Development of a wavelet-based algorithm to detect and determine ...

Development of a wavelet-based algorithm to detect and
determine certain disturbances in power networks
D I P L O M A T H E S I S
Institute for
Electrical Power Systems and High Voltage Engineering
Department of Electrical Power Systems
Graz University of Technology
Head of department: O. Univ.-Prof. Dipl. Ing. Dr.techn. Lothar Fickert
First reviewer: O. Univ.-Prof. Dipl. Ing. Dr. techn. Lothar Fickert
Second reviewer: Dipl. Ing. Robert Schmaranz
Presented by
Stefan Böhler
Graz, October 2003

Mein Dank gilt Herrn Dipl. Ing. Robert Schmaranz für die exzellente Betreuung bei
der Erstellung. Weiters gilt mein Dank Herrn Univ.-Prof. Dipl. Ing. Dr. techn. Lothar
Fickert für die interessante Aufgabenstellung, sowie für die Begutachtung dieser Ar-
beit. Sie waren immer bereit, mir mit ihrem Wissen weiterzuhelfen und haben nie die
Geduld verloren, auch wenn die gesamte Arbeit länger gedauert hat, als vorgesehen.
Widmen möchte ich diese Arbeit meinen Eltern Irmgard und Armin, die mir den nötigen Rück-
halt für das Gelingen dieser Arbeit gaben.

Eidesstattliche Versicherung
Ich versichere an Eides statt durch meine Unterschrift, dass ich die vorstehende Ar-
beit selbständig und ohne fremde Hilfe angefertigt und alle Stellen, die ich wörtlich
oder annähernd wörtlich aus Veröffentlichungen entnommen habe, als solche ken-
ntlich gemacht habe, mich auch keiner anderen als der angegebenen Literatur oder
sonstiger Hilfsmittel bedient habe. Die Arbeit hat in dieser oder ähnlicher Form noch
keiner anderen Prüfungsbehörde vorgelegen.
Schwarzach, den 29. Oktober 2003
Stefan Böhler

Contents
Eidesstattliche Erklärung ii
1 Introduction 1
2 Power Quality Recorder Simeas-R 3
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Fault recorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Power data recorder . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Power and frequency recorder . . . . . . . . . . . . . . . . . . . . 6
2.2.4 Event recorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Common interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.2 Data acquisition units . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Operation modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Software (OSCOP P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5.1 Application range . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
I

2.5.2 Software modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 SIMEAS R in the laboratory 14
3.1 SIMEAS R configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Laboratory use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 IEEE Standard Common Format for Transient Data Exchange (COMTRADE)
for Power Systems 18
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Sources of transient data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Files for data exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.1 Header file (xxx...xxx.HDR) . . . . . . . . . . . . . . . . . . . . . . 20
4.3.2 Configuration file (xxx...xxx.CFG) . . . . . . . . . . . . . . . . . . 21
4.3.3 Data file (xxx...xxx.DAT) . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Differences between the 1991’s and the 1999’s standard . . . . . . . . . . 26
5 Program to read Comtrade files into Matlab 28
5.1 Data structure of data storage in MATLAB . . . . . . . . . . . . . . . . . 31
6 Wavelets 33
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2 Basic ideas of the wavelet transform . . . . . . . . . . . . . . . . . . . . . 35
6.2.1 Representation of signals through transforms . . . . . . . . . . . . 35
6.2.2 Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . 43
6.2.3 Discretization of Continuous Wavelet Transform . . . . . . . . . . 49
II

7 Algorithm to determine certain kind of disturbances 53
7.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Analyzation of the disturbances . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2.1 Detecting and localizing in time a disturbance . . . . . . . . . . . 55
7.2.2 Identifying the disturbance . . . . . . . . . . . . . . . . . . . . . . 57
7.2.3 Generalizing the classification pattern . . . . . . . . . . . . . . . . 58
7.3 MATLAB implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.3.1 Program modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8 Conclusion and Look-Out 63
List of figures 65
List of tables 66
Bibliography 67
III

Chapter 1
Introduction
Electrical power industry worldwide moves toward deregulation and competition. At
the same time, electrical power systems are becoming more complicated. The depen-
dence of modern society on electrical energy is increasing continuously. Even short
interruptions in electrical supply can lead to serious consequences. Aspects of power
quality are getting more important too and are often regulated in contracts. Big facto-
ries with very sensible processes (i.e. paper mills, rolling mills,...) have a big damage
in case of a fault.
This requires the high reliability, accuracy and reaction speed of energy management
systems, control systems and protection systems. The development and operation of
these systems are based on the signals processing, in which the processing of power
transients is one of the most important subjects, because power transients are gener-
ated by state changes and normally concerned with the occurrences of faults, distur-
bances and operations.
This diploma thesis deals with recording of disturbances in power networks by use of
the power quality recorder SIMEAS R. In the second step, this power quality is inte-
grated in the power network simulator at the Department of Electrical Power Systems
for use in the laboratory "Störungen und Schutz in elektrischen Anlagen", so that stu-
dents become acquainted with power quality and its monitoring. This disturbance
1

ecordings from the power quality recorder and some more from different electricity
power suppliers, which are saved in the COMTRADE format are analyzed by use of
wavelet transform. This analyzation should estimate the origin for this disturbance.
The order of the chapters is in the sequence for the analyzation: disturbance record-
ing, import into MATLAB, analyzation with MATLAB. To each step, there is a theo-
retical part about the power quality recorder SIMEAS R, the COMTRADE format and
wavelets.
2

Chapter 2
Power Quality Recorder Simeas-R
2.1 Overview
The shortcut SIMEAS R means "SIemens MEASurement Recorder"
The power quality recorder SIMEAS R is used for quality assurance in power stations,
and extra-high, high-voltage and medium-voltage installations. SIMEAS R records
power faults and power quality and monitors the secondary and primary technical
facilities. It has an integrated digital fault recording system with an integrated fault
recorder, power and frequency recorder, digital recorder and an event recorder system.
The signals are sampling with a high sampling rate (256 times the base frequency with
64-fold oversampling) with a high degree of accuracy (measuring fault < 0,2 %), and a
high resolution (16 bit). With its new transducer concept it is capable to record signals
with DC components. For transfer an configuration it can be equipped with direct
communication interfaces to an analog or digital telephone network, LAN or WAN
using TCP/IP protocol or null modem. Through this communication possibilities it
has the capability for remote diagnosis and full commissioning via modem or network.
3

2.2. FUNCTIONS 4
2.2 Functions
2.2.1 Fault recorder
The analog and binary signals are continually monitored for limit violations. If a trig-
ger occurs, all inputs are recorded in parallel with pre-fault, variable fault progression
and post-fault data. The recorded data can be provided to the diagnostic system for
determining the fault location. An intelligent progression control records the fault only
for the duration of which it is present.
The resolution is 16 bits. The sampling frequency is 256 times the fundamental (12,80kHz
per input at 50Hz). Binary inputs are recorded with a sampling frequency of 2kHz. At
the time of triggering, the real time (in ms) is recorded.
The following triggers are implemented in the software for each input.
For analog inputs:
• Minimum trigger
• Maximum trigger
• Rate of change trigger (positive, negative)
• Positive-sequence system trigger minimum or maximum
• Negative-sequence system trigger minimum or maximum
For binary inputs:
• positive and negative slope
• negative slope
• positive slope
• Logical combination of analog and binary trigger criteria
Triggers are logically combined. For example triggers can be logically combined to
distinguish a fault from intentional disconnection of a supply.

2.2. FUNCTIONS 5
2.2.2 Power data recorder
The power data recorder functionality stores the signals continuously. The averaging
time is freely configurable. The mean values are determined from the calculated r.m.s.
values (which are calculated every cycle). Different averaging times can be configured
for the electrical values.
The following electrical measurements are stored and determined by the evaluation
program:
• Voltage U1, U2, U3, U4
• Current I1, I2, I3, I4 Note: Depending on the type of power network and connec-
tion, either the phase voltages or the phase-to-ground voltages are recorded.
• Active power P1, P2, P3, P4
• Reactive power Q1, Q2, Q3, Q4
• Power factor cos phi 1, 2, 3
• Frequency f
• Positive-sequence system
• Negative-sequence system
• Weighted and unweighted distortion factor
• Harmonics (max. 5 selected for an averaging time up to 10s)
• Harmonics (50 for averaging times of 10s upwards)
• DC signals when the DDAU is used
This function allows the feeders to be continously monitord and an analysis of the
network quality can be performed.

