Zusammenfassung
Inhaltsangabe:Abstract:
The object of this thesis is to design software and hardware to obtain the device parameters of a compensator. This compensator should restore the balance and the power factor of a three-phase three-wire system by using reactive elements only. The derived parameters should be accessible remotely and displayed on a PC.
L. S. Czarnecki recently presented a highly respected approach to derive the device parameters of the compensating susceptances. He defined the admittances Ye and A which represent the conditions in a three-phase system. He also suggested a way to derive these susceptances by measuring two line-to-line voltages and two line currents. The load balancing technique used in this project was based on Czarneckis approach.
The first phase of the project concentrated on understanding and proving the theory behind the project by means of computer simulation.
The second phase of the project involved writing software for the DSP and building an interface to successfully task the requirements set by the theory. The aspect of being able to transfer the data to a PC via a modem-to-modem connection was taken into account too.
In the final stage it is shown that the implemented system is able to derive the necessary parameters in order to balance the currents and restore the power factor as supplied from mains. It was found that even though the supply from the University of Cape Town does not meet the requirements of the theory in terms of harmonic distortion, it is possible to achieve sufficient load balancing and power factor correction.
It was not possible to establish a reliable connection from one modem to the other because of the limitations of the telephone exchange system used at the University of Cape Town. The parts that are necessary for communication, however, were implemented and tested successfully. Therefore it was solely a reliable transmission of data that was unsuccessful and this was due to factors beyond the control or influence of the author.
Inhaltsverzeichnis:Table of Contents:
ERKLÄRUNGII
AcknowledgementsIII
Terms of ReferenceIV
SynopsisV
Table of ContentsVI
List of FiguresX
List of TablesXIII
GlossaryXIV
1Introduction1
1.1The Need for Load Compensation1
1.2The Thesis as a Part of a Project2
1.3Objectives of the Thesis2
2Theory for Balancing a Three-Phase Three-Wire System3
2.1Fictitious Impedance3
2.2Sufficient Condition for Balancing a Three-Phase Load5
2.2.1Compensator to […]
The object of this thesis is to design software and hardware to obtain the device parameters of a compensator. This compensator should restore the balance and the power factor of a three-phase three-wire system by using reactive elements only. The derived parameters should be accessible remotely and displayed on a PC.
L. S. Czarnecki recently presented a highly respected approach to derive the device parameters of the compensating susceptances. He defined the admittances Ye and A which represent the conditions in a three-phase system. He also suggested a way to derive these susceptances by measuring two line-to-line voltages and two line currents. The load balancing technique used in this project was based on Czarneckis approach.
The first phase of the project concentrated on understanding and proving the theory behind the project by means of computer simulation.
The second phase of the project involved writing software for the DSP and building an interface to successfully task the requirements set by the theory. The aspect of being able to transfer the data to a PC via a modem-to-modem connection was taken into account too.
In the final stage it is shown that the implemented system is able to derive the necessary parameters in order to balance the currents and restore the power factor as supplied from mains. It was found that even though the supply from the University of Cape Town does not meet the requirements of the theory in terms of harmonic distortion, it is possible to achieve sufficient load balancing and power factor correction.
It was not possible to establish a reliable connection from one modem to the other because of the limitations of the telephone exchange system used at the University of Cape Town. The parts that are necessary for communication, however, were implemented and tested successfully. Therefore it was solely a reliable transmission of data that was unsuccessful and this was due to factors beyond the control or influence of the author.
