Study of Cables in the Distribution System: Parameters Calculation, Fault Analysis, and Configuration Optimization

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Clemson University
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12-2018
Study of Cables in the Distribution System:
Parameters Calculation, Fault Analysis, and
Configuration Optimization
Bin Sun
Clemson University
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https://tigerprints.clemson.edu/all_dissertations/2281

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i 
STUDY OF CABLES IN THE DISTRIBUTION SYSTEM: PARAMETERS 
CALCULATION, FAULT ANALYSIS, AND CONFIGURATION OPTIMIZATION 
A Dissertation 
Presented to 
the Graduate School of 
Clemson University 
In Partial Fulfillment 
of the Requirements for the Degree 
Doctor of Philosophy 
Electrical Engineering
by 
Bin Sun 
December 2018 
Accepted by: 
Dr. Daniel Noneaker, Committee Chair 
Dr. Elham B. Makram 
Dr. Richard E. Groff 
Dr. Yongqiang Wang 
Dr. John R. Wagner 

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ii 
ABSTRACT 
Underground cables offer more advantages than overhead lines since cables are 
better protected and do not detract from the appearance of urban areas. In recent years, 
more and more electrical utilities are using cables to distribute electric power to their 
customers. However, the cost of installation and maintenance of underground cables is 
very expensive. Thus, the proper design and operation of distribution systems are crucial 
for economic reasons. To design and operate the distribution system properly, analysis of 
the underground cable system is extremely important.
The objective of this dissertation is to analyze the underground cable distribution 
system under both normal and faulted conditions to fully understand an underground 
cable system.
For cables to operate under normal conditions in a distribution system, two 
aspects must be analyzed: firstly, the parameters of different types of cables under normal 
operation, which include impedance matrix, power loss, and voltage drop; and secondly, 
the best configuration of cables in a ductbank based on the total ampacity value. When 
faults occur in the underground cable system, three aspects need to be analyzed: firstly, 
the magnetic force waveforms of cables during different types of faults; secondly, the 
relationship between two types of faults, namely 
low-impedance short-circuit fault and 
high impedance fault
; and thirdly, the impacts of a fault on the configuration 
optimization.
To calculate the parameters of different types of cables, a program with graphical 
user interface was built in MATLAB. Utilizing modeling technology, the magnetic force 
analysis of different types of faults was completed using COMSOL software. The impact 

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iii 
of high impedance fault on ferroresonance was simulated in PSCAD and COMSOL. The 
magnetic force waveforms of cables under different faults were calculated and plotted, 
and the relationship between water tree and ferroresonance was observed. Then an 
optimization program using MATLAB and Yalmip toolbox was undertaken to find the 
best configuration of cables in a ductbank based on the total ampacity value. Based on the 
optimization results, the best configurations of cables in a ductbank under both balanced 
and unbalanced scenarios, even in faulted conditions, were obtained. 

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iv 
ACKNOWLEDGMENTS 
I would like to thank my academic advisor, Dr. Elham Makram, for her guidance 
and support of this research. I appreciate her patient help and valuable time. 
I would like to thank my wife. She has accompanied me these four years and 
supported me all that time. I hope she will find the perfect job and be happy forever. 
I would like to thank my parents for their love and support. I wish them healthy 
and happy lives as they celebrate their diamond wedding anniversary.
I would also like to thank Clemson University Electric Power Research 
Association (CUEPRA) for their financial support. 

