Institute of Energy Technology
TRANSIENT MODELING AND SIMULATION OF A
THREE-PHASE LINE REACTOR IN WIND TURBINE
Conducted by group EPSH3-900
Fall Semester, 2008
TITLE:
Transient modeling and simulation of a three-phase line reactor in wind turbine
SEMESTER:
9th semester – fall 2008
PROJECT PERIOD:
01.09.2008 to 6.01.2009
PROJECT GROUP:
EPSH3-900
SUPERVISORS:
o Claus Leth Bak
o Kenneth Pedersen (assistant supervisor)
STUDENTS:
_______________________________
Domenic Notarnicola
_______________________________
Michal Sztykiel
Copies:
Pages, total:
3
152
SYNOPSIS:
In today’s world there are many sources of
electrical energy. Harnessing the energy from
wind has become the prority of many
governments and companies alike, who are
looking to further expand their share in this
environmentally friendly technology.
This project aims to further aid the advancement
in research currently undertaken by companies
such as Siemens Wind Power.
Developing a model for a three phase reactor,
allows for further research in the field of
interfacable power electronics. The series
reactor plays a vital role in the wind turbine
circuit by improving the quatlity of power fłow
from turbine to end user customer. With an
abundance of papers released about
transformer models, there has been little to no
information regarding line reactor models. It is
for this reason that such a model has been
needed and is the subject of this work.
TABLE OF CONTENTS
1 INTRODUCTION................................................................................................................. 5
1.1
1.2
1.3
2
BRIEF ..................................................................................................................................5
SIEMENS MANUFACTURER BACKGROUND .................................................................................6
PROBLEM DESCRIPTION .........................................................................................................6
OVERALL SYSTEM DESCRIPTION ................................................................................. 9
2.1
2.2
2.3
2.3.1
2.3.2
2.4
2.5
2.6
SYSTEM COMPONENTS AND CONNECTIONS ..............................................................................9
WIND T URBINE – SYNCHRONOUS GENERATOR..........................................................................9
POWER ELECTRONIC - CONVERTERS AND VARIABLE SPEED DRIVES ..........................................13
Power Converters - Voltage-fed Inverters ..........................................................................13
Power Electronics – Variable Speed Drives .......................................................................15
SERIES REACTOR ...............................................................................................................18
SHUNT REACTOR ................................................................................................................19
LINE IMPEDANCE .................................................................................................................19
ANALYSIS OF THE PROBLEM............................................................................................... 20
2.7
2.8
2.9
2.10
2.10.1
2.10.2
2.10.3
2.10.4
2.10.5
3
THEORETICAL ANALYSIS............................................................................................ 23
3.1
3.2
3.2.1
3.2.2
3.3
3.3.1
3.3.2
3.4
4
BRIEF ................................................................................................................................23
ANALYSIS OF A SINGLE COIL ON A SINGLE PHASE ......................................................................24
Analysis of an ideal model of a single coil - model 1...........................................................24
Analysis of a realistic model of a single coil - model 2 ........................................................28
ANALYSIS OF THREE COILS ON A SINGLE PHASE .......................................................................43
Analysis of the ideal model of three coupled coils – model 3 ..............................................43
Analysis of the realistic model of three coupled coils – model 4 ..........................................48
ANALYSIS OF NINE COILS ON THREE PHASES – FINAL MODEL ......................................................53
SIMPLIFIED MODEL..................................................................................................... 56
4.1
4.2
5
BRIEF ................................................................................................................................20
PROBLEM DEFINITION ..........................................................................................................20
SCOPE AND DELIMITATIONS ..................................................................................................20
SOLUTION METHODOLOGIES .................................................................................................21
STEP 1 - Theoretical Analysis ..........................................................................................21
STEP 2 - Experimental Measurement Tests + Simplified Model.........................................21
STEP 3 - Computer Simulation.........................................................................................21
STEP 4 - Experimental Analysis .......................................................................................22
STEP 5 - Verification ........................................................................................................22
BRIEF ................................................................................................................................56
POWERSYS LIBRARY BLOCK DESCRIPTIONS ...........................................................................56
COMPUTER SIMULATION ............................................................................................ 60
5.1
5.2
5.2.1
5.2.2
5.2.3
5.3
5.3.1
BRIEF ................................................................................................................................60
STEADY CONDITIONS ...........................................................................................................62
Case 1: All switches are closed .........................................................................................62
Case 2: Switches A1, A2, A3 are closed...........................................................................64
Case 3: Switch A1 is closed ..............................................................................................66
T RANSIENT CONDITIONS......................................................................................................68
Case 4: All switches close at the same time.......................................................................68
3
5.3.2
5.3.3
5.3.4
5.3.5
5.3.6
5.3.7
5.3.8
5.3.9
5.3.10
6
EXPERIMENTAL ANALYSIS AND VERIFICATION ........................................................ 110
6.1
6.2
6.2.1
6.2.2
6.3
7
Case 5: All switches open at the same time.......................................................................73
Case 6: Switches A1, A2, A3 close at the same time.........................................................78
Case 7: Switches A1, A2, A3 open at the same time.........................................................82
Case 8: Switch A1 closes at certain time ..........................................................................86
Case 9: Switch A1 opens at certain time...........................................................................90
Case 10: Switching mode – normal operation ...................................................................93
Case 11: Switching mode – A phase fault at certain time ..................................................97
Case 12: Switching mode – switch IGBT A1 fault at certain time ..................................... 101
Case 13: Switching mode – normal operation with random switching time delays............ 105
BRIEF .............................................................................................................................. 110
EXPERIMENTAL ANALYSIS - METHODOLOGY .......................................................................... 111
Case 1: Normal operation ............................................................................................... 111
Case 2: Frequency sweep of symmetrical component impedance .................................... 116
VERIFICATION RESULTS...................................................................................................... 122
CONCLUSION............................................................................................................ 129
7.1
7.1.1
7.1.2
7.2
7.2.1
7.2.2
TIME DOMAIN ................................................................................................................... 129
Analysis.......................................................................................................................... 129
Future Work ................................................................................................................... 129
F REQUENCY DOMAIN ......................................................................................................... 130
Analysis.......................................................................................................................... 130
Future Work ................................................................................................................... 130
8
LITERATURE............................................................................................................. 131
9
APPENDIX................................................................................................................. 133
A. EXPERIMENTAL MEASUREMENT T ESTS ......................................................................................... 133
B. VER_1.M FILE SOURCE CODE ..................................................................................................... 143
C. VER_2.M FILE SOURCE CODE ..................................................................................................... 143
D. VER_3.M FILE SOURCE CODE ..................................................................................................... 147
E. CONTENTS OF THE CD-ROM...................................................................................................... 152
4
1 Introduction
1.1 Brief
Wind energy is one of the fastest growing sources of electricity and one of the fastest growing markets
in the world today. These growth trends can be linked to the multi-dimensional benefits associated with
wind energy, such as non polluted, sustainable and affordable source of electric power. As well as being
affordable, wind power is a locally-produced source of electricity that enables communities to keep
energy dollars in their economy. Job creation (manufacturing, service, construction, and operation) and
tax base increase are other economic development benefits for communities utilizing wind energy [1].
Due to gradual depletion of fossil fuels many economists say, that in the nearest future wind energy will
be the most cost-effective source of electrical power. For these reasons many international companies
provide products, technologies and solutions which allow countries to build up wind farms. The most
essential component of these wind farms is called wind turbine.
field of interest
Fig.1.1. Wind turbine components [2].
The wind turbine is a wind energy conversion device that converts wind energy into electrical energy. It
can be either designed for constant or variable speed operation. Variable speed wind turbines are able
to produce 8% to 15% more energy. However, they require power electronic converters to provide a
fixed frequency and fixed voltage to the grid.
5
1.2 Siemens Manufacturer Background
One of the top world manufacturers for variable speed wind turbines is Siemens Wind Power. Siemens
is present in the wind industry for over 25 years and hires there more than 3700 employees. It installed
6597 wind turbines of total power 6,08 MW and cooperates with many companies on the field of electric
power generation. As a power converter in wind farms, Siemens uses proprietary device called net
converter, which is connected between generator and MV transformer [3]. Additional necessary
component (usually built along with power converter assembly) connected between net converter and a
transformer is called a three-phase line reactor.
1.3 Problem Description
A three-phase line reactor is a set of three windings usually set in one assembly. It is a series device,
therefore it is connected in the supply line such that all line current flows through the reactor. Line
reactors are mostly used as a current limiting devices that oppose rapid changes of it mainly because of
their inductance parameter. As a result, they hold down spikes of current and limit any peak currents.
These peaks may appear due to non-sinusoidal current waveform caused by keying of semiconductor
devices in power converter. Therefore, Line Reactors are used to reduce harmonics and prevent grid
from rapid changes of the current. In addition, they can be used to add line impedance or reduce
reflected voltage spike wave, acting like filters.
gearbox
generator
power converter
line reactor
transformer
grid
blades
Fig.1.3. Simplified diagram showing specific location of three-phase line reactor in wind turbine.
Nowadays, power converters are mainly made from semiconductor devices like IGBT transistors or
thyristors, which allow very fast switching (up to kHz) and therefore, provide grid with rapid changes of
current. Since three-phase line reactor in wind turbine is almost always present with power converter,
there is a necessity to analyze its behavior in transient states, especially caused by fast switching
phenomenon.
In other words, the initial problem statement may be formed in a way: how will line reactor respond to
the rapid changes of the current in transient states. In order to find the answer, there is a necessity to
make use of the process called modeling.
6
Modeling process consists of three essential elements shown below:
Problem
Model
Solution
- object
- mathematical
- parameters
- process
- physical
- description
- phenomenon
- mental
- principles etc.
It should be therefore analyzed with other process elements, such as problem and solution. Due to
complexity of many physical phenomena, one does not create general models for a big group of
problems. On the contrary – specific models are being used for specific problems. It is the problem that
defines which model is the most appropriate to be used in order to solve it. Models differ among
themselves mainly from the amount of simplifications and the accuracy of the solution they provide.
Therefore, it is very important to analyze the complexity of the problem and choose the most suitable
model for the solution. In addition, the best solution of the problem is given by individual physical models
which best reflect real state. In conclusion, modeling has enormous meaning in solving problems. That
is why it is said to be the best way to approach the task [4].
As a result, in order to analyze behavior of the line reactor in transient states, it is essential to use
specific model which most accurately corresponds to certain cases that are most significant for analysis.
There are two primary ways to achieve this. First is to create a new model from the beginning, which
would be most suitable for line reactors and analyzed cases. Second is to use already created model
and adopt it, making modifications to the model that would match necessary requirements.
Most common, efficient and accurate models presented in the literature which can be adopted, are
transformer transient models. This is because there are many similarities between line reactors and
transformers as the objects of the model. Transformer transient models are circuit representations of a
transformer which accurately predict its transient voltage response and frequency characteristics when
provoked by the stimulus of interest.
The terminal models allow to study external transient response of transformer in power system and are
mainly used for transient modeling, whereas the detailed models allow to study internal transient
response and are often used by transformer designers.
7
Most suitable linear model for the transient behavior analysis of the line reactor is one of the frequency
dependent model, which includes growing effect of additional phenomena as the frequency changes.
This effect can be predicted in experimental analysis, which consists of measuring specific parameters
affected by changes in frequency and caused by currents flowing.
When the model is created and modified, it needs to be implemented in the simulation software like
PSCad, ORCad , MATLAB etc. By performing simulations, one can then compare them with
experimental results and verify whether model properly reflects line reactor’s transient states or not. If
the simulation results do not fulfill one’s expectations, the model needs to be remodified by taking more
details into consideration, then implemented and verified again. Once the model gives satisfactory
results, one can say that the stated problem is solved and final aim achieved.
Summarizing, it is possible to obtain a specific mathematical model for a 3-phase line reactor and in
addition - also a possibility to verify the results in frequency domain. Further chapters provide detailed
steps and methods how to approach this problem. It must be here stated, that presented approach to
obtain a solution is not only one available. There can be many different methods and this work presents
particularly one of them.
8
2 Overall System Description
2.1 System Components and Connections
Fig. 2.1. Example Wind Turbine Circuit [21].
2.2 Wind Turbine – Synchronous Generator
In general, any type of three-phase generator can be equipped for wind turbines. Connection to the grid
of any of these types of generators is sought via means of a frequency converter. The main types of
generators used in wind turbines are:
Asynchronous (induction) generator:
• squirrel cage induction generator (SCIG)
• wound rotor induction generator (WRIG)
• Doubly-fed induction generator (DFIG)
Synchronous generator:
• wound rotor generator (WRSG)
• permanent magnet generator (PMSG)
Main generators used in wind turbine by Siemens manufacturer are synchronous.
9
The synchronous machine has a three-phase AC winding on the stator and a DC winding on the rotor.
Though the rotor winding, unlike that of an induction generator carries a DC excitation current, which is
usually fed through slip-rings. This produces a magnetic flux wave Br which is stationary with respect to
the rotor. As the machine rotates, three-phase sinusoidal voltages are generated in the stator windings
of frequency f, given by:
f = N⋅p (Hz)
{1}
where:
N - speed in revs/sec
p - number of pole pairs of the magnetic field.
If the stator winding is also excited by three-phase sinusoidal voltages with frequency fs, a magnetic flux
density wave Bs will also be produced, rotating at synchronous speed Ns given by:
Ns = fs ⁄ p (rev/sec)
{2}
Due to the distributed construction of the stator winding, the magnetic flux density wave Bs will also be
sinusoidal. The machine will develop a uniform torque, when the stator and rotor fields Bs and Br are
stationary with respect to each other.It is seen as the rotor magnetic field travels in step with the rotating
stator field. The rotor must not move at any other speed other than Ns, otherwise operation would be
impossible to maintain, because no uniform mean torque would be produced; hence the name
synchronous machine.
The torque produced by the machine is proportional to Bs⋅Brsinθ, where θ is the angle between Bs and
Br at any instant. For operation as a generator, the rotor field Br leads the stator field and the angle θ is
positive. For motoring operation Br is dragged behind the stator field and the direction of torque is
reversed [21].
Synchronous generators are generally more expensive and mechanically more complicated than
induction generators of similar size. However, the need for there not to be a reactive magnetising
current is one clear advantage in comparison with an induction generator.
10
Suited for full power control, connection to the grid through the use of a power electronic converter has
two main goals:
1 - to moderate the power fluctuations caused by gusting winds and as well as limit grid side
transients.
2 - to control synchronisity with grid frequency. Application of a synchronous generator in this
way allows a variable-speed operation of wind turbines.
Permanent magnets can be used to create the magnetic field in a synchronous generator as well as a
conventional DC field winding. Two classical types of synchronous generators have often been used in
the wind turbine industry:
● - wound rotor synchronous generator (WRSG)
● - permanent magnet synchronous generator (PMSG).
2.2.1
Wound Rotor Synchronous Generator
The rotational speed of a wound rotor synchronous generator (WRSG) is fixed by the frequency of the
supply grid, whereby the terminals of the stator windings are directly connected to the grid itself.
Through the use of slip rings and brushes, or a brushless exciter, the rotor winding of a WRSG is
excited using direct current. The synchronous generator, unlike an induction generator, does not need
any reactive power compensation. The rotor winding, through which direct current flows, generates the
exciter field, which rotates with synchronous speed. Here it is the frequency of the rotating field in the
stator winding and the number of pole pairs, which determine the rotor speed of the synchronous
generator.
WRSG can operate without the need of a gearbox, though the WRSG must be of a multipole (lowspeed) design. The disadvantage to such a gearless designs as this is attributed to having a large and
heavy generator, as well as having to include a full-scale power converter to handle the full power of the
system.
2.2.2
Permanent Magnet Synchronous Generator
The application of permanent magnet synchronous generators (PMSG) in wind turbines, allows for
operation at high power factor and a high efficiency because of their property of self-excitation. The
efficiency is higher than in an induction machine, due to the excitation being provided without any
energy supply.
11
Permanent magnets for use in a PMSG are expensive due to the cost of producing and manufacturing
magnetic materials. Also a full scale power converter is required, for use with a PM machines due to the
range of voltage and frequency generated, since excitation occurs at all times. This is an added
expense. However, the benefit is that power can be generated at any speed so as to fit the current
conditions.
The rotor of PMSGs is constructed from a permanent magnet pole system which may have salient
poles. The stator of PMSGs are wound. Salient poles, poles which stick out are used in electrical
machines to concentrate flux into discrete angular sectors, thereby maximising the alignment force
between the fields. Salient poles are most prevalent in slow-speed synchronous machines, and have
many poles. There are different topologies of PM machines. The most common types are the radial flux
machine, the axial flux machine and the transversal flux machine.
Due to synchronisation and voltage regulation the PMSG may cause problems during startup. This is
because the PMSG does not readily provide a constant voltage. Also during an external short circuit, or
if the wind speed is unsteady the PMSG can perform quite stiffly, exposing the magnetic material to
higher than normal temperatures. This could have a disastrous effect on the magnetic material since it is
quite sensitive to temperature. For instance, the magnet could lose its magnetic qualities at high
temperatures. Therefore, the rotor temperature of a PMSG must be managed by a cooling system.
These are some of the costly disadvantages of the PMSG.
12
2.3 Power Electronic - Converters and Variable Speed Drives
Power electronics are necessary to provide the interface between generator and the electrical network.
Often times, generators produce DC voltages that are incompatible with that of fixed frequency three
phase network voltages. Power electronic converters in this case, are used as the interface for those
generators to export energy to the network. At other times, power electronics are used to allow prime
movers to operate at variable speed. Variable speed drive systems control the power flow coming from
wind turbines, and maximise efficiency.
Here the basic principles of operation of the voltage fed inverter are explored. This is the inverter most
commonly used to control power flow from a small electrical generator. It mainly regards operating
characteristics of variable-frequency, variable-voltage PWM drive systems normally used to control the
speed of cage induction machines.
In addition, the examination of two large variable speed drive systems is given, namely the static
Kramer drive used to control the speed of a large wound rotor induction machine, and the load
commutated synchronous machine drive used to control large synchronous machines.
2.3.1
Power Converters - Voltage-fed Inverters
The voltage fed inverter consists of Insulated Gate Bipolar Transistors (IGBTs) as the main switching
devices. The presence of a large filter capacitor on the dc side of the converter is to provide a stiff
voltage source that is not affected by load conditions. Power flow in an inverter, is always from the DC
side of the inverter to the ac side. Hence the name ‘inverter’.
In the figure below, a DC source feeds the inverter, wanting to supply power to the network system via
series reactors and a transformer. From the figure below we could easily say that it could also be
supplying power from a dc source, but rather than being connected to the supply grid, connection can
also be made to an induction machine. This too is the basic power circuit of a variable speed drive.
13
Fig.2.3.1. IGBT voltage-fed inverter [21].
2.3.1.1 Pulse width modulation (PWM)
The AC voltage waveform at the inverter terminals can be controlled and modified by employing pulse
width modulation techniques to control switching. The main power circuit of a PWM voltage-fed inverter
is exactly as shown in the figure for the voltage-fed inverter above. Here the IGBTs are switched on and
off many times during each half cycle to control the ac line-to-line voltage.
The switching points of the power devices are determined by a high frequency triangular carrier wave
(typically between 4 and 10 kHz). This triangular carrier wave is compared with the 50 Hz sinewave
signal, to determine the crossover points. The resulting line voltage from the PWM is shown in the
following figures.
Fig. 2.3.1a. PWM waveforms [21].
14
Fig 2.3.1b PWM line-to-line voltage waveform [21]
The line to line output voltage waveform is sinusoidally modulated by the pulse and notch widths, which
are produced with a fundamental component at 50 Hz. This output waveform also consists of high
frequency harmonics at frequencies related to the carrier frequency. These harmonics are then filtered
out by the supply reactance, where the supply current waveform will be almost sinusoidal.
2.3.2
Power Electronics – Variable Speed Drives
The advantages of variable speed generation when compared with fixed speed induction generation is
that they offer higher energy capture. They do this through maximising the turbine efficiency by
adjusting the speed of the shaft. The basic requirements of variable-frequency, variable-voltage control
commonly are to be described here.
2.3.2.1 Variable-frequency, variable-voltage control
An induction machine is basically just a fixed speed machine, that is operating at just below the
synchronous speed, where the operating speed is a function of the load. Power flow is reversed, and
the machine operates as a generator when the machine is accelerated beyond its synchronous speed.
This is done by the use of by the use of some form of prime mover.
The synchronous speed of an induction machine can be adjusted by varying the frequency of its supply.
This is due to the number of pole pairs of the stator winding, and the fact that the synchronous speed ωs
is a function of the supply frequency.
If you reduce the speed while maintaining the stator voltage at its rated voltage, the machine will begin
to saturate causing excessive stator currents. This is due to machine impedance being lower at the
lower frequencies. If in fact the frequency is increased beyond the rated design value, a loss in torque
capability of the motor will result (because the the machine air gap flux will fall).
By varying the stator voltage with the frequency, the air gap flux is maintained at its rated value. If there
is any decrease in stator frequency, that has to be accompanied by a corresponding reduction in stator
voltage. This is so as to maintain constant flux and torque capability.
15
This voltage/frequency relationship is demonstrated in Fig. 2.3.2a and also shows how, at very low
frequencies, a boost in the stator voltage needed to compensate for the drop in stator resistance.
Fig. 2.3.2a. Constant Volts/Hz ratio control [21]
The frequency and magnitude of the fundamental output voltage are controlled electronically within the
inverter when a PWM inverter drive is used. This is achieved by varying the frequency and amplitude of
the modulating sinewave signal required by the induction machine.
When motoring, power flow is from the dc side into the machine (Fig. 2.3.2b). When generating, power
flow is reversed. A second PWM inverter is then needed to feed this power into the supply network.
Fig 2.3.2b PWM induction motor drive [21]
2.3.2.2 Slip-energy recovery for wound rotor machines (or the Doubly-Fed Induction
Generator DFIG)
There are considerable advantages in using a slip-energy recovery drive for large wind generators, in
the MW range.
The basic form of the system is shown in Fig. 2.3.2b. Through its slip rings and to a three phase diode
bridge rectifier, the wound rotor induction machine is line commutated, using an inverter connecting to
the ac supply via a step up transformer. Mechanical power from the turbine is fed into the mains supply
through the stator, whilst part of it is through the rotor via the dc link frequency changer. Rotor current
and therefore power and speed can be controlled. This is due to the mean dc rectified diode voltage
being approximately proportional to the slip. Thus the inverter firing angle is only to be adjusted.
16
For applications where control is only needed over a limited speed range, that is below or above
synchronous speed substantial reductions in diode and thyristor ratings and costs can be made.
Because the slip power recovery circuit and transformer need only handle slip power and not the full
machine power can these reductions can be obtained.
Fig. 2.3.2c Slip-energy recovery wind generator [21]
Poor drive power factor and highly variable reactive power requirements is one of a number of
drawbacks, that this type of drive suffers from, since this is difficult to correct using power factor
capacitors. The system is also susceptible to supply dips and interruptions.
The use of the force commutated IGBTs since both IGBT converters allow bi-directional power flow,
allows for the possibility of controlling the shape of the rotor current waveform. This means greater
control of power flow at subsynchronous speeds.
17
2.4 Series Reactor
The Series Reactors generally find application in medium and high voltage capacitor installation under
the following circumstances [23]:
•
The short circuit level of the network is high and as a result switching in surges of capacitor
banks are high. Example: Capacitors near a large generating or receiving station.
•
A number of capacitor banks operate in parallel across a common bus. A newly switched in
capacitor bank draws a heavy current from an already charged bank and the mains.
•
The network contains high arcing frequencies and/or is subjected to sudden voltage surges
(Switching or lightning).
•
In modern circuits with large thyristor controls, there is a great deal of generation of harmonics
for which a capacitor bank forms a low or very low impedance path. To prevent the flow of the
harmonics into the bank a filter circuit containing a series reactor is quite often necessary.
•
A properly designed series reactor- capacitor installation can act as a tuned filter circuit. By
