.

An important note is that line energization directly a�ects all phases, whereas the short-circuit af-

25

fects phase 1 only through induction. Therefore, the results of the energization transient re�ect mainly

the accuracy of the models in approximating the diagonal elements of the line matrices (H and Yc or

Zc). As for the short-circuit transient only the o�-diagonal terms of those matrices are taken into account.

This test shows that for common transient conditions, as the ones observed, both FD Line and WB

Line provide accurate and similar approximating responses. On the other hand, the CP Line provides

a poor representation of the line behavior and, therefore, should be used with care and preferably only

for didactic purposes.

3.4.3 Current induced by phase coupling

The last carried out test is one in which all models show signi�cantly di�erent results. Consider �gure

3.10, where the transmission line is short-circuited at all terminals except at the sending end of phase 1,

which is connected to a DC voltage source of 1V, at t = 1 millisecond, by an ideal switch. The current

induced at the sending end of phase 3, ik3(t), is observed.

Figure 3.10: Circuit used to study the phenomena of phase coupling according to the line models

available on the EMTP-RV 2.3, in terms of the current induced in phase 3 by energization of phase 1.

This test results are plotted in �gures 3.11 and 3.12, for the �rst 20 miliseconds and 1 second of

the simulation, respectively. Figure 3.11 plots the very initial transient on the current of phase 3. It is

evident that the responses according to the di�erent line models become considerably di�erent as time

goes by. Again, the results of the CP Line disagree with the responses computed according to the other

two models, which present similar results for the initial period of the transient. Consider, however, a

longer period of the same test, as plotted in �gure 3.12. Now, the di�erence between the models results

is far evident, given that each model presents a di�erent response. However, only the WB Line shows

results physically acceptable.

In fact, due to the source, the circuit will reach a DC steady-state, extinguishing coupling phenomena.

Since phase 3 is grounded, and due to resistivity of line and ground, the current on this phase goes to

zero. The WB Line is the only with a satisfying result in these conditions � the current declines to zero.

The inaccuracy in the models response is due to the problem of �tting the o�-diagonal elements of the

26

line functions, which are related to the coupling between the line phases. Generally, these elements are

very small, compared to the respective diagonal elements. Therefore, if the �tting process is based on

absolute errors, their approximation may be less accurate.

This is a critical test, as perceived by the diverse responses obtained from the various line models.

Nevertheless, it allows to verify that the WB Line is more accurate than the FD Line, an thus, can be

use in a wider variety of transient conditions and still provide physically acceptable results.

Figure 3.11: Current induced by energization of phase 1 � time evolution of the current at the sending

end of phase 3 during the �rst 20 miliseconds of the transient according to the EMTP-RV 2.3 line

models (CP Line � thin, FD Line � dashed, WB Line � bold).

Figure 3.12: Current induced by energization of phase 1 � time evolution of the current at the sending

end of phase 3 during the �rst second of the transient according to the EMTP-RV 2.3 line models (CP

Line � thin, FD Line � dashed, WB Line � bold).

27

3.4.4 Model e�ciency

The model e�ciency is associated both to the processing time required for each step of the simulation

and to the accuracy of the generated results. These two aspects are conditioned by the complexity of the

model. CP Line is by far the fastest model, but this is only possible due to using constant line parameters.

The other two EMTP-RV models consider the frequency dependence of the line functions, which are

�tted by rational functions of frequency. The line functions are thus more accurately represented for a

wide range of frequencies, but the e�ort to compute the convolution integrals on line equations depends

on the order of those approximations. Table 3.3 summarizes the order (that is, the number of poles)

of the approximating functions generated by EMTP-RV for the FD Line. As regards the WB Line,

the same set of poles is used to �t all the columns of Yc(ω). The order of the approximating functions

generated by EMTP-RV are summarized in table 3.4.

Table 3.3: Number poles used by the FD Line, available on the EMTP-RV 2.3,

in the approximation of the propagation parameters of a line

Function Mode 1 Mode 2 Mode 3 Total

Zc(ω) 17 15 18 51

H(ω) 23 23 23 69

Table 3.4: Number of poles used by the WB Line, available on the EMTP-RV 2.3,

in the approximation of the propagation parameters of a line

Function Mode 1 Mode 2 Mode 3 Total

Yc(ω) � � � 11

H(ω) 6 7 8 21

The WB Line uses a total of 11 + 21 = 32 poles compared to the 51 + 69 = 120 poles used by the FD

Line. Furthermore, the test results of this chapter have showed that the WB Line is the most accurate

of the line models provided by the EMTP-RV. Therefore, the WB Line presents a great improvement in

e�ciency, by obtaining better results with less resources. The e�ciency of the WB Line is reinforced by

the fact of being a phase-domain model � there is no need to convert from phase to modal quantities,

and vice-versa, at each simulation step.

3.5 Conclusions

The EMTP-RV 2.3 is a specialized software particularly useful to study transmission systems, including

insulation coordination and switching design. The transmission line models available in the EMTP-RV

provide a summary of the evolution from the constant parameters model, to the more complex frequency

dependent models, that approximate the line functions by analytical expressions in the form of rational

functions of frequency.

28

Generally, transmission line models treat the line as a two-port system, that is, the study of the line

behavior is described in terms of the currents and voltages at the two line terminals. Nevertheless, there

are several aspects that distinguish the line models:

• They may represent the line behavior in terms of incident and re�ected current waves, which

corresponds to using the line functions H and Yc, or in terms of incident and re�ected voltage

waves, by using H and Zc;

• Another di�erentiating aspect of line models is the account for the frequency dependence of line

parameters. Frequency dependent models approximate, within a frequency range of interest, each

line function by a sum of rational terms, which transforms to time domain as a sum of expo-

nential terms. The time domain equations contain convolution integrals which may be computed

recursively [1];

• The line models may also di�er on the domain of solution, whether they described the line in terms

of modal or phase quantities. Given the use of a constant real transformation matrix, the modal

domain approach will be based on approximated modes, which represents a source of inaccuracy in

relation to phase domain models. Furthermore, due to the interaction of the line with the outside

system (which is modeled in phase domain) it is necessary to convert the computed modal variables

to phase domain at each simulation step, increasing the model processing time.