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v 
TABLE OF CONTENTS 
Page 
TITLE PAGE .................................................................................................................... i 
ABSTRACT ..................................................................................................................... ii 
ACKNOWLEDGMENTS .............................................................................................. iv 
LIST OF TABLES ........................................................................................................ viii 
LIST OF FIGURES ........................................................................................................ ix 
CHAPTER 
I. 
INTRODUCTION ......................................................................................... 1 
Research objectives .................................................................................. 1 
Cables in normal condition ...................................................................... 2 
Cables in faulted condition ...................................................................... 4 
Research contribution ............................................................................ 10 
II. 
PARAMETERS CALCULATION USING MATLAB............................... 12 
Introduction ............................................................................................ 12 
Two types of cable ................................................................................. 13 
Concentric neutral cable .................................................................. 14 
Tape shield cable.............................................................................. 15 
Calculation method ................................................................................ 16 
Carson line method for neutral cable ............................................... 16 
Carson line method for shield cable................................................. 17 
Results .................................................................................................... 20 
III. 
FORCE ANALYSIS FOR A THREE-PHASE CABLE 
IN MICROGRID ................................................................................... 22 
Introduction ............................................................................................ 22 
Method of analysis ................................................................................. 23 

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vi 
Table of Contents (Continued) 
Page 
Microgrid design and simulation ..................................................... 24 
Cable model in COMSOL ............................................................... 31 
Results .................................................................................................... 33 
IV. 
IMPACTS OF WATER TREE ON FERRORESONANCE ....................... 38 
Introduction ............................................................................................ 38 
Theoretical principles............................................................................. 38 
Ferroresonance ................................................................................. 38 
Water tree ......................................................................................... 42 
Simulation procedure ............................................................................. 45 
Results .................................................................................................... 47 
V. 
CONFIGURATION OPTIMIZATION OF CABLES IN DUCTBANK .... 53 
Introduction ............................................................................................ 53 
Method of analysis ................................................................................. 55 
Ampacity calculation ....................................................................... 55 
Optimization procedure ................................................................... 57 
Results of optimization .......................................................................... 61 
Configuration optimization for a balanced condition ...................... 61 
Configuration optimization for a special unbalanced example........ 66 
Configuration optimization for general unbalanced condition ........ 68 
VI. 
IMPACTS OF SLGF ON CONFIGURATION OPTIMIZATION 
Introduction ............................................................................................ 70 
Method of simulation ............................................................................. 70 
Results of simulation.............................................................................. 71 
Balanced system............................................................................... 71 
Unbalanced system .......................................................................... 99 
VII. 
CONCLUSIONS........................................................................................ 107 
APPENDICES ............................................................................................................. 110 
A: 
MATLAB programming steps ................................................................... 111 
B: 
Sample code of parameters calculation...................................................... 115 
C: 
Instructions of the software ........................................................................ 125 
D: 
Yalmip toolbox of MATLAB .................................................................... 128 
E: 
Sample code of configuration optimization ............................................... 129 

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vii 
Table of Contents (Continued) 
Page 
F: 
Transfer ampacity calculation of an optimization problem ....................... 135 
G: 
Data of two types of cables ........................................................................ 136 
REFERENCES ............................................................................................................ 137 

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viii 
LIST OF TABLES 
Table Page 
3.1 
Parameters of DGs at each bus .................................................................... 26 
3.2 
Parameters of loads at each bus ................................................................... 27 
3.3 
Parameters of T-lines and cables ................................................................. 27 
5.1 
Detailed results for two balanced cables per phase ..................................... 63 
5.2 
Detailed results for three balanced cables per phase ................................... 66 
5.3 
Detailed results for two unbalanced cables per phase ................................. 67 
5.4 
Detailed results for three unbalanced cables per phase ............................... 68 
6.1 
Impacts of SLGF with zero fault impedance (One cable per phase) ........... 72 
6.2 
Impacts of SLGF with low fault impedance (One cable per phase) ............ 72 
6.3 
Impacts of SLGF with high fault impedance (One cable per phase) ........... 72 
6.4 
Impacts of SLGF with zero fault impedance (Two cables per phase) ......... 73 
6.5 
Impacts of SLGF with low fault impedance (Two cables per phase) .......... 73 
6.6 
Impacts of SLGF with zero fault impedance (Three cables per phase) ....... 74 
6.7 
Impacts of SLGF with low fault impedance (Three cables per phase) ........ 75 
6.8 
Impacts of SLGF with low fault impedance in an unbalanced system ...... 100 