selecting the tuning, one can pass to ground the entire contents of that particular harmonic and
prevent its spread further into computer systems or telecommunication systems. This has to be
done carefully. In advanced countries, laws are being enacted to make this feature compulsory.
Normally a 6% series reactance (6% of the capacitor bank in KVAr) is used. Where warranted, a 3% or
a 1% Series Reactor is also specified. There are number of instances where a series reactor can
entirely be dispensed with.
The advantages of a series reactor are that it limits the surge currents and also high frequency currents
into a capacitor bank. This protects the capacitor bank and reduces the burden on the switchgear
controlling the capacitor bank.
The disadvantages of a series reactor are as follows:
-
A series reactor of specified rate overloads the capacitors with an overvoltage of specified
rate permanently, so the capacitor must be specified for an overvoltage of reactor’s rate.
-
While it helps only during starting or under special circumstances in a normal network, it
performs no continuously useful function; yet it consumes more continuous power than the
capacitor bank itself.
18
2.5 Shunt Reactor
Shunt inductors are used on substation busbars, medium-length and long transmission lines to increase
line loadability and to maintain voltages near rated values. A high voltage reactor is relatively frequently
switched, during the periods of the system operations with low loads it is energized and with the rise of
load it is de-energized again. The inductors absorb reactive power and reduce overvoltages during light
load conditions, also reduce transient overvoltages due to switching and lighting surges [24, 25]. The
shunt reactors can reduce line loadability if they are not removed under full-load conditions. During the
energization, high unsymmetrical currents can occur. At de-energization, a transient recovery voltage
occurs in the breaker contacts with considerable magnitude [26].
The switching overvoltage can be dangerous for the equipment if their peak value exceeds the rated
switching impulse withstand voltage of the equipment [27]. It is very important to know the level of
dielectric stress that occurs during operation in the system in order to avoid insulation failures. Each
interruption involves a complex interaction between the circuit breaker and the source and the
reactor(load side) circuits. This interaction results in overvoltages dependent on system parameters and
characteristics of the load [28].
2.6 Line Impedance
Line impedance is the sum of resistance, inductance, and capacitance found between source and a
load. Common sources of line impedance include copper conductors, transformers, contactors, fuses,
and terminals. Every electrical device contributes a small amount to the total line impedance. As a
result, it causes several power quality problems. Excessive impedance causes voltage sags when
facility loads are energized, especially loads that have high inrush currents. Long term voltage
fluctuations are caused as facility loads are switched on and of during the day. Harmonic distortion and
voltage transients are caused by high frequency currents drawn by electronic loads within the facility.
Line impedance is responsible for many of the power problems within a facility. Close attention to line
impedance during the design phase can result in a large improvement in power quality without
expensive power conditioning devices.
In reality, impedance is better modeled as a resistance and a series inductance. This impedance
becomes much higher at higher frequencies, making impulses, distortion, and high frequency noise
much worse than expected with a 60 Hz or resistive impedance model [29].
.
Fig. 2.6 Simple “resistive” line impedance model  more accurate line impedance model.
19
Analysis of the Problem
2.7 Brief
In order to obtain most appropriate solution and spend time most efficiently, there is a necessity to
deeply analyze and understand the problem statement. By doing so, project work may focus on finding
most proper solution and will accomplish by adding or limiting specific elements in the analysis. In other
words, some realistic elements and phenomena can be omitted in the project work unless they play
significant role in the analyzed problem. As a result, specific elements that play most important role can
be emphasized in a field where problem occurs. This chapter concentrates on formulating problem and
making an introductory preparation for later analysis and results. First, problem is formulated. Second,
specific delimitations are listed. Finally third, brief introduction to solution methodology to each part of
the project is given with regard to formulated problem.
2.8 Problem Definition
Line reactors play an important role in the electrical system. Therefore, it is essential to know the details
on how much they influence the grid from electrical side during fast switching states. This influence is
may be seen in wide range. Main focus in this work is to predict how currents and voltages change
during normal and abnormal operation states and find their magnitudes with time constants, so that it
can be assessed whether installing this type of device fulfill one’s specific expectations. This task
requires analysis to be done in time domain. Another task is to predict when resonances occur and find
conditions under which they can be seen. This requires analysis done in frequency domain.
2.9 Scope and Delimitations
Attained solution to a problem is limited by certain assumptions and simplifications in order to perform
analysis in rational time and complexity. For the analysis of a 3-phase line reactor, below assumptions
are taken into account:
-
analyzed model is designed for frequencies up to 20 kHz.
-
core losses (histeresis and eddie currents) are not taken into account in computer simulations
due to high non-linearity and complexity of their modeling.
-
linear model discard saturation effect of the ferromagnetic core.
20
2.10 Solution Methodologies
The aim of the project is to find and show solutions capable of providing sufficient results. In order to
simplify the approach to tasks previously, four main steps have been formed:
1. Obtain theoretical model of the line reactor, which can be implemented into the simulation
software.
2. Add necessary parameter values into the model taken from measurements.
3. Run computer simulations and show significant plots reflecting currents and voltages responses.
4. Plot sequence impedance in frequency domain from experimental analysis.
5. Compare and verify sequence impedance plots taken from experimental analysis and obtained
theoretical model.
All these steps are necessary to accomplish in order to gain solutions.
2.10.1 STEP 1 - Theoretical Analysis
To perform most understandable and clear solution, deep theoretical analysis has to be taken into
consideration. Therefore one might be able to predict and explain the reason of the realistic behavior of
the model. Moreover, it may also benefit in future, when conditions of the analyzed object are to be
modified. Building model based on theoretical background and performing changes in it would be much
easier when it is known what phenomenon is the result of which process.
Therefore theoretical analysis is done in order to present explanation for further actions in developing
model. It is performed by moving from most simple models to more complex, adding new components
and developing new ideas and conditions. Finally, simplified model is obtained and its specification.
2.10.2 STEP 2 - Experimental Measurement Tests + Simplified Model
Simplified model is obtain from two sources: theoretical analysis and measurements. Measurements are
done to apply specific values of the parameters, that need to be implemented into the model. Values of
these parameters reflect the specific response by model in transient conditions.
2.10.3 STEP 3 - Computer Simulation
Computer simulation is done to obtain plots and values of important parameters, that would reflect
visibly model’s behavior. Simulations are performed in MATLAB software and are necessary to verify
the model and examine how possible changes will influence the model.
21
2.10.4 STEP 4 - Experimental Analysis
To obtain necessary pattern for comparison additional measurements have to be done that most
accurately reflect changes in sequence impedance in frequency domain. Measurements consist of
resulting parameter values reflecting all effects and phenomena that take place in line reactor at once.
2.10.5 STEP 5 - Verification
Final comparison allows to draw conclusions and verify attained model. It can be assessed the quality of
the model based on gained accuracy, simplicity and velocity. One can then determine whether it is
possible to obtain model good enough to reflect realistic transient behavior of the line reactor from this
type of methodology.
22
3 Theoretical Analysis
3.1 Brief
Theoretical analysis will be performed based on realistic structure of a 3-phase line reactor installed in
wind turbine. The line reactor consists of a ferromagnetic core with three columns. Each column has
windings wrapped around it, which represent single phase. Each phase consists of three coils wrapped
around single column, which makes nine coils in total for the line reactor.
Analysis will begin by defining single coil in single phase as an analyzed object. In later steps, analyzed
object will be defined as three coils in single phase attached to the same column of the core, including
their mutual influence on each other. Finally, third analyzed object consists of all nine coils in three
phases, which actually the most precisely reflects the behavior of an analyzed 3-phase line reactor.
Total theoretical analysis consists of 5 different models of the line reactor. The approach is made from
most basic models, and it continues by making more complex ones, ending finally in the most complex
model that includes all physical phenomena described in previous models. Each model is analyzed from
physical and mathematical point of view.
Physical analysis describes phenomena that take place in line reactor. These phenomena eventually
result in creating specific model that reflects them.
Mathematical analysis provides equations and relations between physical parameters of the analyzed
model in transient states, so that the behavior of the model can be predicted from the values of these
parameters. It is therefore an essential tool in order to perform visual simulation and be able to predict
line reactor’s transient behavior and get close to realistic, theoretical transient response of the model.
In addition, some model analyses are provided with examples, showing how specific plots occur due to
presence of the model parameters. Examples show simulations made in computer software in addition
with plots of current and voltage in transient states. Later these plots are compared with realistic plots of
line reactor, giving conclusions on accuracy and quality of performed theoretical analysis.
23
3.2 Analysis of a single coil on a single phase
analyzed
object
coil
core
Fig. 4.2. Simplified sketch of a 3-phase line reactor. Red dashed line shows an analyzed object.
3.2.1 Analysis of an ideal model of a single coil - model 1
PHYSICAL BACKGROUND
Whenever electrons flow through a conductor, a magnetic field will develop around that conductor. This
effect is called electromagnetism. Magnetic fields effect the alignment of electrons in an atom and can
cause physical force to develop between atoms across space just as with electric fields developing force
between electrically charged particles. Like electric fields, magnetic fields can occupy completely empty
space, and affect matter at a distance [5].
Fields can be measured either by a field force, or by a field flux. The field force says how powerful will
be the push of a certain object over a certain distance. The field flux on the other hand, reflects the total
amount of the field through space. There is a deep analogy between field force and voltage, and
between field flux and current, even though flux can exist in vacuum (without electrons) whereas current
can only appear where there are free electrons to move. Both field flux and current have oppositions
that limit their values. Furthermore, just like current can be proportional to the voltage divided by its
opposition (resistance), an amount of field flux is proportional to the field force applied and divided by
the quantity of opposition (space). It is the type of material that dictates specific values of opposition in
both cases. In current case it is conductor and its resistivity. In flux case it is a material that occupies the
space through which a magnetic field force is impressed. This material is known as a core, and its
shape with permeability reflect opposition to the field flux.
An electric field flux allows for an accumulation of free electron charge within conductors. On the other
hand, an electromagnetic field flux allows to accumulate a certain "inertia" by the flow of electrons
through the conductor (current), that produces the field.
24
Coils are components designed to take advantage of this phenomenon. It is done by shaping the length
of a conductive wire in the form of a coil. This shape creates a stronger magnetic field than what would
be produced by a straight wire. Some inductors are formed with wire wound in a self-supporting coil.
Others wrap the wire around a solid core material of some type. Sometimes the core of an inductor will
be straight, and other times it will be joined in a loop (square, rectangular, or circular) to fully contain the
magnetic flux. These design options all have effect on the performance and characteristics of coils [6].
The circuit symbol for a most basic model of a single coil is shown below. This model is described only
by one parameter – inductance.
u(t)
i(t)
L
Fig. 4.2.1. Electric representation of model 1 - an ideal single coil.
Inductance (L, measured in Henries) is an effect which results from the magnetic field that forms
around a current-carrying conductor. Electric current through the conductor creates a magnetic flux
proportional to the current. A change in this current creates a change in magnetic flux that, in turn,
generates an electromotive force (EMF) that acts to oppose this change in current. This phenomenon is
called Faraday’s Law. Inductance is a measure of the amount of EMF generated for a unit change in
current [7].
It is the most important parameter of the coil and it depends only on coil’s physical structure (number of
turns, coil material, shape and diameter) and core material.
MATHEMATICAL BACKGROUND
The following presents mathematical relations between physical parameters of the cylindrical-shaped
coil assuming the following properties of it:
- Area enclosed by each turn of the coil is A [m2]
- Mean path length of the core is l [m]
- Number of turns in the coil is N [-]
- The current flowing through the coil is i [A]
- Permeability of the core is µ [H/m], given by the permeability of free space (µ0) multiplied by a factor,
the relative permeability (µr): µ = µ0 · µr
25
The basic law governing the production of the magnetic field by a current is Ampere’s law:
∫ H ⋅ dI = I
N
{1}
where H is the magnetic flux intensity (measured in ampere-turns per meter) produced by the current IN.
If there is a core composed of ferromagnetic material, then essentially all the magnetic field produced by
the current will remain inside the core, so the path of integration in Ampere’s law is the mean path
length of the core lC. If there is an air gap instead of the core, then the path of integration is assumed to
be the length of the coil, since magnetic field intensity is the strongest there. If a winding consists of N
turns of wire wrapped around the core or air gap, then the current passing within the path of integration
IN is N⋅i, since the coil of wire cuts the path of integration N times while carrying current i. Ampere’s law
thus becomes:
H ⋅l = N ⋅i
{2}
Here H is the magnitude of the magnetic field intensity. Therefore, the magnitude of the magnetic field
intensity in the core due to applied current is:
H=
N ⋅i
l
{3}
The magnetic field intensity H is in a sense a measure of the “effort” that a current is putting into the
establishment of a magnetic field. The strength of the magnetic field flux produced in the core also
depends on the material of the core. The relationship between the magnetic field intensity H and the
resulting magnetic flux density B (measured in teslas, T) produced within material is given by:
B = µ ⋅H
{4}
The actual magnetic flux density produced in a piece of material is thus given by a product of two terms:
• H - representing the effort exerted by the current to establish a magnetic field
• µ - representing the relative ease of establishing a magnetic field in a given material
From equations {3} and {4} the magnetic flux density may be presented as:
B=
N ⋅µ ⋅i
l
{5}
Now, the total flux linkage in the coil, λ (measured in weber-turns, Wb) is given by:
λ = N ⋅ ∫∫ B ⋅ dA
{6}
A
where dA is the differential unit of area. If the flux density vector is perpendicular to a plane of area A,
and if the flux density is constant throughout the area, then this equation reduces to:
λ = N ⋅B ⋅ A
26
{7}
As a result, from {5} and {7}:
λ=
N2 ⋅µ ⋅i ⋅A
l
{8}
The flux linkage in an inductor is therefore proportional to the current, assuming that A, N, l and µ all
stay constant (these parameters are dependant only to the physical structure of the coil and core).
The constant of proportionality is given the name inductance and the symbol L:
λ = L⋅i
{9}
N2 ⋅µ ⋅A
L=
l
{10}
Where:
Taking the derivative with respect to time from equation {9}:
dλ dL di
=
+
dt
dt dt
{11}
dλ
di
=L
dt
dt
{12}
Since L is assumed as time-invariant:
Faraday's Law states that:
−ε = N⋅
dφ dλ
=
dt
dt
{13}
Symbol ε represents the electromotive force (EMF) of the coil, which is opposite to the voltage u across
the inductor, giving from {12} and {13}:
u = L⋅
di
dt
{14}
This means that the voltage across an inductor is equal to the rate of change of the current in the
inductor multiplied by a factor, the inductance. This applies only to ideal inductors as mentioned above,
which do not exist in the real world.
27
3.2.2 Analysis of a realistic model of a single coil - model 2
PHYSICAL BACKGROUND
Real world coils made of physical components present more than just pure inductance. It is especially
seen at high currents and frequencies flowing through the coil. Therefore, in order to accurately reflect
coil’s behavior, model has to be modified by implementing new elements into it which best reflect
inductor’s response. The circuit symbol for an advanced model of a single coil is shown below:
uC = u
iC
C
uL
i
uRDC
L
iRp
uRAC
iR
iL
uRp c
La
R DC
R AC
Rp
Fig. 4.2.2a. Electric representation of model 2 - a realistic single coil .
Direct Current Resistance (RDC, measured in Ohms, Ω ) reflects the resistivity of the material from
which conductive wire is made. It is proven by research that every material, no matter how good
conductor is, has its own resistivity. Therefore, there are inevitable heat losses on wire caused by
currents flowing through it. DC Resistance reflects how big these losses are caused by current. In
addition, DC resistance would be expected to be greater for a longer wire, less for a wire of larger cross
sectional area [8]. Experimentally, dependence upon these properties is a straightforward one for a wide
range of conditions, and the resistance of a wire can be expressed as:
R=
ρ ⋅l
{15}
A
where:
ρ - resistivity [Ω⋅m2 / mm]
l - length [mm]
A - cross sectional area [m2]
28
Alternating Current Resistance (RAC, measured in Ohms, Ω) reflects the effective resistance of the
coil when connected to the AC source. If a conductor is carrying high alternating currents, the
distribution of current is not evenly dispersed throughout the cross section of the conductor. This is due
to phenomenon known as skin effect [9]. AC resistance for coils is dependant upon frequency level and
can be derived from the equation below:
R AC
x S4
=
192 + x S4
for x S2 =
8 ⋅ π ⋅ f ⋅ 10 −7
RDC
{16}
where:
f - frequency, [Hz]
RDC - DC resistance [Ω]
skin effect
The Skin Effect is a physical phenomenon that refers to the
tendency of current flow in a wire to be confined to a layer in the
wire close to its outer surface. At low frequencies the skin effect is
negligible and the distribution of current across the conductor is
uniform. As frequency is increased the depth to which the flow can
penetrate, is reduced. Skin effect occurs because current flow
moves away from those regions of the conductor having the
strongest magnetic field. A consequence of this is that the
Fig. 4.2.2b The skin effect phenomenon [9]
number of flux linkages between turns will be reduced. Therefore skin effect produces a decrease in
inductance; of about 2%, though more if the wire is short [10].
Inter Winding Capatitance (C, measured in Farads, F) reflects the resulting capacitance of former
capacitances between windings in single coil. Since most power inductors are made by winding wire on
a core, the obvious source of these capacitances are from two wires in close proximity. In realistic
multi-winding coils there are always insulation gaps between adjacent windings, which can be reflected
by capacitances, as shown below:
i(t)
C1
C1
...
C2
C
.
Cn
i(t)
i(t)
Cn
Fig. 4.2.2c. Substitution process of a resulting capacitance for single capacitances between each turn.
29
If there are N = n+1 windings on the coil, then there are n capacitances in parallel connection. One may
replace these capacitances with a resulting one. Resulting capacitance C can be calculated from the
equation: C =
n
∑C
i =1
i
.
However, the wires can be adjacent turns that are side by side, or two turns on two different layers that
end up on top of each other or any and all combinations of those two. There can also be capacitance
between the wire and the ferromagnetic core if it is one of those materials that is a reasonably good
electrical conductor. Since analyzed object is the whole coil, the result of modeling one summed
capacitor will be the same; stored energy in the capacitor when there is voltage across the inductor.
The most obvious effect of this capacitance is to cause almost any coil to have a Self-Resonant
Frequency (SRF). SRF frequency (resonance) is the frequency at which imaginary part of the coil’s
impedance is zero. Above this frequency, the inductor behaves like a capacitor and can't be used to
store energy, at least not in the sense of a pure inductor. At one tenth the SRF, the inductor acts pretty
much like an inductor; and the capacitive effect can be effectively ignored [11].
Parallel resistance (Rp, measured in Ohms, Ω) reflects core losses that cause generating heat in the
ferromagnetic core. They consist of eddie current and histeresis losses which are common in
transformers. This parameter is strongly related to the core properties (geometry, structure,
permeability) and is only applicable when there is a core material inside coil. It can be calculated from
the following Legg’s equation [12]:
(
R P = µ ⋅ L ⋅ aB max ⋅ f + e ⋅ f 2
Total loss factor
)
{17}
eddy current loss
where:
µ L a Bmax e f
-
relative permeability of the core [H/m]
inductance of an inductor [H]
histeresis loss coefficient [-]
maximum magnetic induction of the core [T]
eddy current coefficient [-]
frequency [Hz]
hysteresis loss
Coefficients a and b are obtained from experimental tests for each core material and are available in
IEC Publication 60401-3. When a varying magnetic field passes through the core, eddy currents are
induced in it. Joule heat loss by these currents is called eddy current loss. Histeresis loss is due to
irreversible behavior in histeresis curve and equal to the enclosed area of the loop.
30
eddie current losses
A time-changing flux induces a voltage within a
ferromagnetic core (in just the same manner as
would in wire wrapped around that core).
These voltages cause swirls of current in flow
within the core, much like the eddies seen at
the edges of a river. It is the shape of these
currents that gives rise to the name eddy
currents. These eddy currents are flowing in a
Fig. 4.2.2d Eddy Currents phenomenon [13]
resistive material (the iron of the core), so energy is dissipated by them. The lost energy goes into
eating the iron core [14].
histeresis losses
The atoms of iron and similar metals (cobalt, nickel, and some of their alloys) tend to have their
magnetic fields closely aligned with each other. Within the metal, there are many small regions called
domains. In each domain, all the atoms are aligned with their magnetic fields pointing in the same
direction, so each domain within the material acts as a small permanent magnet. The reason that a
whole block of iron can appear to have no flux is that these numerous tiny domains are oriented
randomly within the material [15].
Fig 4.2.2e. An example of the domain structure within a piece of iron – histeresis phenomenon
When an external magnetic field is applied to this block of iron, it causes domains that happen to point
in the direction of the field to grow at the expense of domains pointed in other directions. Domains
pointing in the direction of the magnetic field grow because at their boundaries physically switch
orientation to align themselves with the applied magnetic field. The extra atoms aligned with the field
increase the magnetic flux in the iron, which in turn causes more atoms to switch orientation, further
increasing the strength of the magnetic field.
31
As the strength of the external magnetic field continues to increase, whole domains that are aligned in
the wrong direction eventually reorient themselves as a unit to line up with the field. Finally, when nearly
all the atoms and domains in the iron are lined up with external field, any further increase in the
magnetomotive force can cause only the same flux increase that it would in free space. At this point, the
iron is saturated with flux [16].
The key to histeresis is that when external magnetic field is removed, the domains do not completely
randomize again. It is due to energy they require to turn the atoms within them. Originally, energy was
provided by the external magnetic field to accomplish the alignment; when the field is removed, there is
no source of energy to cause all the domains to rotate back. The fact that turning domains in the iron
requires energy leads to a common type of energy loss in all machines and transformers, which is called
histeresis loss. Summarizing, the histeresis loss in an iron core is the energy required to accomplish the
reorientation of domains during each cycle of the alternating current applied to the core [17].
32
MATHEMATICAL BACKGROUND – SINGLE PHASE
The following presents mathematical relations between physical parameters of the cylindrical-shaped
coil based on the results of mathematical analysis of coil’s basic model and figure 4.2.2f
All parameters are named analogically to those presented on the figure 4.2.2f
Kirchhoff's law #2 states that the sum of the currents entering any node equals the sum of the currents
leaving that node. Based on that statement and on the model of coil one can perform below equations:
= i C + i RD
{18}
i RD = i RA + i L
{19}
i
Kirchhoff's law #1 states that the voltage changes around a closed path in a circuit add up to zero.
Based on that statement and on the model of coil one can perform below equations:
u = RDC ⋅ i RD + u L = u c
{20}
u L = R AC ⋅ i RA
{21}
From the previous analysis of the ideal inductor and from the analogical analysis of the ideal capacitor
there can be formulated these equations:
t2
iC = C ⋅
du C
dt
⇒ uC =
uL = L ⋅
di L
dt
⇒ iL =
33
1
⋅ i C ⋅ dt
C ∫t 1
{22}
t2
1
⋅ u L ⋅ dt
L ∫t 1
{23}
MATHEMATICAL ANALYSIS 1 – circuit with ideal voltage source
uL(t)
iL(t)
L
SWITCH
+
u(t)
Fig. 4.2.2f. Electric circuit model with ideal voltage source for the mathematical analysis of the model 1.
Analyzed circuit is powered by an ideal AC voltage source connected in series with resistor and
inductor. Analyzed situation is when switch is closing the circuit.
Based on previously stated laws and formulas, according to the circuit model, one can write:
uL + u R = u
{24}
where:
di L
dt
{25}
uR = i L ⋅ R
{26}
uL = L ⋅
As a result, the first order differential equation is acquired:
di L
+ iL ⋅R = u
dt
{27}
i (t ) = i transient + i steady
{28}
L⋅
Its solution is provided below:
where:
i transient = A ⋅ e
isteady
R
− ⋅t
L
- transient state current coefficient
- steady state current coefficient flowing through inductor after switching operation
iinitial = i(tic) - initial current flowing through coil at switching time tic
34
When switch is closing:
isteady =
U amplitude
R 2 + (2π fL ) 2
⋅ sin( 2 ⋅ π ⋅ t + ϕU − ϕ Z )
I initial = 0
where:
Uamplitude – voltage source amplitude
φU - initial voltage source phase
ϕ Z = arctan(
2π fL
)
R
Therefore:
i(t ic ) = itransient + i steady = A ⋅ e
R
− ⋅t ic
L
+
U amplitude
R 2 + (2π fL) 2
From this, A parameter can be calculated:
R
⋅t ic
U amplitude
A=−
⋅ sin(2 ⋅ π ⋅ t ic + ϕ U − ϕ Z ) ⋅ e L
R 2 + (2π fL ) 2
⋅ sin( 2 ⋅ π ⋅ t ic + ϕ U − ϕ Z ) = 0 {29}
{30}
Finally, current can be presented as:
R
− ⋅( t −t ic )
U amplitude
U amplitude
i L (t ) = −
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕ Z ) ⋅ e L
+
⋅ sin(2 ⋅ π ⋅ t + ϕ U − ϕ Z )
{31}
R 2 + (2π fL ) 2
R 2 + (2π fL ) 2
Voltage drop on the coil:
R
− ⋅(t −tic )
Uamplitude
Uamplitude
di
uL = L ⋅ L = R ⋅
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕZ ) ⋅ e L
+ 2π fL ⋅
⋅ cos(2 ⋅ π ⋅ t + ϕU − ϕZ )
2
2
dt
R + (2π fL)
R 2 + (2π fL) 2
↑ {32}
35
When switch is opening:
U amplitude
iinitial =
R 2 + ( 2π fL ) 2
⋅ sin( 2 ⋅ π ⋅ t ic + ϕU − ϕ Z ) isteady = 0
where: Uamplitude – voltage source amplitude
φU - initial voltage source phase
2π fL
ϕ Z = arctan(
)
R
Therefore:
i (t ic ) = i transient + i steady = A ⋅ e
R
− ⋅t ic
L
+0=
U amplitude
R 2 + (2π fL ) 2
⋅ sin(2 ⋅ π ⋅ t ic + ϕ U − ϕ Z )
From this, A parameter can be calculated:
R
⋅t ic
U amplitude
A=
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕ Z ) ⋅ e L
R 2 + (2π fL ) 2
Finally, current can be presented as:
R
− ⋅( t −t ic )
U amplitude
i L (t ) =
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕ Z ) ⋅ e L
R 2 + (2π fL ) 2
Voltage drop on the coil:
R
− ⋅(t −t ic )
Uamplitude
di
uL = L ⋅ L = −R ⋅
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕZ ) ⋅ e L
dt
R 2 + (2π fL)2
36
{33}
{34}
{35}
{36}
Example 1
This example contains computer analysis of circuit specified by:
-
AC voltage source:
u(t)
= 690sin(2⋅π⋅60 + 0)
-
Series Resistance:
R
= 1 mΩ
-
Series Inductance:
L
= 74 mH, Initial state: i(tic) = 0 A
-
Time of Switching:
tic
= 0,3 s
Fig. 4.2.2g. Electric circuit diagram of model 1 for simulation in PSCAD software.
Opening operation
Fig. 4.2.2h. Voltage plot when switch opens.
37
Fig. 4.2.2i. Current plot when switch opens.
Closing operation
Fig. 4.2.2j. Voltage plot when switch closes.
Fig. 4.2.2k. Current plot when switch closes.
38
From the curves presented above it can be concluded,
that current cannot change rapidly. At the time switch
should be opening, the current’s is almost at its peak
stage, therefore the delay is necessary so that current
will be equal to zero, and then the state of switch can
change. Theoretically, if switch had been opened with
currents absolute value greater than zero, current’s
curve during opening time would have looked like the
Fig. 4.2.2l. Voltage and current plots when switch opens
one presented on the right graph. Voltage rapidly rise to the level that is sufficient to sustain current after
switch operation. And then both parameters move aperiodically to zero value. The time of moving to
zero is determined by ratio L / R. Consequently, when switch is closing, voltage level changes before
current in the manner to prevent its rapid changes.
MATHEMATICAL ANALYSIS 2 – circuit with ideal voltage source
uC = u
iC
C
uLc
i
L
SWITCH
iRp
uRDC
uRAC
R DC
R AC
iR
iLc
La
uRp c
Rp
+
u(t)
Fig. 4.2.2k Electric circuit model with ideal voltage source for the mathematical analysis of the model 2.
Based on previously stated laws and formulas, according to the circuit model, one can write:
u L + u RD = u C
where: u L = L ⋅
{37}
di L
dt
uRD = RDC ⋅ iRD = RDC ⋅ (i L + i RA) = RDC ⋅ (i L +
u C = u − u C (t = 0)
uL
R
di
L diL
) = RDC ⋅ (i L +
⋅ ) = RDC ⋅ i L + DC ⋅ L ⋅ L
RAC
RAC dt
RAC
dt
{39}
39
← {38}
u is the function of ideal voltage source in time domain. As a result from {28}, {29} , {30} and {31}:
L⋅
di L
R
di
+ R DC ⋅ i L + DC ⋅ L ⋅ L = u
dt
R AC
dt
 L ⋅ (R AC + R DC )  di L
+ R DC ⋅ i L = u