Inhaltsverzeichnis:Table of Contents:
ERKLÄRUNGII
AcknowledgementsIII
Terms of ReferenceIV
SynopsisV
Table of ContentsVI
List of FiguresX
List of TablesXIII
GlossaryXIV
1Introduction1
1.1The Need for Load Compensation1
1.2The Thesis as a Part of a Project2
1.3Objectives of the Thesis2
2Theory for Balancing a Three-Phase Three-Wire System3
2.1Fictitious Impedance3
2.2Sufficient Condition for Balancing a Three-Phase Load5
2.2.1Compensator to […]
Leseprobe
Inhaltsverzeichnis
ID 6836
Beuschel, Andreas: Load Compensation in Three Phase Power Systems
Hamburg: Diplomica GmbH, 2003
Zugl.: Fachhochschule Regensburg, Fachhochschule, Diplomarbeit, 2002
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Printed in Germany
THESIS: Beuschel Andreas
FH REGENSBURG
III
Acknowledgements
The author wishes to thank the following people for their invaluable contribution towards
this project:
Mr Malengret for his support and for making this project and the stay in South Africa
possible.
Prof. Dr.-Ing. Wolf for accepting this thesis in South Africa.
Mr Chris Wozniak for his constant support and his willingness to help in any situation.
Mrs Judy Mackintosh for her helpfulness and for offering accommodation to the author on
his first day of the arrival in South Africa.
Leonè and Sven for their input. The author's housemate, Dirk, for his help and moral
support regarding the thesis and living in a foreign country.
Clinton, Bjorn, Ben, Tristan, Lotten, Graeme and Caxton who always put up with the
author's moods.
Finally, his parents, Rudolf and Elfriede, for their support before the author's arrival in
South Africa and for their help from Schönau, Germany, during the author's stay.
THESIS: Beuschel Andreas
FH REGENSBURG
IV
Terms of Reference
This thesis was commissioned by Prof. Dr.-Ing. Wolf of the Fachbereich Elektrotechnik at
the Fachhochschule Regensburg and supervised by Mr Malengret of the Electrical
Engineering Department at the University of Cape Town.
Mr Malengret's specific instructions were:
1. To become familiar with the TMS320F243 Digital Signal Processor (DSP) in order
to use it for this load compensation project.
2. To become familiar with the theory regarding load balancing, in particularly using
the approach supplied by L. S. Czarnecki
3. To design software for the DSP where the approach of L. S. Czarnecki could be
implemented
4. To use suitable transducers to provide the DSP with the necessary voltages and
currents.
5. To develop remote monitoring via a cellular modem connected to the DSP.
6. To test the load compensation under various load conditions
THESIS: Beuschel Andreas
FH REGENSBURG
V
Synopsis
The object of this thesis is to design software and hardware to obtain the device parameters
of a compensator. This compensator should restore the balance and the power factor of a
three-phase three-wire system by using reactive elements only. The derived parameters
should be accessible remotely and displayed on a PC.
L. S. Czarnecki recently presented a highly respected approach to derive the device
parameters of the compensating susceptances. He defined the admittances Y
e
and A which
represent the conditions in a three-phase system. He also suggested a way to derive these
susceptances by measuring two line-to-line voltages and two line currents. The load
balancing technique used in this project was based on Czarnecki's approach.
The first phase of the project concentrated on understanding and proving the theory behind
the project by means of computer simulation.
The second phase of the project involved writing software for the DSP and building an
interface to successfully task the requirements set by the theory. The aspect of being able
to transfer the data to a PC via a modem-to-modem connection was taken into account too.
In the final stage it is shown that the implemented system is able to derive the necessary
parameters in order to balance the currents and restore the power factor as supplied from
mains. It was found that even though the supply from the University of Cape Town does
not meet the requirements of the theory in terms of harmonic distortion, it is possible to
achieve sufficient load balancing and power factor correction.
It was not possible to establish a reliable connection from one modem to the other because
of the limitations of the telephone exchange system used at the University of Cape Town.
The parts that are necessary for communication, however, were implemented and tested
successfully. Therefore it was solely a reliable transmission of data that was unsuccessful
and this was due to factors beyond the control or influence of the author.