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ix 
LIST OF FIGURES 
Figure Page 
1.1 
Three-phase cables with Carson line ............................................................. 3 
1.2 
Different types of water tree .......................................................................... 5 
1.3 
Water tree in cables: (a) vented water tree; (b) bowtie 
water tree .................................................................................................. 6 
1.4 
Destruction caused by the ferroresonance phenomenon ................................ 7 
1.5 
Physical model and physical field built in COMSOL ................................. 10 
2.1 
Common layer arrangement of a tape shield cable ...................................... 13 
2.2 
Cross section of a concentric neutral cable .................................................. 14 
2.3 
Voltage distribution of a concentric neutral cable ....................................... 14 
2.4 
Cross section of a tape shield cable ............................................................. 15 
2.5 
Voltage distribution of a tape shield cable ................................................... 15 
2.6 
Parameters used in Carson line method for concentric
neutral cable ........................................................................................... 17 
2.7 
Parameters used in Carson line method for tape shield cable ...................... 18 
2.8 
Sample cable arrangement ........................................................................... 18 
2.9 
GUI interface and impedance matrix results ............................................... 21 
3.1 
The roadmap of force analysis ..................................................................... 23 
3.2 
Microgrid system applied to collect current data ......................................... 24 
3.3 
Medium voltage microgrid system benchmark network ............................. 25 
3.4 
Power flow results of POWERWORLD software ....................................... 26 
3.5 
Arrangement of direct burial shield cables .................................................. 28 

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x 
List of Figures (Continued) 
Figure Page 
3.6 
PV array testing system ............................................................................... 30 
3.7 
PV model includes MPPT, converter, inverter and transformer .................. 30 
3.8 
Steam turbine synchronous machine ........................................................... 31 
3.9 
The electrical field of two-dimensions physical model ............................... 32 
3.10 
Magnetic field and Lorentz force of 2D model ........................................... 32 
3.11 
The magnetic field of three-dimensions physical model ............................. 33 
3.12 
The force under capacitor switching and force under
ideal sin-current ..................................................................................... 33 
3.13 
Forces of cables in the x-direction during LLF ........................................... 34 
3.14 
Forces of cables in the x-direction during 2LGF ......................................... 34 
3.15 
Forces of cables in the x-direction during 3PF ............................................ 35 
3.16 
Voltage data collected of cable from 3 to 4 after
capacitor switching ................................................................................ 35 
3.17 
Direct burial method(left) and underground duct method(right) ................. 36 
3.18 
The magnetic field of the cable under direct burial method
after switching ........................................................................................ 36 
3.19 
The magnetic field of the cable under duct burial method
after switching ........................................................................................ 37 
4.1 
Conditions of the ferroresonance phenomenon ........................................... 39 
4.2 
One circuit structure of ferroresonance........................................................ 41 
4.3 
Relative permittivity and electrical conductivity of WT ............................. 44 
4.4 
Equivalent resistance and capacitance during 1mm WT
development ........................................................................................... 45 

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xi 
List of Figures (Continued) 
Figure Page 
4.5 
Ferroresonance circuit including lumped parameter
water tree model ..................................................................................... 46 
4.6 
Ferroresonance responses when WT is located at
different positions of phase a cable ........................................................ 51 
4.7 
WT and single-pole switching occur on the same or
different cables ....................................................................................... 52 
5.1 
Cables in a ductbank for installation ............................................................ 54 
5.2 
Cable ductbank for installation .................................................................... 55 
5.3 
Procedure for configuration optimization of cables
in a ductbank .......................................................................................... 59 
5.4 
The configuration of ductbank simulated in CYMCAP .............................. 60 
5.5 
The optimization result compared with common sense for
two balanced cables per phase ............................................................... 62 
5.6 
The optimization result compared with common sense for
three balanced cables per phase ............................................................. 65 
5.7 
The best configuration for two unbalanced cables per phase 
of a particular example........................................................................... 66 
5.8 
The best configuration for three unbalanced cables per phase
of a particular example........................................................................... 67 
5.9 
The best configuration for general two unbalanced cables
per phase ................................................................................................ 68 
5.10 
The best configuration for general three unbalanced cables
per phase ................................................................................................ 69 
6.1 
SLGF analysis of one cable in POWERWORLD........................................ 71 