2.2. FUNCTIONS 6
2.2.3 Power and frequency recorder
This functionality is used to record the load conditions before, during and after a fault
occurs. Power swinging within the power network is recorded over a long period of
time. The length of the recording time depends on the configuration of the period for
which a mean value is estimated.
The principle of operation is similar to the fault recording mode, however, the signals
are determined every cycle (e.g. every 20ms at 50Hz frequency) and stored depending
on the configuration. The signals are stored at an interval between 1 and 250 cycles. If,
e.g., 100 cycles is configured, the average of the values is determined over this period.
The frequency channel has an accuracy of ± 1mHz. The recording can be initiated
using the start selectors. The trigger is derived by comparing the current value with
the previously determined value. The maximum pre-fault time is, e.g., 40 minutes for
an averaging time of 250 cycles.
The power and frequency progression is recorded for up to 2 hours, independent of the
averaging. If the averaging time is set to one cycle, then 10s of pre-fault are available.
The power and frequency recorder is used especially for recording the primary control
in power stations and swinging turbines in electrical power systems. The frequency
gradient ∆f/∆t reacts if a power station is disconnected from the network, or a large
change in the network power occurs.
Since all the inputs are recorded in parallel, the user can generate a power balance, for
an infeed point in the switchgear. This function is automated in the OSCOP P program
package.
2.2.4 Event recorder
The binary inputs are sampled at 2kHz (0,5ms). State changes are written with a real
time of 1ms into the storage. For each of the 16 binary inputs, 200 state changes can be
stored within 1s. The entire storage can be freely configured. For 5 Mbytes of storage
approximately 120.000 state changes can be stored.

2.3. INTERFACES 7
The binary inputs are assigned exactly to the analog recordings depending on their
real-time values. Furthermore, the output from the OSCOP P program has the form of
a chronologically sorted file, i.e. in the form of an event recorder system. This means
that a separate event recorder system does not need to be installed.
2.3 Interfaces
2.3.1 Common interfaces
The following interfaces are provided for connecting the peripherals and for remote
data transfer
• Printer interface LPT1 (rear)
• PCCARD slot type 2 to accomodate:
– an analog modem V.21, V.22, V.32, V.34, V.90
– Ethernet LAN connection
• COM 1 data interface for: (rear)
– an external digital modem (EURO-ISDN), X.25 pad
• COM 2/COM S maintenance interface for: (front)
– a notebook (on-site configuration)
2.3.2 Data acquisition units
Data acquisition units (DAUs) are provided for connection to the various processes.
The signals are connected via terminals on the rear side. The DAUs contain the com-
plete signal conditioning functions, the analog/digital conversion functions and the
entire downstream digital processing.

2.3. INTERFACES 8
For adaptation to the corresponding process signals, various acquisition units are avail-
able.
VCDAU (Voltage/Current Data Acquisition Unit)
[8 analog (4 x voltage / 4 x current) and 16 binary inputs]
The unit is used to record current / voltage and binary inputs. Typical applications
are the monitoring of feeders or a transformer panel. The calculated measurements are
derived from the input current and voltage.
CDAU (Current Data Acquisition Unit)
[8 analog (8 x current) and 16 binary inputs]
The unit is used to record current and binary inputs. A typical application is the record-
ing of two feeders.
VDAU (Voltage Data Acquisition Unit)
[8 analog (8 x voltage) and 16 binary inputs]
The unit is used to record voltage and binary inputs. A typical application is the record-
ing of busbar voltages.
BDAU (Binary Data Acquisition Unit)
[32 binary inputs]
This unit is used whenever 16 binary inputs per analog acquisition module are insuffi-
cient. This allows the user, for example to combine 8 analog and 48 binary inputs.
DDAU (DC Data Acquisition Unit)
[8 analog (8 x process measurands) and 16 binary inputs]
The unit is used to record process measurements, such as 4 to 20 mA, or DC voltages,
such as 10 V DC. In conjunction with a transducer, voltages of up to 400 V can be pro-

2.4. OPERATION MODES 9
cessed.
Figure 2.1 shows the front and the back of the power quality recorder SIMEAS R. On
the front side there are some status LED’s. They can be assigned to the different status
signals from SIMEAS R. There are a few buttons also to block the device or to start
a manual recording. Under the power switch there is a serial interface to configure
the device and to transfer the recordings. On the back side there is the power supply
connector and the data acquisition units. There a parallel interface to connect a printer
and a serial interface to connect a modem for example. It is possible to attach the
SIMEAS R with an internal modem or ISDN card also.
Figure 2.1: Connections of SIMEAS R
The SIMEAS R in the laboratory is equipped with the data acquisition units VCDAU
and VDAU. The VCDAU is used in the laboratory for measuring voltage and current
in the power network simulator and VDAU is measuring the voltage in the power
supply.
2.4 Operation modes
The SIMEAS R has three basic operation modes:

2.5. SOFTWARE (OSCOP P) 10
Normal operation In normal operation, all functions and triggers are active. Depend-
ing on the configured averaging time, recordings are performed continuously .
If a trigger limit is violated, a recording begins with prefault values, fault pro-
gression and post-fault values. The cause of the trigger violation is stored with
the header data. Depending on the configuration, any SIMEAS R connected via
the LAN network is also triggered.
Blocked operation In blocked operation, no recording is initiated when a trigger con-
dition occurs. All triggers are inactive, and the signalling relay is not energized.
The device is only active in the "digital recorder" mode.
This operating mode is selected when maintenance work is carried out on the
affected feeders.
Test operation In test operation, as in normal operation, all functions and triggers are
active, however, the recorded events always have "Test" entered as their cause.
The signalling relay "Event currently being recorded" is not energized. This op-
erating mode is used to test the SIMEAS R.
The operating modes can be selected via the keyboard. Remote control using OSCOP
P (see chapter 5) is also possible.
2.5 Software (OSCOP P)
2.5.1 Application range
OSCOP P is a program package for transmission and diagnosis of measured values,
which have been recorded with power quality recorder SIMEAS R.
The user is able to control the transmission and diagnosis in manual or automatic op-
eration. The results are send to the printer.

2.5. SOFTWARE (OSCOP P) 11
2.5.2 Software modules
The program package OSCOP P consists of several different program modules. This
modules are described below. Not all programs are necessary for data transmission.
Main program modules
Parameterize PC The "Parameterize PC" program module is used to set the basic sys-
tem parameters which are: name and passwords of the authorized users of OS-
COP P, standard interfaces of the PC, number and parameters of the respective
SIMEAS Rs. Once these settings have been entered, later modification is not re-
quired unless the user wishes to provide an extension to the existing SIMEAS R
unit (additional DAU modules or new SIMEAS R systems, modification of the
passwords of existing SIMEAS R systems or new access authority for OSCOP P).
Parameterize devices Parameterize module, where the registration device of type SIMEAS
R is parameterized.
Transfer Communication module, which transfers the recorded disturbances to the
evaluation PC.
Evaluate Graphical evaluation module, where the disturbances are displayed graphi-
cally
Additional program modules
Simple Diagnosis Diagnosis module which can determine kind and location of failure
in the controlled branch.
Complex Diagnosis Diagnosis module which can determine kind and location of fail-
ure in controlled branch. In addition to simple Diagnosis, the effect to the higher-
ranking network is evaluated.
Statistic module SICARO PQ Diagnosis module which makes an analysis about the
power quality. The recordings are evaluated according to the norms EN50160

2.5. SOFTWARE (OSCOP P) 12
and IEC1000. The results are shown in diagrams.
These screenshots shown here are from OSCOP P. Figure 2.2 shows the parameterizing
page. It is the channel for the voltage in phase L1 on the power network simulator.
The trigger levels for maximum voltage, minimum voltage and the rate of change of
the voltage. The page for currents is similar to this page.
Figure 2.2: OSCOP P parameterizing
Figure 2.3 shows the visualization of a recorded fault. RMS values are choosed to show.
It can be seen, that there was a ground failure in phase L1. The RMS values are given
in each phase an at the different segments.
In figure 2.4 the same fault as in figure 2.3 can be seen. This view shows the fault in
analog values. In this view it is possible to recognize the transient phenomenom at the
beginning and the end of the fault.