In order to characterize the model e�ciency, it is necessary to analyze how it is implemented, and

speci�cally, which computation techniques are used. Section 3.2.2 gives an introduction to the technique

of recursive convolutions, which allows a very e�cient computation of the integrals in the line equations.

As concerns the rational approximation of the line functions, Asymptotic Fitting and Vector Fitting

are described in section 3.2.3 as two alternative techniques. Vector Fitting approximates both real and

imaginary part of the original function, and it allows real or complex conjugate pairs of poles. On the

other hand, Asymptotic Fitting approximates only the magnitude of the original function, which means

considering it as a minimum phase shift function � generally, an approximation. Furthermore, Asymp-

totic Fitting allows only for real poles. Generally, Vector Fitting achieves more accurate results with

lower order approximations than Asymptotic Fitting. Furthermore, Vector Fitting allows to pre-establish

the number of approximating poles, and is therefore the preferable technique to be used by line models

in-line with the target of real-time simulations.

After giving an overview on some line modeling issues, section 3.3 presents a description of the line

models available on the EMTP-RV 2.3. The CP Line is a modal domain model which approximates

the line parameters as constant. The model represents the losses on the line as lumped resistances

inserted at particular points of the line. The FD Line is a modal domain frequency dependent line

model, which �ts the line functions using the technique of Asymptotic Fitting, thus originating robust

high order approximations. The WB Line is another frequency dependent model. Though this is a

phase domain model, it approximates the elements of the propagation matrix H using the poles and de-

29

lays de�ned by the modes, and computed by applying Vector Fitting to the modal propagation functions.

Finally, section 3.4 tests and compares the line models CP Line, FD Line andWB Line through a set

of frequency and time domain simulations. The �rst of the tests regards the short-circuit and open-end

frequency responses of the line represented in appendix, according to the three models. The CP Line

shows high relative errors for both conditions, whereas the FD Line and the WB Line provide accept-

able and similar results in any conditions. The second test simulates line energization and single-phase

short-circuit conditions. Again, the FD Line and the WB Line generate practically the same results,

whereas the CP Line provides unrealistic time responses, namely in what concerns representing distor-

tion phenomena. The last test concerns the study of coupling phenomena between line phases. For this

test, all line models provide di�erent responses. Nevertheless, only the WB Line generates physically

acceptable results, as regards approximating the steady-state condition of the line.

The described tests show that the CP Line should be avoided for its inaccurate results in most

of the cases observed. As regards the FD Line, it provides very accurate results for several typical

transient conditions. However, it should be avoided for simulating more complex transient conditions,

specially those concerning coupling phenomena. The WB Line proved to be the most accurate of the

tested models, even in approximating the coupling phenomena between the line phases. Furthermore,

the WB Line uses lower order approximations than the FD Line, and is therefore the most e�cient of

the EMTP-RV line models.

30

Chapter 4

Wide-band model for real-time

simulations � RT_WB Line

4.1 Introduction

The goal of this work is to establish adequate numerical techniques for approximating the propagation

parameters for transmission line modeling, allowing real-time simulations. This requires an e�cient use

of reduced modeling resources. In order to ensure additional accuracy, it is necessary to introduce some

optimization procedures. The resulting model is called RT_WB Line, as it is a reformulation of the

EMTP-RV model WB Line, in-line with the real-time simulation target.

The real-time requirement means that during the digital simulation, each time step should have a

processing time never greater than the period represented. This possibility depends on the order of the

model and on the speci�c computer processing capacity � a faster computer can perform real-time simu-

lations with higher order models. To accomplish its function in any conditions, that is, in any computer,

it is assumed that the correct order of the model for real-time performance, that is, the number of poles

to use in the approximation of the line functions, is pre-de�ned.

In order to test the developed model, it must be able to interface with the EMTP-RV. This is ensured

by writing the model data into an output �le using the same template of the WB Line. This allows

testing the applications of the developed model in the EMTP-RV environment as if they had been com-

puted by the software itself.

The following text covers all the steps taken for the de�nition and implementation of the RT_WB

Line: section 4.2 starts by introducing the theoretical formulation of the model; then, section 4.3 presents

the optimization procedures introduced to ensure increased accuracy with reduced order approximations;

�nally, section 4.4 presents the conceptual structure of the routine developed to compute the applications

of the RT_WB Line, with a summary of its subroutines, main variables, input and output data.

31

4.2 Model formulation

The developed line model, called RT_WB Line, uses an approach similar to the WB Line provided by

the EMTP-RV 2.3. It is a phase domain model based on the rational approximations of the propagation

matrix H and characteristic admittance Yc, which are computed in frequency domain using the line

characteristic parameters:

H = e(−√

YZ)d (4.1)

Yc = Z−1√

ZY (4.2)

The elements of the characteristic admittance matrix are all �tted by the same set of poles, through

Vector Fitting. For the ijth element of Yc:

Ycij (ω) ≈ y0ij+

Ny∑n=1

ynij

jω − pn(4.3)

where Ny is the number of poles used to �t the characteristic admittance matrix. Generally, all the

�tting parameters in (4.3) are real quantities.

The elements of the propagation matrix H are �tted by the poles and delays de�ned by the approxi-

mated modes, obtained through a constant real transformation matrix evaluated at 1 kHz1. For the ijth

element of H:

Hij(ω) ≈n∑k=1

(Nk∑m=1

cmkijjω − pmk

)e−jωτk (4.4)

where the poles pmk and residues cmkij are real quantities or come as complex conjugate pairs.

The �rst step to �t the elements of the propagation matrix is to gather the modal data, that is, the

delays and poles de�ned by the approximation of the modal propagation functions. Section 4.3.1 presents

the method used to compute the propagation delays τk. After extracting a constant propagation delay

to each of the modes, the correspondent modal propagation function becomes approximately a minimum

phase shift function, being approximated through Vector Fitting, so as to obtain the modal poles pmk.