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xii 
List of Figures (Continued) 
Figure Page 
6.2 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 50 percent
loading condition, zero impedance SLGF ............................................. 77 
6.3 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 50 percent
loading condition, low impedance SLGF .............................................. 78 
6.4 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 50 percent
loading condition, high impedance SLGF ............................................. 79 
6.5 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 80 percent
loading condition, zero impedance SLGF ............................................. 80 
6.6 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 80 percent
loading condition, low impedance SLGF .............................................. 81 
6.7 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 80 percent
loading condition, high impedance SLGF ............................................. 82 
6.8 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 100 percent
loading condition, zero impedance SLGF ............................................. 83 
6.9 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 100 percent
loading condition, low impedance SLGF .............................................. 84 
6.10 
The temperature of optimization (a) and common sense (b)
configuration of one cable per phase under 100 percent
loading condition, high impedance SLGF ............................................. 85 
6.11 
The temperature of optimization (a) and common sense (b)
configuration of two cables per phase under 50 percent
loading condition, zero impedance SLGF ............................................. 86 

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xiii 
List of Figures (Continued) 
Figure Page 
6.12 
The temperature of optimization (a) and common sense (b)
configuration of two cables per phase under 50 percent
loading condition, low impedance SLGF .............................................. 87 
6.13 
The temperature of optimization (a) and common sense (b)
configuration of two cables per phase under 80 percent
loading condition, zero impedance SLGF ............................................. 88 
6.14 
The temperature of optimization (a) and common sense (b)
configuration of two cables per phase under 80 percent
loading condition, low impedance SLGF .............................................. 89 
6.15 
The temperature of optimization (a) and common sense (b)
configuration of two cables per phase under 100 percent
loading condition, zero impedance SLGF ............................................. 90 
6.16 
The temperature of optimization (a) and common sense (b)
configuration of two cables per phase under 100 percent
loading condition, low impedance SLGF .............................................. 91 
6.17 
The temperature of optimization (a) and common sense (b)
configuration of three cables per phase under 50 percent
loading condition, zero impedance SLGF ............................................. 92 
6.18 
The temperature of optimization (a) and common sense (b)
configuration of three cables per phase under 50 percent
loading condition, low impedance SLGF .............................................. 93 
6.19 
The temperature of optimization (a) and common sense (b)
configuration of three cables per phase under 80 percent
loading condition, zero impedance SLGF ............................................. 94 
6.20 
The temperature of optimization (a) and common sense (b)
configuration of three cables per phase under 80 percent
loading condition, low impedance SLGF .............................................. 95 
6.21 
The temperature of optimization (a) and common sense (b)
configuration of three cables per phase under 100 percent
loading condition, zero impedance SLGF ............................................. 96 

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xiv 
List of Figures (Continued) 
Figure Page 
6.22 
The temperature of optimization (a) and common sense (b)
configuration of three cables per phase under 100 percent
loading condition, low impedance SLGF .............................................. 97 
6.23 
The temperature of optimization (a) and common sense (b)
configuration under 50 percent loading condition,
unbalanced condition, fault at lowest phase ........................................ 101 
6.24 
The temperature of optimization (a) and common sense (b)
configuration under 80 percent loading condition,
unbalanced condition, fault at lowest phase ........................................ 102 
6.25 
The temperature of optimization (a) and common sense (b)
configuration under 100 percent loading condition,
unbalanced condition, fault at lowest phase ........................................ 103 
6.26 
The temperature of optimization (a) and common sense (b)
configuration under 50 percent loading condition,
unbalanced condition, fault at highest phase ....................................... 104 
6.27 
The temperature of optimization (a) and common sense (b)
configuration under 80 percent loading condition,
unbalanced condition, fault at highest phase ....................................... 105 
6.28 
The temperature of optimization (a) and common sense (b)
configuration under 100 percent loading condition,
unbalanced condition, fault at highest phase ....................................... 106 