⋅
R AC

 dt
{40}
⋅
1
R DC
 L ⋅ (R AC + R DC )  di L
u
+ iL =

⋅
R DC
 R AC ⋅ R DC  dt
{41}
{42}
This is first-order ordinary differential equation. Its solution will be:
i L (t ) = i transient + i steady
where: i transient = A ⋅ e
−
t
λ
- transient state current coefficient, λ =
{33}
L ⋅ (R AC + R DC )
R AC ⋅ R DC
isteady = steady state current coefficient flowing through inductor after switching
iinitial = i(tic) – initial current flowing through coil at switching time tic
When switch is closing:
i steady = M 2 + N 2 ⋅ sin( 2 ⋅ π ⋅ t + ϕI )
i initial = 0
where:


Rp
−1
M = real U ⋅ (Z + (− j ⋅ X C−1 )) ⋅

R p + j ⋅ X L 


Rp

−1
N = imag U ⋅ (Z + ( − j ⋅ X C−1 )) ⋅

R p + j ⋅ X L 

R ⋅ jX L
−1
Z =( P
+ R DC + R AC ) −1 + ( − jX C ) −1
R P + jX L
1
XC =
2π f ⋅ C
X L = 2π f ⋅ L
U = U amplitude ⋅ e jϕU
φU - initial voltage source phase
N 
ϕ I = arctan 
M 
Therefore:
i (t ic ) = i transient + i steady = A ⋅ e
1
− ⋅t ic
λ
+ M 2 + N 2 ⋅ sin(2 ⋅ π ⋅ t ic + ϕ I ) = 0
40
{34}
From this, A parameter can be calculated:
1
A = − M 2 + N 2 ⋅ sin(2 ⋅ π ⋅ t ic + ϕ I ) ⋅ e λ
⋅t ic
{35}
Finally, current can be presented as:
i L (t ) = − M 2 + N 2 ⋅ sin(2 ⋅ π ⋅ t ic + ϕ I ) ⋅ e
1
− ⋅( t − t ic )
λ
+ M 2 + N 2 ⋅ sin(2 ⋅ π ⋅ t + ϕ I )
{36}
Voltage drop on the coil:
1
− ⋅( t −t ic )
di
L
uL = L ⋅ L = ⋅ M 2 + N 2 ⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕZ ) ⋅ e λ
+ 2π fL ⋅ M 2 + N 2 ⋅ cos(2 ⋅ π ⋅ t + ϕI )
dt λ
When switch is opening:
i steady = 0
i initial =
U amplitude
R 2 + (2π fL ) 2
⋅ sin( 2 ⋅ π ⋅ t ic + ϕ U − ϕ Z )c
where:
Uamplitude - voltage source amplitude
- initial voltage source phase
φU
2π fL
ϕ Z = arctg (
)
R
Therefore:
i (t ic ) = i transient + i steady = A ⋅ e
R
− ⋅t ic
L
+0=
U amplitude
R 2 + (2π fL ) 2
⋅ sin(2 ⋅ π ⋅ t ic + ϕ U − ϕ Z ) {38}
From this, A parameter can be calculated:
R
⋅t ic
U amplitude
A=
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕ Z ) ⋅ e L
2
2
R + (2π fL )
Finally, current can be presented as:
R
− ⋅( t −t ic )
U amplitude
i L (t ) =
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕ Z ) ⋅ e L
R 2 + (2π fL ) 2
Voltage drop on the coil:
R
− ⋅(t −t ic )
Uamplitude
diL
uL = L ⋅
= −R ⋅
⋅ sin(2 ⋅ π ⋅ t ic + ϕU − ϕZ ) ⋅ e L
dt
R 2 + (2π fL)2
41
{39}
{40}
{41}
{37}
Example 3
This example contains computer analysis of circuit specified by:
-
AC voltage source:
u(t)
= 690sin(2⋅π⋅1000000 + 0)
-
Series Resistance RDC:
R
= 1 mΩ
-
Series Resistance RAC:
R
= 0,929 Ω
-
Parallel Resistance:
R
= 582 kΩ
-
Series Inductance:
L
= 74 mH, Initial state: i(tic) = 0 A
-
Parallel Capacitance:
C
= 1 nF, initial state: u(tic) = 0 V
-
Time of Switching:
tic
= 0,3 s
Resistance and capacitance parameter values above are characteristic for a single phase inductor with
a fully flux-coupled ferrite 61 core [18].
.
Fig. 4.2.2m. Electric circuit diagram of model 2 for simulation in PSCAD software.
Opening operation
Fig. 4.2.2n. Currents and voltages in transient conditions for model 2 when switch opens.
Closing operation
Fig. 4.2.2o. Currents and voltages in transient conditions for model 2 when switch closes.
42
3.3 Analysis of three coils on a single phase
Fig. 4.3. Simplified sketch of a 3-phase line reactor. Red dashed line shows an analyzed object.
3.3.1 Analysis of the ideal model of three coupled coils – model 3
PHYSICAL BACKGROUND
On the previous analysis of a single coil, Faraday’s law was mentioned. It states that due to changes in
current flowing through the single coil magnetic flux changes, which in fact results in inducing
electromotive force (EMF). This force nullifies changes in current, so that magnetic flux remains
constant. This phenomenon is known as Lenz’s law. It is important to state here, that changing current
causes changing flux, and changing flux causes changing current in the coil, inducing in both cases
electromotive force in opposite directions. In single coil model there is only one element present, which
is also the source of magnetic field. Therefore, created magnetic field does not influence any other
component. On the other hand, in three coils model there are three components (coils) sensitive to
magnetic field, which may in addition behave as a magnetic field sources. In each coil occurs same
phenomena explained in single coil model. However, there are additional phenomena caused by the
interference of other two magnetic fields created by coupling coils. If there is a current flowing through
the single coil, it induces magnetic field in it, which lines cut across two other coils. As a result, if current
changes, flux of the magnetic field is forced to change as well. And to prevent this change,
electromotive force occurs in all three coils. Since the density of the magnetic flux (B parameter) is
different in each coil , there are different EMF’s induced in each of them. Therefore one can divide
inductances in coils as:
Self inductances (Lii ; i ∈ {1,2,3} ; measured in Henries) reflect the amount of EMF generated in coil for
a unit change in current flowing through that coil (caused by changes in magnetic field created by coil
itself ).
43
Mutual inductances (Lij ; i,j ∈ {1,2,3} ∧ {i ≠ j} ; measured in Henries) reflecting the amount of EMF
generated in coil for a unit change in current flowing through the coupled coil (caused by changes in
magnetic field created by coupled coil).
The circuit symbol for a most basic model of a three coils is shown below. This model is described by a
matrix of one parameter – inductance.
u1(t)
i1(t)
L12
i2(t)
L13
L23
i3(t)
L
11
u2(t)
22
u3(t)
L21
L
L31
L32
L
33
Fig. 4.3.1. Electric representation of model 3 - ideal three coupled coils.
Self inductances:
L11 – proportionality between the EMF generated in coil 1 to the change in current in coil 1 which produced it.
L22 – proportionality between the EMF generated in coil 2 to the change in current in coil 2 which produced it.
L33 – proportionality between the EMF generated in coil 2 to the change in current in coil 2 which produced it.
Mutual inductances:
L12 – proportionality between the EMF generated in coil 2 to the change in current in coil 1 which produced it
L13 – proportionality between the EMF generated in coil 3 to the change in current in coil 1 which produced it
L21 – proportionality between the EMF generated in coil 1 to the change in current in coil 2 which produced it
L23 – proportionality between the EMF generated in coil 3 to the change in current in coil 2 which produced it
L31 – proportionality between the EMF generated in coil 1 to the change in current in coil 3 which produced it
L32 – proportionality between the EMF generated in coil 2 to the change in current in coil 3 which produced it
As a result, an inductance matrix is given:
L11
L[] = L 21
L 31
L12
L 22
L 32
44
L13 
L23 
L33 
{43}
MATHEMATICAL BACKGROUND
The following presents mathematical relations between physical parameters of three cylindrical-shaped
coils. Magnetic flux density and intensity are the results of Ampere’s law analysis presented in
subchapter 4.2.2: “Analysis of the ideal model of a single coil - Mathematical background”. The only
difference is that there are three values of these parameters responding to three coils in the model.
Therefore there are three values of each parameter reflecting physical properties of coil are presented
below:
Symbol
SI unit
A 1  A2  A3
[m2]
Area enclosed by each turn of coil 1  2  3
l1  l2  l 3
[m]
Mean path length of the core of coil 1  2  3
N1  N2  N3
[-]
Number of turns in coil 1  2  3
µ1  µ2  µ3
i1  i2  i 3
Description
[H / m] Permeability of the core of coil 1  2  3
[A]
Current flowing through coil 1  2  3
Mathematical approach and taken steps are analogical to the one of single coil analysis. Therefore, the
resulting magnetic flux densities inside each coil, B, are given by:
B1 =
N1 ⋅ µ ⋅ i 1
l1
{43}
B2 =
N2 ⋅ µ ⋅ i2
l1
{44}
B3 =
N3 ⋅ µ ⋅ i3
l1
{45}
The flux linkages in each coil, λ, represent resulting flux density in each coil. Resulting flux density is a
total of densities created by all coils. Therefore each flux linkage from the resulting flux density is given
by:
λ1 = N 1 ⋅ (B1 + B2 + B3 ) ⋅ A1
{46}
λ2 = N 2 ⋅ (B1 + B 2 + B3 ) ⋅ A2
{47}
λ3 = N 3 ⋅ (B1 + B 2 + B3 ) ⋅ A3
{48}
45
As a result from equations {43} to {48}:
λ1 =
N12 ⋅ µ ⋅ A1 ⋅ i 1 N1 ⋅ N 2 ⋅ µ ⋅ A2 ⋅ i 2 N1 ⋅ N 3 ⋅ µ ⋅ i 3 ⋅ A3
+
+
l1
l2
l3
{46}
λ2 =
N 2 ⋅ N1 ⋅ µ ⋅ A1 ⋅ i 1 N 22 ⋅ µ ⋅ A2 ⋅ i 2 N 2 ⋅ N 3 ⋅ µ ⋅ i 3 ⋅ A3
+
+
l1
l2
l3
{47}
λ3 =
N 3 ⋅ N1 ⋅ µ ⋅ A1 ⋅ i 1 N 3 ⋅ N 2 ⋅ µ ⋅ A2 ⋅ i 2 N 32 ⋅ µ ⋅ i 3 ⋅ A3
+
+
l1
l2
l3
{48}
Assuming that parameters A, N, l and µ of each coil stay constant, there is a visible proportionality
between currents flowing through each coil and the resulting flux linkage. These proportionalities may
be defined as inductances:
λ1 = L11 ⋅ i1 + L12 ⋅ i 2 + L13 ⋅ i 3
{49}
λ2 = L21 ⋅ i1 + L22 ⋅ i 2 + L23 ⋅ i 3
{50}
λ3 = L31 ⋅ i 1 + L32 ⋅ i 2 + L33 ⋅ i 3
{51}
These three equations are equal to one matrix equation presented below:
 λ1  L11
λ  = L
 2   21
λ3  L31
L12
L22
L32
L13   i 1 
L 23  × i 2  ⇔ λ[] = L[] ⋅ i []
L 33  i 3 
{52}
where:
L ij =
Ni ⋅N j ⋅ µ ⋅ Aj
lj
; i , j ∈ {1,2,3}
{53}
From the equation {53}, one can see that mutual inductances between two coils are equal to each other
(Lij = Lji; i,j ∈ {1,2,3} ∧ {i ≠ j}). As a result, L matrix is symmetrical. This is an important conclusion for
further calculations.
46
Now taking the derivative with respect to time:
dλ [] dL[] di []
=
+
dt
dt
dt
{54}
Since L matrix is time-invariant (assuming there are not any changes in physical structure of analyzed
object):
dλ []
di []
= L[]
dt
dt
{55}
Based on Faraday's Law, one can perform analogical equations as described in subchapter 4.2.3:
“Analysis of the ideal model of a single coil - Mathematical background” It can be stated, that:
− ε [] = N [] ⋅
dφ [] dλ []
=
dt
dt
{56}
where:
ε 1 
ε [] = ε 2 
ε 3 
N1 
N [] = N 2 
N 3 
Symbol ε represents the electromotive force (EMF) matrix of each coil, and it is opposite to the voltage
matrix u across the inductors, giving:
u 1  L11 L12
di []
u [] = L[] ⋅
⇔ u 2  = L 21 L 22
dt
u 3  L 31 L 32
L13 
d
L 23  ⋅
dt
L 33 
 i1 
i 
 2
i 3 
{57}
This means that the voltage matrix across inductors is equal to the rate of change of the current matrix
in the inductors multiplied by a factor, the inductance matrix. This applies only to ideal coupled inductors
as mentioned above, which do not exist in the real world. One can see that there is total analogy in
results with single ideal coil. The only difference is that numerical equation is replaced by analogous
matrix equation.
47
3.3.2 Analysis of the realistic model of three coupled coils – model 4
PHYSICAL BACKGROUND
Real world coupled coils made of physical components present more than just pure inductances. It is
especially seen at high currents and frequencies flowing through these coils. Therefore, in order to
accurately reflect coils behavior, model 3 has to be modified by implementing new elements into it,
which would best reflect inductors response. The circuit symbol for an advanced model of a triple
coupled coils, shown below, is analogical to the one of a single coil:
uC1 = u1
iC1
C1
uL
i1
iL1
iRp1
iR1
L11c
uRp1
R
L31
i2
iL2
C2
iR2
L
La
uRp122
c
iRp2
1
R AC
1
2
L 12
uRA2C
uRD2
R DC
1
Rp
L 32
C
uC2= u2
1
`uL2
uRA11
C
R DC
1
p
iC2
L21
La
uRD11
2
2
L 23
uC3 = u3
iC3
C3
uL3
i3
iL3
iR3
L33c
La
uRp3
iRp3
uRA3C
uRD3
R DC
1
3
R AC
3
Rp
3
Fig. 4.3.2a. Electric representation of model 4 - realistic three coupled coils.
Inter-winding Capacitances:
C1 - reflects resulting insulation gap between windings in coil 1
C2 - reflects resulting insulation gap between windings in coil 2
C3 - reflects resulting insulation gap between windings in coil 3
Self Inductances:
L11 - reflects the voltage level on coil 1 made by changes of current flowing through coil 1
L22 - reflects the voltage level on coil 2 made by changes of current flowing through coil 2
L33 - reflects the voltage level on coil 3 made by changes of current flowing through coil 3
48
L 13
R AC
Mutual Inductances:
L12 - reflects the voltage level on coil 2 made by changes of current flowing through coil 1
L13 - reflects the voltage level on coil 3 made by changes of current flowing through coil 1
L21 - reflects the voltage level on coil 1 made by changes of current flowing through coil 2
L23 - reflects the voltage level on coil 3 made by changes of current flowing through coil 2
L31 - reflects the voltage level on coil 1 made by changes of current flowing through coil 3
L32 - reflects the voltage level on coil 2 made by changes of current flowing through coil 3
Alternating Current Resistances:
Parallel Resistances:
Direct Current Resistances:
RAC1 - reflects skin effect in coil 1
RP1 – reflects core losses in coil 1
RDC1 - reflects copper losses in coil 1
RAC2 - reflects skin effect in coil 2
RP2 – reflects core losses in coil 2
RDC2 - reflects copper losses in coil 2
RAC3 - reflects skin effect in coil 3
RP3 – reflects core losses in coil 3
RDC3 - reflects copper losses in coil 3
Model 4 presented above is a consolidation of model 2 with model 3. It consists of three realistic models
of single coupled coils. Simultaneously physical phenomena known from these models take place here.
Main purpose of creating this complex model is to show how all previously described physical
phenomena affect each other at the same time.
MATHEMATICAL BACKGROUND
In order to simplify calculations, these matrices have been created from:
-
Self parameters:
C1 
C [] = C 2 
C 3 
-
L 32
L13 
L23 
L33 
 R AC 1 
R AC [] = R AC 2 
R AC 3 
R DC 1 
R DC [] = R DC 2 
R DC 3 
i RA
 i RA1 
= i RA 2 
i RA 3 
i RD
 i RD 1 
= i RD 2 
i RD 3 
 i C1 
i C [] = i C 2 
i C 3 
Voltage drops on model elements:
 u L1 
u L [] = u L 2 
u L 3 
-
L12
L 22
Currents flowing through the model elements:
 i L1 
i L [] = i L 2 
i L 3 
-
L11
L[] = L 21
L 31
u RA
u RA1 
= u RA 2 
u RA 3 
u RD
u RD1 
= u RD 2 
u RD 3 
Sources of voltages powering each coil:
u C 1 
u C = u C 2 
u C 3 
u1 
u [] =  u 2 
 u 3 
49
The following presents mathematical relations between physical parameters of the cylindrical-shaped
coils based on the results of mathematical analysis of coil’s advanced model and figure 4.12. Since
there is more than one coil, matrix equations have to be created. They are replaced with single value
equations:
Kirchhoff's law #2 states that the sum of the currents entering any node equals the sum of the currents
leaving that node. Based on that statement and on the model of coils one can perform below matrix
equations:
i [] = i C [] + i RD []
{58}
i RD [] = i RA [] + i L []
{59}
Kirchhoff's law #1 states that the voltage changes around a closed path in a circuit add up to zero.
Based on that statement and on the model of coils one can perform below equations:
u [] = R DC [] ⋅ i RD [] + u L [] = u C []
{60}
u L [] = R AC [] ⋅ i RA []
{61}
From the previous analysis of the ideal inductor and from the analogical analysis of the ideal capacitor
there can be formulated these equations:
i C [] = C[] ⋅
du C []
⇒ u C [] = C[] −1 ⋅ ∫ i C [] ⋅ dt
dt
{62}
u L [] = L[] ⋅
di L []
dt
{63}
⇒ i L [] = L[] −1 ⋅ ∫ u L [] ⋅ dt
50
MATHEMATICAL ANALYSIS 5 – circuit with ideal voltage source
u 1 (t)
I
III
i2
L
SWITCH 2
II
L
SWITCH 3
+
L21 = L12
L31 = L13
22
L23 = L32
u L3
iL3
i3
11
u L2
iL2
+
u 3 (t)
L
SWITCH 1
+
u 2 (t)
uL1
iL1
i1
33
Fig. 4.3.2b. Electric circuit model with ideal voltage source for the mathematical analysis of the model 3.
Based on previously stated laws and formulas, according to the circuit model, one can write these
equations:
I:
u 1 = u R 1 + u L1
{64}
II:
u2 = uR 2 + u L2
{65}
III:
u 3 = u R 3 + u L3
{66}
Equations {21}, {22} and {23} also can be presented in this way:
di 1
di 2
di 3

 u1 = R1 ⋅ i 1 + L11 ⋅ dt + L21 ⋅ dt + L31 ⋅ dt

di 1
di 2
di
+ L32 ⋅ 3
u 2 = R 2 ⋅ i 2 + L12 ⋅ + L22 ⋅
dt
dt
dt

di
di
di
1
2
u = R ⋅ i + L ⋅ + L ⋅
+ L33 ⋅ 3
3 3
13
23
 3
dt
dt
dt
{67}
{68}
{69}
This is a set of three first-order differential equations. Its set of solutions is provided below:
R
R
R
− 1 ⋅t
− 1 ⋅t
− 1 ⋅t

L11
L21
L31
+ B1 ⋅ e
+ C1 ⋅ e
+ i 1steady (t )
 i 1 (t ) = A1 ⋅ e
R2
R2
R2

− ⋅t
− ⋅t
− ⋅t

L
L
L
i 2 (t ) = A2 ⋅ e 12 + B 2 ⋅ e 22 + C 2 ⋅ e 32 + i 2 steady (t )
R
R
R

− 3 ⋅t
− 3 ⋅t
− 3 ⋅t
i 3 (t ) = A3 ⋅ e L13 + B 3 ⋅ e L23 + C 3 ⋅ e L33 + i 3 steady (t )

{70}
{71}
{72}
A,B,C parameters may then be calculated depending on switching operations:
A 1 = ( i 1initial − i 1steady (t ic )) ⋅ e
B1 = i 2 initial ⋅ e
C 1 = i 3 initial ⋅ e
R1
⋅t ic
L 21
R1
⋅ t ic
L 31
R
1
L
11
⋅t
ic
A2 = i 1initial ⋅ e
R2
R2
⋅t ic
L12
A 2 = i 1 initial ⋅ e L 12
R2
B2 = (i 2initial − i2steady(t ic )) ⋅ e L22
C 2 = i 3 initial ⋅ e
R3
⋅t ic
L 32
51
⋅t ic
R3
B 3 = i 3 initial ⋅ e L 23
⋅t ic
⋅t ic
C 3 = ( i 3 initial − i 3 steady ( t ic )) ⋅ e
R3
⋅t ic
L 33
Based on example 2 and 3, one can write matrix equation:
u L [] + u RD [] = u C []
{73}
where:
u L = L[] ⋅
di L []
dt
uRD [] = RDC ⋅ i L [] +
RDC
di []
⋅ L[]⋅ L
RAC
dt
u C [] = u [] − u C [](t ic )
u is the matrix function of ideal voltage source in time domain. As a result from {28}, {29} , {30} and {31}:
P [] ⋅ L[] ⋅
di L []
u []
+ i L [] =
dt
R DC []
{74}
where:
P [] = [P1 P2
 R + R DC 1
P3 ] =  AC1
 R AC 1 ⋅ R DC 1
R AC 2 + R DC 2
R AC 2 ⋅ R DC 2
R AC 3 + R DC 3 

R AC 3 ⋅ R DC 3 
{75}
Matrix equation can be presented as a set of first-order ordinary differential equations:
di
di
di 
u

P1 ⋅  L11 ⋅ 1 + L 21 ⋅ 2 + L31 ⋅ 3  + i 1 = 1
dt
dt
dt 
R DC 1

{76}
di
di
di 
u

P2 ⋅  L12 ⋅ 1 + L 22 ⋅ 2 + L32 ⋅ 3  + i 2 = 2
dt
dt
dt 
R DC 2

{77}
di 
u
di
di

P3 ⋅  L13 ⋅ 1 + L 23 ⋅ 2 + L 33 ⋅ 3  + i 3 = 3
dt
dt
dt 
R DC 3

{78}
Its solution will be:
P
P
P

− 1 ⋅t
− 1 ⋅t
− 1 ⋅t
L11
L21
L31
+ B1 ⋅ e
+ C1 ⋅ e
+ i 1steady (t )
 i 1 (t ) = A1 ⋅ e
P2
P2
P2

− ⋅t
− ⋅t
− ⋅t

L
L
L
i 2 (t ) = A2 ⋅ e 12 + B2 ⋅ e 22 + C 2 ⋅ e 32 + i 2 steady (t )
P
P
P

− 3 ⋅t
− 3 ⋅t
− 3 ⋅t
i 3 (t ) = A3 ⋅ e L13 + B3 ⋅ e L23 + C 3 ⋅ e L33 + i 3 steady (t )