After obtaining the modal poles and delays, these are used to compute the residues of each element

of H. To do so, it is necessary to write (4.4) for several frequencies, so as to obtain an overdetermined

linear matrix equation of the form A X = B, where X contains the unknown residues. Each row in A

and B corresponds to a frequency point, and each column in X and B corresponds to an element of H.

The equation A X = B is solved as a linear least squares problem.

1The modal poles di�er slightly from the accurate ones. However, this has little impact on the �nal approximation (4.4)

since a small displacement of the poles will be compensated by a small displacement of the corresponding residues.

32

4.3 Optimized �tting of the propagation function

The RT_WB Line model is based on the rational approximations of the line matrices H and Yc. The

elements of Yc are generally smooth functions of frequency and can easily be �tted by low order func-

tions. The �tting of H is a more challenging task, due to the contribution of the various line modes, all

with a di�erent frequency dependent propagation delay. Furthermore, it must respect the limited order

of the model, pre-de�ned by real-time requirements.

This section presents a set of optimization procedures used on the �tting process of H in order to

ensure increased accuracy of approximation for the pre-de�ned model order.

4.3.1 Optimal modal delay identi�cation

This optimization process regards the computation of the modal propagation delays, necessary for the

process of approximating the phase domain propagation matrix H. Consider the propagation function

of the kth line mode:

Hk(ω) = e−(αk(ω)d+jωτk(ω)) (4.5)

where τk(ω) is the propagation delay of the kth mode. Each function Hk(ω) may be approximated

by a rational function and a constant time delay factor e−jωτ′k :

Hk(ω) = H ′k(ω)e−jωτ′k ≈

(Nk∑n=1

cnjω − pn

)e−jωτ

′k (4.6)

where Nk is the number of poles used to �t the kth mode. The extraction of a constant propagation

delay ensures a more accurate �tting of Hk(ω), using the same number of poles.

The constant delay τ ′k may be computed so that H ′k(ω) becomes approximately a minimum phase

shift function. According to [12], this is done through:

τ ′k ≈ τk(ω) +1

ω

(π

2

d ln|Hk(ω1)|d ln ω1

∣∣ω1=ω

)(4.7)

computed for a frequency ω such that |Hk(ω)| = 0.1.

However, according to [12], the computed delay may not correspond to the most accurate �tting of

Hk(ω). Therefore, the process of optimization suggested regards �nding the modal delays leading to the

most accurate rational approximation obtained with (4.6). Several tests involving di�erent lines have

showed that a good estimation for τ ′k can be found in the interval [ 0.9 τ ′k ; 1.1 τ ′k ], where τ ′k is given by

(4.7). Tests have further showed that generally it is enough to search with an iteration of 1% of the base

modal delay.

Though the accuracy in the approximating modes is not directly related to the phase domain results

[12], this optimization routine has a positive impact on the accuracy of the approximating propagation

matrix H, in phase domain. This is con�rmed in the following numerical example.

33

Numerical Example: Consider the line described in appendix. Table 4.1 shows the average error

of the approximating propagation functions, in modal and phase domain, using a simple modal delay

computed through (4.7) or using an optimized modal delay.

Table 4.1: E�ect of using optimized modal delays � average error of approximating propagation

functions according to di�erent order applications of the RT_WB Line, in mode and phase domain.

Modal Poles Modal Delay Mode domain error Phase domain error

1 Not optimized 3.8524× 10−2 1.7446× 10−2

Optimized 3.5934× 10−2 1.5711× 10−2

9 Not optimized 1.0959× 10−3 3.7699× 10−4

Optimized 1.0875× 10−3 3.6292× 10−4

13 Not Optimized 1.1505× 10−3 4.2359× 10−4

Optimized 9.6221× 10−4 3.3057× 10−4

The table shows that, for certain orders of the approximating functions, the use of optimized modal

delays leads to increased accuracy both in modal and phase domain, when compared to the approximating

functions obtained by simply using the lossless modal delays. Nevertheless, it must be noted that the

improvements introduced by this process vary with the particular line and with the order of the model.

4.3.2 Optimal modal poles assignment

The RT_WB Line model approximates the propagation matrix of a line using the poles and delays de-

�ned by the modes, as expressed in (4.4). In order to respect the pre-de�ned order of the model, the sum

of the poles assigned to each mode, Nk, must be equal to the maximum number of poles allowed for H,

that is,∑Nk = Nmax. The simplest would be to assign an equal number of poles to each mode. How-

ever, practical tests have showed that this choice generally does not lead to the most accurate �tting of H.

Therefore, it is advantageous to optimize the number of poles assigned to each modal propagation

function. This is done by trying all the possible distributions of the available number of poles among

the modes, with the requirement that the total number of poles must respect the pre-de�ned order of

the model. The following numerical example illustrates and justi�es this procedure.

Numerical example: Consider the line described in appendix. Table 4.2 shows the average error of

the approximating propagation matrix considering an equal or optimized distribution of modal poles, for

several applications of the RT_WB Line, which di�er the order of the approximations.

As table 4.2 shows, the optimized assignment of modal poles has a positive impact on the accuracy

of the approximating propagation matrix, with a reduction of the approximation error of at least 6 % in

relation to the equal distribution of modal poles, for all the tested conditions.

34

Table 4.2: Average error of the approximation of H, according to di�erent order applications of the

RT_WB Line. Use of equal (E) or optimized (O) distribution of the modal poles

Total poles Distribution of poles Phase domain error Di�erence (%)

6 E = [2− 2− 2] 7.5693× 10−3

� O = [2− 3− 1] 5.4463× 10−3 −28%

9 E = [3− 3− 3] 3.8117× 10−3

� O = [4− 4− 1] 3.5990× 10−3 −6%

12 E = [4− 4− 4] 2.5625× 10−3

� O = [5− 0− 7] 1.1446× 10−3 −55%

15 E = [5− 5− 5] 1.4659× 10−3

� O = [5− 1− 9] 8.3372× 10−4 −43%

18 E = [6− 6− 6] 7.9743× 10−4

� O = [8− 1− 9] 6.7828× 10−4 −15%

Another interesting aspect showed in table 4.2 is that, for a low number of approximating poles, the

optimized distribution tends to assign more poles to modes 1 and 2, whereas for a higher approximating

order, modes 1 and 3 are preferred. Nevertheless, only in exceptional cases one mode is neglected, being

assigned zero �tting poles, as showed in table 4.2 for a total of 12 poles. Therefore, there is not an

explicit tendency of optimal pole distribution that allows to de�ne a single strategy that works both for

low and high orders of approximation. For example, if one mode is not very signi�cant for the phase

�tting, how can it be de�ned whether it should be assign 1 or 0 poles, without checking the phase error

obtained in the two cases?