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1 
CHAPTER ONE 
INTRODUCTION 
1.1 
Research Objectives 
The objective of this dissertation is to analyze the underground cable distribution 
system under both normal and faulted conditions. 
Nowadays, more and more electrical power is being distributed to customers by 
underground cables rather than overhead lines due to their advantages of better protection 
and less disruptive appearance. Cables also have significantly reduced electromagnetic 
field emissions because of their copper shielding. But cables are deeply buried in the soil 
and thus difficult to monitor and repair. So engineers need to fully understand all 
properties and potential problems that might happen during the long operation life of 
cables. 
To fully understand the underground cable systems, two conditions should be 
focused on: normal condition and faulted condition. In normal condition, two aspects are 
extremely critical: firstly, parameters during operation, such as voltage drop and power 
loss; and secondly, the best arrangement of cables to ensure the largest total ampacity 
value. Similarly, in a faulted condition, three factors need to be analyzed: firstly, the 
change of magnetic forces of cables during a fault; secondly, the impact of one type of 
fault on the other type of fault; and thirdly, the impact of a fault on the configuration 
optimization results. In this dissertation, all five of these factors are discussed in details, 
and the results are shown in each chapter. 

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1.2 
Cables in Normal Condition 
Underground cables are deeply buried below the ground surface and protected by 
the surrounding soil. Therefore, cables usually operate under a healthy and normal 
condition, which means no fault occurs in the cable system. When cables are operating 
normally, the parameters of cables, including the impedance matrix, voltage drop, power 
loss and ampacity value of these cables, must be calculated. Based on these values, the 
steady-state and transient analysis of cables can be completed [1]. 
Nowadays, many methods are employed to calculate the impedance matrix of 
different types of cables. Dr. Kodzo Obed Abledu proposed that the impedance of cables 
could be calculated by subdivision of the conductors in 1976 [2]. A faster computation 
method to calculate the impedance matrix of cables was proposed in 2014 [3], which 
includes the skin and proximity effects. Using the vector impedance meter to measure 
parameters of cables has also been proposed in [4]. The impedance matrix of cables can 
also be calculated by the finite element method [5][6].
The Carson line method is the most widely used method [7][8][9][10] to calculate 
the impedance matrices of cables. This method is an empirical formula, and it is 
explained in detail [11]. It assumes a fictitious line that is laid at a depth below the 
ground surface, which is shown in Fig 1.1. It shows Carson line as a returning conductor 
of these three-phase cables [12]. The voltage drop on these three-phase conductors and 
one returning conductor can be described using the impedance matrix. Then the formula 
of self and mutual impedance, after eliminating Carson line, can be formed using the 

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3 
Kron reduction method. With the impedance matrix, the power loss and voltage drop of 
cables can be easily calculated. 
Figure 1.1. Three-phase cables with Carson line. 
Voltage drops on these four lines: 
(1-1) 
Using Kron reduction method on these equations: 
(1-2) 
In the meantime, several types of software can be used to calculate the parameters 
of cables as well, such as CYME [13] and POWER WORLD [14]. After the correct 