2.5. SOFTWARE (OSCOP P) 13
Figure 2.3: OSCOP P evaluation RMS values shows a
earth-fault in L1
Figure 2.4: OSCOP P evaluation analog value shows
the same earth-fault in L1 as in Figure 5

Chapter 3
SIMEAS R in the laboratory
As described in chapter 1, the SIMEAS R was used in the laboratory "Störungen und
Schutz in elektrischen Anlagen" to show possibilities monitor power quality in power
networks. The SIMEAS R was integrated into the power network model at the Institute
for Electrical Power Systems and High Voltage Engineering, Department for Electrical
Power Systems at Graz University of Technology.
3.1 SIMEAS R configuration
For use of the SIMEAS R in the laboratory, the measuring device has to be prepared.
There were many old disturbance recordings on the internal hard-disk. Asking the
Siemens helpdesk how to clear this recordings the answer was that it is only possi-
ble to clear old recordings after removing the PCMCIA hard-disk from SIMEAS R and
building it in a PCMCIA slot of a notebook. All files on the hard-disk, except the fol-
lowing should be deleted. It is important to let these files untouched, because SIMEAS
R is out of function without them.
This files must not be deleted!
• bootromsys
• VXworks
14

3.1. SIMEAS R CONFIGURATION 15
• DAU.bin
• Parm.txt
• Slot1 - 4.txt
After that, the hard-disk is to remount into the SIMEAS R. After switching it on, the
LED’s on the front are beginning to flash. This means that the SIMEAS R is preparing
its memory areas, which are defined through Oscop P. During this process the SIMEAS
R should not be switched off.
To test the proper function and to make some real disturbance recordings, a test-run
was performed. In this test-run the SIMEAS R was connected to the public power net-
work over a few weeks. The recordings were viewed with Oscop P. It could be seen
that the threshold values were very narrow, so the SIMEAS R had many recordings
with very small under- and overvoltages.
Figure 3.1: SIMEAS R in the testing setup with laptop

3.2. LABORATORY USE 16
Figure 3.1 shows the SIMEAS R in the test setup for testing the proper function. The
laptop is used to configure it and to transfer the disturbance recordings and is connect
to the service interface on the front side.
For an easy use together with the power network model, the SIMEAS R was equipped
with cables and banana plugs to connect it to the power network model. There are
cables to measure three currents together with four voltages and additionally three
voltages.
3.2 Laboratory use
The exercises in the laboratory exist of setting the correct parameters (limiting values
for voltages and currents) with the configuring program OSCOP P which is described
in chapter 2 and the proper connection and integration into a power network simulator.
Figure 3.2 shows this integration into this simulator.
Figure 3.2: SIMEAS R in the laboratory use at the De-
partment for Electrical Power Systems
A description of the SIMEAS R was written for the students. It is an abstract about the
functions of the SIMEAS R, the handling of the SIMEAS R and the use of the program
OSCOP P to paramterize the SIMEAS R and to transfer the disturbance recordings.

3.2. LABORATORY USE 17
Then the SIMEAS R is connected to the power network model in the laboratory. At
least the plugs for the current in all three phases and the plugs for the voltage in all
three phases have to be plugged in. As an option the fourth voltage input can be
connected to the neutral conductor. There are four additional voltage inputs and some
digital inputs. So it is possible to record status information from switches or distance
relays.
After that some failures (single phase and three-phase short circuits, earth faults) were
simulated with the power network model with different neutral earthing (solidly earthed,
isolated and inductive earthed system). If the parameters were set correctly, the SIMEAS
R records them. With Oscop P the recordings are viewed and analyzed. In addition to
the SIMEAS R, the disturbances were also recorded with a digital distance relay. The
recordings of these two devices were compared and analyzed.

Chapter 4
IEEE Standard Common Format for
Transient Data Exchange
(COMTRADE) for Power Systems
4.1 Introduction
The rapid evolution and implementation of digital devices for fault and transient record-
ing and testing in the electric utility industry has generated the need for a standard
format for the exchange of such data for use with various devices to enhance and auto-
mate the analysis, testing, evaluation and simulation of the power system and related
protection schemes during fault and disturbance conditions.
The COMTRADE standard defines a common format for the data files and exchange
medium needed for the interchange of various types of fault, test, or simulation data.
The proliferation and implementation of digital devices for the acquisition, analysis,
simulation and testing of power - system equipment has made available a profusion
and variety of data that has not been readily available in the past. Since these data
may come from a variety of sources, from different manufacturers using proprietary
or other standard formats, a common format standard is necessary to facilitate the
exchange of such data between devices with diverse applications but that have the
18

4.2. SOURCES OF TRANSIENT DATA 19
capability of utilizing digital data from other devices.
4.2 Sources of transient data
There are several possible sources of transient data for exchange:
• Digital fault recorders
Digital fault recorders for monitoring power-system voltages, currents and dig-
ital events are supplied by several manufacturers. Typical recorders monitor 16
to 64 analog channels and a comparable number of event (contact status) inputs.
Sampling rates, analog-to-digital converter resolution, record format, and other
parameters have not been standardized.
• Digital protective relays
New relay designs using microprocessors are currently being developed and
marketed. Some of these relays have the ability to capture and store relay input
signals in digital form and transmit these data to another device. In performing
this function, they are similar to digital fault recorders, except that the nature
of the recorded data may be influenced by the needs of the relaying algorithm.
As with the digital fault recorders, record format and other parameters have not
been standardized.
• Transient simulation programs
Unlike the above devices which record actual power system events, transient
simulation programs generate transient data based on mathematical models of
power systems. Because this analysis is carried out by a digital computer, the
results are inherently in digital form suitable for digital data dissemination. An
example for such a simulation program is the SABER simulator [2].
While originally developed for the evaluation of transient overvoltages in power
systems, these programs are finding increasing usage in other types of studies, in-
cluding test cases for digital relaying algorithms. Because of the ease with which

4.3. FILES FOR DATA EXCHANGE 20
the input conditions of the study can be changed, transient simulation programs
can provide many different test cases for a relay.
• Analog simulators
Analog simulators model power system operations and transient phenomena,
with scaled values of resistance, inductance and capacitance, operating at greatly
reduced values of voltages and current. The components usually are organized
with similar line segments that can be connected together to form longer lines.
The frequency response of the analog simulator primarily is limited by the equiv-
alent length of the model segment and typically ranges from 1 - 5 kHz. The
analog output of the simulator could be converted to digital records with appro-
priate filtering and sampling. The power network simulator at the Department
for Electrical Power Systems is such an analog simulator.
4.3 Files for data exchange
The COMTRADE data format consists of three files, the configuration file, the data file
and the header file (the header file is defined in [4] only). The configuration file and
the header file are ASCII files, the data file is an ASCII or binary file, as it is configured
in the configuration file.
The extension of the file name is used to identify the type of file, ".HDR" for header,
".CFG" for configuration, and ".DAT" for data file.
4.3.1 Header file (xxx...xxx.HDR)
The header file is created by the originator of the fault data using a word processor
program. The data is intended to be printed and read by the user. The creator of the
header file can include any information in any format desired. The header files are
free-form ASCII files of any length. ASCII files can be read by simple editors and word
processors.

4.3. FILES FOR DATA EXCHANGE 21
4.3.2 Configuration file (xxx...xxx.CFG)
The intent of the configuration file is to provide the information necessary for a com-
puter program to read and interpret the data values in the associated data files. Some
of these information might be duplicated in the header file (e.g., number of channels)
to be compatible with CIGRE.
Content
The configuration file has the following information:
1. Station name and identification
2. Number and type of channels
3. Channel names, units, and conversion factors
4. Line frequency
5. Sample rate and number of samples at this rate
6. Date and time of first data value
7. Date and time of trigger point
8. File type
Format
The configuration file is a standard ASCII file of a fixed or predetermined format. It
must be included on every disturbance recording.
The file is divided into lines. Each line is terminated by a carriage return and line feed
. Commas are used to separate elements within a line. For missing data items,
the separator commas are retained with no spaces between the commas.
The information in the file must be listed in the exact order and fixed-order format
shown after.

4.3. FILES FOR DATA EXCHANGE 22
• Station name and identification
The configuration file begins with the station name and the station identification.
station_name,id
where:
station_name =the unique name of the recorder
id =the unique number of the recorder
• Number and type of channels
This statement contains the number and type of channels as they occur in each
data record in the data file:
TT,nnt,nnt
where:
TT= total number of channels
nnt= number of channels of
t= types of input (A=analog/D=status).
• Channel information
This is a group of lines containing channel information. There is one line for each
channel as follows:
nn,id,p,cccccc,uu,a,b,skew,min,max
...
nn,id,p,cccccc,uu,a,b,skew,min,max
nn,id,m
nn,id,m
where:
nn= channel number
id= channel name

4.3. FILES FOR DATA EXCHANGE 23
p= channel phase identification
cccccc= circuit/component being monitored
uu= channel units (kV, kA, etc.)
a= real number (see below )
b= real number. Channel conversion factor is ax+b
[i.e., a recorded value of x corresponds to (ax+b) in units uu
specified above]
skew= real number. Channel time skew (in µs) from start of sample period
min= an integer equal to the minimum value (lower limit of
sample range) for samples of this channel
max= an integer equal to the maximum value (upper limit of sample range)
for samples of this channel
m= (0 or 1) the normal state of this channel (applies to digital
channels only)
This "nn,id,p,cccccc,uu,a,b,skew,min,max" sequence is necessary to provide the
channel names, phase, units, and conversion factors for each of the channels in
the order in which they occur in the data. Phase, circuit component, and conver-
sion factors are not necessary for digital channels and thus are omitted.
In the above record format specification, the line seperated by dashs represent
analog channels and the last two lines represent digital channels.
The real value for the analog value is calculated as follows:
value = x · a + b
• Line frequency
The line frequency is listed on a seperate line in the file:
lf
where:
lf= line frequency in Hz (50 or 60)