THESIS: Beuschel Andreas
FH REGENSBURG
VI
Table of Contents
ERKLÄRUNG II
Acknowledgements III
Terms of Reference
IV
Synopsis V
Table of Contents
VI
List of Figures
X
List of Tables
XIII
Glossary XIV
1 Introduction 1
1.1 The Need for Load Compensation
1
1.2 The Thesis as a Part of a Project
2
1.3 Objectives of the Thesis
2
2 Theory for Balancing a Three-Phase Three-Wire System
3
2.1 Fictitious Impedance
3
2.2 Sufficient Condition for Balancing a Three-Phase Load
5
2.2.1 Compensator to Balance Currents
6
2.2.2 Compensator with Reactive Susceptances
8
2.3 Single-Phase Load as an Extreme Case of Imbalance
9
2.4 Resistive Single-Phase Load
11
2.5 Resistive and Reactive Load
13
2.6 Extension from an One-Phase to a Two-Phase Load
15
2.7 Further Equations
16
2.8 Conclusion 17
3 Computer Simulations
18
3.1 Orcad 18
THESIS: Beuschel Andreas
FH REGENSBURG
VII
3.2 Matlab m-file
21
3.3 Simulink 22
3.3.1 Results of the Simulations
23
a. Single
Resistor
23
b. Load with Three Resistors (unbalanced / symmetric)
25
c. Load with Resistors and Reactances
27
4 Hardware Description
30
4.1 DSP TMS320F243 from Texas Instruments
30
4.1.1 Interrupts
30
4.1.2 Event Manager (EV2)
32
4.1.3 Digital Input / Output Ports (I/O)
33
4.1.4 Interfaces for Communication
35
a. Serial Communication Interface (SCI)
35
b. Serial Peripheral Interface (SPI)
36
4.1.5 Analog to Digital Converter (ADC)
37
4.2 Intelligent Liquid Crystal Display (LCD)
39
4.3 DSP Board
41
4.3.1 Digital to Analog Converter (DAC)
42
4.4 Interface Board
43
4.5 LEM-box 45
4.6 Phase Shifting Circuitry
49
4.7 Modems and their Usage
50
4.7.1 The Hyper Terminal
51
4.7.2 The AT-Command Set
52
5 Software Description
53
5.1 The File System to Program the DSP
53
5.2 The EditPlus Editor
55
5.3 Description of the Functions Implemented in the Program
56
5.3.1 Function
arctan 56
THESIS: Beuschel Andreas
FH REGENSBURG
VIII
5.3.2 Function
decimal 59
5.3.3 Function
displaydigits 60
5.3.4 Function
DACoutput 60
5.3.5 Function
byteTX 61
5.3.6 Function
strInRXBuf 61
5.3.7 Function
waitRXOK 62
5.4 The Main Function
63
5.4.1 Settings
and
Initializations
64
a. LCD
64
b. Interrupts
64
c. Timer
65
d. SPI
66
e. SCI
66
f. Cellular
Modem
66
5.4.2 Infinite
Loop
67
a. Calculations
67
b. Presentation of the Results on the LCD
68
c. Communication
Part
68
5.5 Interrupt Service Routines (ISRs)
69
5.5.1 GPT1_Periodmatch Interrupt
69
a. Processing the Reading of Current I
1
70
b. Processing of Current I
1
and Voltage V
13
70
c. Processing after 2048 Readings
71
5.5.2 SCIRXTX Interrupt
71
6 Results and Discussion
72
6.1 Balancing a Resistive Single-Phase Load
72
6.2 Balancing a resistive and reactive Single-Phase Load
75
6.3 Balancing a Resistive Three-Phase Load
76
6.4 Balancing a Three-Phase Load with Reactances
80
6.5 Communications 83
THESIS: Beuschel Andreas
FH REGENSBURG
IX
7 Conclusions 86
Bibliography 87
Internet Links
88
Appendices 89
Appendix A
89
Appendix B
91
Appendix C
97
Appendix D
101
Appendix E
125
Appendix F
126
Appendix G
129
Appendix H
130
Appendix I
131
THESIS: Beuschel Andreas
FH REGENSBURG
X
List of Figures
Figure 1: The complete project ... 2
Figure 2: Three-phase system with unknown load... 3
Figure 3: Delta connected admittances ... 4
Figure 4: Two-phase load equivalent to Figure 3, phase 3 is taken as the reference ... 4
Figure 5: Unequal admittances add up to sum = 0; a* conjugate of a (equals