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4 
models of underground cables are built in the simulation software, the impedance matrix 
of cables can be calculated by the software using similar equations. 
To calculate the ampacity value of cables, understanding the definition of 
ampacity is the first step. Ampacity is the maximum current limitation that allows the 
cable to operate under maximum allowable temperature [15]. 
Several publications 
proposed different methods to calculate cables’ ampacities for both single and multiple 
cable configurations [16][17][18][19]. Among these methods, two of them are widely 
used: the Neher and McGrath method [20] and IEC Standards 287-3-2 [21].
In practice, several cables are generally installed in some compact ductbanks in 
order to provide convenient installation of multiple cables in a concrete space [22]. The 
total ampacity calculation of all these cables is usually completed by the iteration method, 
since the problem includes a set of interrelated equations. But Dr. Moutassem 
recommended a more efficient method [23], which involves converting the ampacity 
calculation problem to an optimization problem. With the development of optimization 
technology, Yalmip toolbox of MATLAB can quickly build the optimization model and 
solve this optimization problem using Gurobi [15].
1.3 
Cables in Faulted Condition 
Two types of faults can occur in an underground cable distribution system: low-
impedance fault and high-impedance fault. Low-impedance fault includes four types: 
three-phase fault (3PF), single-line-to-ground fault (SLGF), line-to-line fault (LLF) and 
double-line-to-ground fault (2LGF). The SLGF is the most common fault that occurs in a 
power system. In fact, more than 85 percent of faults in a power system are SLGF [11]. 

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5 
Water tree is a very common high-impedance fault of underground cables in 
power systems [24]. Water tree is a fault phenomenon that occurs in the insulation layer 
of cables. It normally occurs when 
the humidity of the surrounding soil is higher than 65 
percent [25]. The soil humidity at a depth of one meter remains 100 percent for most of 
the year. So water tree is prone to occur in underground cables.
There are two types of water tree, which are shown in Fig. 1.2 and Fig. 1.3. 
Water 
tree forms from some small voids and grows by increasing the surrounding voltage stress. 
Then fractures occur and are filled with water. The water tree forms and grows in a tree 
shape until it reaches the conductor layer of cables and the high-impedance fault takes 
place. This type of fault doesn’t cause significan
t voltage or current change due to the 
high impedance of the water tree. It is, therefore, hard to be detected, but it causes 
damage, even cable failure eventually, after certain periods of operating time. 
Figure 1.2. Different types of water tree.

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6 
(a) 
(b) 
Figure 1.3. Water tree in cables: (a) vented water tree; (b) bowtie water tree. 

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7 
Ferroresonance is another fault phenomenon that more often happens in 
underground cable systems compared with overhead lines, since cables have larger shunt 
capacitance per unit. Ferroresonance is a highly nonlinear process that is caused by 
nonlinear electric elements [26]. 
It can result in either a short transient or continuous 
overvoltage and overcurrent that can reach up to 4 to 6 times the normal values. Thermal 
problems in electrical equipment as well as loud noises can also result [50]. Destruction 
caused by ferroresonance as collected by the ABB company is shown in Fig. 1.4.
Figure 1.4. Destruction caused by the ferroresonance phenomenon. 

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8 
To complete the fault analysis of underground cable distribution systems, various 
faults need to be modeled and analyzed in different simulation software. To simulate a 
distribution power system, software such as MATLAB, Power World, PSS/E, PSLF, 
DigSilent, OpenDSS and PSCAD[14] can be selected. The software could be divided into 
two classes: power system analysis modeling, and economic and forecast modeling [28]. 
To complete steady-state power flow and transient stability analysis, Power World and 
PSS/E are normally selected. To finish the electromagnetic transient (EMT) study, the 
different electrical elements need to be modeled in detail, and PSCAD is the most 
common choice [29]. EMTDC is the engine of PSCAD, which is used to solve the 
differential equations of the power system in time domain. 
Most simulation programs include fault modules, which can be used directly in 
fault analysis. But the high-impedance fault, such as water tree (WT) fault, does not have 
a predefined model, and the detailed model must be built. To simulate the WT fault in 
cables, there are different methods to model it. The most common method is using a 
lumped parameter model. 
This model involves a parallel resistor and capacitor [30] 
[31][32][33].
The equivalent resistance and capacitance are calculated by COMSOL, 
based on relative permittivity and electrical conductivity value of WT. 
To simulate ferroresonance, four conditions need to be met [27]. Firstly, the 
system should include a medium level voltage source. Secondly, the system should 
include the electrical elements that can cause ferroresonance, which are capacitor and 
inductor. Normally, the capacitor is the shunt capacitor of long-distance underground 
cables, and the nonlinear inductor is the saturable iron core of the transformer. Thirdly, 