4.3. FILES FOR DATA EXCHANGE 24
• Sampling rate information
This section contains the total number of sample rates followed by a list contain-
ing each sample rate and number of the last sample at the given rate.
nrates
sssss1,endsamp1
sssss2,endsamp2
...
sssssn,endsampn
where:
nrates= number of different sample rates in the data file
sssss1 - sssssn= the sample rate in Hz
endsamp1 - endsampn= last sample number at this rate
• Date/time stamps
There are two date/time stamps: the first one is for the first data value in the data
file and the second one is for the trigger point. They have the following format:
mm/dd/yy,hh:mm:ss.ssssss
mm/dd/yy,hh:mm:ss.ssssss
where:
mm= month (01-12)
dd= day of month (01-31)
yy= last two digits of year
hh= hours (00-24)
mm= minutes (00-59)
ss.ssssss=seconds (from 0 sec to 59.999999 sec)
• File type
The data file type is identified as being an ASCII file by the file type identifier ft.

4.3. FILES FOR DATA EXCHANGE 25
ft
where:
ft= ASCII
Example configuration file
The following example shows a configuration file from a digital protection device. The
first line contains the station name (UAB_110kV_PN 7S513_1294). The second line
contains the number of channels, in this example there are 8 analog channels and 1
digital channel. The next few lines are explaining each analog and the digital chan-
nel (see former section for detailed information) The last lines contain information
about the power frequency (50 Hz), number of sampling frequencies (1), sample rate
(999,000999), number of samples (240), time and type of data file (BINARY).
UAB_110kV_PN 7SA513_1294„1997
9,8A,1D
1,IL1„1,I/In,4.3234,-4.3859,0,-32767,32767,1,1,S
2,IL2„2,I/In,4.3636,8.7719,0,-32767,32767,1,1,S
3,IL3„3,I/In,4.3368,-8.7719,0,-32767,32767,1,1,S
4,IE„4,I/In,1.3385,-4.3859,0,-32767,32767,1,1,S
5,UL1„5,U/Un,2.5189,-1.0152,0,-32767,32767,1,1,S
6,UL2„6,U/Un,2.5235,-1.5228,0,-32767,32767,1,1,S
7,UL3„7,U/Un,2.5018,-1.5228,0,-32767,32767,1,1,S
8,Uen„8,U/Un,2.6335,-3.5532,0,-32767,32767,1,1,S
1,Bin 1„,0
50.0
1
999.000999,240
27/11/2001,14:44:00.006000
27/11/2001,14:44:00.006000
BINARY
1.0

4.4. DIFFERENCES BETWEEN THE 1991’S AND THE 1999’S STANDARD 26
4.3.3 Data file (xxx...xxx.DAT)
The data file contains the value of each sample of each input channel. The number
stored for a sample is usually the number produced by the device that samples the
input waveform.
The stored data may be either zero based, or it may have a zero offset. Zero-based data
goes from a negative number to a positive number, e.g., -2000 to +2000. Zero-offset
numbers are all positive with a positive number chosen to represent zero, e.g., 0 to
4000 with 2000 representing zero. Conversion factors specified in the configuration file
define how to convert the data values to engineering units.
In addition to data representing analog inputs, inputs that represent on/off signals are
also frequently recorded. These are often referred to as digital inputs, digital channels,
digital subchannels, event inputs, logic inputs, binary inputs, contact inputs or status
inputs. In this standard, this type of input is referred to as a status input. The state of
a status input is represented by a number "1" or "0" in the data file.
4.4 Differences between the 1991’s and the 1999’s stan-
dard
In 1999 was a revision of this COMTRADE Standard. Their are a few differences be-
tween the 1991’s and the 1999’s standard, which are listed below.
1. The Header (.HDR) file is explicitly defined as optional.
2. The Configuration (.CFG) file has been modified. A field containing the COM-
TRADE standard revision year has been added to distinguish files made under
this or future revisions of the standard. If this field is not present, the data is
assumed to comply with the COMTRADE standard C37.111-1991 [3]. A field
for a Time Stamp Multiplication Factor has been added to meet the need for
long duration files. To assist in conversion of the data, three new scaling fields
(primary, secondary, and primary-secondary) are added, defining the instrument

4.4. DIFFERENCES BETWEEN THE 1991’S AND THE 1999’S STANDARD 27
transformer winding rations, and whether the recorded data is scaled to reflect
primary or secondary values. Configuration fields for Status (Digital) Channel
Information habe been expanded to five fields to allow the same level of defini-
tion as for analog channels. Line Frequency is now defined as a floating point
field. Support for Event Triggered data has been added by the addition of a new
mode for Sampling Rate Information when the sampling rate is variable. The
Date/Time Stamps format has been modified with the day of month preceding
the month entry field, and the year field now has all four numbers of the year. A
requirement that at least one leading space shall be tolerated in the data fields, in-
cluding those fields for which no data is available (previously specified as comma
delimiters with no spaces in between) has also been added.
3. A new format for a binary data (.DAT) file has been specified and the requirement
for a supplied conversion program has been eliminated.
4. A new optional Information file (.INF) has been added to provide for transmis-
sion of extra public and private information in computer-readable form.
5. All field descriptions are explicitly defined with respect to: criticality, format,
type, minimum/maximum length, and minimum/maximum value.

Chapter 5
Program to read Comtrade files into
Matlab
Because of writing the wavelet analyzation algorithm in MATLAB it is necessary to
import all signal, which are recorded and saved by the power quality recorder SIMEAS
R in COMTRADE format, into MATLAB. This import routine was written in Matlab
too, so it can be integrated into the calculating algorithm.
In addition to the program to convert Comtrade files, a program to read other data files
of recorded disturbances [10] was written. This files are in Matlab format already and
have to be loaded and converted into my data structure only .
This program for Comtrade input has two parts. The first part is the reading of the
config file and the second part is the reading of the data file. The structure chart of the
first part is shown in figure 5.1, the structure chart from the second part is shown in
figure 5.2.
The statement variable = line in figure 5.2 means that one line in the config file is read
and saved into variable. The lines to be read are in a strict order. The first variable =
line statement is reading the first line, the second statement is reading the second line
and so on. That’s why, the order must not be changed.
There is only one decision concerning to the revision year (see 4.3.2) in the first part,
because the time-multiplication is new in the 1999’s standard. In the second part, there
28

Figure 5.1: Structure chart of the first program part for
reading the configuration file
29

Figure 5.2: Structure chart of the second program part
for reading the data file
30

5.1. DATA STRUCTURE OF DATA STORAGE IN MATLAB 31
is also one decision concerning to the revision year, because the ratio is also new in the
1999’s standard. To have this decision only, the program is calculating with ratio=1 if
the revision year is not 1991. Otherwise the ratio is being read from the config file.
The minor difference of the input routine for the date file between the two different
types (ASCII or binary) is not explained in the structure chart. The difference is only in
the step reading file and saving.
5.1 Data structure of data storage in MATLAB
The following table lists the variables, which are used in my data storage in MATLAB
and their meanings. For the details of the meanings look at 4.3.2.