{79}
{80}
{81}
A,B,C parameters may then be calculated depending on switching operations:
A1 = ( i1initial − i1steady (t ic )) ⋅ e
P1
B1 = i 2 initial ⋅ e L21
C1 = i 3 initial ⋅ e
⋅t ic
P1
⋅t
L 31 ic
P1
⋅t ic
L11
A2 = i1initial ⋅ e
P3
P2
⋅t ic
L12
A3 = i 1initial ⋅ e L13
B2 = ( i 2 initial − i 2steady (t ic )) ⋅ e
C 2 = i 3 initial ⋅ e
P3
⋅t
L32 ic
52
P2
⋅t ic
L22
B 3 = i 3 initial ⋅ e
⋅t ic
P3
⋅t ic
L 23
P3
ć
C 3 = ( i 3 initial − i 3 steady ( t ic )) ⋅ e L 33
⋅t ic
3.4 Analysis of nine coils on three phases – final model
Coil A1
LA1
LC1
LA2
LA3 LB3 LB2
LB1
LC3
LC2
Fig. 4.4a. Simplified sketch of a 3-phase line reactor. Red dashed line shows analyzed object.
Final model consists of nine single coils wrapped around ferromagnetic core. Core itself consists of
three legs with three coils on each leg. These coils correspond to one phase, which makes it in total a
three-phase device. Coil’s terminals of each phase from one side are connected together, and from the
other are connected to switches which open and close at the same time for each phase. Additional
parameters that are taken into account in this model are ground capacitances and capacitances
between coils. Capacitance itself is an ability of a body to hold an electrical charge [19]. This parameter
plays an important role in very high frequencies, when current starts flowing through them instead of
flowing directly through inductive parts. As a result one may divide all capacitances into three groups:
- self capacitances, which are presented on Fig. 4.2.2a as inter-winding capacitances, (●)
- mutual capacitances of each pair of coils, (●)
- resulting ground capacitances of each coil, (●)
- resulting capacitances between each coil an a core, (●)
A1
A2
A3
B1
C1
B2
C2
B3
C3
Fig 4.4b Capacitances which occur in model 5
53
All capacitances can be modeled as capacitors and can be calculated if the distance between coils and
the dielectric properties of the insulator between them are known. A proper equation is derived as
follows:
C = εr ⋅ε0 ⋅
A
d
{82}
where: A - area of each coil [m2]
ε0 - permittivity of free space, where ε0= 8.854⋅10-12 F/m
εr - the relative static permittivity of the material between coils [-],
d – distance between coils [m]
From the equation above one may conclude, that the most significant role will play capacitances
between adjacent coils and between coils and a core, since a distance (d parameter) will be the lowest
in these groups.
Finally, resulting linear model can be presented with its parameters responsible for modeling various
physical phenomena described previously. Due to complexity of the resulting model, next step is to
implement it into computer software that is capable of providing simulations and analysis of its
response.
54
55
4 Simplified Model
4.1 Brief
In order to analyze model’s behavior in certain conditions and states, it is essential to implement it into
computer software capable of calculating complex mathematical equations explained earlier in
„Theoretical Analysis” chapter. By doing so, one may simulate it and obtain specific and necessary
results needed for further research. Simulations done numerically require though specific values and
functions of all parameters that the model includes.
For this reason there are made measurements of all self parameters of the line reactor, so that there is
a possibility to put its results into computer software. Numerical modeling is provided in many software
available on the market.
This chapter provides MATLAB Simulink model description created to analyze the 3-phase series
reactor. Model is comprised of blocks from the SimPowerSystems library of the Simulink analysis
application in MATLAB. MATLAB software is chosen mainly for its user-friendly interface and wide
range of possibilities to perform simulations in different conditions. Whole model consists of all
parameters mentioned in previous chapter except for paralell resistances, which are neglected due to
high complexity and non-linearity of core losses.
4.2 PowerSys Library Block Descriptions
DC resistance
DC resistance of each coil is modeled simply by placing nine Series R Branch blocks and putting there
RDC values obtained from measurements.
AC resistance
AC resistance is non-linearly dependant of frequency. Therefore its function first needs to be linearized
to implement it into MATLAB model since the model is said to be linear. Linearization is done by
performing calculations of RAC values from equation {8} in “Theoretical Analysis” chapter. Calculations
for each coil are made for frequency range from 60Hz to 2000Hz with chosen step of 20Hz (it makes
1000 iterations for a single coil). Next, through obtained RAC values one may linearize its characteristics
and gain linear functions of it with sufficient approximation. Linearization is done using Euler’s method.
56
An example of linearized function for A1 coil is presented below:
AC Resistance in function of frequency - coil A1
0,6
0,5
Rac [uOHM]
y = 2,207E-05x + 8,422E-06
0,4
0,3
0,2
0,1
0,0
60
2060
4060
6060
8060
10060
12060
14060
16060
18060
f [Hz]
Fig. 5.2a. AC resistance in function of frequency (● – discrete plot ; ● - approximation plot)
Calculated RAC functions with respect to frequency are as follows:
-
coil A1: R AC = 2,207 ⋅ 10 −5 + 8,442 ⋅ 10 −6 ⋅ f
-
coil A2: R AC = 2,207 ⋅ 10 −5 + 8,442 ⋅ 10 −6 ⋅ f
-
coil A3: R AC = 2,878 ⋅ 10 −5 + 1,319 ⋅ 10 −5 ⋅ f
-
coil B1: R AC = 2,639 ⋅ 10 −5 + 1,128 ⋅ 10 −5 ⋅ f
-
coil B2: R AC = 3,155 ⋅ 10 −5 + 1,566 ⋅ 10 −5 ⋅ f
-
coil B3: R AC = 3,155 ⋅ 10 −5 + 1,566 ⋅ 10 −5 ⋅ f
-
coil C1: R AC = 2,639 ⋅ 10 −5 + 1,128 ⋅ 10 −5 ⋅ f
-
coil C2: R AC = 2,027 ⋅ 10 −5 + 7,402 ⋅ 10 −6 ⋅ f
-
coil C3: R AC = 2,406 ⋅ 10 −5 + 9,668 ⋅ 10 −6 ⋅ f
Modeling these equations in MATLAB requires to use Series RL Branch blocks, where resistance and
inductance values are equal to:
R=b
L=
a
2 ⋅π
where:
a,b – designated constants of calculated RAC function: RAC = a⋅f +b
In this manner nine RL Branch blocks provide resulting linear AC resistances of each coil.
57
Inductance
Resulting inductance is obtained by using Mutual Coils block and filling in inductance matrix values of
measured self- and mutual inductances. Specified number of windings is set to nine, which reflects nine
coils in total. Inductance matrix has 81 components and is symmetrical. Resistance matrix is set to zero.
As a result, there is one block with nine inputs and nine outputs for all branches.
Inter-winding capacitance
Inter-winding capacitances are simply made from parallel C Branch blocks connecting them across each
branch. There are nine C Branch blocks in total.
Ground capacitance
Ground capacitances are simply added by placing pararallel C Branch blocks on the left and right side
of the inductor. There are eighteen blocks in total, two for each winding. The other node of block is
grounded with Ground block.
After implementing all values into mentioned blocks and connecting them accordingly to the theoretical
model, one may provide necessary simulations using additional blocks such as specific source-, switchand measurement blocks. After all blocks have been added and connected, acquired MATLAB model is
ready to start simulation analysis in different conditions.
58
5 Computer Simulation
5.1 Brief
Creation of MATLAB model provides possibilities to simulate certain states and conditions, which can be
used to reflect how model will react and respond. From simulation results one may take necessary
conclusions regarding many technical aspects of the conditions that have to be fulfilled, i.e by
measuring current levels flowing through the reactor at certain voltage one may conclude what kind of
insulation is necessary to make reactor working correctly, or how high currents can be . This and other
aspects have to be taken into account while installing inductor in the wind turbine.
Simulation results have been made in certain conditions and states, so that line reactor’s behavior may
be analyzed from different possible situations. .These conditions have been divided in cases presented
below:
STEADY STATE
- Case 1: All switches are closed
Simulation is made in order to show how model behaves when it is simply attached to the source
and how high are currents flowing through each phase and single branch.
- Case 2: Switches A1, A2, A3 are closed
Simulation is made in order to show how currents in single phase of the analyzed object affect
voltages in other phases due to mutual coupling phenomenon.
- Case 3: Switch A1 is closed
Simulation is made in order to show how current in single branch of the analyzed object affects
voltages in other branches due to mutual coupling phenomenon.
TRANSIENT STATE
- Case 4: All switches close at the same time
Simulation is made in order to show time constant and shape of currents and voltages in each
phase when all branches affect each other due to mutual coupling phenomenon. In addition results
may be assessed based on “theoretical analysis” chapter.
- Case 5: All switches open at the same time
Simulation is made in order to show time constant and shape of currents and voltages in each
phase when all branches affect each other due to mutual coupling phenomenon. In addition results
may be assessed based on “theoretical analysis” chapter.
60
- Case 6: Switches A1, A2, A3 close at the same time
Simulation is made in order to show how current’s transient response in single phase affects
voltages in other branches due to mutual coupling phenomenon.
- Case 7: Switches A1, A2, A3 open at the same time
Simulation is made in order to show how current’s transient response in single phase affects
voltages in other branches due to mutual coupling phenomenon.
- Case 8: Switch A1 closes at certain time
Simulation is made in order to show how current in single branch of the analyzed object affects
voltages in other phases due to mutual coupling phenomenon.
- Case 9: Switch A1 opens at certain time
Simulation is made in order to show how current in single branch of the analyzed object affects
voltages in other phases due to mutual coupling phenomenon.
- Case 10: Switching mode – normal operation
Simulation is made in order to show currents and voltages under expected operation state in 3phase line reactor when all switching devices are fully operational. From here breaker models are
replaced by IGBT devices to gain more accurate results of the functionality of the model.
- Case 11: Switching mode – A phase fault at certain time
Simulation is made in order to show currents and voltages under normal operation state in 3-phase
line reactor affected by single phase drop.
- Case 12 Switching mode – switch IGBT A1 fault at certain time
Simulation is made in order to show currents and voltages under normal operation state in 3-phase
line reactor affected by single branch drop.
- Case 13: Switching mode – normal operation with random switching time delays
Simulation is made in order to show currents and voltages under expected operation state in 3phase line reactor when all switching devices are fully operational including additional phenomenon
of time lags between switching periods. Results from this simulation are the most important for the
manufacturer.
61
5.2 Steady Conditions
5.2.1 Case 1: All switches are closed
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase A:
- amplitude: 800 V
- phase: -1200
- frequency: 60 Hz
Voltage phase A:
- amplitude: 800 V
- phase:
1200
- frequency: 60 Hz
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block(Simulink library)
Table 6.2.1. Steady values of currents and voltages from simulation process.
Phase Currents
Phase Voltages
phase
RMS value [A]
phase [0]
RMS value [V]
phase [0]
A
2,145E+04 -86.72°
5,657E+02 0.02°
B
1,464E+04 152.18°
5,657E+02 -119.99°
C
2,126E+04 34.23°
5,657E+02 120.02°
branch
A1
A2
A3
B1
B2
B3
C1
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
7,57E+03
6,862E+03
-86,53°
-86,21°
5,657E+02
0.02°
5,657E+02
0.02°
7,014E+03
-87,43°
5,657E+02
0.02°
6,418E+03
3,190E+03
155,13°
143,71°
5,657E+02
-119.98°
5,657E+02
-119.98°
5,072E+03
153,77°
5,657E+02
-119.98°
7,205E+03
34,22°
5,657E+02
120.02°
62
phase [0]
C2
C3
7,152E+03
34,47°
5,657E+02
120.02°
6,899E+03
33,99°
5,657E+02
120.02°
Fig. 6.2.1.a. Phase currents [A] and voltages [V] (● – A phase  ● – B phase  ● – C phase).
Fig. 6.2.1.b. Branch currents [A] and voltages [V] (● – A1 branch  ● – A2 branch  ● – A3 branch).
Fig. 6.2.1.b. Branch currents [A] and voltages [V] (● – B1 branch  ● – B2 branch  ● – B3 branch).
Fig. 6.2.1.c. Branch currents [A] and voltages [V] (● – C1 branch  ● – C2 branch  ● – C3 branch).
63
5.2.2 Case 2: Switches A1, A2, A3 are closed
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
- amplitude: 800 V
- phase: -1200
- frequency: 60 Hz
Voltage phase C:
- amplitude: 800 V
- phase:
1200
- frequency: 60 Hz
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library)
Table 6.2.2. Steady values of currents and voltages from simulation process.
Phase Currents
Phase Voltages
phase
0
RMS value [A]
phase [ ]
RMS value [V]
phase [0]
2,449E+04
-85,84°
5,657E+02
0,02°
A
0,000E+00
0,00°
5,657E+02
-120,00°
B
0,000E+00
0,00°
5,657E+02
120,00°
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
8,819E+03
-84,78°
5,657E+02
0,03°
7,35E+03
8,325E+03
-87,18°
-85,79°
5,657E+02
0,03°
5,657E+02
0,03°
0,000E+00
0,00°
1,127E+02
-175,41°
0,000E+00
0,00°
1,154E+02
-176,06°
0,000E+00
0,00°
1,269E+02
-175,98°
0,000E+00
0,00°
8,095E+01
-175,65°
0,000E+00
0,00°
6,628E+01
-175,79°
0,000E+00
0,00°
8,271E+01
-175,88°
64
Fig. 6.2.1.a. Phase currents [A] and voltages [V] (● – A phase  ● – B phase  ● – C phase).
Fig. 6.2.2.b. Branch currents [A] and voltages [V] (● – A1 branch  ● – A2 branch  ● – A3 branch).
Fig. 6.2.2.c. Branch currents [A] and voltages [V] (● – B1 branch  ● – B2 branch  ● – B3 branch).
Fig. 6.2.2.c. Branch currents [A] and voltages [V] (● – C1 branch  ● – C2 branch  ● – C3 branch).
5.2.3 Case 3: Switch A1 is closed
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
- amplitude: 800 V
- phase: -1200
- frequency: 60 Hz
Voltage phase C:
- amplitude: 800 V
- phase:
1200
- frequency: 60 Hz
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library)
Table 6.2.3. Steady values of currents and voltages from simulation process.
phase
A
B
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Phase Currents
Phase Voltages
phase [0]
RMS value [A]
phase [0]
RMS value [V]
1,333E+04
-83,11°
5,657E+02
0,01°
0,000E+00
0,00°
5,657E+02
-120,00°
0,000E+00
0,00°
5,657E+02
120,00°
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
1,333E+04
-83,11°
5,657E+02
0,01°
0,000E+00
0,00°
2,249E+02
6,89°
0,000E+00
0,00°
1,088E+02
6,89°
0,000E+00
0,00°
1,128E+02
-173,11°
0,000E+00
0,00°
4,868E+01
-173,11°
0,000E+00
0,00°
2,175E+01
-173,11°
0,000E+00
0,00°
3,229E+01
-173,11°
0,000E+00
0,00°
6,015E+01
-173,11°
0,000E+00
0,00°
3,671E+01
-173,11°
RMS value [V]
66
phase [0]
Fig. 6.2.3.a. Phase currents [A] and voltages [V] (● – A phase  ● – B phase  ● – C phase)
Fig. 6.2.3.b. Branch currents [A] and voltages [V] (● – A1 branch  ● – A2 branch  ● – A3 branch).
Fig. 6.2.3.b. Branch currents [A] and voltages [V] (● – B1 branch  ● – B2 branch  ● – B3 branch).
Fig. 6.2.3.c. Branch currents [A] and voltages [V] (● – C1 branch  ● – C2 branch  ● – C3 branch).
5.3 Transient Conditions
5.3.1 Case 4: All switches close at the same time
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
- amplitude: 800 V
- phase: -1200
- frequency: 60 Hz
All breakers close at time: 2 ms
Voltage phase C:
- amplitude: 800 V
- phase:
1200
- frequency: 60 Hz
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library) and Timer block (SimPowerSystems library)
Table 6.3.1. Steady values of currents and voltages before switching operation from simulation process.
Phase Currents
Phase Voltages
phase
RMS value [A]
phase [0]
RMS value [V]
phase [0]
0,000E+03
00,00°
5,659E+02
0,00°
A
0,000E+03
00,00°
5,659E+02
-120,00°
B
0,000E+03
00,00°
5,659E+02
120,00°
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
68
Fig. 6.3.1.a. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.1.b. B-phase current [A] and voltage [V] in transient states.
Fig. 6.3.1.c. C-phase current [A] and voltage [V] in transient states.
Fig. 6.3.1.d. Phase currents [A] in steady states.
69
Fig. 6.3.1.e. A1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.f. A2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.g. A3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.h. A-branch currents [A] in steady states.
70
Fig. 6.3.1.i. B1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.j. B2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.k. B3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.l. B-branch currents [A] in steady states.
71
Fig. 6.3.1.m. C1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.n. C2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.o. C3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.1.p. C-branch currents [A] in steady states.
72
5.3.2 Case 5: All switches open at the same time
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
- amplitude: 800 V
- phase: -1200
- frequency: 60 Hz
All breakers open at time: 2 ms
Voltage phase C:
- amplitude: 800 V
- phase:
1200
- frequency: 60 Hz
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library) and Timer block (SimPowerSystems library)
Table 6.3.2. Steady values of currents and voltages before switching operation from simulation process.
phase
A
B
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Phase Currents
Phase Voltages
RMS value [A]
phase [0]
RMS value [V]
phase [0]
2,145E+04
-86,72°
5,657E+02
0,02°
1,464E+04
152,18°
5,657E+02
-119,99°
2,126E+04
34,23°
5,657E+02
120,02°
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
7,573E+03
-86,53°
5,657E+02
0,02°
6,862E+03
-86,21°
5,657E+02
0,02°
7,014E+03
-87,43°
5,657E+02
0,02°
6,418E+03
155,13°
5,657E+02
-119,98°
3,190E+03
143,71°
5,657E+02
-119,98°
5,072E+03
153,77°
5,657E+02
-119,98°
7,205E+03
34,22°
5,657E+02
120,02°
7,152E+03
34,47°
5,657E+02
120,02°
6,899E+03
33,99°
5,657E+02
120,02°
73
Fig. 6.3.2.a. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.2.b. B-phase current [A] and voltage [V] in transient states.
Fig. 6.3.2.c. C-phase current [A] and voltage [V] in transient states.
Fig. 6.3.2.d. Phase currents [A] in steady states.
74
Fig. 6.3.2.e. A1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.f. A2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.g. A3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.h. A-branch voltages [V] in steady states.
75
Fig. 6.3.2.i. B1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.j. B2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.k. B3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.l. B-branch voltages [V] in steady states.
76
Fig. 6.3.2.m. C1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.n. C2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.o. C3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.2.p. C-branch voltages [V] in steady states.
77
5.3.3 Case 6: Switches A1, A2, A3 close at the same time
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
- phase: -1200
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
A-phase breakers close after 2 ms delay
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library) and Timer block (SimPowerSystems library)
Table 6.3.3. Steady values of currents and voltages before switching operation from simulation process.
phase
A
B
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Phase Currents
Phase Voltages
RMS value [A]
phase [0]
RMS value [V]
phase [0]
0,000E+03
00,00°
5,659E+02
0,00°
0,000E+03
00,00°
5,659E+02
-120,00°
0,000E+03
00,00°
5,659E+02
120,00°
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
78
Fig. 6.3.3.a. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.3.b. B-phase current [A] and voltage [V] in transient states.
Fig. 6.3.3.c. C-phase current [A] and voltage [V] in transient states.
Fig. 6.3.3.d. Phase currents [A] in steady states.
79
Fig. 6.3.3.e. A1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.3.f. A2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.3.g. A3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.3.h. A-branch currents [A] in steady states.
80
Fig. 6.3.3.i. B1-branch voltage [V] in transient states  steady states.
Fig. 6.3.3.j. B2-branch voltage [V] in transient states  steady states.
Fig. 6.3.3.k. B3-branch voltage [V] in transient states  steady states.
Fig. 6.3.3.l. C1-branch voltage [V] in transient states  steady states.
Fig. 6.3.3.m. C2-branch voltage [V] in transient states  steady states.
Fig. 6.3.3.n. C3-branch voltage [V] in transient states  steady states.
81
5.3.4 Case 7: Switches A1, A2, A3 open at the same time
Input data for MATLAB model:
Voltage phase A:
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
- amplitude: 800 V
- phase:
00
- phase: -1200
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
- frequency: 60 Hz
A-phase breakers open after 2 ms delay when current reaches zero value
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library) and Timer block (SimPowerSystems library)
Table 6.3.4. Steady values of currents and voltages before switching operation from simulation process.
Phase Currents
Phase Voltages
phase
RMS value [A]
phase [0]
RMS value [V]
phase [0]
2,449E+04
-85,84°
5,657E+02
0,02°
A
0,000E+00
0,00°
5,657E+02
-120,00°
B
0,000E+00
0,00°
5,657E+02
120,00°
C
branch
Branch Currents [A]
RMS value [A]
A1
A2
A3
B1
B2
B3
C1
C2
C3
phase
Branch Voltages [V]
[0]
RMS value [V]
phase [0]
8,819E+03
-84,78°
5,657E+02
0,03°
7,347E+03
-87,18°
5,657E+02
0,03°
8,325E+03
-85,79°
5,657E+02
0,03°
0,000E+03
00,00°
1,127E+02
-175,41°
0,000E+03
00,00°
1,154E+02
-176,06°
0,000E+03
00,00°
1,269E+02
-175,98°
0,000E+03
00,00°
8,095E+01
-175,65°
0,000E+03
00,00°
6,628E+01
-175,79°
0,000E+03
00,00°
8,271E+01
-175,88°
82
Fig. 6.3.4.a. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.4.b. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.4.c. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.4.d. Phase currents [A] in steady states.
83
Fig. 6.3.4.e. A1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.4.f. A2-branch current [A] and voltage [V] in transient states.
Fig. 6.3.4.g. A3-branch current [A] and voltage [V] in transient states.
Fig. 6.3.4.h. A-branch voltages [V] in steady states.
84
Fig. 6.3.4.i. B1-branch voltage [V] in transient states  steady states.
Fig. 6.3.4.j. B2-branch voltage [V] in transient states  steady states.
Fig. 6.3.4.k. B3-branch voltage [V] in transient states  steady states.
Fig. 6.3.4.l. C1-branch voltage [V] in transient states  steady states.
Fig. 6.3.4.m. C2-branch voltage [V] in transient states  steady states.
Fig. 6.3.4.o. C3-branch voltage [V] in transient states  steady states.
5.3.5 Case 8: Switch A1 closes at certain time
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
- phase: -1200
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
A1 breaker closes at time after 2 ms delay
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library) and Timer block (SimPowerSystems library)
Table 6.3.5. Steady values of currents and voltages before switching operation from simulation process.
Phase Currents
Phase Voltages
phase
RMS value [A]
phase [0]
RMS value [V]
phase [0]
0,000E+03
00,00°
5,659E+02
0,00°
A
0,000E+03
00,00°
5,659E+02
-120,00°
B
0,000E+03
00,00°
5,659E+02
120,00°
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
0,000E+03
00,00°
86
Fig. 6.3.5.a. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.5.b. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.5.c. A-phase current [A] and voltage [V] in transient states.
Fig. 6.3.5.d. A-phase currents [A] in steady states.
87
Fig. 6.3.5.e. A1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.5.f. A1-branch current [A] in steady state.
Fig. 6.3.5.g. A2-branch voltage [V] in transient states  steady states.
Fig. 6.3.5.h. A3-branch voltage [V] in transient states  steady states.
Fig. 6.3.5.i. B1-branch voltage [V] in transient states  steady states.
Fig. 6.3.5.j. B2-branch voltage [V] in transient states  steady states.
88
Fig. 6.3.5.k. B3-branch voltage [V] in transient states  steady states.
Fig. 6.3.5.l. C1-branch voltage [V] in transient states  steady states.
Fig. 6.3.5.m. C2-branch voltage [V] in transient states  steady states.
Fig. 6.3.5.n. C3-branch voltage [V] in transient states  steady states.
89
5.3.6 Case 9: Switch A1 opens at certain time
Input data for MATLAB model:
Voltage phase A:
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
- amplitude: 800 V
- phase:
00
- phase: -1200
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
- frequency: 60 Hz
A1 breaker opens after 2 ms delay when current reaches zero value
Breaker parameters:
- Internal resistance [Ω]:
1E-06
- Snubber resistance [Ω]:
infinite
- Snubber capacitance [F]:
infinite
Breaker controllers: Constant block (Simulink library) and Timer block (SimPowerSystems library)
Table 6.3.6. Steady values of currents and voltages before switching operation from simulation process.
Phase Currents
Phase Voltages
phase
0
RMS value [A]
phase [ ]
RMS value [V]
phase [0]
1,333E+04
-83,11°
5,657E+02
0,01°
A
0,000E+03
00,00°
5,657E+02
-120,00°
B
0,000E+03
00,00°
5,657E+02
120,00°
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
1,333E+04
-83,11°
5,657E+02
0,01°
0,000E+03
00,00°
2,249E+02
6,89°
0,000E+03
00,00°
1,088E+02
6,89°
0,000E+03
00,00°
1,128E+02
-173,11°
0,000E+03
00,00°
4,868E+01
-173,11°
0,000E+03
00,00°
2,175E+01
-173,11°
0,000E+03
00,00°
6,015E+01
-173,11°
0,000E+03
00,00°
3,671E+01
-173,11°
0,000E+03
00,00°
3,229E+01
-173,11°
90
Fig. 6.3.6.a. A1-branch current [A] and voltage [V] in transient states.
Fig. 6.3.6.b. A1-branch voltage VA] in steady state.
Fig. 6.3.6.c. A2-branch voltage [V] in transient states  steady states.
Fig. 6.3.6.d. A3-branch voltage [V] in transient states  steady states.
Fig. 6.3.6.e. B1-branch voltage [V] in transient states  steady states.
Fig. 6.3.6.f. B2-branch voltage [V] in transient states  steady states.
91
Fig. 6.3.6.g. B3-branch voltage [V] in transient states  steady states.
Fig. 6.3.6.h. C1-branch voltage [V] in transient states  steady states.
Fig. 6.3.6.i. C2-branch voltage [V] in transient states  steady states.
Fig. 6.3.6.j. C3-branch voltage [V] in transient states  steady states.
92
5.3.7 Case 10: Switching mode – normal operation
A-phase IGBT
B-phase IGBT
C-phase IGBT
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
0
- phase: -120
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
Switching frequency is set to 2,5 kHz
Normal operation starts with 1,2 ms delay to simulation run.
IGBT parameters:
- Internal resistance [Ω]:
1E-03
- Internal inductance [H]:
infinite
- Forward voltage [V]:
1E+00
- Current 10% fall time [s]:
1E-06
- Current tail time [s]:
2E-06
- Initial current [A]:
0
- Snubber resistance [Ω]:
1E+05
- Snubber capacitance [F]:
infinite
IGBT controllers: controllers are made by creating subsystem that provides switching frequency shown
o diagram above.
Fig. 6.3.7.a. Switch controller diagram used for simulations
93
Table 6.3.7. Steady values of currents and voltages before switching operation from simulation process.
Phase Currents
Phase Voltages
phase
0
RMS value [A]
phase [ ]
RMS value [V]
phase [0]
1,697E-02
-0,00°
5,657E+02
0,00°
A
1,697E-02
-120,00°
5,657E+02
-120,00°
B
1,697E-02
120,00°
5,657E+02
120,00°
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
5,657E-03
-0,00°
4,463E-04
88,86°
5,657E-03
-0,00°
4,631E-04
89,60°
5,657E-03
-0,00°
4,677E-04
90,39°
5,657E-03
-120,00°
5,781E-04
-32,51°
5,657E-03
-120,00°
6,723E-04
-32,13°
5,657E-03
-120,00°
6,393E-04
-32,27°
5,657E-03
120,00°
4,590E-04
-156,24°
5,657E-03
120,00°
4,680E-04
-156,87°
5,657E-03
120,00°
4,757E-04
-157,12°
Fig. 6.3.7.b. A –phase currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.c. B –phase currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.d. C –phase currents [A] and voltages [V] in transient states steady states.
94
Fig. 6.3.7.e. A1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.f. A2-branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.g. A3–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.h. B1-branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.i. B2-branch currents [A] and voltages [V] in transient states steady states.
95
Fig. 6.3.7.j. B3-branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.k. C1-branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.7.l. C2-branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.6.m. C3-branch voltage [V] in transient states  steady states.
96
5.3.8 Case 11: Switching mode – A phase fault at certain time
A-phase IGBT
B-phase IGBT
C-phase IGBT
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
IGBT parameters:
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
- phase: -1200
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
Switching frequency is set to 2,5 kHz
Normal operation starts with 1,2 ms delay to simulation run.
A phase drop occurs at 5 ms from simulation run.
IGBT parameters:
- Internal resistance [Ω]:
1E-03
- Internal inductance [H]:
infinite
- Forward voltage [V]:
1E+00
- Current 10% fall time [s]:
1E-06
- Current tail time [s]:
2E-06
- Initial current [A]:
0
- Snubber resistance [Ω]:
1E+05
- Snubber capacitance [F]:
infinite
IGBT controllers: controllers are made by creating subsystem that provides switching frequency shown
on diagram above.
Fig. 6.3.8.a. Switch controller diagram used for simulations
97
Table 6.3.8. Steady values of currents and voltages before switching operation from simulation process.
Phase Currents
Phase Voltages
phase
RMS value [A]
phase [0]
RMS value [V]
phase [0]
1,697E-02
-0,00°
5,657E+02
0,00°
A
1,697E-02
-120,00°
5,657E+02
-120,00°
B
1,697E-02
120,00°
5,657E+02
120,00°
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
5,657E-03
-0,00°
4,463E-04
88,86°
5,657E-03
-0,00°
4,631E-04
89,60°
5,657E-03
-0,00°
4,677E-04
90,39°
5,657E-03
-120,00°
5,781E-04
-32,51°
5,657E-03
-120,00°
6,723E-04
-32,13°
5,657E-03
-120,00°
6,393E-04
-32,27°
5,657E-03
120,00°
4,590E-04
-156,24°
5,657E-03
120,00°
4,680E-04
-156,87°
5,657E-03
120,00°
4,757E-04
-157,12°
Fig. 6.3.8.b. A –phase currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.c. B –phase currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.d. C –phase currents [A] and voltages [V] in transient states steady states.
98
Fig. 6.3.8.e. A1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.f. A2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.g. A3–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.h. B1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.i. B2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.j. B3–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.k. C1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.l. C2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.8.m. C3–branch currents [A] and voltages [V] in transient states steady states.
100
5.3.9 Case 12: Switching mode – switch IGBT A1 fault at certain time
A1 IGBT
A2, A3 IGBT
B-phase IGBT
C-phase IGBT
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
IGBT parameters:
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
- phase: -1200
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
Switching frequency is set to 2,5 kHz
Normal operation starts with 1,2 ms delay to simulation run.
A1 IGBT fault occurs at 5 ms from simulation run.
IGBT parameters:
- Internal resistance [Ω]:
1E-03
- Internal inductance [H]:
infinite
- Forward voltage [V]:
1E+00
- Current 10% fall time [s]:
1E-06
- Current tail time [s]:
2E-06
- Initial current [A]:
0
- Snubber resistance [Ω]:
1E+05
- Snubber capacitance [F]:
infinite
IGBT controllers:
● Controller for A1 IGBT is analogous to one presented in case 11.
● Controllers for other transistors are analogous to ones presented in case 10.
101
Table 6.3.9. Steady values of currents and voltages before switching operation from simulation process.
phase
A
B
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Phase Currents
Phase Voltages
RMS value [A]
phase [0]
RMS value [V]
phase [0]
1,697E-02
-0,00°
5,657E+02
0,00°
1,697E-02
-120,00°
5,657E+02
-120,00°
1,697E-02
120,00°
5,657E+02
120,00°
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
5,657E-03
-0,00°
4,463E-04
88,86°
5,657E-03
-0,00°
4,631E-04
89,60°
5,657E-03
-0,00°
4,677E-04
90,39°
5,657E-03
-120,00°
5,781E-04
-32,51°
5,657E-03
-120,00°
6,723E-04
-32,13°
5,657E-03
-120,00°
6,393E-04
-32,27°
5,657E-03
120,00°
4,590E-04
-156,24°
5,657E-03
120,00°
4,680E-04
-156,87°
5,657E-03
120,00°
4,757E-04
-157,12°
Fig. 6.3.9.a. A –phase currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.b. B–phase currents [A] and voltages [V] in transient states steady states.
102
Fig. 6.3.9.c. C –phase currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.d. A1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.e. A2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.f. A3–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.g. B1–branch currents [A] and voltages [V] in transient states steady states.
103
Fig. 6.3.9.h. B2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.i. B3–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.j. C1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.k. C2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.9.l. C3–branch currents [A] and voltages [V] in transient states steady states.
104
5.3.10
Case 13: Switching mode – normal operation with random
switching time delays
A-phase IGBT
B-phase IGBT
C-phase IGBT
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase B:
Voltage phase C:
- amplitude: 800 V
- amplitude: 800 V
- phase: -1200
- phase:
1200
- frequency: 60 Hz
- frequency: 60 Hz
Switching frequency is set to 2,5 kHz
Normal operation starts with 1,2 ms delay to simulation run.
Random time delay before / after each switching operation is set to 5 µs
IGBT parameters:
IGBT parameters:
- Internal resistance [Ω]:
1E-03
- Internal inductance [H]:
infinite
- Forward voltage [V]:
1E+00
- Current 10% fall time [s]:
1E-06
- Current tail time [s]:
2E-06
- Initial current [A]:
0
- Snubber resistance [Ω]:
1E+05
- Snubber capacitance [F]:
infinite
IGBT controllers:
Controllers are made by upgrading existing subsystem from case 10 with Bernoulli binary generator
blocks which generate random signal for each IGBT switch dependant upon initial seed value.
105
Fig. 6.3.10.a. Switch controller diagram used for simulations.
Table 6.3.10. Steady values of currents and voltages before switching operation from simulation
process.
Phase Currents
Phase Voltages
phase
RMS value [A]
phase [0]
RMS value [V]
phase [0]
1,697E-02
-0,00°
5,657E+02
0,00°
A
1,697E-02
-120,00°
5,657E+02
-120,00°
B
1,697E-02
120,00°
5,657E+02
120,00°
C
branch
A1
A2
A3
B1
B2
B3
C1
C2
C3
Branch Currents [A]
Branch Voltages [V]
RMS value [A]
phase [0]
RMS value [V]
phase [0]
5,657E-03
-0,00°
4,463E-04
88,86°
5,657E-03
-0,00°
4,631E-04
89,60°
5,657E-03
-0,00°
4,677E-04
90,39°
5,657E-03
-120,00°
5,781E-04
-32,51°
5,657E-03
-120,00°
6,723E-04
-32,13°
5,657E-03
-120,00°
6,393E-04
-32,27°
5,657E-03
120,00°
4,590E-04
-156,24°
5,657E-03
120,00°
4,680E-04
-156,87°
5,657E-03
120,00°
4,757E-04
-157,12°
106
Fig. 6.3.10.b. A –phase currents [A] and voltages [V].
Fig. 6.3.10.b. B –phase currents [A] and voltages [V].
Fig. 6.3.10.b. C –phase currents [A] and voltages [V].
107
Fig. 6.3.10.c. A1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.10.d. A2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.10.e. A3–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.10.f. B1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.10.g. B2–branch currents [A] and voltages [V] in transient states steady states.
108
Fig. 6.3.10.h. B3–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.10.i. C1–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.10.j. C2–branch currents [A] and voltages [V] in transient states steady states.
Fig. 6.3.10.k. C3–branch currents [A] and voltages [V] in transient states steady states.
109
6 Experimental Analysis and Verification
6.1 Brief
In order to verify functionality and quality of the designed theoretical model one should investigate
specific cases that mostly correspond to expectations from the model in comparison with real object’s
behavior. Therefore, the most essential case studies which were chosen for analysis and comparison
are as follows:
- Case 1: Normal operation
Under normal symmetrical operation at 60 Hz, the resulting positive sequence inductance should
equal the value on the specification plate, that is 74 µH.
- Case 2: Frequency sweep of symmetrical component impedance
It can give an idea about resonance frequencies, which should be avoided. Therefore, this case
requires to plot the impedance of each sequence – positive, negative and zero – as a function of
frequency. Plots are analysed in the frequency range up to 20 kHz which corresponds to
switching times in real usage.
- Case 3: Frequency sweep of symmetrical component matrix during fall out of one phase
This type of verification can provide if the model is possible to use for fault’s analysis when line
reactor works in unwanted states.
110
6.2 Experimental Analysis - Methodology
6.2.1 Case 1: Normal operation
For normal operation, verification experiments have to be made to confirm data stated on the nameplate
of the line reactor. The way to approach this task is to measure mutual and self resulting impedances of
each phase in nominal frequency – 60 Hz. Then, through calculation transform these values into
sequence impedances. In order to measure phase impedances, measurement method is presented
below.
Vaa
Vba
Vca
Fig . 7.2.1a Measurement setup for acquiring resulting parameters in normal operation.
To simulate normal symmetrical operation of a 3-phase line reactor with a single phase power supply,
there is a necessity to short-circuit all windings of each phase, treating them as a resulting load per
phase. Next, by injecting AC current to each phase and measuring all phase voltages with current one
can obtain all nine impedances, that is three self impedances and six mutual. Measurements have been
made using oscilloscope to track each waveform and read time lags between current and voltages. In
addition, amplifier was used to reduce possible errors made by very low value of voltage.
1st set of experiments – obtaining Z aa, Z ba, Zca
V
Iaa
+
Vaa
V
+
Vba
V
Vca
+
A+
+
PHASE A
PHASE B
PHASE C
Fig . 7.2.1b. Measurement setup for 1st set of experiments current and voltages plots on oscilloscope.
111
Table 7.2.1a. Measured values from 1st set of experiments.
IAA
VAA
VBA
VCA
Magnitude
Phase
Magnitude
Phase
Magnitude
Phase
Magnitude
Phase
[V]
[o]
[mV]
[o]
[mV]
[o]
[mV]
[o]
7,50
0,00
194
64,04
30
-96,06
27
-96,06
Impedances can be calculated from equations below:
Z AA =
VAA 0,194 ⋅ e j 64,04
=
= 25,87 ⋅ 10 −3 ⋅ e j 64,04 Ω
j0
I AA
7,5 ⋅ e
{1}
Z BA =
VBA 0,030 ⋅ e − j 96,06
=
= 4,00 ⋅ 10 −3 ⋅ e − j 96,06 Ω
j0
I AA
7,5 ⋅ e
{2}
ZCA =
VCA 0,027 ⋅ e − j 96 ,06
=
= 3,60 ⋅ 10 − 3 ⋅ e − j 96 ,06 Ω
j0
I AA
7,5 ⋅ e
{3}
2nd set of experiments – obtaining Zbb, Zab, Zcb
V
V
+
Vbb
+
V
+
Vab
Vac
PHASE B
PHASE C
Ibb
A+
PHASE A
+
Fig . 7.2.1c. Measurement setup for 2nd set of experiments current and voltages plots on oscilloscope.
Table 7.2.1b. Measured values from 2nd set of experiments.
IBB
VAB
VBB
Magnitude
Phase
Magnitude
Phase
Magnitude
[A]
[o]
[mV]
[o]
[mV]
7,50
0,00
39
-99,34
251
Impedances can be calculated from equations below:
VCB
Phase
[o]
57,01
Magnitude
[mV]
32
Z AB =
VAB 0,039 ⋅ e − j 99,34
=
= 5,20 ⋅ 10 −3 ⋅ e − j 99,34 Ω
I BB
7,5 ⋅ e j 0
{4}
Z BB =
VBA 0,251⋅ e j 57,01
=
= 33,47 ⋅ 10 −3 ⋅ e j 57,01 Ω
I AA
7,5 ⋅ e j 0
{5}
ZCB =
VCA 0,032 ⋅ e − j 99,34
=
= 4,27 ⋅ 10 −3 ⋅ e − j 99,34 Ω
j0
I AA
7,5 ⋅ e
{6}
112
Phase
[o]
-99,34
3rd set of experiments – obtaining Z cc, Zac, Zbc
V
V
+
Vac
V
+
Vbc
+
Vcc
Icc
A+
PHASE A
PHASE B
+
PHASE C
Fig . 7.2.1d. Measurement setup for 3rd set of experiments current and voltages plots on oscilloscope.
Table 7.2.1c. Measured values from 3rd set of experiments.
IC
VAC
VBC
Magnitude
Phase
Magnitude
Magnitude
Phase
o
o
[A]
[]
[mV]
[]
[mV]
7,50
0,00
21
-101,93
34
Impedances can be calculated from equations below:
V
0,021⋅ e − j 101,93
Z AC = AC =
= 2,80 ⋅ 10 −3 ⋅ e − j 101,93 Ω
IC
7,5 ⋅ e j 0
VCC
Phase
[o]
-101,93
Magnitude
[mV]
215
{7}
Z BC =
VBC 0,034 ⋅ e − j 101,93
=
= 4,53 ⋅ 10 −3 ⋅ e − j 101,93 Ω
j0
IC
7,5 ⋅ e
{8}
ZCC =
VCC 0,215 ⋅ e j 46,65
=
= 28,67 ⋅ 10 −3 ⋅ e j 46,65 Ω
IC
15 ⋅ e j 0
{9}
From these measurements impedance matrix is obtained in the form presented below:
Z AA
Z ABC [] = Z BA
Z CA
Z AB
Z BB
Z CB
Z AC  25,87 ⋅ e j 64 ,04