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9 
the system must be a low loss system. If significant losses existed in the system, then the 
ferroresonance can be damped out. Finally, single-pole switch occurs in the system to 
cause the ferroresonance.
The current and voltage data can be collected from power system simulation 
software. But the physical field can’t be calculated directly using this type of software. 
To model the physical fields around cables, many other types of software could be 
selected such as CYMCAP [13], SPICE [34] and COMSOL [35]. COMSOL 
Multiphysics simulation software is a common choice. The AC/DC module of COMSOL 
could be used to simulate the low-frequency electromagnetic phenomena [36]. This 
module includes essential electrical elements such as resistances, capacitors, inductors 
and coils. It can be coupled with the Lorentz Force calculation module and heating 
module. As long as the correct physical model and materials are built, this software can 
calculate the electrical field, magnetic field, and Lorentz force based on Maxwell 
equations. Fig. 1.5 is a physical model of three-phase cables directly buried underground 
and built by COMSOL. Detailed data was collected from the Okonite company 
[43].

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10 
(a) 
(b) 
Figure 1.5. Physical model and physical field built in COMSOL. 
1.4 
Research Contribution 
This dissertation used different methods and software for modeling and 
simulation of the proposed research. The detailed steps of programming software to 
calculate parameters of different types of cables are summarized in Chapter 2. In Chapter 
3, the magnetic force analysis during different types of faults using COMSOL and 

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11 
PSCAD is described. Chapter 4 focuses on impacts of high-impedance fault on 
ferroresonance; the general relationship between water tree and ferroresonance is 
examined in this chapter. The configuration optimization of cables in a ductbank based 
on their total ampacity value is discussed in Chapter 5, and the best and worst 
configuration of cables in a ductbank under both balanced and unbalanced condition are 
proposed. The impact of SLGF on the configuration optimization results is analyzed in 
Chapter 6. A concluding chapter summarizes the dissertation in Chapter 7. 

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12 
CHAPTER TWO 
PARAMETERS CALCULATION USING MATLAB
2.1 
Introduction 
With the development of technology, more and more options are available for 
insulated cables in distribution systems. Different cable construction types have different 
parameters, electrical field, and magnetic field. In order to efficiently design these cable 
systems and adequately model them for system analysis, engineers should be able to 
calculate the parameters and understand the electrical/magnetic field of different types of 
cables.
Although the methods, which include calculations of impedance matrix, power 
losses, and voltage drop, are very mature [11], few types of software are used to calculate 
all these parameters for different cables [13]. The objective of this section is to build a 
user-friendly software to calculate these parameters with greater flexibility even if the 
users are not familiar with the methods of estimating the parameters of different types of 
cables. A software that is used to calculate the parameters of different cross-sections of 
cables is built by combining the method of estimating these parameters with the graphical 
user interface (GUI). Using this software, customers can input or choose any type of 
cables, and calculate the parameters they need.
In this dissertation, COMSOL is used to generate physical models for insulated 
underground power cables and multi-phase power lines in order to study the time-domain 
electrical field, magnetic field, and induced forces of underground cables in distribution 
systems [35]. 

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13 
2.2 
Two Types of Cable 
Two different types of cables are commonly used in the distribution system: 
concentric neutral cable and tape shield cable [11]. Before calculating the parameters of a 
cable, the composition of cables has to be studied first. Generally, there are five levels of 
different materials. They are a copper conductor, EPR for fixation, insulation EPR, 
neutral copper conductors, and a jacket for physical protection as shown in Fig. 2.1. 
However, for different types of cables, there are different conductor configurations. 
Figure 2.1. Common layer arrangement of a tape shield cable. 