2
a )... 6
Figure 6: Compensating admittances: T
12
, T
23
, T
31
... 7
Figure 7: Single-phase load Y
12
=0, Y
31
=0 ... 9
Figure 8: Two-element compensator T
23
=0 mS ... 10
Figure 9: Phasor Diagram of balancing, balanced and load currents... 12
Figure 10: Phasor Diagram of balancing, balanced and load currents
23
3
B
I
IB
=
... 14
Figure 11: (left) Balanced resistive load ... 19
Figure 12: (right) Equivalent two-phase Load to Figure 11... 19
Figure 13: Resulting line currents of previous figures ... 19
Figure 14: (left) Unbalanced load with reactances... 20
Figure 15: (right) Equivalent two-phase load to Figure 14 ... 20
Figure 16: Resulting line currents of previous two figures ... 20
Figure 17: Schematics of the simulation ... 22
Figure 18: Line currents single-phase load ... 23
Figure 19: Balanced line currents single-phase load... 23
Figure 20: Power unbalanced single-phase load ... 24
Figure 21: Power balanced single-phase load ... 24
Figure 22: Unbalanced currents resistive three-phase load... 25
Figure 23: Balanced line currents resistive three-phase load ... 25
Figure 24: Unbalanced power resistive three-phase load... 26
Figure 25: Power balanced resistive three-phase load ... 26
Figure 26: Unbalanced line currents reactive three-phase load ... 27
Figure 27: Balanced line currents reactive three-phase load... 27
Figure 28: Power unbalanced reactive three-phase load ... 28
Figure 29: Power balanced ... 28
THESIS: Beuschel Andreas
FH REGENSBURG
XI
Figure 30: Interrupts low and high priority ... 31
Figure 31: General interrupts and peripheral interrupts; example for priorities ... 31
Figure 32: Flowchart of a double ADC-conversion... 38
Figure 33: LCD-module ... 40
Figure 34: DSP-board... 41
Figure 35: DAC frame... 42
Figure 36: Interface board ... 43
Figure 37: Signal without offset will destroy ADC ... 44
Figure 38: Signal with offset will not destroy ADC ... 44
Figure 39: Mains voltage measured from LEM ... 45
Figure 40: Voltage output from voltage LEM... 45
Figure 41: Schematics of the LEM-box ... 48
Figure 42: Interior of the LEM-box ... 49
Figure 43: Output LEM-box voltages V
13
and V
23
... 49
Figure 44: Schematics of the phase shifting circuit ... 50
Figure 45: Compile-Link-Program routine ... 54
Figure 46: EditPlus window ... 56
Figure 47: Division of the circle into 8 sections, conditions for each section ... 57
Figure 48: Flowchart of the main program ... 63
Figure 49: Performance of Timer1... 65
Figure 50: Flowchart of the GPT1_periodmatch interrupt... 69
Figure 51: Periods where simple subtraction is correct and incorrect ... 71
Figure 52: Single-phase load unbalanced line currents... 73
Figure 53: Single-phase load balanced line currents... 73
Figure 54: (left) Current in the capacitive compensator branch... 73
Figure 55: (right) Current in the inductive compensator branch... 73
Figure 56: Resistive three-phase load; unbalanced line currents ... 77
Figure 57: Resistive three-phase load; balanced line currents ... 77
Figure 58: Resistive three-phase load; unbalanced line currents; supply: generator set 1.. 79
Figure 59: Resistive three-phase load; balanced line currents; supply: generator set 1... 79
Figure 60: Three-phase load with reactances; unbalanced line currents... 81
Figure 61: Three-phase load with reactances; balanced line currents... 81
THESIS: Beuschel Andreas
FH REGENSBURG
XII