Z BC  =  5,20 ⋅ e − j 96,6
Z CC   2,80 ⋅ e − j 96,6
4,00 ⋅ e − j 99,3
33,47 ⋅ e j 96,1
4,53 ⋅ e
113
− j 99 ,3
3,60 ⋅ e − j 101,9 

4,27 ⋅ e − j 101,9  [mΩ ]
28,67 ⋅ e j 96,06 
{10}
Phase
[o]
46,65
MATHEMATICAL ANALYSIS
Acquiring resulting sequence impedances of the line reactor under symmetrical operation requires to
make necessary assumptions, so that three-phase voltage source is symmetrical:
VABC [] = [VA VB
[
VC ] = C C ⋅ e − j 120
C ⋅ e j 120
0
0
]
{11}
where:
C = RMS / amplitude value of the source
It can be referred from the circuit diagram, that voltage drops on each phase are equal to voltage phase
sources. Next step is to calculate phase currents flowing through each phase from a set of equations:
VA = I A ⋅ Z AA + IB ⋅ Z BA + IC ⋅ ZCA

VB = I A ⋅ Z AB + IB ⋅ Z BB + IC ⋅ ZCB
V = I ⋅ Z + I ⋅ Z + I ⋅ Z
 C A AC B BC C CC
{12}
Equations above can be replaced with matrix equation:
I ABC [] = VABC [] × Z
−1
ABC
[] = [VA VB
Z AA
VC ]× Z BA
ZCA
Z AC 
Z BC 
ZCC 
Z AB
Z BB
ZCB
−1
I A 
= I B 
IC 
{13}
Having phase currents calculated, there is now possibility to make a transformation from phase- values
(A-B-C) to sequence- values (0-1-2) out of currents and voltages:
1 1 1 
1
1
V012 [] = ⋅ VABC [] ⋅ A = ⋅ [VA VB VC ] ⋅ 1 a a 2  =
3
3
2
a 
1 a
1
1
I012 [] = ⋅ I ABC [] ⋅ A = ⋅ [I A
3
3
IB
1 1
IC ] ⋅ 1 a
1 a 2
1
a 2  =
a 
V0 
V 
 1
V2 
I0 
I 
 1
I2 
{14}
{15}
where:
a = 1⋅ e j 120
0
As a result of symmetrical source, there will be only positive sequence voltage with value different than
zero:
V0 =
C
⋅ (1 + a + a 2 ) = 0
3
{16}
C
⋅ (1 + a 0 + a 0 ) = C
3
C
V2 = ⋅ (1 + a 3 + a 4 ) = 0
3
V1 =
{17}
{18}
114
Finally, resulting sequence impedance values are available from:

V0
Z0 = = 0
I0

V1 C

Z1 = =
I1 I1

V2

Z2 = I = 0

2
{19}
Eventually, constant value C will reduce, which means that sequence impedance values are
independent from the magnitude value of voltage:
∀
Z 012 [] = const
{20}
C≥0
From this point, resulting positive-sequence inductance can be derived from:
L1 =
img {Z1 }
2 ⋅π ⋅f
{21}
where:
f = 60 Hz (frequency)
Calculations of positive-sequence impedance have been performed in MATLAB software. All data and
equations used in calculations had been put in a VER_1.m file, which internal structure is presented in
the Appendix 2. For calculations, voltage magnitude value C is set to 1 V.
Simulation results:
1. Phase voltage vector [V]:
[
VABC [] = 1 e − j 120
0
e j 120
0
]
{22}
2. Sequence voltage vector [V]:
V012 [] = [0 1 0]
{23}
3. Phase current vector [A]:
[
I ABC [] = 34,509 ⋅ e − j 64,918
0
26,508 ⋅ e − j 176,856
0
30,580e j 72,445
4. Sequence current vector [A]:
[
I 012 [] = 2,014 ⋅ e − j 134 ,966
0
30,189 ⋅ e − j 57,918
0
5,235e − j 108,227
0
0
]
{24}
]
{25}
5. Sequence impedance vector [mΩ ]:
[
Z 012 [] = 82,66 ⋅ 10 −15 ⋅ e j 134 ,966
0
33,12 ⋅ e j 57,918
115
0
44,52 ⋅ 10 −6 ⋅ e j 108,227
0
] {26}
6. Positive sequence impedance:
Z 1 = 33,12 ⋅ 10 −3 ⋅ e j 57,918 = (17,593 + 28,065 ⋅ j ) ⋅ 10 −3
0
{27}
7. Positive sequence inductance:
L1 =
Relative error: δ L =
img {Z 1 } 28,065 ⋅ 10 −3
=
= 74,445 µH
2 ⋅π ⋅ f
2 ⋅ π ⋅ 60
LR − L1
LR
⋅ 100 =
74 − 74,445
74
{28}
⋅ 100 = 0,601 %
{29}
6.2.2 Case 2: Frequency sweep of symmetrical component impedance
The measurement procedure which is used to acquire sequence impedance plots is the same as
presented in case 1. The only difference is, that instead of AC Source with fixed frequency there is a
necessity to implement frequency generator with the maximum frequency range not lower than 20 kHz.
By using frequency generator one obtains voltage and current values in different frequencies.
After performing the same calculations for zero, positive and negative sequence impedances for all
measured frequency points as presented in study case 1, there is a possibility to acquire transfer
function Z012(s) for each sequence impedance and present it on Bode diagram.
Measurement results for phase A
Table 7.2.2a Measured values of currents, voltages and EMF’s induced by A phase current.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Current IA
magnitude
[A]
7,500
10,300
10,400
9,600
9,800
7,500
5,000
2,500
2,600
0,860
0,550
0,143
0,147
0,098
0,101
phase
[0]
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
Voltage VAA
magnitude
[mV]
194
367
684
1483
2938
4572
5868
4068
5616
2376
1872
605
677
490
569
phase
[0]
64,04
71,93
77,70
83,44
84,45
87,88
88,70
90,11
85,18
87,58
89,89
96,07
89,90
92,36
90,72
116
Voltage V BA
magnitude
[mV]
30
70
76
364
734
1166
1476
1030
1426
598
468
177
169
121
143
phase
[0]
-96,06
-87,75
-90,65
-90,63
-91,61
-93,64
-93,31
-90,97
-93,24
-90,45
-89,89
-88,99
-92,20
-92,36
-90,72
Voltage VCA
magnitude
[mV]
27
46
101
246
526
893
1166
835
1166
504
396
85
143
105
122
phase
[0]
-96,06
-87,39
-86,33
-86,31
-87,32
-89,32
-87,55
-88,38
-90,94
-89,01
-86,43
-86,97
-93,36
-93,66
-92,16
Table 7.2.2b Calculated values of self and mutual impedances induced by A phase current.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Impedance ZAA
magnitude
[mΩ]
25,92
35,65
65,77
154,50
299,76
609,60
1173,60
1627,20
2160,00
2762,79
3403,64
4229,37
4604,08
4995,92
5631,68
Impedance ZBA
magnitude
[mΩ]
3,97
6,82
7,34
37,88
74,94
155,52
295,20
411,84
548,31
694,88
850,91
1238,60
1148,57
1234,29
1411,49
phase
[0]
64,04
71,93
77,70
83,44
84,45
87,88
88,70
90,11
85,18
87,58
89,89
96,07
89,90
92,36
90,72
phase
[0]
-96,06
-87,75
-90,65
-90,63
-91,61
-93,64
-93,31
-90,97
-93,24
-90,45
-89,89
-88,99
-92,20
-92,36
-90,72
Impedance ZCA
magnitude
[mΩ]
3,65
4,47
9,69
25,65
53,63
119,04
233,28
334,08
448,62
586,05
720,00
594,13
969,80
1072,65
1211,88
phase
[0]
-96,06
-87,39
-86,33
-86,31
-87,32
-89,32
-87,55
-88,38
-90,94
-89,01
-86,43
-86,97
-93,36
-93,66
-92,16
Measurement results for phase B
Table 7.2.2c Measured values of currents, voltages and EMF’s induced by B phase current.
frequency
Current IB
[Hz]
magnitude
phase
[A]
[0]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
7,500
10,200
10,200
10,200
9,800
7,500
4,900
2,350
2,500
0,920
0,510
0,142
0,145
0,093
0,098
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
Voltage VAB
magnitude
phase
[mV]
[0 ]
39
75
150
385
734
1130
1440
979
1361
612
432
150
150
130
148
-99,34
-87,49
-87,84
-90,72
-50,40
-81,26
-91,08
-93,46
-98,00
-96,38
-91,62
-97,73
-103,68
-97,28
-94,85
117
Voltage V BB
magnitude
phase
[V]
[0]
251
518
914
2059
3708
5544
6732
4464
6120
2736
1872
749
763
562
648
57,01
65,98
72,00
79,20
45,60
83,42
85,32
83,08
79,55
86,31
86,43
83,63
81,79
84,31
80,48
Voltage VCB
magnitude
phase
[mV]
[0]
32
75
153
392
742
1166
1490
1001
1404
634
432
173
184
130
151
-99,34
-87,49
-90,72
-92,16
-50,40
-93,49
-93,39
-93,46
-96,85
-94,95
-91,62
-95,72
-99,07
-95,98
-96,29
Table 7.2.2d Calculated values of self and mutual impedances induced by B phase current.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Impedance ZAB
magnitude
phase
[Ω]
[0]
5,18
7,34
14,68
37,76
74,94
150,72
293,88
416,68
544,32
665,22
847,06
1057,18
1037,79
1393,55
1506,12
Impedance ZBB
magnitude
phase
[Ω]
[0]
-99,34
-87,49
-87,84
-90,72
-50,40
-81,26
-91,08
-93,46
-98,00
-96,38
-91,62
-97,73
-103,68
-97,28
-94,85
33,41
50,82
89,65
201,88
378,37
739,20
1373,88
1899,57
2448,00
2973,91
3670,59
5273,24
5263,45
6038,71
6612,24
Impedance ZCB
magnitude
phase
[Ω]
[0]
57,01
65,98
72,00
79,20
45,60
83,42
85,32
83,08
79,55
86,31
86,43
83,63
81,79
84,31
80,48
4,22
7,34
14,96
38,47
75,67
155,52
304,16
425,87
561,60
688,70
847,06
1216,90
1266,21
1393,55
1542,86
-99,34
-87,49
-90,72
-92,16
-50,40
-93,49
-93,39
-93,46
-96,85
-94,95
-91,62
-95,72
-99,07
-95,98
-96,29
Measurement results for phase C
Table 7.2.2e Measured values of currents, voltages and EMF’s induced by C phase current.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Current IC
magnitude
[A]
7,500
10,300
10,300
10,200
9,600
7,800
5,250
2,800
2,850
0,980
0,485
0,141
0,150
0,095
0,094
phase
[0]
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
Voltage VAC
magnitude
[V]
21
50
101
266
518
886
1217
972
1274
598
367
151
155
112
112
phase
[0]
-101,93
-80,56
-83,55
-84,96
-89,10
-87,84
-61,68
-88,32
-92,09
-90,63
-92,89
-98,85
-95,52
-96,00
-96,48
118
Voltage V BC
magnitude
[V]
34
76
154
392
742
1210
1606
1238
1606
720
446
180
187
134
137
phase
[0]
-101,93
-90,63
-92,20
-90,72
-94,85
-93,60
-64,97
-93,52
-95,54
-94,95
-94,19
-96,83
-93,22
-94,70
-95,04
Voltage VCC
magnitude
[V]
215
410
702
1606
2966
4860
6264
4932
6300
2916
1786
619
749
526
554
phase
[0]
46,65
60,78
72,03
82,08
81,92
84,96
63,32
88,32
84,03
83,44
86,41
86,75
87,47
86,92
86,40
Table 7.2.2f Calculated values of self and mutual impedances induced by C phase current.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Impedance ZAC
magnitude
[Ω]
2,74
4,82
9,79
26,12
54,00
113,54
231,77
347,14
447,16
609,80
757,11
1072,34
1032,00
1182,32
1194,89
Impedance ZBC
magnitude
[Ω]
4,51
7,41
14,96
38,47
77,25
155,08
305,83
442,29
563,37
734,69
920,41
1276,60
1248,00
1409,68
1455,32
phase
[0]
-101,93
-80,56
-83,55
-84,96
-89,10
-87,84
-61,68
-88,32
-92,09
-90,63
-92,89
-98,85
-95,52
-96,00
-96,48
phase
[0]
-101,93
-90,63
-92,20
-90,72
-94,85
-93,60
-64,97
-93,52
-95,54
-94,95
-94,19
-96,83
-93,22
-94,70
-95,04
Impedance ZCC
magnitude
[Ω]
28,61
39,84
68,16
157,41
309,00
623,08
1193,14
1761,43
2210,53
2975,51
3681,65
4391,49
4992,00
5532,63
5897,87
phase
[0]
46,65
60,78
72,03
82,08
81,92
84,96
63,32
88,32
84,03
83,44
86,41
86,75
87,47
86,92
86,40
MATHEMATICAL ANALYSIS - example
An example, showing the functionality of verification process is made for an unbalanced voltage source
values:
VA = 690 ⋅ e j 10 V