Therefore, in order to achieve the optimal approximation of the propagation matrix H, the optimiza-

tion procedure tries all the possible assignements of modal poles. This is not the most e�cient process,

but it certainly reaches the most accurate result, based on the average error of approximating H.

4.4 Computer program

This section presents the structure of the program aimed at transmission line modeling for real-time

simulation. It is a MATLAB routine which is original, except for the use of an external function called

vect�t3.m. This is a free accessMATLAB routine available on the Internet [7], which computes a rational

expression to approximate a function of frequency using the technique of Vector Fitting [4, 5, 6].

The RT_WB Line program is formed by several subroutines which represent the main steps of a

transmission line modeling process. The structure of the program, as well as its input and output, are

illustrated in �gure 4.1. The expected input to the program are the location of the �le containing the

information about the speci�c transmission line to be modeled and the prescribed order of the model,

35

that is, the number of poles allowed to �t the characteristic admittance Yc and the number of poles to

�t the propagation matrix H of the line.

Figure 4.1: General structure of the program developed to compute applications of the RT_WB Line

model in-line with the real-time simulation target, with respective input and output data.

Sections 4.4.1 to 4.4.5 provide a description of the main program and each of its subroutines, namely

its objective and expected input and output.

4.4.1 Main program

• Objective:

� Computing an application of the RT_WB Line using pre-de�ned orders for approximating

the line propagation parameters Yc and H.

• Input Data:

� Line_data_rv.lig � location of the EMTP-RV �le containing the number and value of the

frequency samples to consider, the line parameters per unit length Z and Y computed for

those samples, and the constant real transformation matrix.

� Ny � order of the approximating characteristic admittance matrix of the line, that is, the total

number of poles used to �t the matrix Yc.

� Nh � order of the approximating propagation matrix of the line, that is, the total number of

poles used to �t the matrix H.

36

• Output Data:

� model.dat � location of the generated �le, containing the description of the RT_WB Line

application computed. The template of this �le follows that of an EMTP-RV WB Line

model.

4.4.2 Propagation parameters computation

• Objective:

� The computation of the frequency samples of the propagation parameters of the line to model,

namely, the characteristic admittance Yc, the propagation matrix H, the modal propagation

functions Hk and the corresponding propagation delays and τk, for k = 1, ..., n (n is the

number of line modes).

The computation of these functions is based on the longitudinal impedance matrix Z, on the

transversal admittance matrix Y, as well as on the transformation matrix, as explained on

chapter 2, concerning line theory. These matrices are provided by the input �le Line_data_rv.lig.

• Input Data:

� Line_data_rv.lig � location of the EMTP-RV �le containing the number and value of the

frequency samples to consider, the line parameters per unit length Z and Y computed for

those samples, and the constant real transformation matrix.

• Output Data:

� f � the vector containing the set of sampling frequencies considered.

� Yc � the characteristic admittance matrix of the line, evaluated at all the sampling frequencies.

� H � the propagation matrix of the line, evaluated at all the sampling frequencies.

� Hk, for k = 1, · · · , n � the set of n modal propagation functions of the line, evaluated at all

the sampling frequencies.

� τk, for k = 1, · · · , n � the set of n modal propagation delays of the line, computed through

equation (4.7).

4.4.3 Yc �tting

• Objective:

� The computation of the approximating rational functions of frequency that �t the elements

of the characteristic admittance matrix of the line using a pre-de�ned number of poles.

As explained earlier on this chapter the elements of Yc are all �tted together by the same set

of poles, using the technique of Vector Fitting. The �tting parameters are the poles used to

�t the whole matrix, and for each element of the matrix, the set of residues corresponding to

those poles (see equation (4.3)).

37

• Input Data:

� f � the set of sampling frequencies considered.

� Ny � order of the rational approximation of the elements of the characteristic admittance

matrix of the line, that is, the total number of poles used to �t the matrix Yc.

� Yc � the characteristic admittance matrix of the line, evaluated at all the sampling frequencies.

• Output Data:

� pn, for n = 1, · · · , Ny � the set of poles used to �t the matrix Yc.

� y0ij� the value used to �t the ijth elements of Yc when s→∞.

� ynijfor n = 1, · · · , Ny � the residues used to �t each of the ijth elements of Yc.

These �tting parameters are all included in a structure called Yc_data.

4.4.4 H �tting

• Objective:

� The computation of approximating rational functions of frequency that �t the elements of the

propagation matrix of the line using a pre-de�ned number of poles.

The elements of H are �tted by the poles and delays de�ned by the modes through the solution

of a least squares problem (see equation (4.4)). The modal delays are optimized so as to allow

the most accurate modal representation, as described in section 4.3.1. After extracting the

optimal modal delay, each modal propagation function is approximated by Vector Fitting,

de�ning the modal poles. The number of poles assigned to each mode is that leading to the

most accurate �tting of the propagation matrix H, as described in section 4.3.2.

• Input Data:

� f � the set of sampling frequencies considered.

� Nh � order of the rational approximation of the elements of the propagation matrix of the

line, that is, the total number of poles used to �t the matrix H.

� H � the characteristic admittance matrix of the line, evaluated for all the sampling frequencies.

� Hk, for k = 1, · · · , n � the set of n modal propagation functions of the line, evaluated for all

the sampling frequencies.

� τk, for k = 1, · · · , n � the set of n modal propagation delays of the line, computed through

equation (4.7), which are a base for the corresponding optimization process.