Figure 62: (left) Unbalanced line currents; supply: generator set 2; 500 Hz filter on ... 82
Figure 63: (right) Unbalanced line currents; supply: generator set 2; 500 Hz filter off... 82
Figure 64: (left) Balanced line currents; supply: generator set 2; 500 Hz filter on... 82
Figure 65: (right) Balanced line currents; supply: generator set 2; 500 Hz filter off... 82
Figure 66: (left) Balanced line currents; supply: generator set 2; 500 Hz filter on... 82
Figure 67: (right) Balanced line currents; supply: generator set 2; 500 Hz filter off... 82
Figure 68: Hyper Terminal window; communication history... 83
Figure 69: Faxmodem U. S. Robotics dialing to cellular modem... 84
Figure 70: Cell modem; incoming call bearer and type of incoming call... 84
THESIS: Beuschel Andreas
FH REGENSBURG
XIII
List of Tables
Table 1: Pin out of LCD ... 40
Table 2: Features of current and voltage LEM-Modules ... 46
Table 3: Files and their contents... 54
Table 4: Example of a look-up table ... 59
Table 5: LCD compared to Yokogawa Power Meter... 75
THESIS: Beuschel Andreas
FH REGENSBURG
XIV
Glossary
AC Alternating
Current
ADC
Analog to Digital Converter
ADCSOC
ADC Start Of Conversion
ANSI
American National Standars Institution
ASCII
American Standard Code for Information Interchange
ASIC
Application Specific Integrated Circuit
bps
bits per second
CLIP
Calling Line Identification Presentation
CLK Clock
CLP Routine
Compile-Link-Programe Routine
CMOS
Complementary Metal-Oxide Semiconductor
DAC
Digital to Analog Converter
DC Direct
Current
DSP
Digital Signal Processor
EV2 Event
Manager
GPIO
General Purpose I/O
GPT
General Purpose Timer
GUI
Graphical User Interface
I/O
Input / Output
INT Interrupt
ISR
Interrupt Service Routine
LCD
Liquid Christal Display
LEM
Suisse company
LS(B)
Least Significant (Bit)
MS(B)
Most Significant (Bit)
NRZ
No Return Zero
OCRA
Output Control Register A
PC Personal
Computer
PIE
Programable Intrrupt Expansion
PWM
Pulse Width Modulation
PXDATDIR
Port X Data Direction Register
QEP
Quadrature Encoder Pulse
RAM
Random Access Memory
RNG Range
SCI
Serial Communications Interface
SPI
Serial Peripheral Interface
TI Texas
Instruments
UART
Universal Asynchronous Receiver Transmitter
VREFHI
Referece Voltage High
VREFLOW
Referece Voltage Low
XINT2
External Interrupt 2
THESIS: Beuschel Andreas
Chapter 1
FH REGENSBURG
1
1 Introduction
1.1 The Need for Load Compensation
Due to the world-wide rise in economic
competition, companies are forced to improve on
efficiency in terms of operation costs if they want to succeed in the global market.
Unbalanced and reactive currents are factors directly influencing operating costs for both
the power-supplying and power-consuming industries.
It is a known fact that line losses are directly proportional to the square of the current. It
follows then that the power dissipated in a transmission line due to warming is greater in
an unbalanced case than a balanced case due to the excess negative sequence currents.
Similarly, reactive currents add additional losses to the supplying company.
Moreover, the torque of a generator is proportional to the power it delivers. Unbalanced
loads result in power ripples and the torque is therefore not constant as in the desired case.
This will decrease the lifetime of the generator sets because the steady rise and fall of the
load adds wear and tear to the mechanical parts.