5.1. DATA STRUCTURE OF DATA STORAGE IN MATLAB 32
name meaning
station Station name and identification as given in the configuration file
rev_year Revision year of the standard (1991 or newer)
T_analog Time information for each sample in each analog channel
number_analog Number of analog channels
number_digital Number of digital channels
number_all Number of all channel (analog and digital)
AnalogChannel Channel information for each analog channel
DigitalChannel Channel information for each digital channel
Frequency Frequency as given in the config file
number_samples Number of different sampling rates
Sample Sampling rate and number of samples with this sampling rate
Time_begin Time when the recording begins
Time_trigger Time when the recording devices was triggering
Typ Type of data file (ASCII or binary)
timemult Time-multiplication factor (in 1991’s standard timemult=1)
ratio Ratio of the current- or voltage - transformer
skew Channel time skew from start of sample-period for each analog
channel
Table 5.1: Data structure of the calculating algorithm

Chapter 6
Wavelets
6.1 Introduction
The wavelet transform is a new and developing branch of mathematics. Recently it
has become one of the hot points in the research of engineering applications. This is
neither a novelty nor a fashion. The wavelet transform leads to the signal processing
techniques with special characteristics that have been expected for a long time. [6]
The term "wavelet" literally mens little wave because a wavelet decays quickly (lit-
tle) with oscillations (wave) (see figure 6.1). Although countless functions may be little
waves, the item wavelet is reserved for those little waves that are associated with a par-
ticular choice: narrow-band in frequency. Simply, wavelets are window functions not
only in time domain but also in frequency domain. The wavelet transform represents a
signal through wavelets. It is a linear transformation much like the Fourier transforma-
tion but with important differences: the wavelet transform can localize simultaneously
in time and in frequency, adjust the window widths according to frequency automati-
cally and allow the flexible selection of wavelets to match different applications.
The wavelet transform as a new item was proposed by Morlet for geophysics signal
processing in 1982 [8], but there was a long time preparation for its development. The
wavelet transform was developed as a new theory in mathematics by Meyer, Gross-
mann and Daubechies during 1984-1988. In 1989, Mallat proposed the multiresolution
33

6.1. INTRODUCTION 34
analysis theory and developed the fast algorithms of the wavelet transform [7]. Since
then, the wavelet transform has been spreading in different scientific areas because of
its perfection in theory and suitability for a wide range of applications specially for
processing transient signals.
As a transient signal, its amplitude, frequency or both do not stay constant when time
changes. Up to now, the Fourier transform based methods are generally employed
for the processing of power transients, despite that the Fourier transform is only suit-
able for periodical signals. This is why there are still many unsatisfactory results and
nonunified definitions concerning the Fourier transform applications in power sys-
tems, although great deal of concerning work has been done. There is always the
requirement for a more suitable method to process power transients.
With the development of the wavelet transform, it is naturally to explore and see how
this promising approach works in power systems. Since 1994, the wavelet transform
has been introduced into power systems and applied for the analysis and the classifica-
tion of power quality events [9], protections, data compression, cable partial discharge
measurement and so on. There is a sharp increasing tendency in the presentation about
the wavelet transform applications in power systems. The wavelet transform is becom-
ing a well-known method and is drawing more attention of electrical engineers.
Through the published papers, it is known that the wavelet transform is suitable for the
analysis of power transients and has a big potential area in power system applications.
However, it is noticed that the wavelet transform applications in power systems are
still quite rough and limited:
• They are mainly on the introduction of the basic conceptions and theory of the
wavelet transform but lack of detail explanations and practical application re-
sults.
• They are mainly with qualitative analysis but short of rationing criteria and suf-
ficient verifications.
• They are mainly on the application of basic wavelet algorithm but hardly on the
deep or extension theories such as wavelet singularity theory and improved al-

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 35
gorithms etc.
Therefore, the situation is that the wavelet transform applications in power systems
are very meaningful but just at the beginning, and there is a great deal of concerning
research to do.
The wavelet transform with its special characteristics, such as the time-frequency lo-
cation, the auto-adaptivity to frequency, the flexible selection and the fast algorithms
etc., opens a door towards a lot of possibilities and large developing field in power
systems. However, it is not very easy to start understanding and using the wavelet
transform because the wavelet transform is built up on the theories of modern math-
ematics and modern signal processing [7] and the most the of published books about
wavelet transform are the contributions of mathematicians.
In this chapter, the first section gives the basic ideas of the wavelet transform. In the
second section, the continuous wavelet transform (CWT) is introduced. The discrete
wavelet transform (DWT) as the key point will be introduced in the third section. The
forth section displays how the wavelet transform works through an example. The fifth
section then describes and discusses the general problems and corresponding solutions
in practical applications.
6.2 Basic ideas of the wavelet transform
6.2.1 Representation of signals through transforms
The idea in mathematics and in engineering is to represent and to analyze a signal
or a system in different domains. The changes from on domain to another are called
transforms.
An useful transform is to decompose a signal into elementary functions for a space.
These elementary functions should be chosen with some care to be sure that the trans-
form is invertible. Generally, they should form an orthogonal base or a base for the
respective space.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 36
Suppose that a sequence {ϕn(t)} is a base for a space, and then any functions f(t) in
the space can be represented with this base. Namely, there is Equation 6.1.
f(t) = � Inϕn(t) n ∈ Z (6.1)
Where In is a scalar sequence generated through the inner product of f(t) and ϕn(t), as
shown in equation 6.2.
{In} = 〈f(t), ϕn(t)〉 =
Where ϕ ∗ n(t) is the complex conjugation of the function ϕn(t).
�
+∞
−∞
f(t)ϕ ∗ n(t)dt (6.2)
Equations 6.1 and equation 6.2 form a pair of transforms. Equation 6.2 is called the (for-
ward) transform or the decomposition because it breaks a function into pieces through
the base for the space. Equation 6.1 is called the inverse transform or the reconstruction
because it puts the pieces back together to retrieve the original signal. It is clear that
the property of a base is determinate for the transform.
A transform theory always involves how to break a function apart (transform) and
then to put it back together (inverse transform). The reason for doing this is that sig-
nificant insights, clear patterns, and efficiencies can be obtained through the operation
on the pieces rather than the original function. Electrical engineers are familiar to many
transforms: Fourier transform, Laplace transform, z transform, symmetric components
transform and so on. These transforms all have own well working areas.
The wavelet transform is nothing else but another transform. It represents signals
through the space base that have the expected characteristics.
Representation of signals through the wavelet transform
Wavelets and the admissible condition: A wavelet belongs to one kind of functions
which have the following characteristics:
• The amplitude quickly decays to zero in both directions.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 37
• The function is oscillatory.
• Average value is zero
The first characteristic makes a function "little" whereas the second one makes it "wavy"
and hence a wavelet. These two characteristics must be simultaneously satisfied for a
function to be a wavelet. A sinusoid never decays (i.e. wavy but not little) so that can
not be a wavelet. A decay function (little but not wavy, e.g. Gauss function) also can
not be a wavelet. However, the product of a wavy and a decay function can lead a
wavelet. Figure 6.1 shows, as an example, how a wavelet is created: the product of
a sinusoid (6.1(a)) and a Gauss function (6.1(b)) makes a wavelet (6.1(c)) - the famous
Morlet wavelet [8].
Figure 6.1: Wavelet
In mathematics, any function ψ(t) can be a wavelet if it satisfies the following condi-
tion.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 38
�
+∞
−∞
Equation 6.3 is called the admissible condition.
ψ(t)dt = 0 (6.3)
The admissible condition is quite lenient so that there are a lot of wavelets, which can
be discrete or continuous, real or complex functions. Figure 6.2 shows three different
wavelets as examples. The Haar wavelet (the simplest wavelet) and the Db2 wavelet
(one of Daubechies wavelets) are discrete while the Mexican Hat wavelet is continuous.
The continuous wavelets are described with formulas, but the discrete wavelets nor-
mally have no direct formulas and are given through discrete filters.
Figure 6.2: Examples of wavelets
Mother wavelet, wavelet family and wavelet domain: Any wavelet can be taken
as a mother wavelet. A mother wavelet generates a wavelet family by scaling and
translating. All members in one wavelet family have the same energy and the similar
waveform as the mother wavelet.
Let ψ(t) denote a mother wavelet, the corresponding wavelet family ψab(t) is defined
as Equation 6.4:
ψab(t) = a −1/2 � �
t − b
· ψ
a
(6.4)
Where a represents "scale" and b represents "translation". Figure 6.3 displays the re-
lation between a mother wavelet (Morlet wavelet) and it’s family. It can be seen that

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 39
translation b is just a representation of time and scale a influences the duration so that
has a direct relation with frequency.
Figure 6.3: Mother wavelet and wavelet family
A wavelet family forms a base or an orthogonal base of the general space L 2 (R). There-
fore, nearly every interesting function could be represented as a linear combination of
a wavelet family and also could be reconstructed. A space base formed by a wavelet
family is also called a wavelet base. Such representation with a wavelet base is called
the wavelet transform.
Special characteristics of a wavelet family: As other transforms, the characteristic
of the wavelet transform is decided by the properties of the space base formed by the
wavelet family, i.e. the wavelet base.
Figure 6.4(a) shows an orthogonal wavelet base in the time-frequency plane. The
wavelet base is like a music note, as shown in Figure 6.4(b), which tells when and
which tone (frequency) is played. the wavelet base forms such windows, which are