• Output Data:

� (τk)opt for k = 1, · · · , n � the optimized propagation delay of each modal propagation function;

38

� pmk for m = 1, · · · , Nk � the set of poles used to �t the kth modal propagation function, where

Nk is the optimal number of poles assigned to the kth mode.

� cmkij � the residues corresponding to the poles pmk used to �t each of the ijth elements of

the propagation matrix H.

These �tting parameters are all included in a structure called H_data.

4.4.5 Output generation

• Objective:

� Write the parameters of the generated application of the RT_WB Line model into a �le,

using the same template of the WB Line model, generated by the EMTP-RV.

The �tting parameters to write are those included in the structures Yc_data and H_data.

• Input Data:

� Yc_data � the �tting parameters used to approximate the elements of the characteristic ad-

mittance matrix of the line to model.

� H_data � the �tting parameters used to approximate the elements of the propagation matrix

of the line to model.

• Output Data:

� model.dat � location of the generated �le, containing the description of the developed model

computed, following the template of a WB Line �le, as computed by the EMTP-RV.

4.5 Conclusions

The goal of this work is to establish adequate numerical techniques for approximating the propagation

parameters for transmission line modeling, allowing real-time simulations. This requires an e�cient use

of reduced modeling resources. In order to ensure additional accuracy, it is necessary to introduce some

optimization procedures. The resulting model is called RT_WB Line, as it is a reformulation of the

EMTP-RV 2.3 model WB Line, in-line with the real-time simulation target.

The RT_WB Line is a phase domain model based on the rational approximations of the character-

istic admittance and propagation matrix. The elements of the characteristic admittance matrix are all

�tted by the same set of poles, whereas the elements of the propagation matrix are �tted by the poles

and delays de�ned by the approximated modes.

The elements of Yc are generally smooth functions of frequency and can easily be �tted by low order

functions. The �tting of H is a more challenging task, due to the contribution of the various line modes,

39

each with di�erent frequency dependent propagation delays. Furthermore, it must respect a limited

order of the model pre-de�ned by real-time requirements.

In order to ensure increased accuracy of the approximating propagation function for the pre-de�ned

order of the model, the RT_WB Line takes a set of optimization procedures regarding the computation

of the modal delays and the assignment of the poles to the line modes. The �rst optimization process

is based on the fact that the extraction of the lossless delays from the modal propagation functions may

not lead to the most accurate modal �tting. Therefore, it is necessary to search within a given interval

around the lossless delay for an ideal value. The second optimization procedure concerns the number of

poles assigned to each mode � tests have showed that, generally, assigning the same number of poles to

all modes does not lead to the most accurate approximation of the elements of H. The tests have also

showed that there is not an explicit logic that allows to de�ne a strategy to decide which modes should be

preferred and which should be neglected, or whether a mode with little in�uence on the phase quantities

should be assign zero or one pole, in order to achieve the most accurate phase �tting. Therefore, the

optimization process consists of searching within all the possible assignments of poles to the modes for

the one leading to the most accurate �tting of H.

40

Chapter 5

RT_WB Line model validation

5.1 Validation perspective

The purpose of the present chapter is the validation of the RT_WB Line, which is a reformulation of

the EMTP-RV model WB Line, in-line with the real-time simulation target. The formulation of the

developed model, as well as the optimizations introduced to ensure additional accuracy with low order

approximations, are presented in chapter 4.

The validation process consists of frequency and time domain simulations in the EMTP-RV 2.3

environment, using an application of the RT_WB Line, which performance is compared to that of theWB

Line, computed by the EMTP-RV and taken as a reference of accuracy. The order of the approximating

line functions used for the two models, that is, the number of poles used to �t the propagation matrix

H and the characteristic admittance matrix Yc, is presented in table 5.1. As already mentioned, the

elements of H are �tted by the delays and poles de�ned by the approximated modes, whereas the poles

used to �t the characteristic admittance matrix Yc are the same for all of its elements.

Table 5.1: Number of poles used for the approximation of the propagation parameters of a line, for the

applications of tested models � WB Line and RT_WB Line

Model H � Mode 1 H � Mode 2 H � Mode 3 Yc Total

WB Line 6 7 8 11 32

RT_WB Line 4 4 1 9 18

The application of the RT_WB Line assigns 9 poles to each line function. The distribution of modal

poles to �t H is optimized as described in section 4.3. The total number of poles used by this applica-

tion is in-line with the examples in real-time line modeling literature, namely [8, 9]. It is demonstrated

throughout this chapter that it is possible to achieve very accurate simulation results even using this

order for the approximating line functions.

41

The next sections, dedicated to the tests, are organized in the following way: section 5.2 presents a

study of the frequency response in short-circuit and open-end conditions, according to the two models;

then, section 5.3 concerns two typical conditions in transmission line transients studies: line energization

and single-phase short-circuit; �nally, section 5.4 refers a critical test concerning the current induced by

phase coupling, for which line models usually show substantially di�erent results. Finally, section 5.5

presents a summary of the analysis of the tests results.

5.2 Frequency response � short-circuited and open-ended line

This test evaluates the accuracy of the line models by comparing their frequency responses with the

expected behavior, according to analytical expressions computed with the exact line functions, which are

rewritten, respectively, as:

Ik_short = Yc

(I− H2

)−1 (I + H2

)Vs (5.1)

Vm_open =(I− H2

)−1H 2Vs (5.2)

For the short-circuit condition, the quantity observed is the current at the sending end of the line.

In open-end, the line is studied in terms of the receiving end voltage. Figure 5.1 illustrates these two

situations. In both cases, the line is connected to a symmetrical sinusoidal voltage source of 1V�RMS

voltage. Given the symmetry of the problem, it is su�cient to analyze one phase of the line (phase 1

was chosen).

Figure 5.1: Circuits used to study the short-circuit and open-end frequency responses according to the

WB Line and RT_WB Line models.

Figure 5.2 plots the approximating short-circuit responses according to the WB Line and to the

RT_WB Line, as well as the expected value for those functions of frequency, according to expression

(5.1).