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14 
2.2.1 Concentric Neutral Cable 
For a common concentric neutral cable, the layer arrangement is shown in Fig. 
2.2. The inner layer is normally an aluminum conductor and the second layer is the EPR 
insulation. The circles around the insulation layer are several symmetrical copper 
conductors named neutral line. The outermost layer is a rubber jacket for physical 
protection. When the conductors are connected with a certain source such as 15 kV 
voltage, the voltage distribution is shown in Fig. 2.3. 
Figure 2.2. Cross section of a concentric neutral cable. 
Figure 2.3. Voltage distribution of a concentric neutral cable. 

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15 
2.2.2 Tape Shield Cable 
For a standard tape shield cable, the layer arrangement is shown in Fig. 2.4. The 
inner layer is a copper conductor and the second layer is the EPR insulation. The thin 
layer around the insulation layer is a copper shield that is normally grounded. The 
outermost layer is a rubber jacket for physical protection. When the conductors are 
connected with a 15 kV voltage source, the voltage distribution is shown in Fig. 2.5. 
Figure 2.4. Cross section of a tape shield cable. 
Figure 2.5. Voltage distribution of a tape shield cable. 

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16 
2.3 
Calculation Method of Cable Parameters 
For medium voltage insulated cables, two types of construction that are 
mentioned in Chapter 2 are analyzed: concentric neutral cable and tape shield cable. 
Carson line method is used to estimate the impedance matrix and calculate the parameters 
of these two types of cables [11]. 
2.3.1 Carson Line Method for Concentric Neutral Cable 
For a concentric neutral cable, equations 2.1, 2.2, 2.3, 2.4, 2.5, 2.6 below are used 
to estimate the parameters[11]. 
(2.1) 
(2.2) 
(2.3) 
(2.4) 
(2.5) 
(2.6) 
where
for 
average earth; 
is the resistance of the conductor 
in
is the resistance of Carson line; 
is the 
nominal diameter of the cable in inches; 
is the diameter of the neutral conductor in 
inches; 
is the geometric mean radius of the neutral conductor in 
; is the 

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17 
resistance of neutral conductor in 
; is the earth constant resistance coefficient; 
K is the number of concentric neutral strands; R is the radius of a circle passing through 
the center of the concentric neutral strands; 
is the equivalent resistance of the 
concentric neutral. These parameters are shown in Fig. 2.6 below. 
Figure 2.6. Parameters used in Carson line method for concentric neutral cable. 
Equation 2.1 and 2.2 are used to calculate self-impedance and mutual impedance 
separately. 
2.3.2 Carson Line Method for Tape Shield Cable 
For a tape shield cable, all equations are the same except equations 2.3 and 
2.5[11]. 
(2.7) 
(2.8) 

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18 
Figure 2.7. Parameters used in Carson line method for tape shield cable. 
Using the above equations combined with some other basic electrical equations, 
the impedance matrix of any cross-sections can be calculated. For example, assume the 
cross-section below is applied: 
Figure 2.8. Sample cable arrangement. 

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19 
Using equation 2.1 and 2.2, a 13 x 13 impedance matrix could be developed with 
the corresponding conductor positions. 
In order to simplify, number all the conductors: 
Top Row:
A1 = “1”, C1 = “2”, B1 = “3”, B2 = “4”, C2 = “5”, A2 = “6” 
Bottom Row: A1 = “7”, C1 = “8”, B1 = “9”, B2 = “10”, C2 = “11”, A2 = “12” 
Ground: 
G = “13” 
The output impedance matrix is as follows:
13
13
13
,
13
12
,
13
1
,
13
13
,
12
12
,
12
1
,
12
13
,
1
12
,
1
1
,
1