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 40
Figure 6.4: Property of a wavelet base
rectangles with the same area but different shapes. Hence, the windows adjust auto-
matically narrow at high frequency and wide at low frequency. Just these properties
make the wavelet transform work in a special way.
From Fourier Transform to Wavelet Transform
The wavelet transform does not appear by chance but is a good result from the long
time researches. The motivation is to overcome the drawbacks of the Fourier transform
in analyzing transient signals.
Fourier Transform: Nowadays, the Fourier transform is the most well known trans-
form. Since 1822 Fourier created the theory and especially since 1965 the fast Fourier
transform (FFT) was proposed, the Fourier transform has been applied in nearly all the
engineering fields.
In mathematics, the Fourier transform of a function f(t) is defined as:
F (ω) = 〈f(t)e jωt 〉 =
�
+∞
−∞
f(t)e −jωt dt (6.5)
It represents a signal with an orthogonal base {e jωt }, which includes the sinusoids of
different frequencies. In this way, the Fourier transform changes our view from time
domain to frequency domain.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 41
The Fourier transform is perfect and quite simple in mathematics. However, many
people are still working on it and there are many concerned definitions, such as tran-
sient frequency and harmonics etc., still under the discussion in engineering. The rea-
son is that the practical applications requires more than the theory has.
The Fourier transform transfers a signal into the frequency domain, the information
evolving with time is lost completely. For example, two different signals have the
same result through the Fourier transform, as shown in Figure 6.5. It is impossible
for the Fourier transform to tell when a particular frequency took place. Besides, the
Figure 6.5: Different signals but same Fourier trans-
form representation
Fourier transform needs the information in an infinite amount of time, i.e. all the past,
present and future information of a signal, just to evaluate the spectrum at a single
frequency. Strictly, the Fourier transform is only suitable for handling periodic or time-
invariant signals in practical applications. Unfortunately, most of signals in practice are
non-periodic and time-variant and transient characteristics such as a sudden change,
beginnings and ends of events are often the most important part of signals.
Therefore, it remains a challenge in practical applications how to analyze transient
signals and how to obtain time information as well as frequency information simulta-

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 42
neously.
Figure 6.6: Wavelet Transform of two different sinu-
soids
Wavelet Transform The wavelet transform has the capability of time location as well
as frequency location, similar as the Short Time Fourier Transform (STFT) in a certain
degree. Unlike other transforms, it leads the expected flexible windows. Namely, it
uses automatically wide windows at low frequency and narrow windows at high fre-
quency. These properties are just what have been searched for. Besides, the wavelet
transform is perfect hence fully supported by mathematics.
The wavelet transform represents a signal with two new parameters, i.e. scale a and
translation b. Because translation b has the same meaning as time, the plane formed
by translation and scale is called the time-scale domain or the wavelet domain. Notice
that scale is related with frequency, the time-scale plane then is related with the time-
frequency plane.
Figure 6.6 displays the wavelet transforms of two different sinusoids, which have the
same Fourier transform result (see Figure 6.5). It can be seen that the wavelet transform
can distinguish between these two signals clearly and tell simultaneously when and
which scale or frequency concerning with the signals.
A Physical Explanation of Wavelet Transform
The wavelet transform can be roughly explained with the function of a lens, as shown
in Figure 6.7. If a signal f(t) corresponds to an observed object, a wavelet base ψab(t)
to a lens, then changing translation b is similar to make the lens move parallel with the
object and the change of scale a to make the lens move forwards or backwards, i.e. to

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 43
Figure 6.7: Wavelet transform, illustration of scale and
translation
change the lens resolution. Different a corresponds to different resolution. It can be
seen that:
• The wavelet transform has the multiresolution characteristic so that a signal can
be analyzed from larger features to finer features.
• The wavelet transform has the ability of the simultaneous localization in time
and in scale (or frequency) so that local properties of a signal can be analyzed.
This is the reason that the wavelet transform is praised as a "mathematical microscope"
in signal processing, as it is illustrated in figure 6.7.
6.2.2 Continuous Wavelet Transform
Definition of Continuous Wavelet Transform
The continuous wavelet transform (CWT) of a function f(t) ⊂ L 2 (R) is given by Equa-
tion 6.6.
Wψf(a, b) =
�
+∞
−∞
f(t)ψ ∗ ab(t)dt (6.6)

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 44
Where Wψf(a, b) are wavelet coefficients ψab(t) represents a wavelet family created by
a mother wavelet ψ(t), a and b are the scale and the translation respectively, and the
superscript " ∗ " denotes complex conjugation because wavelets can be complex.
The continuous wavelet transform transfers one-dimension functions into the two di-
mension plane, called the time-scale plan or simply the wavelet domain. A wavelet
coefficient shows the correlation degree between the function and the wavelet at a par-
ticular scale and translation. The set of all wavelet coefficients is the wavelet domain
representation of the function f(t) with respect to the wavelet family.
The CWT is invertible. The inverse continuous wavelet transform (ICWT) is defined
by Equation 6.7.
Where the coefficient cψ =
f(t) = W −1
ψ f(a, b) = c−1
ψ
+∞ �
0
�
+∞
0
a −2 �
da
|ψ(ω)| 2
dω < +∞
ω
+∞
−∞
Wψf(a, b)ψab(t)db (6.7)
The inverse continuous wavelet transform reconstructs the original function by sum-
ming appropriately weighted wavelet family. the weights are the wavelet coefficients
Wψf(a, b).
Explanations about Continuous Wavelet Transform
More about a Wavelet Family: A wavelet family can be produced from a mother
wavelet:
ψ(t)ψab(t) = a−1/2 � �
t − b
ψ
a
Wavelet families are only different versions of the mother wavelet. They have a similar
shape, maybe some longer or some higher, but they have the same energy or norm as
the mother wavelet, i.e. ||ψab(t)||2 = ||ψ(t)||2.
It is very important to keep a unified energy in one wavelet family so that a unified
base for energy measurement can be kept during the wavelet transform.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 45
Scale and Frequency Scaling a function simply means stretching or compressing it.
Figure 6.3 shows the effect of the scale a. The larger scales correspond to the more
"stretched" waves. It is clear that, for a sinusoidal function, the scale a is inverse pro-
portion to the frequency ω.
There is a similar relation between the scale and the frequency in the wavelet trans-
form. The more stretched the wavelet, the longer the portion of the signal will be
analyzed, and thus the coarser or lower frequency features of the signal will be mea-
sured by the wavelet coefficients. Therefore, there is a corresponding relation between
the wavelet scale and frequency as following:
• Small scale ⇒ compressed wavelet ⇒ rapidly changing detail ⇒ high frequency.
• Large scale ⇒ stretched wavelet ⇒ slowly changing, coarse feature ⇒ low fre-
quency.
In other words, frequency is inverse proportional to scale. However, the proportion
ratio depends on the mother wavelet, sampling frequency and the wavelet transform
type (CWT or DWT). Figure 6.8(a) shows the relation between frequency and scale
with a given sampling frequency. The sampling frequency leads to the corresponding
frequency band in different scale levels. Figure 6.8(b) is an example for a sampling
frequency of 4,8 kHz. Of course the sampling frequency should fulfil the Nyquist
criteria which depends on the highest frequency of the signal.
For obtaining a fixed relation, there is a simple method. [9] The relation can be derived
for a particular wavelet function by making the wavelet transform of a cosine wave
function with a known frequency and computing the scale a at which the wavelet co-
efficient reaches its maximum. Following this method, the relation between frequency
and scale is obtained for the Db4 wavelet in form ψ(t) = e −t2 /2 cos(5t):
1
f = kfs
a
1
= 0, 8fs
a
(6.8)
Where f and a are corresponding to frequency and scale, fs denotes sampling fre-
quency in Hz and k is a constant which changes solely with function form of each

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 46
Figure 6.8: Scale level and frequency band
mother wavelet (for the Db4 wavelet k = 0, 8).
More about the Admissible Condition The admissible condition shown in Equation
6.3 can be represented in frequency domain:
�+∞
0
|ψ(ω)| 2
dω < +∞ (6.9)
ω
Where ω denotes the radian frequency and Ψ(ω) is the Fourier transform representa-
tion of the function ψ(t). Equation 6.8 is not a new condition but just the same condition
described in a different domain.
The requirement on coefficient cψ in the inverse continuous wavelet transform, shown
in Equation 6.7, is just the admissible condition. This is not by chance but means that
the admissible condition keeps the wavelet transform invertible.
Any function can be a mother wavelet if it satisfies the admissible condition. Most
natural signals satisfy this condition, thus they could be mother wavelets in theory. In
practice, a mother wavelet should be chosen carefully to meet the needs of applica-
tions.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 47
Calculation of the Continuous Wavelet Transform
The calculation of the continuous wavelet transform is a quite simple process. There
are five steps:
1. Chose a wavelet and locate it at the start section of the original signal.
Figure 6.9: Step 1 of the continuous wavelet transform
2. Calculate a wavelet coefficient according to Equation 6.6.
3. Shift the wavelet to the right and repeat step 1 and 2 until the whole signal is
covered.
Figure 6.10: Step 3 of the continuous wavelet trans-
form
4. Scale the wavelet and repeat step 1 through 3
5. Repeat step 1 through 4 for all scales.
With the help of the wavelet transform, a one-dimensional signal can be represented
in two-dimensional domain. Figure 6.12 shows, as an example, the wavelet domain