42

Figure 5.2: Short-circuit frequency response (0.1 Hz - 1 MHz) � analytical (bold), WB Line (thin) and

RT_WB Line (dashed). Current at the sending end of phase 1.

The approximating frequency responses are specially innacurate for range of frequencies up to 50 Hz,

for which the developed model is particularly bad. However, for higher frequencies, including the range

from 100 Hz to 1 kHz (relevant for switching transients studies) the approximations of both models tend

to be very accurate.

Figure 5.3: Relative error of the short-circuit frequency response (0.1 Hz - 1 MHz) � WB Line (bold)

and RT_WB Line (thin). Current at the sending end of phase 1.

43

Figure 5.4: Detailed relative error of the short-circuit frequency response (700 Hz - 10 kHz) � WB Line

(bold) and RT_WB Line (thin). Current at the sending end of phase 1.

For a closer analysis of the models performance, �gure 5.3 shows the relative error of the approx-

imating short-circuit responses, according to the two models. It is now clear that the RT_WB Line

is inadequate for transients studies involving low frequencies, specially under the 50 Hz, for which the

approximating errors go over the 10 %. The WB Line is also not very adequate for this range of fre-

quencies, though the correspondent errors are far lower. On the other hand, for higher frequencies, both

models provide accurate approximations, except when approximating the various peaks in the expected

frequency response.

Figure 5.4 shows a zoom of the relative errors of the approximating frequency responses from 700

Hz to 10 kHz, which includes the switching transients frequencies. For this range, the RT_WB Line is

more accurate than the WB Line, despite using lower order approximations.

The open-end response according to the developed model and to the WB Line, as well as its

expected value, are plotted in �gure 5.5. At �rst sight, both models seem very accurate, even for the low

frequencies.

Taking a closer look at the relative errors of the approximating frequency responses, as �gure 5.6

shows, it is clear that both the WB Line and the RT_WB Line are very accurate for low frequencies.

However, their approximating errors tend to be higher for the frequency points correspondent to the

voltage peaks in the open-end response. The errors of the RT_WB Line approximation tend to increase

with frequency. Nevertheless, apart from the peaks of the open-end response, the developed model is

very accurate for the range of the switching transients frequencies.

44

Figure 5.7 shows a zoom into the relative errors of the approximating open-end responses, from 700

Hz to 10 kHz. As observed in the short-circuit scan, the RT_WB Line is more accurate than the WB

Line for this range of frequencies. Except for the �rst peak observed, the relative approximation error

of the developed model is always below the 2 %, whereas the WB Line may reach a correspondent value

of 10 %.

Figure 5.5: Open-end frequency response (100 Hz - 1 MHz) � analytical(bold), WB Line (thin) and

RT_WB Line (dashed). Voltage at the receiving end of phase 1.

Figure 5.6: Relative error of the open-end frequency response (100 Hz - 1 MHz) � WB Line (bold) and

RT_WB Line(thin). Voltage at the receiving end of phase 1.

45

Figure 5.7: Detailed relative error of the open-end frequency response (700 Hz - 10 kHz) � WB Line

(bold) and RT_WB Line (thin). Voltage at the receiving end of phase 1.

5.3 Line energization and single-phase short-circuit

This test evaluates how the models represent the behavior of the line under typical transient conditions.

Figure 5.8 illustrates the circuit used, where the line is connected by ideal switches to a three-phase

symmetrical source of 1V peak voltage and 50 Hz. The line energization occurs at t = 20 milliseconds,

with the closure of the switches connecting the line to the source. After reaching steady-state, a new

transient is originated, at t = 180 milliseconds, by closing the receiving end switch (short-circuit on phase

3). The voltage at the receiving end of phase 1 is observed.

Figure 5.8: Circuit used to study the response to line energization followed by single-phase

short-circuit, according to the WB Line and to the RT_WB Line applications.

The results of this test are plotted in �gures 5.9 and 5.10, regarding the energization transient and

the short-circuit of phase 3, respectively.

46

Both �gures show very good agreement between the two models performance. A di�erence is perceived

only in the energization condition, where the approximating line response according to the RT_WB Line

denotes a slightly weaker attenuation of the voltage at the receiving end of phase 1.

Figure 5.9: Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)

and RT_WB Line (thin).

Figure 5.10: Line energization: voltage at the receiving end of phase 1, according to WB Line (bold)

and RT_WB Line (thin).

5.4 Current induced by phase coupling

The last test presented is a "hard" test, in the sense that generally all model applications generate

considerably di�erent results. Figure 5.11 illustrates the circuit used, where the line is short-circuited at

all terminals, except at the sending end of phase 1, connected to a 1V�DC voltage source by an ideal

switch, 1 millisecond after the simulation start. The current at the sending end of phase 3 is observed.

47

Figure 5.11: Circuit used to study the phenomena of phase coupling according to the WB Line and to

the RT_WB Line applications, in terms of the current induced in phase 3 by energization of phase 1.

The energization of phase 1 introduces a transient condition on the system. Since phase 2 and phase

3 are grounded, they present a current which is induced by the time varying electric quantities in phase

1. After reaching a steady-state condition, the whole system must be in DC, to be in accordance with

the voltage source. Therefore, the coupling phenomena is extinguished, and the current on phase 2 and

phase 3 decline to zero. The simulation results for these conditions are plotted in �gures 5.12 and 5.13.

As �gures show, the time responses according to the two models agree only for the very initial period

of the transient. Furthermore, the response according to the RT_WB Line presents two undesired

peaks, as plotted in �gure 5.13. The induced current in steady-state is another important aspect, which

approximation is more accurately computed using the WB Line. Therefore, this test is an example of

transient conditions for which the use of the developed RT_WB Line is not particularly adequate.

Figure 5.12: Current at the sending end of phase 3, induced by energization of phase 1, at the �rst 20

miliseconds, according to the WB Line (bold), and to the RT_WB Line (thin).

48

Figure 5.13: Current at the sending end of phase 3, induced by energization of phase 1, at the �rst

second of simulation, according to the WB Line (bold), and to the RT_WB Line (thin).