− j 1200
V
VB = 670 ⋅ e
0
115
 V = 695 ⋅ e
V
 C
0
{30}
Table 7.2.2g. Sequence impedance results taken from VER_2.m file for unbalanced voltage source.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Zero sequence Z0
magnitude
[Ω]
2,219E-02
1,608E-02
1,921E-02
4,451E-02
8,085E-02
1,812E-01
6,451E-01
5,545E-01
7,275E-01
1,462E+00
1,771E+00
1,893E+00
1,711E+00
1,665E+00
1,809E+00
phase
[0]
65,413
78,132
77,789
89,610
130,950
87,320
18,063
114,788
99,688
92,828
106,134
144,324
125,182
126,385
117,291
Positive sequence Z1
magnitude
[Ω]
1,305E-02
1,861E-02
5,706E-02
2,193E-01
2,230E-01
1,181E+00
5,956E-01
1,883E+01
1,003E+01
4,711E+00
6,279E+00
3,581E+00
1,269E+01
1,117E+01
1,902E+01
119
phase
[0]
-155,816
-143,847
-136,965
-101,353
-35,322
-118,221
-131,934
-223,798
-329,362
-225,915
-231,059
-95,034
-247,103
-155,611
-90,150
Negative sequence Z2
magnitude
[Ω]
2,157E-02
3,043E-02
5,528E-02
1,325E-01
2,544E-01
5,239E-01
9,949E-01
1,412E+00
1,829E+00
2,335E+00
2,905E+00
3,725E+00
3,970E+00
4,435E+00
4,875E+00
phase
[0]
57,407
69,054
75,162
83,116
78,760
86,314
80,987
87,389
83,445
85,776
87,561
90,590
86,421
87,264
86,048
From values presented in table 7.2.2g one may create four vectors, each consisting of 15 components:
= [60 100 ... 20000]
- frequency vector
Z0
= [2,22·exp(65,4i) ... 180,9·exp(65,4i)]
- zero sequence impedance vector
Z1
= [1,31·exp(-155,8i) ... 1902·exp(-90,2i)]
- positive sequence impedance vector
Z2
= [2,16·exp(57,4i) ... 487,5·exp(86,1i)]
- negative sequence impedance vector
f
These data are made in a VER_2.m file, which is extended version of the .m file presented in case 1
and therefore performs additional functions. First, it calculates sequence impedance values out of phase
values for each frequency level (it uses the same algorithm as VER_1.m file) with specified 3-phase
voltage source, which may be balanced or unbalanced (results of calculations are presented in table
7.2.2g). Second, it creates four vectors out of obtained values in addition with frequency vector. After
establishing these values, command invfreqs is invoked. This command is able to “find a continuous
time transfer function that corresponds to the complex frequency response” (ie it can convert the
magnitude and phase data collected in the field, into a transfer function).
Invfreqs uses a non-linear least square (NLS) method [20] to minimize the squared norm of error
between the estimated model and the measured frequency response data Gk as shown below:
N −1
min ( A, B)∑
k =0
B(θ k )
− Gk
A(θ k )
2
{31}
where:
Gk = G(e jθ k ) = Gk' + η k , η k being a measurement error – 1,58 · 10-6
The routine returns the coefficients of the numerator and denominator in vectors A and B such that a
transfer function is yielded in the form:
Z ( s) =
B (s ) b(1) ⋅ s nb + b(2) ⋅ s ( nb −1) + ... + b(nb + 1)
=
A(s ) a(1) ⋅ s na + a(2) ⋅ s ( na −1) + ... + a(na + 1)
{32}
Invoking the invfreqs command requires the specification by the user, of the orders of the A and B
polynomials, i.e the number of poles and zeroes associated with the model in question.
Once the routine has been delivered a suitable transfer function, it is possible to plot the frequency
response of the sequence impedances. Plots are made by using bode function with vectors A and B for
each sequence as its arguments. In this manner, plots are successfully made and are ready to compare
with simulation plots taken from the model [22].
120
Fig. 7.2.2a. Obtained Bode plot of zero-sequence impedance.
Fig. 7.2.2b. Obtained Bode plot of positive-sequence impedance.
Fig. 7.2.2c. Obtained Bode plot of zero-sequence impedance.
121
6.3 Verification results
Verification is made for simplified model obtained from measurements in chapter 5. It consists of three
cases, which are taken into consideration as the most significant for Siemens company.
Case 1 - positive sequence inductance under normal operation.
Case 2 - frequency sweep of symmetrical component matrix under normal operation.
Case 3 - frequency sweep of symmetrical component matrix during fall out of one phase.
Verification is made in MATLAB software by implementing additional blocks, which transform phase
currents and voltages of the model into sequence values. It requires necessary blocks such as:
-
3-phase measurement block - necessary for measuring phase currents and voltages of the
model during simulation process (●).
3-phase sequence analyzer blocks - responsible for transforming current and voltage
phase values into their sequence equivalents (●).
divide blocks – responsible for calculating sequence impedances and/or inductances (●).
display blocks – responsible for presenting actual resulting values during simulation (●).
Fig. 7.3a Measurement setup for sequence impedances in MATLAB model.
User has to define phase voltage source values such as: amplitude, phase, frequency and perform
certain type of connection. Under these defined conditions one may obtain sequence impedance values
of the model.
122
Case 1: positive sequence inductance under normal operation
Input data for MATLAB model:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: 60 Hz
Voltage phase A:
- amplitude: 800 V
- phase: -1200
- frequency: 60 Hz
Voltage phase A:
- amplitude: 800 V
- phase:
1200
- frequency: 60 Hz
Fig. 7.3b Measurement setup for positive-sequence inductance in MATLAB model.
From the simulation process, resulting positive-sequence inductance of the model is 78,567 µH.
Relative error:
ρ=
74 − 78,567
74
⋅ 100 = 6,17%
Case 2: frequency sweep of symmetrical component matrix under normal operation
Input data for MATLAB model and VER_2.m file:
Voltage phase A:
- amplitude: 800 V
- phase:
00
- frequency: [60...20000] Hz
Voltage phase A:
- amplitude: 800 V
- phase: -1200
- frequency: 60 Hz
Voltage phase A:
- amplitude: 800 V
- phase:
1200
- frequency: 60 Hz
In order to obtain sequence impedance values, there is a new block created especially for project
requirements called “programmable frequency 3-phase voltage source”, which changes its frequency
from 60 Hz to 2 kHz with certain steps. In this way, gathering data process can be done automatically. It
is simply done by switching different frequency sources at certain time steps for every phase. Diagram
below presents internal structure of the created block.
Fig. 7.3c. Variable Frequency Supply – internal structure
123
Table 7.3a. Sequence impedance results taken from VER_2.m file.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Zero sequence Z0
magnitude
[Ω]
7,05E-17
1,09E-16
2,38E-17
2,99E-16
3,16E-01
1,73E-15
1,52E-15
2,94E-15
5,91E-15
7,82E-15
7,60E-15
4,84E-15
7,22E-15
7,16E-15
9,26E-15
phase
[0]
134,97
64,61
24,81
41,01
71,57
29,27
-138,15
105,05
91,09
173,35
158,43
148,61
122,82
116,87
105,28
Positive sequence Z1
magnitude
[Ω]
3,31E-02
4,66E-02
6,99E-02
2,01E-01
3,09E-01
7,99E-01
1,54E+00
2,15E+00
2,78E+00
3,55E+00
4,41E+00
5,71E+00
6,04E+00
6,76E+00
7,42E+00
phase
[0]
57,92
69,67
68,08
83,36
81,92
86,43
82,46
87,62
83,57
86,21
87,78
90,42
86,74
87,51
86,08
Negative sequence Z2
magnitude
[Ω]
4,45E-08
6,14E-08
7,06E-08
3,96E-07
3,09E-01
1,77E-06
2,34E-06
7,10E-06
9,70E-06
1,61E-05
2,07E-05
9,02E-06
2,61E-05
1,83E-05
1,88E-05
phase
[0]
108,23
119,41
67,80
150,50
81,92
150,40
129,35
158,00
163,22
125,78
136,35
155,09
148,64
144,05
160,67
Table 7.3b Sequence impedance results taken from MATLAB model.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Zero sequence Z0
magnitude
[Ω]
5,39E-16
1,46E-15
2,61E-15
2,48E-14
1,84E-14
4,64E-14
7,36E-13
1,02E-13
2,52E-13
2,10E-13
5,96E-13
6,55E-13
1,65E-12
1,50E-13
1,68E-12
phase
[0]
25,83
28,22
30,26
31,39
31,75
31,95
32,04
32,07
32,08
32,07
32,07
32,05
32,04
32,03
32,01
Positive sequence Z1
magnitude
[Ω]
2,58E-02
4,93E-02
9,85E-02
2,46E-01
4,93E-01
9,85E-01
1,97E+00
2,96E+00
3,95E+00
4,95E+00
5,95E+00
6,66E+00
7,98E+00
9,00E+00
1,00E+01
124
phase
[0]
87,67
88,59
89,30
89,72
89,86
89,93
89,97
89,98
89,98
89,99
90,00
89,99
89,99
90,00
89,99
Negative sequence Z2
magnitude
[Ω]
3,63E-08
2,55E-15
2,88E-15
1,55E-14
7,01E-15
2,20E-13
1,01E-12
6,34E-10
4,88E-13
2,17E-13
1,01E-08
8,88E-09
1,38E-12
5,34E-08
2,56E-12
phase
[0]
142,80
145,00
146,20
147,10
147,40
147,51
147,59
147,64
147,65
147,68
147,76
147,80
147,70
147,85
147,86
Fig. 7.3d. Bode plots of positive-sequence impedance (● – experimental  ● – MATLAB model)
Fig. 7.3e. Bode plots of negative-sequence impedance (● – experimental  ● – MATLAB model).
Fig. 7.3f. Bode plots of zero-sequence impedance (● – experimental  ● – MATLAB model).
125
Case 3: frequency sweep of symmetrical component matrix during fall out of one phase
Input data for MATLAB model and VER_3.m file:
Voltage phase A:
- amplitude: 0 V
- phase:
00
- frequency: [60...20000] Hz
Voltage phase A:
- amplitude: 800 V
- phase:
-1200
- frequency: [60...20000] Hz
Voltage phase A:
- amplitude: 800 V
- phase:
1200
- frequency: [60...20000] Hz
Current flowing through phase A: 0 A
Verification is made between model and measurements by switching off A phase. In VER_3.m file this is
done by ascribing zero value to the current flowing through phase A. In this manner sequence currents
and voltages reflect unwanted and unbalanced state when one of phases is cut off either due to
breaker’s failure or too high current level from the power converter. MATLAB model can be simply
upgraded by switching off all breakers in phase A. Then currents and voltages can be measured and
sequence impedances calculated in the same manner as they are in case 2.
Fig. 7.3g. Switching setup in MATLAB model for frequency sweep verification with A phase fault.
126
Table 7.3c. Sequence impedance results taken from VER_3.m file.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Zero sequence Z0
magnitude
[Ω]
1,31E-17
1,98E-17
3,53E-17
9,18E-17
1,32E-16
3,42E-16
4,95E-16
1,11E-15
1,34E-15
1,46E-15
1,91E-15
2,71E-15
3,07E-15
3,53E-15
3,91E-15
phase
[0]
-121,90
-101,98
-69,37
-81,89
-98,08
-81,72
-100,81
-92,09
-93,29
-99,11
-99,11
-99,28
-99,60
-96,70
-92,94
Positive sequence Z1
magnitude
[Ω]
5,27E-02
7,64E-02
1,41E-01
3,21E-01
4,63E-01
1,25E+00
2,29E+00
3,43E+00
4,33E+00
5,43E+00
6,81E+00
9,16E+00
9,63E+00
1,10E+01
1,19E+01
phase
[0]
51,87
65,60
79,18
81,99
81,92
85,26
76,59
84,65
81,36
83,37
84,38
81,94
83,15
83,22
82,21
Negative sequence Z2
magnitude
[Ω]
2,59E-17
3,4E-17
5,03E-17
1,26E-16
1,98E-16
5,24E-16
1,42E-15
1,3E-15
1,69E-15
2,45E-15
2,93E-15
3,76E-15
3,69E-15
4,22E-15
4,45E-15
phase
[0]
-136,36
-128,66
-130,56
-112,77
-98,08
-108,07
-108,38
-97,90
-103,14
-93,85
-92,03
-96,92
-94,64
-96,85
-101,45
Table 7.3d. Sequence impedance results taken from MATLAB model.
frequency
[Hz]
60
100
200
500
1000
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
Zero sequence Z0
magnitude
[Ω]
5,15E-16
7,21E-15
9,68E-16
2,27E-14
6,8E-15
5,1E-15
6,54E-15
1,31E-14
4,72E-14
2,41E-13
2,65E-14
4,11E-14
1,91E-13
1,43E-13
4,66E-14
phase
[0]
-80,73
-79,46
-78,50
-77,93
-77,74
-77,64
-77,59
-77,56
-77,54
-77,52
-77,51
-77,48
-77,44
-77,42
-77,37
Positive sequence Z1
magnitude
[Ω]
4,28E-02
7,13E-02
1,43E-01
3,56E-01
7,13E-01
1,43E+00
2,85E+00
4,28E+00
5,72E+00
7,16E+00
8,61E+00
1,01E+01
1,15E+01
1,30E+01
1,45E+01
phase
[0]
87,51
88,51
89,25
89,70
89,85
89,93
89,96
89,98
89,98
89,99
89,99
89,98
89,99
89,99
89,99
Negative sequence Z2
magnitude
[Ω]
3,04E-12
2,7E-15
1,24E-14
2,58E-14
1,36E-13
1,28E-13
1,46E-13
2,97E-10
5,84E-14
1,2E-13
4,78E-09
4,19E-09
5,91E-14
2,53E-08
1,22E-13
phase
[0]
-114,14
-113,33
-112,72
-112,42
-112,23
-112,17
-112,16
-112,17
-112,21
-112,25
-112,29
-112,38
-112,45
-112,54
-112,64
Fig. 7.3h. Bode plots of positive-sequence impedance
127 (● – experimental  ● – MATLAB model)
Fig. 7.3i. Bode plots of negative-sequence impedance (● – experimental  ● – MATLAB model)
Fig. 7.3j. Bode plots of zero-sequence impedance (● – experimental  ● – MATLAB model)
128
7 Conclusion
Conclusions are divided with respect to analyses performed in time and frequency domain. . In closing
of the project, the following remarks can be drawn:
7.1 Time Domain
7.1.1 Analysis
MATLAB simulation plots from “Computer Simulation” chapter have expected shapes as discussed
in “Theoretical Analysis” chapter. Changes of current and voltage values in transient conditions are
characterized with time-constants and suppress to the steady states. However, values of these
parameters in steady states are far higher than expected from the specification of the 3-phase line
reactor. Therefore it may be concluded that parameters implemented into the model were not
measured with designed precision and rate. This is caused either by methodology of
measurements (too many simplifications) or accuracy of equipment used for measurements, since
values of measured parameters are relatively low.
Comparison between plots taken in short times and long times shows, that some parameters
require lower sample times in simulation for long times in order to prevent from too large
differences in magnitudes of initial values during switching operations. It is because steady values
in short and long times do not match.
Simulations in steady states (Figures 6.2.1a – 6.2.3c) show significant changes of current
magnitudes in B phase in comparison to other phases. This can be explained by influence of
magnetizing flux flowing through the core which affects mutual couplings between each branch.
Middle phase is most influenced because distances between it and other phases are lowest. There
can be seen lower changes of currents in middle branches as well.
7.1.2 Future Work
Analysis method used in this project can be upgraded by performing simulations in other computer
software that uses different and more complex algorithms for more accurate results. Measurement
procedure can be upgraded by using more accurate meters and taking wire impedances into account
129
7.2 Frequency Domain
7.2.1 Analysis
Bode plots (figures 7.3d-f and 7.3h-j) show that resulting sequence impedances of MATLAB model
in lower frequencies is much more inductive than ones taken from verification procedure, although
positive-sequence inductance values almost perfectly correspond to the nominal value (74 µH) in
both cases. This can be explained by influence of additional resistances of wires used in
verification procedure. Specific branches had to be connected together through these wires to
measure resulting impedances. In parameter measurements all branches were separated, which
resulted in lower influence of resistive wires.
Obtained plots show that there are significant drops of impedance phase in frequency around 150
Hz and 1 kHz. This can be a starter for deeper searching resonances that may occur near these
values.
7.2.2 Future Work
Instead of invfreqs routine, there can be used a vector fitting method algorithm for more accurate plots
which was invented by B. Gustavsen and A. Semlyen (http://www.energy.sintef.no/produkt/VECTFIT/).
This however requires different measurement procedure to obtain admittance matrix through specific
terminal connection combinations.
Remarks:
•
All circuit models were made in “Tina for Windows – the Complete Electronics Lab” software
made by DesignSoft:
http:// www.designsoftware.com
•
Three-phase line reactor model was adapted from online MATLAB library – 3-winding 3-phase
transformer model:
http://www.kxcad.net/cae_MATLAB/toolbox/physmod/powersys/ref/threephasetransformerinductancematrixtypethreewinding
s.html
•
Invfreqs description was adapted from online MATLAB library – invfreqs:
http://www.mathworks.com/access/helpdesk/help/toolbox/signal/index.html?/access/helpdesk/help/toolbox/signal/invfreqs.ht
mlwww.quadtech.com
•
Verification procedure was based on paper:
“Identification of High Frequency Transformer Equivalent Circuit Using Matlab from Frequency Domain Data” –
S. Islam K. Coates' G. Ledwich' , http://ieeexplore.ieee.org/iel3/4939/13803/00643049.pdf
130
8 Literature
[1] NC Wind Energy - Introduction to Wind Power:
http://www.wind.appstate.edu/windpower/windpower.php
[2] Worlds of David Darling – The encyclopedia of Alternative Energy and Sustainable Living
http://www.daviddarling.info/encyclopedia/M/AE_mean_power_output_wind_turbine.html
[3] Siemens Wind Power GmBH
http://www.windfair.net/siemenswindpower/welcome.html
[4] “Wind Turbine Operations in Electric Power Systems” - Z.Lubosny
[5] All About Circuits – “Magnetic fields and inductance” chapter
http://www.allaboutcircuits.com/vol_1/chpt_15/1.html
[6] “Design Science License” - R. Kuphaldt
http://www.web-books.com/eLibrary/Engineering/Circuits/DC/DC_15P1.htm
[7] “Inductor” - NationMaster - Encyclopedia:
http://www.nationmaster.com/encyclopedia/Inductor.html
[8] “Resistivity and Conductivity” – R.Nave.
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html
[9] “Power losses in wound components” – R.Clarke, University of Surrey
[10] “Power losses in wound components” – R.Clarke, University of Surrey
[11] “ITEM: Optimizing EMI Filters Using Circuit Simulation” – C. Debraal
http://www.interferencetechnology.com/articles/articles/article/optimizing-emi-filters-using-circuit-simulation.html
[12] “AC Resistance, Skin & Proximity Effect” – General Cable co.
http://www.generalcable.co.nz/Technical/10.3.2.1.pdf
[12] “Powder Cam” - Arnold Magnetics Limited.
www.arnoldmagnetics.com/products/powder/pdf/MPP_en_2006_Rev2_intro.pdf
[13] “Electric Machinery Fundamentals” - Stephen J. Chapman
[14] “Electric Machinery Fundamentals” - Stephen J. Chapman
[15] “Electric Machinery Fundamentals” Stephen J. Chapman
[16] “Electric Machinery Fundamentals” Stephen J. Chapman
[17] “Electric Machinery Fundamentals” Stephen J. Chapman
[18] “Maximization of the Q of ferrite-rod inductors, contra wound inductors” – Ben H.Tongue
http://www.bentongue.com/xtalset/29mxqfl/29mxqfl.html
[19] “Capacitance” - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Capacitance
[21] “Distribution Generation - Course Notes”, Dr.Bashar Zahawi.
[22] “Identification of High Frequency Transformer Equivalent Circuit Using Matlab from Frequency Domain Data” –
S. Islam K. Coates' G. Ledwich' http://ieeexplore.ieee.org/iel3/4939/13803/00643049.pdf
[23] “General Applications for Series Reactors” - Madhav Capacitors Pvt Ltd.
[24] "Comparison of Transient Phenomena when Switching Shunt Reactors on the Line’s Two Terminals and Station
Busbar," - Ching-Yin Lee, Chang-Jhih Chen, Chao-Rong Chen Yen-Feng Hsu presented at POWERCON 2004.
131
[25] “Power System Analysis and Design” - J. D. Glover, M. S. Sarma,
[26] “Electrical Transients in Power Systems” - A. Greenwood,
[27], "Potential Risk of Failures in Switching EHV Shunt Reactors in Some One-and-a-half Breaker Scheme Substations," B. Khodabakhchian, J. Mahseredjian, M.-R. Sehati, M.Mir-Hosseini, presented at International Conference on Power
Systems Transients,