We have mentioned factors that affect the power supplier, but the effect on power
consumers should be considered as well. Unbalanced currents result in unbalanced
voltages due to unequal volt drops along the line impedance. Also, more current in one line
leads to greater losses as the line gets warmer and the resistance of the line increases with
its temperature. An increasing resistance produces an increasing voltage drop along the
line. The resulting difference in the voltages for connected three-phase devices may be
small, but it is not negligible. For example, in the case of three-phase induction motors
even a small voltage unbalance can lead to a severe degradation in terms of motor
efficiency and lifespan.
These facts should point out why it is desirable for suppliers as well as customers to
operate with balanced systems and to compensate the reactive power as good as possible.
THESIS: Beuschel Andreas
Chapter 1
FH REGENSBURG
2
1.2 The Thesis as a Part of a Project
This thesis is a part of a project to implement a complete system which is able to balance
the line currents and improve the power factor of a three-phase three-wire system. The
system should display the data locally as well as making the data accessible from a remote
site via a modem-to-modem connection. The remote connection should allow the end-user
to read data and also to interact with the system. Figure 1 shows an overview of the
complete project.
Power
Supply
GRID
DSP
CURRENT /
VOLTAGE
TRANSDUCER
Compensator
PC
Modem
Modem
LCD
LOAD
Figure 1: The complete project
1.3 Objectives of the Thesis
This thesis concentrates on implementing a basic and simplified version of the above
described project in order to enable further development towards a product which meets all
the requirements implied by Figure 1.
The objectives of this project were:
- To build an interface which contains current and voltage transducers
- To implement software on the DSP to obtain the data for the compensator
- To display the data for the compensator on an LCD
- To implement communication between the DSP and a modem
- To set up a modem-to-modem connection transferring data from the DSP to a PC
- To set up tests with manually connected compensator reactances
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
3
2 Theory for Balancing a Three-Phase Three-Wire
System
The basic theory necessary for this project will be introduced in this chapter. It is based on
the achievements of L. S. Czarnecki, who published several science papers regarding load
balancing. Most of the theory was gathered with support from the author's supervisor M.
Malengret.
2.1 Fictitious Impedance
Measuring the load impedance is a main object of this project. If one wants to obtain the
load impedance anywhere in the grid it is never known what "construction" the load is.
Moreover, there will not be only one load connected but several different loads. Some kind
of black box has to be assumed which contains the load, but where the actual circuitry is
not known.
V3
1
V23
V12
I2
I1
I3
LOAD
Figure 2: Three-phase system with unknown load
The key to simplifying the load is basically the same principle as the two-wattmeter
method used for measuring real power in a three-phase system. In this method, there are
only two currents and two voltages measured, but the power presented by the sum of the
two displays shows the power absorbed by the whole load. This is possible because the
three-phase system was actually reduced to a two-phase system by using one of the phases
as the reference. By measuring the currents of the phases, excluding the one taken as the
reference, all information is now available. This also allows one to look at the load in a
different way as shown in Figure 4.