Though this test is not relevant for switching transients studies, it must be noted that the inaccuracy

of the results is a consequence of the low order used for the approximating line functions of the RT_WB

Line application. To demonstrate this, consider including in the test another application of the developed

model, which order of the approximating functions H and Yc is the same as that used by the WB Line

application, generated by the EMTP-RV. The new results are plotted in �gures 5.14, 5.15 and 5.16.

Figure 5.14: Current at the sending end of phase 3, induced by energization of phase 1, at the �rst 20

miliseconds, according to the WB Line (bold), and to the RT_WB Line applications (low order � thin;

high order � dashed).

49

Figure 5.15: Current at the sending end of phase 3, induced by energization of phase 1, at the �rst

second of simulation, according to the WB Line (bold), and to the RT_WB Line applications (low

order � thin; high order � dashed).

Figure 5.16: Current at the sending end of phase 3, induced by energization of phase 1, at the �rst 50

second of simulation (reaching steady-state), according to the WB Line (bold), and to the RT_WB

Line applications (low order � thin; high order � dashed).

Figure 5.14 shows that it is possible to reach a good agreement with the WB Line time responses, by

using the higher order application of the developed model. Furthermore, the accentuated peaks observed

in the approximating response of the low order RT_WB Line application are practically eliminated when

using the higher order approach.

A very important aspect in this test is the correct approximation of the steady-state condition, where

50

the induced current must decline to zero. Figure 5.16 shows the test results after a long simulation

period, and it is possible to observe that the best approximation is the one computed using the higher

order application of the RT_WB Line, though it uses the same order for the approximating line functions

as the WB Line application.

5.5 Conclusions

The purpose of the present chapter is the validation of the model RT_WB Line, which is a reformulation

of the EMTP-RV model WB Line, in-line with the real-time simulation target. The formulation of the

developed model, as well as the optimizations introduced to ensure additional accuracy with low order

approximations, are presented in chapter 4.

The validation process consists of frequency and time domain simulations in the EMTP-RV 2.3 en-

vironment, using an application of the RT_WB Line, which performance is compared to that of the

WB Line, computed by the EMTP-RV and taken as a reference of accuracy. The WB Line application

uses 21 + 11 = 32 poles to �t the propagation matrix H and the characteristic admittance matrix Yc,

whereas the RT_WB Line uses 9 + 9 = 18, respectively. The total number of poles used by this de-

veloped model application is in-line with the examples in real-time line modeling literature, namely [8, 9].

Section 5.2 presents the �rst test, concerning the approximation of the line frequency response un-

der short-circuit and open-end conditions. As regards the short-circuit frequency response both

models generated inaccurate approximations for the range of frequencies up to 50 Hz, particularly the

RT_WB Line application. However, for higher frequencies, both models provide reasonable approxima-

tions. Speci�cally for the range from 700 Hz to 10 kHz (including the range of switching transients),

the RT_WB Line provides the most accurate results. Regarding the open-end frequency response,

both models provide good results for low frequencies. The approximating errors are more pronounced

for the voltage peaks of the analytical open-end response and for very high frequencies, which are not

relevant for switching transient studies. Once again, the RT_WB Line generates the most accurate

approximations for the range from 700 Hz to 10 kHz.

Section 5.3 concerns two typical transmission line transients: line energization and single-phase short-

circuit. In both cases, there is a good agreement on the performance of two models. A di�erence is

perceived only in the energization condition, where the approximating line response according to the

RT_WB Line denotes a slightly weaker attenuation of the voltage at the receiving end of phase 1.

Finally, section 5.4 refers a critical test concerning the current induced by phase coupling, for which

line models usually show substantially di�erent results. The response provided by the RT_WB Line is

inaccurate both for the initial period of the transient and to the steady-state condition, for which the

induced current is expected to decline to zero. Though this test is not relevant for switching transients

51

studies, it must be noted that this result is a consequence of the low order used by the approximating

functions of the RT_WB Line application. This is observed by including in the test another application

of the developed model, which order of the approximating functions H and Yc is the same as that used

by the WB Line application, generated by the EMTP-RV. The new results show a good agreement

between the new approach and the WB Line. Furthermore, the high order RT_WB Line application

provides the most accurate approximation of the induced current in steady-state, even though it uses

the same number of poles as the WB Line.

52

Chapter 6

Conclusions

6.1 Introduction

This chapter presents the conclusions of the work in this dissertation, whose objective is to establish

adequate numerical techniques for approximating the propagation parameters for transmission line mod-

eling, allowing real-time simulations, which requires an e�cient use of reduced modeling resources. In

order to ensure additional accuracy, it is necessary to introduce some optimization procedures. The re-

sulting model is called RT_WB Line, as it is a reformulation of the EMTP-RV model WB Line, in-line

with the real-time simulation target.

The applications of the developed model are computed by a MATLAB program speci�cally created

in this work. The real-time requirement imposes a limited order for the model, which varies according

to the processor. Therefore, the order of approximations to use in computed applications is assumed

as a pre-de�ned input to the program. The validation of the developed model consists of testing in the

EMTP-RV 2.3 environment a set of applications of the RT_WB Line, covering both steady-state and

transient conditions.

EMTP-RV is not a real-time simulator, so it is not possible to test the speed of the models appli-

cations using this program. Therefore, the analysis is made mainly from the point of view of accuracy.

The real-time requirement has been enforced by using orders for the model that are usually adequate for

this type of simulation (see examples in [8, 9]).

This chapter presents a summary of the conclusions, concerning the performance of the developed

modeling program, and as well it presents a set of proposals for future improvements on transmission

line modeling in-line with the real-time simulation target.

53

6.2 Completion of proposed objectives

The accurate representation of a transmission line requires the use of its frequency dependent param-

eters. This poses a challenge on the de�nition of an adequate line model. The goal of this work is to

establish adequate numerical techniques for approximating the propagation parameters for transmission

line modeling, allowing real-time simulations.

The study of existing line models, namely those provided by the EMTP-RV 2.3, leads to the conclu-

sion that the WB Line, that is, the Universal Model [3] approach, presents an increased e�ciency when

compared to modal domain frequency dependent models, by obtaining better results with less resources.