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 48
Figure 6.11: Step 4 of the continuous wavelet trans-
form
Figure 6.12: The continuous wavelet transform (Here:
f=800/a, t=b)
representations of a power signal. figure 6.12(a) displays the signal which contains 3rd
harmonic component. The signal is transferred into 32 scales through Morlet wavelet,
where the relation between scale and frequency is: frequency = 800/scale. Figure
6.12(b) displays the wavelet representation on the time-scale plane in 2D, where the
dark degree is proportional to the absolute value of the wavelet coefficients. It can be
seen that the wavelet transform locates the energy around scale 16 and 5.34 (i.e. 50 Hz
and 150 Hz), which corresponds to the fundamental frequency and the 3rd harmonics
in the signal, and shows the time information of the signal at the same time. Creating
the wavelet coefficients at every possible scale leads to a large quantity of the calcula-
tion because the continuous wavelet transform is great redundant. Because the basic
calculation of the CWT is an inner product or a convolution, there are some faster algo-

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 49
Figure 6.13: The dyadic lattice in the time-scale plane
rithms that concern the convolution theorem. For example, it is considerably faster to
do the calculation in the frequency domain with the help of the Fourier transform: one
can calculate the wavelet coefficients for a given scale at all sections simultaneously
(step 2 and 3 at the same time) and efficiently.
6.2.3 Discretization of Continuous Wavelet Transform
As mentioned above, to carry out the continuous wavelet transform is a fair amount
of work and it generates large amount of data. Therefore, the continuous wavelet
transform is primarily employed to derive properties and the discrete form is necessary
for reducing redundancy and for most computer implementations.
In general, the discrete forms of the continuous wavelet transforms are generated by
the sampling from the corresponding continuous wavelet transform. For example, the
scale is sampled by a = a j
0, and the translation by b = nb0a j
0, where a0 and b0 are the
discrete scale and the translation step sizes respectively and j ∈ Z. If a0 = 2 and b0 = 1,
then the scales and positions will be based on the power of two, i.e. a = 2 j and b = n2 j .
The wavelet coefficients then correspond to the dyadic lattice points in the time-scale
plane, as shown in Figure 6.13. This lattice can be indexed by two integers: j and n,
where j corresponds to the discrete scale levels and n to the discrete translation steps.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 50
For a function f(t) and a mother wavelet ψ(t), the corresponding discrete form of the
continuous wavelet transform is then:
Wψf(j, n) = 2 −j/2
�
f(t)ψ ∗
� j t − 2 n
2j �
dt (6.10)
In such a way, the discretization of the continuous wavelet transform is done. Since
only the scale and the translation are discrete in the power of two and the time variable
is continuous, the discrete transform in Equation 6.10 is called the continuous time
wavelet series or the dyadic wavelet transform.
A discrete form of the continuous wavelet transform is specified by the sampling
choice of the scale and the translation in the time-scale plane and the choice of the
mother wavelet. It is clear that there might be infinite such choices but certainly that
the choices cannot be made arbitrarily. The limit is that such the discretization should
satisfy a basic property, namely, invertibility.
This very practical filtering algorithm yields a fast wavelet transform - a box into which
a signal passes, and out of which wavelet coefficients quickly emerge. Let’s examine
this in more depth.
One-Stage Filtering: Approximations and Details
For many signals, the low-frequency content is the most important part. It is what gives
the signal its identity. The high-frequency content, on the other hand, imparts nuance.
It is like the human voice. Removing of high-frequency components, causes the voice
sounding different, but it can be understand. However, enough removing of the low-
frequency components, causes the voice not to be understood. In wavelet analysis, this
parts are called approximations and details. The approximations are the high-scale,
low-frequency components of the signal. The details are the low-scale, high-frequency
components.
The one stage filtering process, at its most basic level, looks like this:
To get as much samples as the input signals, the approximated and the detailed part

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 51
are downsampled by two.
Multiple-level decomposition
Figure 6.14: One - stage - filtering
The decomposition process can be iterated, with successive approximations being de-
composed in turn, so that one signal is broken down into many lower resolution com-
ponents. This is called the wavelet decomposition tree. Since the analysis process is
iterative, in theory it can be continued indefinitely. In reality, the decomposition can
proceed only until the individual details consist of a single sample or pixel.

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 52
Figure 6.15: Multiple - level - decomposition

Chapter 7
Algorithm to determine certain kind of
disturbances
The discrete wavelet transform, which was described in the former chapter, is now
used to detect and to determine certain kind of disturbances in power networks.
7.1 Theory
For the theory of the discrete wavelet transform is fully explained in chapter 6. The
discrete wavelet transform together with multiple level decomposition is used for the
analyze of disturbance recordings.
By using the DWT and observing the particular features of the several decomposition
levels of a signal, some important conclusions can be drawn. These information can
be used to detect and to identify the disturbance. A digital program is developed and
written with the wavelet toolbox in MATLAB.
Each disturbance occurred in the power network results in specific frequencies in the
voltage waveform. Thus only the voltages have to be analyzed.
The program works in five steps as follows:
Step 1: Evaluation of the square of the wavelet coefficients of the recorded distur-
53

7.1. THEORY 54
bance.
Step 2: Evaluation of the square of the wavelet coefficients found at step 1.
Step 3: Calculation of the distorted signal energy, in each wavelet coefficient level.
The "energy" mentioned above is based on the Parseval’s theorem: "the energy that
a time domain function contains is equal to the sum of all energy concentrated in
the different resolution levels of the corresponding wavelet transformed signal".
This can be mathematically expressed as:
Where:
N�
|f(n)| 2 =
n=1
N�
|aJ(n)| 2 +
n=1
f(n): Time domain signal in study
J�
j=1 n=1
N: Total number of samples of the signal
N�
|f(n)| 2 : Total energy of the f(n) signal
n=1
N�
|dj(n)| 2
N�
|aJ(n)| 2 : Total energy concentrated in the level "j" of the approximated
n=1
J�
j=1 n=1
version of the signal.
N�
|dj(n)| 2 :Total energy concentrated in the detailed version of the signal,
from levels "1" to "j".
(7.1)
Step 4: In this stage the steps 1, 2 and 3 are repeated for the corresponding "pure sinu-
soidal version" of the signal in study.
Step 5: The total distorted signal energy of the signal in study (found in step 3) is com-
pared to the corresponding one of the pure sinusoidal signal version (evaluated
in step 4). The result of this comparison is a deviation that can be evaluated by
equation 7.2.
� �
energy(j) − reference_energy(j)
deviation_energy(j)(%) =
· 100% (7.2)
reference_energy(7)
where:

7.2. ANALYZATION OF THE DISTURBANCES 55
j: wavelet decomposition level
deviation_energy: Deviation between the energy distributions of the signal in
study and its corresponding pure sinusoidal wave signal, at
each wavelet decomposition level
energy: energy distribution of the signal in study
reference_energy: energy distribution of the pure sinusoidal signal version
reference_energy(7): energy concentrated at the level 7 (which concentrates the
highest energy) of the pure sinusoidal signal version.
As will be shown in the next section, the deviation of the distributed energy curve of
the signal in study and the pure sinusoidal version of every particular power quality
disturbance has an unique pattern that can be used to identify the origin of a voltage
disturbance.
7.2 Analyzation of the disturbances
7.2.1 Detecting and localizing in time a disturbance
The wavelet which is used for this study was the "Daub4", which is shown in figure 7.1
The figure ?? shows a voltage which was distorted by a capacitor switching.
Figure ??(a) is the voltage which is analyzed. Figure ??(b) is the wavelet approximated
coefficient for level 5, (c) (d) (e) (f) and (g)are the the detailed versions for levels 5,4,3,2
and 1. The detailed level 1 (figure ??(g)) of the transformed signal clearly shows a peak
at t=400ms. The other wavelet levels have also experienced variations at the same time.
This implies that some transient phenomena has occurred at this time. Therefore, it can
be said that the disturbance has been detected and located in time. However this figure
brings no sufficient evidences of what kind of disturbance occurred to this signal.