[28] “Transient Analysis of Shunt Reactor Switching” - Ariel Rivera-Colón, Juan L. Vargas-Figueroa, Lionel R. OramaExclusa,
[29] “Line Impedance” – Application Note AN-10, TEAL Electronics Corporation
http://www.teal.com/newsletter/AppsNote10.pdf
132
9 Appendix
A. Experimental Measurement Tests
A1
B1
C1
A2
B2
C2
A3
B3
C3
Fig.10.A.1. Three-phase line
CA1
+
L
A1
Measured parameters:
RA1
CG LA1-A2
CG
LA2
R - DC resistance
C - inter-winding capacitance
L - self inductance of single coil
Lm - mutual inductance to other coils
CG – ground capacitances
Fig. 10.A.2. Single coil with measured parameters.
10.A.1 Brief
The experimental tests will be used to discover individual parameters of the line reactor. By determining
these parameters one will then be able to input these into our model, in an attempt to simulate the
behavior of the line reactor.
Each individual coil will be examined for its DC resistance (R), parasitic capacitance (C), self inductance
(Ls), and mutual inductance (Lm) in relation to the other coils in this three phase reactor. Obtaining
these parameters in proper measurement tests, will give an insight to the behavior of this series
inductor. All measurements are done in steady states.
Methods used to determine these parameters via experimental means are described below. Each of the
experimental setups are described diagrammatically.
133
Diagram above describes the initial setup of the analyzed three phase inductor.
10.A.2 Measuring Accuracy
In order to obtain most accurate results of measured parameters, these conditions have to be fulfilled
during measurements:
a)
Inductance measurements
•
Because measured inductor is used in power supply and needs a large current supply (2200 A),
in order to test it under actual (real life) conditions for current flowing through it, DC bias current
has to be applied in addition with AC current. DC bias current provides a way of biasing the
inductor to normal operating conditions where the inductance can then be measured with a
normal LCR meter. The bottom line is that the measured inductance is dependent on the
current flowing through the inductor.
•
LCR meter has to be switched in series mode due to low value of inductance (74 µH per phase)
and higher value of series resistance ( RDC ), which is more significant than the core losses.
•
Constant voltage (voltage leveling) measurement mode. A voltage leveling circuit would monitor
the voltage across the inductor and continually adjust the programmed source voltage in order
to keep the voltage across the inductor constant. It is made mainly because impedance is low.
Due to low value of impedance, injecting specific value of current would make voltage across
the inductor much smaller than the current. Very small signal of voltage is difficult for LCR meter
to measure and can lead to inaccurate results. Therefore it is better to measure current with
known constant level of voltage.
•
Measurements have to be done in high frequencies, so that impedance may be significant
enough to make accurate results, since reactive part of impedance is straight proportional to
frequency.
134
(b) Capacitance measurements
•
LCR meter has to be switched in a parallel mode due to low value of capacitance and in
addition low value of resistance and inductance in the lead wires and nodes. Therefore reactive
part of impedance is much greater than the real part ( reactance is inversely proportional to
capacitance).
•
Constant voltage (voltage leveling) measurement mode. A voltage leveling circuit would monitor
the voltage across the inductor and continually adjust the programmed source voltage in order
to keep the voltage across the inductor constant. It is made mainly because impedance is low.
Because impedance is low, injecting specific value of current would make voltage across the
inductor much smaller than the current. Very small signal of voltage is difficult for LCR meter to
measure and can lead to inaccurate results. Therefore it is better to measure current with
known constant level of voltage.
•
Measurements have to be done in low frequencies, so that impedance may be significant
enough to make accurate results, since reactive part of impedance is inversely proportional to
frequency.
c)
Resistance measurements
•
LCR meter has to be switched in series mode due to low value of inductance and higher value
of series resistance ( RDC ), which is more significant than the core losses.
•
It is mostly expected to use LCR meter with DC resistance (DCR) capability. If not,
measurements has to be done in low frequencies because copper losses are inversely
proportional to frequency, which means as frequency increase, DC resistance decrease.
d) LCR meter suitability
For the accurate measurements of the analyzed object, LCR meter has to suitable for:
-
Parallel / Series Measurement Modes available
-
Frequency range: 20 Hz – 1 kHz
-
High DC bias current source ( not lower than 1 mA )
-
Constant Voltage Mode available
-
DC resistance measurements ( not higher than 1 mΩ )
-
Basic accuracy: 0.1 %
135
10.A.3 DC Resistance Test (Rdc parameter):
CA1
+
L
R
A1
CG LA1-A2
A1
CG
L A2
(a) Using LCR Meter
LCR Meter is a multi-frequency impedance measuring instrument capable of measuring resistance,
capacitance, and inductance between specific ranges and accuracy.
With use of an LCR meter, we apply it to the terminals of each of the coils, one by one, to find the DC
resistance. As per the diagram below:
A1
B1
C1
A2
B2
C2
A3
B3
C3
Fig. 10.A..3. Measurement via use of the LCR meter.
Measurements are done by connecting LCR meter to nodes of each single coil (in total 9
measurements). Configuration of LCR meter is to show resistance in DC mode (by injecting DC current
to the coil). Since current is constant (magnetic field is constant as well), there are no visible effects of
other parameters. Resistance is automatically calculated from voltage drop and current (Ohm’s Law).
Table 10.3a. Results of DC resistance using LCR meter.
DC Resistance (Ω)
A
B
C
1
3,39·10-3
5,41·10-3
4,13·10-3
2
2,49·10-3
8,95·10-3
2,37·10-3
3
3,85·10-3
7,64·10-3
4,53·10-3
136
(b)
Injecting 2A DC into Coil
With the use of a DC current source, we apply 2A DC to each of the coils in the reactor individually,
measuring the voltage. There are nine measurements in total. After each measurement single coil is
grounded in order to dissipate energy from electric field stored in all capacitances in single coil.
A1
B1
C1
A2
B2
C2
A3
B3
C3
Fig 10.A.4. DC Resistance Test using DC Current Source.
The incumbent DC resistance is to be derived from Ohm’s Law:
U = I ⋅R ⇒ R =
U
I
{1}
Table 10.3b. Measured values of current and voltage in addition with calculated resistance.
A
B
C
coil number
U [mV]
I [A]
U [mV]
I [A]
U [mV]
I [A]
1
308
4
306
4
306
4
2
308
4
304
4
309
4
3
305
4
304
4
307
4
DC RESISTANCE RESULTS (Ω)
coil number
A
B
C
1
3,75·10-3
3,24·10-3
3,24·10-3
2
3,75·10-3
2,75·10-3
4,00·10-3
3
3,00·10-3
2,75·10-3
3,50·10-3
137
10.A.4 Inter-Winding Capacitance Test (C parameter):
C
L
+
A1
R A1
A1
CG LA1-A2
CG
L
A2
Using LCR Meter
LCR meter is applied the terminals of the each of the coils one by one, to measure the total
accumulated parasitic capacitance, as illustrated in figure 3 (p. 2). There are nine measurements in
total.
Table 10.4a. Measured values of capacitance using LCR meter.
Capacitance (pF)
A
B
C
1
0,530
0,609
0,283
2
0,390
0,952
0,480
3
0,291
0,263
0,303
C A1
10.A.5 Self Inductance Test (LS parameter)
L A1
+
CG LA1-A2
R
A1
CG
(a) Using LCR Meter – No Load Test
L A2
With the use of an LCR meter, we apply it to the terminals of each of the coils one by one, to measure
their self inductance, as illustrated in figure 3 (p. 2).
There are nine measurements in total. During measurements the rest of coils are open (to prevent from
possible disturbances made by coupling effect). After each measurement coils are grounded or shortcircuited to dissipate the energy stored in their electric fields (through capacitance) made by EMF
induced in them while current was changing in other coil.
Table 10.4b. Measured values of inductance using LCR meter.
Inductance (µH)
A
B
C
1
111,76
131,71
113,92
2
97,91
133,66
97,71
3
116,01
146.99
117,48
138
10.A.6 MUTUAL INDUCTANCE TEST (Lm parameter)
C A1
+
L A1
R
A1
CG LA1-A2
(a) Using LCR Meter – No Load Test
CG
L A2
Each pair of windings in the line reactor is put in series in two different ways, corresponding to the two
possible ways of coupling (see Figure 6 below). This gives us two measurement results, which we
denote as La and Lb:
La = Ls1 + Ls 2 + 2 ⋅ Lm
Lb = Ls1 + Ls 2 − 2 ⋅ Lm
Lb parameter
{2}
{3}
La parameter
O
O
O
O
Fig. 10.A.5. Measuring inductances La and Lb of each pair of windings;
a) inverse series connection, Lb;
b) series connection, La.
We check the no load test results of L1 and L2 by the equation:
L + Lb
Ls 1 + Ls 2 = a
{4}
2
If the equality is met, then the no load test measured values of L1 and L2 (Chapter 3a, p. 4) are correct.
Using the measured values of the inductances La and Lb, we can find the mutual inductance Lm between
each pair of coils from three equations:
L a − Lb
4
L a − ( L s1 + L s 2 )
Lm =
2
( L s1 + L s 2 ) − L b
Lm =
2
Lm =
{5}
{6}
{7}
139
By obtaining these equations, not only we may calculate values of all mutual inductances, but also we
can verify results of self inductance measurements performed in chapter 3 (p. 4).
Final value of the mutual inductance is the arithmetical average of results from equations {5} {6} and {7}.
In order to obtain all possible mutual inductances between coils, seventy two measurements have to be
done (there are thirty six mutual inductances within the reactor with two parameters for one mutual
inductance makes in total seventy two).
Table 10.5a. Measured values of series connected pairs of coils - parameter La.
A
L a [µH]
L A1
LA2
B
L A3
LB1
L B2
C
LB3
L C1
LC2
LC3
LA1
LA2
122,07
LA3
186,44
123,52
LB1
290,64
249,08
259,40
LB2
269,00
268,97
278,30
137,24
LB3
269,62
267,42
319,71
212,36
137,92
LC1
250,54
227,55
243,76
293,71
271,73
272,15
LC2
224,92
209,50
230,45
249,34
268,70
267,27
121,91
LC3
243,22
231,52
260,89
261,11
279,37
320,77
187,31
124,24
Table 10.5b. Measured values of inverse series connected pairs of coils - parameter Lb.
A
Lab [µH]
L A1
LA2
B
L A3
LB1
L B2
C
LB3
L C1
LC2
LA1
LA2
301,09
LA3
273,02
308,54
LB1
200,84
214,97
241,00
LB2
230,25
202,99
230,50
407,12
LB3
252,31
227,02
211,91
354,43
439,41
LC1
202,66
197,90
217,46
201,26
230,52
253,13
LC2
195,70
183,91
199,50
213,99
202,15
226,15
303,63
LC3
217,51
201,62
209,08
242,26
231,54
213,08
277,37
140
308,772
LC3
Table 10.5c. Calculated values of mutual inductances between each pair of coils - parameter Lm.
A
L m [µH]
L A1
LA2
B
L A3
LB1
C
L B2
LB3
L C1
LC2
LC3
LA1
LA2
44,75
LA3
21,64
46,26
LB1
-22,45
-8,53
-4,60
LB2
-9,69
-16,49
-11,95
67,47
LB3
-4,33
-10,10
-26,95
35,52
75,37
LC1
-11,97
-7,41
-6,57
-23,11
-10,30
-4,75
LC2
-7,31
-6,40
-7,74
-8,84
-16,64
-10,28
45,43
LC3
-6,43
-7,47
-12,95
-4,71
-11,96
-26,92
22,52
46,13
C
10.A.7 GROUND CAPACITANCE TEST (Cgr parameter)
+
L
R
A1
CG LA1-A2
Using LCR Meter
A1
A1
CG
L
A2
With the use of an LCR meter, its nodes are applied between terminal of the coil and a ground, both to
right and left side of each coil. All coils apart from the one which is currently measured are grounded
from both sides. There are eighteen measurements in total.
Fig. 10.A.6. Test setup for ground capacitance measurements.
Table 10.6. Measured values of ground capacitances using LCR meter.
Capacitance [pF]
A-L
A-R
B-L
B-R
C-L
C-R
1
3,82
3,51
4,92
4,92
4,41
4,41
2
3,92
3,92
4,31
4,31
3,49
3,49
3
3,41
3,41
3,78
3,78
3,30
3,30
141
10.A.8 COIL IMPEDANCE TESTS (ZCoil parameter)
The impedance measurement process of the device under test (DUT), will measure the voltage across
the DUT and the phase angle between the measured V and I. The four terminals of the LCR meter, two
current and two for voltage will be connected as per the diagram below.
Fig. 10.A.7. Test circuit for individual coil impedance.
A typical LCR meter has four terminals labeled IH, IL, PH and PL. The IH/IL pair is for the generator and
current measurement and the PH/PL pair is for the voltage measurement. LCR meter used in tests has
function that automatically displays impedance values. They are gathered in the table below:
Table 10.7. Measured values of impedances using LCR meter.
A
B
Impedance
magnitude
angle
magnitude
angle
ZCoil (Ω)
(mΩ)
(0)
(mΩ)
(0)
C
magnitude
(mΩ)
angle
(0)
1
46,3
85,8
54,8
54,6
38,8
83,9
2
40,5
86,5
55,8
55,4
40,6
86,7
3
48,38
85,4
61,6
61,1
36,8
83,5
Reference:
[1] Van den Bossche, Alex. Cekov Valchev, Vencislav. Inductors and Transformers For Power Electronics. Boca
Raton, FL, USA: Taylor & Francis Group
[2] All circuit models were made in “Tina for Windows – the Complete Electronics Lab” software made by
DesignSoft:
http:// www.designsoftware.com
[3] Three-phase line reactor model was adapted from online MATLAB library – 3-winding 3-phase transformer
model:
http://www.kxcad.net/cae_MATLAB/toolbox/physmod/powersys/ref/threephasetransformerinductancematrixtypeth
reewindings.html
[4] Quad Tech. LCR Primer Measurement – Instruction Manual. Massachusetts USA.
http:// www.quadtech.com
142
B. VER_1.m file source code
% THREE-PHASE VOLTAGE SOURCE VECTOR
% Values (amplitude and phase) for this vector should be adequate to the ones used in model
simulation
V = 1*[1 1*exp(-2*pi*i/3) 1*exp(2*pi*i/3)];
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
%CONSTANTS
%..........................................................................
%impedance matrix components:
% 60 Hz
Z60(1,1)=0.007486000+0.015374325*i;
Z60(1,2)=-0.000555275-0.003374875*i;
Z60(1,3)=-0.000373350-0.001766050*i;
Z60(2,1)=-0.000276925-0.002607270*i;
Z60(2,2)=0.011999925+0.018487000*i;
Z60(2,3)=-0.000615600-0.002912225*i;
Z60(3,1)=-0.000254125-0.002393050*i;
Z60(3,2)=-0.000452200-0.002749775*i;
Z60(3,3)=0.012956575+0.013723225*i;
%..........................................................................
% Symmetrical component matrix:
A = [1 1 1; 1 1*exp(2*pi*i/3) 1*exp(-2*pi*i/3); 1 1*exp(-2*pi*i/3) 1*exp(2*pi*i/3)];
% CURRENT CALCULATIONS - frequency range: 60Hz - 20kHz
I60 = V*Z60^-1;
%VOLTAGE TRANSFORMATION FROM PHASE- TO SEQUENCE COMPONENTS
V0_1_2 = (1/3)*V*A;
%CURRENT TRANSFORMATION FROM PHASE- TO SEQUENCE COMPONENTS
I0_1_2_60 = (1/3)*I60*A;
%SEQUENCE IMPEDANCE CALCULATION - USING EQUATION: Z = V/I
%zero-sequence impedance:
Z0_60 = V0_1_2(1,1)/I0_1_2_60(1,1);
%negative-sequence impedance:
Z1_60 = V0_1_2(1,3)/I0_1_2_60(1,3);
%positive-sequence impedance:
Z2_60 = V0_1_2(1,2)/I0_1_2_60(1,2);
%..........................................................................
L= Z2_60/(120*pi);
%.........................................................................
C. VER_2.m file source code
% THREE-PHASE VOLTAGE SOURCE VECTOR
% Values (amplitude and phase) for this vector should be adequate to the ones used in model
simulation
V = 800*[1*exp(0*pi*i/180) 1*exp(-120*pi*i/180) 1*exp(120*pi*i/180)];
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
%CONSTANTS
%..........................................................................
%impedance matrix components:
% 60 Hz
Z60(1,1)=0.007486000+0.015374325*i;
143
Z60(1,2)=-0.000555275-0.003374875*i;
Z60(1,3)=-0.000373350-0.001766050*i;
Z60(2,1)=-0.000276925-0.002607270*i;
Z60(2,2)=0.011999925+0.018487000*i;
Z60(2,3)=-0.000615600-0.002912225*i;
Z60(3,1)=-0.000254125-0.002393050*i;
Z60(3,2)=-0.000452200-0.002749775*i;
Z60(3,3)=0.012956575+0.013723225*i;
% 100 Hz
Z100(1,1)=0.007296000+0.022359200*i;
Z100(1,2)=0.000212325-0.004838350*i;
Z100(1,3)=0.000522025-0.003138800*i;
Z100(2,1)=0.000176225-0.004493025*i;
Z100(2,2)=0.013650550+0.030625150*i;
Z100(2,3)=-0.000053675-0.004888225*i;
Z100(3,1)=0.000134425-0.002948325*i;
Z100(3,2)=0.000212325-0.004838350*i;
Z100(3,3)=0.012832600+0.022941550*i;
% 200 Hz
Z200(1,1)=0.009244925+0.042393275*i;
Z200(1,2)=0.000365275-0.009679550*i;
Z200(1,3)=0.000724850-0.006415350*i;
Z200(2,1)=-0.000054625-0.004841200*i;
Z200(2,2)=0.018276100+0.056247600*i;
Z200(2,3)=-0.000378100-0.009861475*i;
Z200(3,1)=0.000408975-0.006381150*i;
Z200(3,2)=-0.000123975-0.009871925*i;
Z200(3,3)=0.013872850+0.042769950*i;
% 500 Hz
Z500(1,1)=0.011650800+0.101259075*i;
Z500(1,2)=-0.000313025-0.024912325*i;
Z500(1,3)=0.001513825-0.017163650*i;
Z500(2,1)=-0.000274550-0.024985475*i;
Z500(2,2)=0.024956500+0.130827350*i;
Z500(2,3)=-0.000318725-0.025377825*i;
Z500(3,1)=0.001087750-0.016886725*i;
Z500(3,2)=-0.000956650-0.025361675*i;
Z500(3,3)=0.014309375+0.102857450*i;
% 1000 Hz
Z1000(1,1)=0.019114475+0.196829075*i;
Z1000(1,1)=0.031513400-0.038093100*i;
Z1000(1,1)=0.000558600-0.035620725*i;
Z1000(1,1)=-0.001389375-0.049419475*i;
Z1000(1,1)=0.174648000+0.178344925*i;
Z1000(1,1)=-0.004309200-0.050780825*i;
Z1000(1,1)=0.001656800-0.035343800*i;
Z1000(1,1)=0.031822625-0.038466925*i;
Z1000(1,1)=0.028666250+0.201828450*i;
% 2000 Hz
Z2000(1,1)=0.014911200+0.401890375*i;
Z2000(1,2)=0.015104525-0.098279400*i;
Z2000(1,3)=0.002822925-0.074850500*i;
Z2000(2,1)=-0.006509400-0.102393375*i;
Z2000(2,2)=0.055882800+0.484454400*i;
Z2000(2,3)=-0.006423900-0.102106000*i;
Z2000(3,1)=0.000938125-0.078527950*i;
Z2000(3,2)=-0.006241975-0.102410000*i;
Z2000(3,3)=0.036111875+0.409468525*i;
% 4000 Hz
Z4000(1,1)=0.017511825+0.774051925*i;
Z4000(1,2)=-0.003657025-0.193843225*i;
Z4000(1,3)=0.072539625-0.134602650*i;
Z4000(2,1)=-0.011251325-0.194424625*i;
Z4000(2,2)=0.074011175+0.903350725*i;
Z4000(2,3)=0.085368425-0.182811825*i;
Z4000(3,1)=0.006573525-0.153759400*i;
Z4000(3,2)=-0.011854100-0.200312725*i;
Z4000(3,3)=0.353385750+0.703357675*i;
% 6000 Hz
Z6000(1,1)=-0.002029200+1.073498100
Z6000(1,2)=-0.016597450-0.274392300*i;
Z6000(1,3)=0.006705100-0.228919600*i;
Z6000(2,1)=-0.004622225-0.271660575*i;
Z6000(2,2)=0.151055700+1.244054450*i;
Z6000(2,3)=-0.017903225-0.291235800*i;
Z6000(3,1)=0.006248150-0.220311175*i;
Z6000(3,2)=-0.016963675-0.280444750*i;
144
Z6000(3,3)=0.034021400+1.161555500*i;
% 8000 Hz
Z8000(1,1)=0.119740375+1.419960250*i;
Z8000(1,2)=-0.049967150-0.355606850*i;
Z8000(1,3)=-0.010739750-0.294804475*i;
Z8000(2,1)=-0.020428325-0.361153425*i;
Z8000(2,2)=0.292878825+1.588221400*i;
Z8000(2,3)=-0.035878175-0.369930950*i;
Z8000(3,1)=-0.004830750-0.295922150*i;
Z8000(3,2)=-0.044160750-0.367859000*i;
Z8000(3,3)=0.151708825+1.450420575*i;
% 10000 Hz
Z10000(1,1)=0.077048325+1.821045025*i;
Z10000(1,2)=-0.048794375-0.436137875*i;
Z10000(1,3)=-0.004418925-0.402271800*i;
Z10000(2,1)=-0.003589575-0.458416325*i;
Z10000(2,2)=0.126141950+1.957897275*i;
Z10000(2,3)=-0.041781000-0.482889750*i;
Z10000(3,1)=0.006659975-0.386570675*i;
Z10000(3,2)=-0.039165175-0.452656475*i;
Z10000(3,3)=0.224378125+1.950144325*i;
% 12000 Hz
Z12000(1,1)=0.004234150+2.245450400*i;