This e�ciency is reinforced by the fact that the WB Line is a phase-domain model � there is no need to

convert from phase to modal quantities, and vice-versa, at each simulation step.

The line model developed in this work, called RT_WB Line, is a reformulation of the WB Line, in-

line with the real-time simulation target. Therefore, the RT_WB Line is a phase domain model which

�ts the propagation matrix H using the poles and delays de�ned by the modes. To ensure additional

accuracy with reduced �tting resources, two optimizations are introduced, regarding the computation of

the modal delays and the assignment of the modal poles.

The applications of the RT_WB Line are computed by a MATLAB program, speci�cally built for

this dissertation, which enforces the real-time requirement by assuming the order of the approximating

line functions as a pre-de�ned input. The developed program must additionally receive the location of

the �le generated by the EMTP-RV 2.3, containing the line characteristic parameters Z and Y, com-

puted for a set of frequency samples. These parameters are used to compute the original line functions

to be �tted � H and Yc. The output of the program is a �le containing the modeling parameters of the

computed application.

The validation of the developed model consists of frequency and time domain simulations in the

EMTP-RV environment, using an application of the RT_WB Line, which performance is compared to

that of the WB Line, computed by the EMTP-RV and taken as a reference of accuracy. The total

number of poles used by the developed model application is in-line with the examples in real-time line

modeling literature, namely [8, 9].

The developed model presents some weaknesses, for example in approximating the low frequencies of

the short-circuit and open-end responses, plotted in �gures 5.3 and 5.6, or in approximating the current

induced by phase coupling, as the plots in �gures 5.12 and 5.13 show.

These less accurate results are not due to a problem in the model computing routine, but a conse-

quence of the low order used for the approximating functions. This is proved by considering another

54

application of the developed model which approximating functions have the same order as those of the

WB Line application. In fact, this new application provides an approximation of the steady-state in-

duced current which is more accurate than that of the WB Line application, as plotted in �gure 5.16.

Furthermore, these tests for which the RT_WB Line is not particularly adequate, are not the appli-

cations in which the real-time is more relevant. On the other hand, real-time is of utmost importance

for the study of switching transients, and the tests show that the numerical techniques and optimization

procedures introduced in the RT_WB Line allow to produce low order applications that generate ac-

curate simulation results in switching transient conditions. Examples of this good performance are the

short-circuit and open-end frequency scans for the range of 700 Hz to 10 kHz, plotted in �gures 5.4 and

5.7, respectively. The test of line energization followed by single-phase short-circuit, illustrated by the

plots of �gures 5.9 and 5.10, is another case of good agreement between the performance of the the WB

Line and RT_WB Line applications, despite the di�erence on the order of approximations.

6.3 Proposals for further improvements in line modeling

The �rst proposal for further improvements in line modeling is motivated by the test presented in sec-

tion 5.4, regarding the approximation of the current induced by coupling between phases according to

an application of the RT_WB Line, and taking as a reference the application of the WB Line computed

by the EMTP-RV 2.3. The results of the test show several problems with the developed model perfor-

mance, namely the existence of accentuated peaks in the current response of the line and the inaccurate

approximation of the steady-state condition.

Section 5.4 demonstrates that the inaccuracy of the RT_WB Line application for this speci�c test

is a consequence of the low order of its approximations, which is imposed by the real-time target. The

described problems are overcome by using higher order approximations. However, to ensure a real-time

performance alternative strategies must be found.

One of the reasons for the model inaccuracy is that the referred test concerns coupling phenomena.

Therefore, the quantities observed are computed using the o�-diagonal elements of the approximating

line matrices H and Yc, which generally have a magnitude lower than the correspondent diagonal ele-

ments. The RT_WB Line is based on the approximation of these functions using the Universal Model

scheme [3], which computes the �tting parameters by solving a least-squares problem. This technique is

based on the absolute deviation between original and approximating functions. Therefore, by enforcing

a given maximum deviation, the relative error is higher for the functions of reduced magnitude, which

is the case for the o�-diagonal elements of the line matrices. A possible strategy to tackle this problem

and still use the least-squares technique is to de�ne a scaling strategy for the elements of H and Yc that

assures the relative error is the same for all approximating functions.

55

The other point in the test of section 5.4 is the inaccurate approximation of the steady-state condition,

which is due to a bad approximation of the low frequency samples of the line functions. One possible

solution is to use more samples in the low frequencies. Another alternative is to de�ne an adequate

weighting scheme that concentrates more e�ort in approximating the low frequency samples, without

neglecting the frequencies of switching transients.

Another proposal for an improvement in line modeling concerns the optimization presented in section

4.3.2. The procedure regards the approximation of H, which uses the poles and delays de�ned by the

modes. The objective of this optimization is to compute the number of poles to assign to each mode,

given a total number of poles, in order to minimize the error of approximation. Section 4.3.2 provides an

example of several tests performed in order to de�ne a searching strategy. Given the disparity of results

for di�erent orders of approximation tested, it is not possible, within the range of this work, to de�ne

that strategy. Therefore, the approximation procedure used to compute the RT_WB Line searches all

the possible distributions of modal poles. This motivates a deeper study, including a wider number of

lines, in order to de�ne a strategy that may represent a major time saving in the pre-processing of the

model �tting parameters.

56

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58

Appendix A

Transmission line used in model testing

This dissertation presents several tests using di�erent line models. One set of tests compares the line

models provided by the EMTP-RV 2.3. The other compares the EMTP-RV model WB Line with two

applications of the RT_WB Line model, developed for this work.

The line represented in these tests is a three-phase line with a spacial con�guration as illustrated in

�gure A.1. Other characteristics of this line and of near conductive ground are:

• Length: 100 km

• Number of phases: 3

• Number of ground wires: 0

• DC resistance of each conductor: 0.168228 Ohm/km

• Conductor relative permeability (µr): 1

• Ground resistivity: 100 Ohm/km

Figure A.1: Spacial con�guration of the transmission line used throughout this dissertation.

59