7.2. ANALYZATION OF THE DISTURBANCES 56
Figure 7.1: Daubechies4 wavelet
Figure 7.2: Detecting a disturbance, using
Daubechies4 wavelet - wavelet coefficients squared

7.2. ANALYZATION OF THE DISTURBANCES 57
Figure 7.3: Distribution of energy in 10 wavelet coeffi-
cient levels
7.2.2 Identifying the disturbance
In order to try to identify the type of disturbance of this voltage signal, the step 3
(which was explained in chapter 7.1 has to be work out. In this step, through equation
7.1, the energy concentrated in 10 wavelet coefficient levels is calculated and plotted.
The results are in figure 7.3(a). The 7th level holds the biggest part of the signal energy.
The 6th level keeps also an important parcel of it. The remaining levels practically do
not add any important parcel to the signal energy. After this, in step 4, the fundamental
component of figure ??(a) voltage is evaluated as well as the corresponding energy dis-
tribution. This is illustrated in figure 7.3(b). After this, if a visual comparison was made
between figures 7.3(a) and (b), no important differences could be observed. However,
these figures are not equal. In order to spot these differences, the equation 7.2 is used.
This equation allows the calculation of the deviation between the energy distributions
of the signal in study and its corresponding fundamental sinusoidal wave signal, at
each wavelet decomposition level. The result of this is the deviation curve, illustrated
in figure . This curve shows the deviations of the signal with disturbance from the
corresponding pure sinusoidal one. The curve of figure 7.3(c) is within the "pattern"
for capacitor bank switching.

7.2. ANALYZATION OF THE DISTURBANCES 58
Figure 7.4: Some voltage signals with power quality
disturbances
7.2.3 Generalizing the classification pattern
The analysis of the capacitor bank switching, shown in the previous section, can be
expanded to other power quality disturbances. Figure 7.4(a) shows the perfect fun-
damental voltage waveform and is used as a reference for the other seven voltage
waveforms. Figures 7.4(b) to (h) show seven voltage waveforms that represent typ-
ical power quality disturbances: a short-circuit fault 7.4(b), a notching distortion 7.4(c),
a harmonic distortion 7.4(d), a voltage sag 7.4(e), a voltage swell 7.4(f), a capacitor bank
switching 7.4(g) and a inductive-resistive load switching 7.4(h). All this voltage wave-
forms leads to characteristic deviations of energy distribution pattern. So the five steps
described above were used to get these pattern. The pattern for this seven voltage
waveforms are shown in figure 7.5. Figure 7.5(a) does not show any deviation because
it refers exactly to the pure sinusoidal and there is no deviation. This pattern can be
calculated for each disturbance which can occur in power networks. Of course these
pattern are not always exactly the same, but in a small range.

7.2. ANALYZATION OF THE DISTURBANCES 59
Figure 7.5: Deviation in distributed energy related to
figure 7.4

7.3. MATLAB IMPLEMENTATION 60
Figure 7.6: Structure chart of the program module
which calculates the reference energy
7.3 MATLAB implementation
7.3.1 Program modules
The theory in the section above is now implemented in a Matlab program. This pro-
gram is splitted into three small modules. The first module, which imports the distur-
bance recordings into Matlab was already described in chapter 5. Two versions of this
module are able to import disturbances recorded in Comtrade format (see chapter 4)
as well as disturbances stored in MATLAB [10].
The second module has to be run after the import of the disturbance data and calculates
the reference signal, the multiple wavelet decomposition and the energy distribution
over ten wavelet decomposition levels. This energy distribution is used as the reference
energy distribution for the second module to identify the disturbance. Figure 7.6 shows
the structure chart of this module.
The third small program calculates the disturbed energy distribution of the signal in
study and the deviation from the reference energy distribution. The result is the devi-
ation of energy distribution pattern. The function of this program is illustrated in the
structure chart in figure 7.7. The shadowed part of the program module is finished but
not tested. It can be tested if the pattern are calculated for 50Hz and the corresponding
sampling frequency.

7.3. MATLAB IMPLEMENTATION 61
7.3.2 Summary
Figure 7.7: Structure chart of the program module
which identifies the disturbance
It is possible to add new pattern for other disturbances. The calculation is the same as
with already integrated disturbances. It is not necessary to find new criteria to detect
these new disturbances, as it is necessary if the identifying is estimated with changing
of the RMS values (for example for a short circuit the criterias are: decreasing voltage
and increasing current, but this can be a load switching also, thus additional criterias
are need for this identifying).
The characteristic pattern described are calculated for a voltage frequency of 60Hz at
an unknown sampling rate. So these pattern have to be adopted for 50Hz and a known
sampling frequncy. Through simulations with Matlab it is known that for getting the
same pattern as described, a sampling rate of 8333,33Hz is necessary. Because this is
not possible, the new pattern have to be calculated. To calculate them, it is necessary
to have very clear disturbance recordings. Of course this is possible with a simula-
tor (power network model or computer based simulator). It is very interesting to do
this simulation with a sampling rate of 12,8kHz, because the Simeas R disturbance
recorder can be used to get the disturbance recordings or 6,4kHz because the distur-
bance recording can be downsampled from the recordings from the Simeas R. The
pattern for several equal disturbances are not the same exactly but in a small range. So
several disturbances for each fault have to be calculated to get the range of the pattern.
In the time for this diploma thesis, it was not possible to make these simulations.
It is obvious to use this small peaks, which are described above for localizing the dis-

7.3. MATLAB IMPLEMENTATION 62
turbance to detect also involving faults. So they can be separated and each one can
be calculated for its own. The result is a higher accuracy in the identification of the
disturbances.
Also possible is the adaptation of the Matlab programs for a very fast digital signal
processor. So the disturbances can be detected very fast. And it can be integrated in a
digital measuring or recording device. It is only necessary to store the pattern for each
disturbance. Each pattern has ten values only, so their is a very small need of memory.

Chapter 8
Conclusion and Look-Out
The identifying of disturbances with the use of wavelets is a very individual method.
As a result of this Diploma thesis, the power quality recorder could be integrated in
the laboratory and in the power network simulator. It is still in use and monitors the
power supply the hole time. It is used by the students too. Another usage of this power
quality recorder is in different companies, which have problem with power quality, to
monitor disturbances and determine the origin of this disturbances.
As a second step the COMTRADE format was analyzed and a program was written to
import this recording in COMTRADE format into MATLAB for further calculations.
The extensive theory of wavelets could be introduced to the students and the staff at
the institute. The use of wavelets to analyze disturbances in power networks is a very
new and growing area in electrical engineering. This diploma thesis is a contribution
to this field. As a result of missing disturbance recordings with the correct sampling
frequency, it was not possible to test the identifying algorithm completely. But it is a
base to carry on this work.
63

List of Figures
2.1 Connections of SIMEAS R . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 OSCOP P parameterizing . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 OSCOP P evaluation RMS values shows a earth-fault in L1 . . . . . . . . 13
2.4 OSCOP P evaluation analog value shows the same earth-fault in L1 as
in Figure 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 SIMEAS R in the testing setup with laptop . . . . . . . . . . . . . . . . . 15
3.2 SIMEAS R in the laboratory use at the Department for Electrical Power
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.1 Structure chart of the first program part for reading the configuration file 29
5.2 Structure chart of the second program part for reading the data file . . . 30
6.1 Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Examples of wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.3 Mother wavelet and wavelet family . . . . . . . . . . . . . . . . . . . . . 39
6.4 Property of a wavelet base . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.5 Different signals but same Fourier transform representation . . . . . . . 41
6.6 Wavelet Transform of two different sinusoids . . . . . . . . . . . . . . . . 42
6.7 Wavelet transform, illustration of scale and translation . . . . . . . . . . 43
64

LIST OF FIGURES 65
6.8 Scale level and frequency band . . . . . . . . . . . . . . . . . . . . . . . . 46
6.9 Step 1 of the continuous wavelet transform . . . . . . . . . . . . . . . . . 47
6.10 Step 3 of the continuous wavelet transform . . . . . . . . . . . . . . . . . 47
6.11 Step 4 of the continuous wavelet transform . . . . . . . . . . . . . . . . . 48
6.12 The continuous wavelet transform (Here: f=800/a, t=b) . . . . . . . . . . 48
6.13 The dyadic lattice in the time-scale plane . . . . . . . . . . . . . . . . . . 49
6.14 One - stage - filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.15 Multiple - level - decomposition . . . . . . . . . . . . . . . . . . . . . . . . 52
7.1 Daubechies4 wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.2 Detecting a disturbance, using Daubechies4 wavelet - wavelet coeffi-
cients squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Distribution of energy in 10 wavelet coefficient levels . . . . . . . . . . . 57
7.4 Some voltage signals with power quality disturbances . . . . . . . . . . . 58
7.5 Deviation in distributed energy related to figure 7.4 . . . . . . . . . . . . 59
7.6 Structure chart of the program module which calculates the reference
energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.7 Structure chart of the program module which identifies the disturbance 61

List of Tables
5.1 Data structure of the calculating algorithm . . . . . . . . . . . . . . . . . 32
66

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68