Z12000(1,2)=-0.015804675-0.558600000*i;
Z12000(1,3)=-0.025225825-0.498847375*i;
Z12000(2,1)=0.001058775-0.561362600*i;
Z12000(2,2)=0.150593050+2.416881700*i;
Z12000(2,3)=-0.044376875-0.605592700*i;
Z12000(3,1)=0.029539300-0.474080400*i;
Z12000(3,2)=-0.015804675-0.558600000*i;
Z12000(3,3)=0.151924950+2.424109775*i;
% 14000 Hz
Z14000(1,1)=-0.294921325+2.774579500*i;
Z14000(1,2)=-0.093846225-0.691104575*i;
Z14000(1,3)=-0.108854325-0.699021875*i;
Z14000(2,1)=0.014421000-0.817005700*i;
Z14000(2,2)=0.386144600+3.457376325*i;
Z14000(2,3)=-0.100213600-0.836215175*i;
Z14000(3,1)=0.020743725-0.391408550*i;
Z14000(3,2)=-0.079984775-0.798822700*i;
Z14000(3,3)=0.164400350+2.892494925*i;
% 16000 Hz
Z16000(1,1)=0.005346125+3.037410300*i;
Z16000(1,2)=-0.161920375-0.665232750*i;
Z16000(1,3)=-0.065542400-0.677671100*i;
Z16000(2,1)=-0.029144100-0.757177550*i;
Z16000(2,2)=0.495746575+3.436843500*i;
Z16000(2,3)=-0.046282575-0.822031675*i;
Z16000(3,1)=-0.037458500-0.638698300*i;
Z16000(3,2)=-0.131713225-0.824895450*i;
Z16000(3,3)=0.145489175+3.290118375*i;
% 18000 Hz
Z18000(1,1)=-0.135588750+3.293128450*i;
Z18000(1,2)=-0.116495650-0.911943950*i;
Z18000(1,3)=-0.081532325-0.775727250*i;
Z18000(2,1)=-0.033498425-0.813596150*i;
Z18000(2,2)=0.395045150+3.964236000*i;
Z18000(2,3)=-0.076246525-0.926869400*i;
Z18000(3,1)=-0.045155400-0.706211000*i;
Z18000(3,2)=-0.095822700-0.914347450*i;
Z18000(3,3)=0.196184025+3.644723925*i;
% 20000 Hz
Z20000(1,1)=-0.046687275+3.715053375*i;
Z20000(1,2)=-0.084013250-0.990064350*i;
Z20000(1,3)=-0.088964650-0.783261700*i;
Z20000(2,1)=-0.011701150-0.931114475*i;
Z20000(2,2)=0.721551600+4.302155750*i;
Z20000(2,3)=-0.084346700-0.956394450*i;
Z20000(3,1)=-0.030133525-0.798936700*i;
Z20000(3,2)=-0.111471575-1.011734800*i;
Z20000(3,3)=0.244315300+3.883279375*i;
%..........................................................................
% Symmetrical component matrix:
A = [1 1 1; 1 -0.5+0.866025*i -0.5-0.866025*i; 1 -0.5-0.866025*i -0.5+0.866025*i];
% CURRENT CALCULATIONS - frequency range: 60Hz - 20kHz
145
I60 = V*Z60^-1;
I100 = V*Z100^-1;
I200 = V*Z200^-1;
I500 = V*Z500^-1;
I1000 = V*Z1000^-1;
I2000 = V*Z2000^-1;
I4000 = V*Z4000^-1;
I6000 = V*Z6000^-1;
I8000 = V*Z8000^-1;
I10000 = V*Z10000^-1;
I12000 = V*Z12000^-1;
I14000 = V*Z14000^-1;
I16000 = V*Z16000^-1;
I18000 = V*Z18000^-1;
I20000 = V*Z20000^-1;
%VOLTAGE TRANSFORMATION FROM PHASE- TO SEQUENCE COMPONENTS
V0_1_2 = (1/3)*V*A;
%CURRENT TRANSFORMATION FROM PHASE- TO SEQUENCE COMPONENTS
I0_1_2_60 = (1/3)*I60*A;
I0_1_2_100 = (1/3)*I100*A;
I0_1_2_200 = (1/3)*I200*A;
I0_1_2_500 = (1/3)*I500*A;
I0_1_2_1000 = (1/3)*I1000*A;
I0_1_2_2000 = (1/3)*I2000*A;
I0_1_2_4000 = (1/3)*I4000*A;
I0_1_2_6000 = (1/3)*I6000*A;
I0_1_2_8000 = (1/3)*I8000*A;
I0_1_2_10000 = (1/3)*I10000*A;
I0_1_2_12000 = (1/3)*I12000*A;
I0_1_2_14000 = (1/3)*I14000*A;
I0_1_2_16000 = (1/3)*I16000*A;
I0_1_2_18000 = (1/3)*I18000*A;
I0_1_2_20000 = (1/3)*I20000*A;
%SEQUENCE IMPEDANCE CALCULATION - USING EQUATION: Z = V/I
%zero-sequence impedance:
Z0_60 = V0_1_2(1,1)/I0_1_2_60(1,1);
Z0_100 = V0_1_2(1,1)/I0_1_2_100(1,1);
Z0_200 = V0_1_2(1,1)/I0_1_2_200(1,1);
Z0_500 = V0_1_2(1,1)/I0_1_2_500(1,1);
Z0_1000 = V0_1_2(1,1)/I0_1_2_1000(1,1);
Z0_2000 = V0_1_2(1,1)/I0_1_2_2000(1,1);
Z0_4000 = V0_1_2(1,1)/I0_1_2_4000(1,1);
Z0_6000 = V0_1_2(1,1)/I0_1_2_6000(1,1);
Z0_8000 = V0_1_2(1,1)/I0_1_2_8000(1,1);
Z0_10000 = V0_1_2(1,1)/I0_1_2_10000(1,1);
Z0_12000 = V0_1_2(1,1)/I0_1_2_12000(1,1);
Z0_14000 = V0_1_2(1,1)/I0_1_2_14000(1,1);
Z0_16000 = V0_1_2(1,1)/I0_1_2_16000(1,1);
Z0_18000 = V0_1_2(1,1)/I0_1_2_18000(1,1);
Z0_20000 = V0_1_2(1,1)/I0_1_2_20000(1,1);
%positive-sequence impedance:
Z1_60 = V0_1_2(1,3)/I0_1_2_60(1,3);
Z1_100 = V0_1_2(1,3)/I0_1_2_100(1,3);
Z1_200 = V0_1_2(1,3)/I0_1_2_200(1,3);
Z1_500 = V0_1_2(1,3)/I0_1_2_500(1,3);
Z1_1000 = V0_1_2(1,3)/I0_1_2_1000(1,3);
Z1_2000 = V0_1_2(1,3)/I0_1_2_2000(1,3);
Z1_4000 = V0_1_2(1,3)/I0_1_2_4000(1,3);
Z1_6000 = V0_1_2(1,3)/I0_1_2_6000(1,3);
Z1_8000 = V0_1_2(1,3)/I0_1_2_8000(1,3);
Z1_10000 = V0_1_2(1,3)/I0_1_2_10000(1,3);
Z1_12000 = V0_1_2(1,3)/I0_1_2_12000(1,3);
Z1_14000 = V0_1_2(1,3)/I0_1_2_14000(1,3);
Z1_16000 = V0_1_2(1,3)/I0_1_2_16000(1,3);
Z1_18000 = V0_1_2(1,3)/I0_1_2_18000(1,3);
Z1_20000 = V0_1_2(1,3)/I0_1_2_20000(1,3);
%negative-sequence impedance:
Z2_60 = V0_1_2(1,2)/I0_1_2_60(1,2);
Z2_100 = V0_1_2(1,2)/I0_1_2_100(1,2);
146
Z2_200 = V0_1_2(1,2)/I0_1_2_200(1,2);
Z2_500 = V0_1_2(1,2)/I0_1_2_500(1,2);
Z2_1000 = V0_1_2(1,2)/I0_1_2_1000(1,2);
Z2_2000 = V0_1_2(1,2)/I0_1_2_2000(1,2);
Z2_4000 = V0_1_2(1,2)/I0_1_2_4000(1,2);
Z2_6000 = V0_1_2(1,2)/I0_1_2_6000(1,2);
Z2_8000 = V0_1_2(1,2)/I0_1_2_8000(1,2);
Z2_10000 = V0_1_2(1,2)/I0_1_2_10000(1,2);
Z2_12000 = V0_1_2(1,2)/I0_1_2_12000(1,2);
Z2_14000 = V0_1_2(1,2)/I0_1_2_14000(1,2);
Z2_16000 = V0_1_2(1,2)/I0_1_2_16000(1,2);
Z2_18000 = V0_1_2(1,2)/I0_1_2_18000(1,2);
Z2_20000 = V0_1_2(1,2)/I0_1_2_20000(1,2);
%..........................................................................
%SEQUENCE IMPEDANCE AND FREQUENCY VECTORS:
Z0 = [Z0_60 Z0_100 Z0_200 Z0_500 Z0_1000 Z0_2000 Z0_4000 Z0_6000 Z0_8000 Z0_10000 Z0_12000
Z0_14000 Z0_16000 Z0_18000 Z0_20000];
Z1 = [Z1_60 Z1_100 Z1_200 Z1_500 Z1_1000 Z1_2000 Z1_4000 Z1_6000 Z1_8000 Z1_10000 Z1_12000
Z1_14000 Z1_16000 Z1_18000 Z1_20000];
Z2 = [Z2_60 Z2_100 Z2_200 Z2_500 Z2_1000 Z2_2000 Z2_4000 Z2_6000 Z2_8000 Z2_10000 Z2_12000
Z2_14000 Z2_16000 Z2_18000 Z2_20000];
f = [60 100 200 500 1000 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000];
w = 2*pi*f;
%MODEL VALUES:
Z0_M_mag = [5.394e-16 1.459e-15 2.608e-15 2.480e-14 1.837e-14 4.637e-14 7.356e-13 1.021e-13
2.519e-13 2.098e-13 5.956e-13 6.547e-13 1.653e-12 1.499e-13 1.675e-12];
Z1_M_mag = [0.0258 0.04928 0.09853 0.2463 0.4925 0.9852 1.971 2.960 3.952 4.948 5.951 6.659
7.977 9.004 10.04];
Z2_M_mag = [3.628e-8 2.548e-15 2.881e-15 1.546e-14 7.005e-15 2.199e-13 1.005e-12 6.336e-10
4.881e-13 2.169e-13 1.014e-8 8.882e-9 1.377e-12 5.338e-8 2.564e-12];
Z0_M_angle = [2.583e1 2.822e1 3.026e1 3.139e1 3.175e1 3.195e1 3.204e1 3.207e1 3.208e1 3.2074e1
3.2072e1 3.205e1 3.204e1 3.20309e1 3.2011e1]*pi/180;
Z1_M_angle = [87.67 88.59 89.3 89.72 89.86 8.9929e1 8.9965e1 8.9978e1 8.9982e1 8.9986e1
8.9997e1 8.999199298994489e1 8.999101e1 8.99974e1 8.999298e1]*pi/180;
Z2_M_angle = [1.428e2 1.450e2 1.462e2 1.471e2 1.474e2 1.4751e2 1.4759e2 1.4764e2 1.4765e2
1.4768e2 1.4776e2 1.478e2 1.477e2 1.47850e2 1.47859e2]*pi/180;
Z0_M = abs(Z0_M_mag).*exp(j*Z0_M_angle);
Z1_M = abs(Z1_M_mag).*exp(j*Z1_M_angle);
Z2_M = abs(Z2_M_mag).*exp(j*Z2_M_angle);
%INVOKING INVFREQS() FUNCTION:
[b0, a0] = invfreqs(Z0, w,'complex', 30, 28);
[b1, a1] = invfreqs(Z1, w,'complex', 30, 29);
[b2, a2] = invfreqs(Z2, w,'complex', 30, 28);
[b0_M, a0_M] = invfreqs(Z0_M, w,'complex', 30, 28);
[b1_M, a1_M] = invfreqs(Z1_M, w,'complex', 30, 28);
[b2_M, a2_M] = invfreqs(Z2_M, w,'complex', 30, 28);
%COMPUTING TRANSFER FUNCTIONS:
Z0_f = tf(b0,a0);
Z1_f = tf(b1,a1);
Z2_f = tf(b2,a2);
Z0_f_M = tf(b0_M,a0_M);
Z1_f_M = tf(b1_M,a1_M);
Z2_f_M = tf(b2_M,a2_M);
%PRESENTING BODE PLOTS:
bode(Z0_f, Z0_f_M)
bode(Z1_f, Z1_f_M)
bode(Z2_f, Z2_f_M)
%..........................................................................
D. VER_3.m file source code
% THREE-PHASE VOLTAGE SOURCE VECTOR
% Values (amplitude and phase) for this vector should be adequate to the ones used in model
simulation
V = 800*[0*exp(0*pi*i/180) 1*exp(-120*pi*i/180) 1*exp(120*pi*i/180)];
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
147
%CONSTANTS
%..........................................................................
%impedance matrix components:
% 60 Hz
Z60(1,1)=0.0113472 + 0.02330424i;
Z60(1,2)=-0.00084168 - 0.0051156i;
Z60(1,3)=-0.00056592 - 0.00267696i;
Z60(2,1)=-0.00041976 - 0.00395208i;
Z60(2,2)=0.01818936 + 0.0280224i;
Z60(2,3)=-0.00093312 - 0.00441432i;
Z60(3,1)=-0.0003852 - 0.00362736i;
Z60(3,2)=-0.00068544 - 0.00416808i;
Z60(3,3)=0.01963944 + 0.02080152i;
% 100 Hz
Z100(1,1)=0.0110592 + 0.03389184i;
Z100(1,2)=0.00032184 - 0.00733392i;
Z100(1,3)=0.00079128 - 0.00475776i;
Z100(2,1)=0.00026712 - 0.00681048i;
Z100(2,2)=0.02069136 + 0.04642128i;
Z100(2,3)=-0.00008136 - 0.00740952i;
Z100(3,1)=0.00020376 - 0.00446904i;
Z100(3,2)=0.00032184 - 0.00733392i;
Z100(3,3)=0.01945152 + 0.03477456i;
% 200 Hz
Z200(1,1)=0.01945152 + 0.03477456i;
Z200(1,2)=0.00055368 - 0.01467216i;
Z200(1,3)=0.00109872 - 0.00972432i;
Z200(2,1)=-0.0000828 - 0.00733824i;
Z200(2,2)=0.02770272 + 0.08525952i;
Z200(2,3)=-0.00057312 - 0.01494792i;
Z200(3,1)=0.00061992 - 0.00967248i;
Z200(3,2)=-0.00018792 - 0.01496376i;
Z200(3,3)=0.02102832 + 0.06483024i;
% 500 Hz
Z500(1,1)=0.01766016 + 0.15348744i;
Z500(1,2)=-0.00047448 - 0.03776184i;
Z500(1,3)=0.00229464 - 0.02601648i;
Z500(2,1)=-0.00041616 - 0.03787272i;
Z500(2,2)=0.0378288 + 0.19830672i;
Z500(2,3)=-0.00048312 - 0.03846744i;
Z500(3,1)=0.0016488 - 0.02559672i;
Z500(3,2)=-0.00145008 - 0.03844296i;
Z500(3,3)=0.02169 + 0.15591024i;
% 1000 Hz
Z1000(1,1)=0.02897352 + 0.29835144i;
Z1000(1,1)=0.04776768 - 0.05774112i;
Z1000(1,1)=0.00084672 - 0.05399352i;
Z1000(1,1)=-0.002106 - 0.07490952i;
Z1000(1,1)=0.2647296 + 0.27033336i;
Z1000(1,1)=-0.00653184 - 0.07697304i;
Z1000(1,1)=0.00251136 - 0.05357376i;
Z1000(1,1)=0.0482364 - 0.05830776i;
Z1000(1,1)=0.043452 + 0.30592944i;
% 2000 Hz
Z2000(1,1)=0.02260224 + 0.6091812i;
Z2000(1,2)=0.02289528 - 0.14897088i;
Z2000(1,3)=0.00427896 - 0.1134576i;
Z2000(2,1)=-0.00986688 - 0.1552068i;
Z2000(2,2)=0.08470656 + 0.73433088i;
Z2000(2,3)=-0.00973728 - 0.1547712i;
Z2000(3,1)=0.001422 - 0.11903184i;
Z2000(3,2)=-0.00946152 - 0.155232i;
Z2000(3,3)=0.054738 + 0.62066808i;
% 4000 Hz
Z4000(1,1)=0.02654424 + 1.17329976i;
Z4000(1,2)=-0.00554328 - 0.29382552i;
Z4000(1,3)=0.1099548 - 0.20402928i;
Z4000(2,1)=-0.01705464 - 0.2947068i;
Z4000(2,2)=0.11218536 + 1.36928952i;
Z4000(2,3)=0.12940056 - 0.27710424i;
Z4000(3,1)=0.00996408 - 0.23306688i;
Z4000(3,2)=-0.01796832 - 0.30363192i;
Z4000(3,3)=0.5356584 + 1.06614216i;
% 6000 Hz
Z6000(1,1)=-0.00307584 + 1.62719712i;
Z6000(1,2)=-0.02515824 - 0.41592096i;
148
Z6000(1,3)=0.01016352 - 0.34699392i;
Z6000(2,1)=-0.00700632 - 0.41178024i;
Z6000(2,2)=0.22896864 + 1.88572464i;
Z6000(2,3)=-0.02713752 - 0.44145216i;
Z6000(3,1)=0.00947088 - 0.33394536i;
Z6000(3,2)=-0.02571336 - 0.4250952i;
Z6000(3,3)=0.05156928 + 1.7606736i;
% 8000 Hz
Z8000(1,1)=0.1815012 + 2.1523608i;
Z8000(1,2)=-0.07573968 - 0.53902512i;
Z8000(1,3)=-0.0162792 - 0.44686152i;
Z8000(2,1)=-0.03096504 - 0.54743256i;
Z8000(2,2)=0.44394264 + 2.40740928i;
Z8000(2,3)=-0.05438376 - 0.56073744i;
Z8000(3,1)=-0.0073224 - 0.44855568i;
Z8000(3,2)=-0.0669384 - 0.5575968i;
Z8000(3,3)=0.22995864 + 2.19853224i;
% 10000 Hz
Z10000(1,1)=0.11678904 + 2.76032088i;
Z10000(1,2)=-0.073962 - 0.6610932i;
Z10000(1,3)=-0.00669816 - 0.60975936i;
Z10000(2,1)=-0.00544104 - 0.69486264i;
Z10000(2,2)=0.19120464 + 2.96776008i;
Z10000(2,3)=-0.0633312 - 0.7319592i;
Z10000(3,1)=0.01009512 - 0.58595976i;
Z10000(3,2)=-0.05936616 - 0.68613192i;
Z10000(3,3)=0.34011 + 2.95600824i;
% 12000 Hz
Z12000(1,1)=0.00641808 + 3.40363008i;
Z12000(1,2)=-0.02395656 - 0.84672i;
Z12000(1,3)=-0.03823704 - 0.7561476i;
Z12000(2,1)=0.00160488 - 0.85090752i;
Z12000(2,2)=0.22826736 + 3.66348384i;
Z12000(2,3)=-0.067266 - 0.91795104i;
Z12000(3,1)=0.04477536 - 0.71860608i;
Z12000(3,2)=-0.02395656 - 0.84672i;
Z12000(3,3)=0.23028624 + 3.67444008i;
% 14000 Hz
Z14000(1,1)=-0.44703864 + 4.2056784i;
Z14000(1,2)=-0.14225112 - 1.04756904i;
Z14000(1,3)=-0.16500024 - 1.05957i;
Z14000(2,1)=0.0218592 - 1.23840864i;
Z14000(2,2)=0.58531392 + 5.24065464i;
Z14000(2,3)=-0.15190272 - 1.26752616i;
Z14000(3,1)=0.03144312 - 0.59329296i;
Z14000(3,2)=-0.12124008 - 1.21084704i;
Z14000(3,3)=0.24919632 + 4.38441336i;
% 16000 Hz
Z16000(1,1)=0.0081036 + 4.60407456i;
Z16000(1,2)=-0.2454372 - 1.0083528i;
Z16000(1,3)=-0.09934848 - 1.02720672i;
Z16000(2,1)=-0.04417632 - 1.14772176i;
Z16000(2,2)=0.75144744 + 5.2095312i;
Z16000(2,3)=-0.07015464 - 1.24602696i;
Z16000(3,1)=-0.0567792 - 0.96813216i;
Z16000(3,2)=-0.19964952 - 1.25036784i;
Z16000(3,3)=0.22053096 + 4.9871268i;
% 18000 Hz
Z18000(1,1)=-0.205524 + 4.99168944i;
Z18000(1,2)=-0.17658288 - 1.38231504i;
Z18000(1,3)=-0.12358584 - 1.1758392i;
Z18000(2,1)=-0.05077656 - 1.23324048i;
Z18000(2,2)=0.59880528 + 6.0089472i;
Z18000(2,3)=-0.11557368 - 1.40493888i;
Z18000(3,1)=-0.06844608 - 1.0704672i;
Z18000(3,2)=-0.14524704 - 1.38595824i;
Z18000(3,3)=0.29737368 + 5.52463416i;
% 20000 Hz
Z20000(1,1)=-0.07076808 + 5.631238799999999i;
Z20000(1,2)=-0.1273464 - 1.50072912i;
Z20000(1,3)=-0.13485168 - 1.18725984i;
Z20000(2,1)=-0.01773648 - 1.41137352i;
Z20000(2,2)=1.09372032 + 6.521162399999999i;
Z20000(2,3)=-0.12785184 - 1.44969264i;
Z20000(3,1)=-0.04567608 - 1.21101984i;
Z20000(3,2)=-0.16896744 - 1.53357696i;
Z20000(3,3)=0.37033056 + 5.886234i;
149
%..........................................................................
% Symmetrical component matrix:
A = [1 1 1; 1 1*exp(120*pi*i/180) 1*exp(-120*pi*i/180); 1 1*exp(-120*pi*i/180)
1*exp(120*pi*i/180)];
% CURRENT CALCULATIONS - frequency range: 60Hz - 20kHz
I60 = V*Z60^-1;
I100 = V*Z100^-1;
I200 = V*Z200^-1;
I500 = V*Z500^-1;
I1000 = V*Z1000^-1;
I2000 = V*Z2000^-1;
I4000 = V*Z4000^-1;
I6000 = V*Z6000^-1;
I8000 = V*Z8000^-1;
I10000 = V*Z10000^-1;
I12000 = V*Z12000^-1;
I14000 = V*Z14000^-1;
I16000 = V*Z16000^-1;
I18000 = V*Z18000^-1;
I20000 = V*Z20000^-1;
%VOLTAGE TRANSFORMATION FROM PHASE- TO SEQUENCE COMPONENTS
V0_1_2 = (1/3)*V*A;
%CURRENT TRANSFORMATION FROM PHASE- TO SEQUENCE COMPONENTS
I0_1_2_60 = (1/3)*I60*A;
I0_1_2_100 = (1/3)*I100*A;
I0_1_2_200 = (1/3)*I200*A;
I0_1_2_500 = (1/3)*I500*A;
I0_1_2_1000 = (1/3)*I1000*A;
I0_1_2_2000 = (1/3)*I2000*A;
I0_1_2_4000 = (1/3)*I4000*A;
I0_1_2_6000 = (1/3)*I6000*A;
I0_1_2_8000 = (1/3)*I8000*A;
I0_1_2_10000 = (1/3)*I10000*A;
I0_1_2_12000 = (1/3)*I12000*A;
I0_1_2_14000 = (1/3)*I14000*A;
I0_1_2_16000 = (1/3)*I16000*A;
I0_1_2_18000 = (1/3)*I18000*A;
I0_1_2_20000 = (1/3)*I20000*A;
%SEQUENCE IMPEDANCE CALCULATION - USING EQUATION: Z = V/I
%zero-sequence impedance:
Z0_60 = V0_1_2(1,1)/I0_1_2_60(1,1);
Z0_100 = V0_1_2(1,1)/I0_1_2_100(1,1);
Z0_200 = V0_1_2(1,1)/I0_1_2_200(1,1);
Z0_500 = V0_1_2(1,1)/I0_1_2_500(1,1);
Z0_1000 = V0_1_2(1,1)/I0_1_2_1000(1,1);
Z0_2000 = V0_1_2(1,1)/I0_1_2_2000(1,1);
Z0_4000 = V0_1_2(1,1)/I0_1_2_4000(1,1);
Z0_6000 = V0_1_2(1,1)/I0_1_2_6000(1,1);
Z0_8000 = V0_1_2(1,1)/I0_1_2_8000(1,1);
Z0_10000 = V0_1_2(1,1)/I0_1_2_10000(1,1);
Z0_12000 = V0_1_2(1,1)/I0_1_2_12000(1,1);
Z0_14000 = V0_1_2(1,1)/I0_1_2_14000(1,1);
Z0_16000 = V0_1_2(1,1)/I0_1_2_16000(1,1);
Z0_18000 = V0_1_2(1,1)/I0_1_2_18000(1,1);
Z0_20000 = V0_1_2(1,1)/I0_1_2_20000(1,1);
%positive-sequence impedance:
Z1_60 = V0_1_2(1,3)/I0_1_2_60(1,3);
Z1_100 = V0_1_2(1,3)/I0_1_2_100(1,3);
Z1_200 = V0_1_2(1,3)/I0_1_2_200(1,3);
Z1_500 = V0_1_2(1,3)/I0_1_2_500(1,3);
Z1_1000 = V0_1_2(1,3)/I0_1_2_1000(1,3);
Z1_2000 = V0_1_2(1,3)/I0_1_2_2000(1,3);
Z1_4000 = V0_1_2(1,3)/I0_1_2_4000(1,3);
Z1_6000 = V0_1_2(1,3)/I0_1_2_6000(1,3);
Z1_8000 = V0_1_2(1,3)/I0_1_2_8000(1,3);
Z1_10000 = V0_1_2(1,3)/I0_1_2_10000(1,3);
Z1_12000 = V0_1_2(1,3)/I0_1_2_12000(1,3);
Z1_14000 = V0_1_2(1,3)/I0_1_2_14000(1,3);
150
Z1_16000 = V0_1_2(1,3)/I0_1_2_16000(1,3);
Z1_18000 = V0_1_2(1,3)/I0_1_2_18000(1,3);
Z1_20000 = V0_1_2(1,3)/I0_1_2_20000(1,3);
%negative-sequence impedance:
Z2_60 = V0_1_2(1,2)/I0_1_2_60(1,2);
Z2_100 = V0_1_2(1,2)/I0_1_2_100(1,2);
Z2_200 = V0_1_2(1,2)/I0_1_2_200(1,2);
Z2_500 = V0_1_2(1,2)/I0_1_2_500(1,2);
Z2_1000 = V0_1_2(1,2)/I0_1_2_1000(1,2);
Z2_2000 = V0_1_2(1,2)/I0_1_2_2000(1,2);
Z2_4000 = V0_1_2(1,2)/I0_1_2_4000(1,2);
Z2_6000 = V0_1_2(1,2)/I0_1_2_6000(1,2);
Z2_8000 = V0_1_2(1,2)/I0_1_2_8000(1,2);
Z2_10000 = V0_1_2(1,2)/I0_1_2_10000(1,2);
Z2_12000 = V0_1_2(1,2)/I0_1_2_12000(1,2);
Z2_14000 = V0_1_2(1,2)/I0_1_2_14000(1,2);
Z2_16000 = V0_1_2(1,2)/I0_1_2_16000(1,2);
Z2_18000 = V0_1_2(1,2)/I0_1_2_18000(1,2);
Z2_20000 = V0_1_2(1,2)/I0_1_2_20000(1,2);
%..........................................................................
%SEQUENCE IMPEDANCE AND FREQUENCY VECTORS:
Z0 = [Z0_60 Z0_100 Z0_200 Z0_500 Z0_1000 Z0_2000 Z0_4000 Z0_6000 Z0_8000 Z0_10000 Z0_12000
Z0_14000 Z0_16000 Z0_18000 Z0_20000];
Z1 = [Z1_60 Z1_100 Z1_200 Z1_500 Z1_1000 Z1_2000 Z1_4000 Z1_6000 Z1_8000 Z1_10000 Z1_12000
Z1_14000 Z1_16000 Z1_18000 Z1_20000];
Z2 = [Z2_60 Z2_100 Z2_200 Z2_500 Z2_1000 Z2_2000 Z2_4000 Z2_6000 Z2_8000 Z2_10000 Z2_12000
Z2_14000 Z2_16000 Z2_18000 Z2_20000];
f = [60 100 200 500 1000 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000];
w = 2*pi*f;
%MODEL VALUES:
Z0_M_mag = [5.394e-16 1.459e-15 2.608e-15 2.480e-14 1.837e-14 4.637e-14 7.356e-13 1.021e-13
2.519e-13 2.098e-13 5.956e-13 6.547e-13 1.653e-12 1.499e-13 1.675e-12];
Z1_M_mag = [0.0258 0.04928 0.09853 0.2463 0.4925 0.9852 1.971 2.960 3.952 4.948 5.951 6.659
7.977 9.004 10.04];
Z2_M_mag = [3.628e-8 2.548e-15 2.881e-15 1.546e-14 7.005e-15 2.199e-13 1.005e-12 6.336e-10
4.881e-13 2.169e-13 1.014e-8 8.882e-9 1.377e-12 5.338e-8 2.564e-12];
Z0_M_angle = [2.583e1 2.822e1 3.026e1 3.139e1 3.175e1 3.195e1 3.204e1 3.207e1 3.208e1 3.2074e1
3.2072e1 3.205e1 3.204e1 3.20309e1 3.2011e1]*pi/180;
Z1_M_angle = [87.67 88.59 89.3 89.72 89.86 8.9929e1 8.9965e1 8.9978e1 8.9982e1 8.9986e1
8.9997e1 8.999199298994489e1 8.999101e1 8.99974e1 8.999298e1]*pi/180;
Z2_M_angle = [1.428e2 1.450e2 1.462e2 1.471e2 1.474e2 1.4751e2 1.4759e2 1.4764e2 1.4765e2
1.4768e2 1.4776e2 1.478e2 1.477e2 1.47850e2 1.47859e2]*pi/180;
Z0_M = abs(Z0_M_mag).*exp(j*Z0_M_angle);
Z1_M = abs(Z1_M_mag).*exp(j*Z1_M_angle);
Z2_M = abs(Z2_M_mag).*exp(j*Z2_M_angle);
%INVOKING INVFREQS() FUNCTION:
[b0, a0] = invfreqs(Z0, w,'complex', 30, 28);
[b1, a1] = invfreqs(Z1, w,'complex', 30, 27);
[b2, a2] = invfreqs(Z2, w,'complex', 30, 28);
[b0_M, a0_M] = invfreqs(Z0_M, w,'complex', 30, 28);
[b1_M, a1_M] = invfreqs(Z1_M, w,'complex', 30, 28);
[b2_M, a2_M] = invfreqs(Z2_M, w,'complex', 30, 28);
%COMPUTING TRANSFER FUNCTIONS:
Z0_f = tf(b0,a0);
Z1_f = tf(b1,a1);
Z2_f = tf(b2,a2);
Z0_f_M = tf(b0_M,a0_M);
Z1_f_M = tf(b1_M,a1_M);
Z2_f_M = tf(b2_M,a2_M);
%PRESENTING BODE PLOTS:
bode(Z0_f, Z0_f_M)
bode(Z1_f, Z2_f_M)
bode(Z2_f, Z1_f_M)
%..........................................................................
151
E. Contents of the CD-ROM
Folders:
-
Articles: contains articles that were used in project as source of informationavailable on the
internet.
-
M-files: contains verification files: VER_1.m, VER_2.m and VER_3.m
-
MATLAB_sim: contains all MATLAB file models used for simulations saved in .mdl format.
File number corresponds to the simulation case of “Computer Simulation” Chapter.
-
MATLAB_plots: contains all simulation plots saved in .png format.
Folder number corresponds to the simulation case of “Computer Simulation” Chapter.
Files:
-
Project.pdf – contains all contents of this work in .pdf format
-
Project.doc – contains all contents of this work in .doc format
152