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
4
I1
2
I2
3
Y1
2
V31
V2
3
V1
2
I2
I1
Y23
I3
1
Y3
1
I3
Figure 3: Delta connected admittances
I2
3b
V13
V2
3
I2b
I1b
Y2
3b
I3
1b
Y3
1b
I3b
Figure 4: Two-phase load equivalent to Figure 3, phase 3 is taken as the reference
Figure 4 shows an equivalent two-phase load to Figure 3. It represents exactly the same
conditions but with one less impedance. To get from the circuitry in Figure 3 to the setup
in Figure 4, two formulae had to be derived, as these were not found in any of the available
books. The following formulae were derived for the fictitious impedance of the load (see
Appendix A):
(
)
31
12
12
31
1
)
(
120
1
1
Z
Z
Z
Z
b
+
-
-
-
=
Equation 1
(
)
23
12
12
23
1
)
(
120
1
1
Z
Z
Z
Z
b
+
-
-
=
Equation 2
Hence the admittances are:
b
b
Z
Y
31
31
1
=
b
b
Z
Y
23
23
1
=
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
5
2.2 Sufficient Condition for Balancing a Three-Phase Load
A three-phase, three-wire system can be represented by three admittances as shown in
Figure 3. Is the load in a star configuration then it will be converted to an equivalent delta
connected load. The three admittances are assumed to consist of linear elements such as
resistors, capacitors and inductances. It is required that the supply currents are of equal
magnitude and at 120 degrees from each other then:
3
2
2
1
I
a
I
a
I
=
=
where
2
3
2
1
j
a
+
-
=
and
2
3
2
1
2
j
a
-
-
=
0
2
1
=
- I
a
I
and
0
3
2
2
=
-
I
a
I
a
But since
0
3
2
1
=
+
+
I
I
I
0
)
(
2
1
2
2
=
-
-
-
I
I
a
I
a
which also gives
0
2
1
=
- I
a
I
Hence, a sufficient condition is that:
0
2
1
=
- I
a
I
Substituting
31
12
1
I
I
I
-
=
and
12
23
2
I
I
I
-
=
)
(
12
23
31
12
I
I
a
I
I
-
-
-
gives:
0
23
2
31
12
=
+
+
I
a
I
a
I
Assuming balanced voltage:
0
12
23
2
2
12
31
12
12
=
+
+
V
Y
a
a
V
Y
aa
V
Y
Therefore:
0
31
2
23
12
=
+
+
Y
a
Y
a
Y
Equation
3
or:
0
12
2
31
23
=
+
+
Y
a
Y
a
Y
As a result, the satisfying condition for a three-phase balanced voltage to have balanced
currents, is that the sum of the space vector of the three-phase admittances be equal to
zero. A space vector is the result of a mathematical transformation called Park
transformation. Note that the above implies that the three admittances do not necessarily
have to be equal. The only necessary condition is that the space vector sum of the three
impedances be zero. This is illustrated in Figure 5, where the admittances are shown in the
complex plane with their respective real and imaginary components. The real part is the
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
6
conductance G and the imaginary part is the susceptance B of each branch of the delta
load. This Figure 5 represents how the phasors a and a
2
behave.
Re
Im
aY
23
Y12
(a*)
Y31
Y23
Y31
(a*)
Y31
aY
23
-120
120
Figure 5: Unequal admittances add up to sum = 0; a* conjugate of a (equals
2
a
)
2.2.1 Compensator to Balance Currents
Should the space vector sum
31
2
23
12
Y
a
Y
a
Y
+
+
not be zero
,
then the load will be
unbalanced. In order to restore the balance, three susceptances T
12
, T
23
and T
31
can be
added in parallel to each of the load admittances as seen in Figure 6. The use of
susceptances enables compensation without the loss of energy (with ideal capacitors/
inductors). The total admittance per branch must be considered and Equation 3 is used to
obtain supply current balance:
0
)
(
)
(
)
(
31
31
2
23
23
12
12
=
+
+
+
+
+
jT
Y
a
jT
Y
a
jT
Y
This has to be solved for T
12
, T
23
and T
31
:
0
)
(
31
2
23
12
31
2
23
12
=
+
+
+
+
+
T
a
ajT
jT
Y
a
Y
a
Y
Czarnecki defines A to be the unbalanced admittance, where
12
2
31
23
Y
a
Y
a
Y
A
-
-
-
=
In order to balance the supply:
0
31
2
31
23
=
+
+
+
-
jT
a
ajT
jT
A
0
2
3
2
1
2
3
2
1
31
31
23
=
-
-
+
+
-
+
+
-
jT
j
jT
j
jT
A
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
7
0
2
1
2
1
)
Im(
2
3
2
3
)
Re(
12
31
23
12
31
=
-
-
+
-
+
+
-
-
T
T
T
A
j
T
T
A
IP1
IP2
IN
1
IN
2
IN
3
V31
IP3
V2
3
V1
2
I3
I2
T12
T23
T31
Y2
3
Y1
2
Y31
I1
Figure 6: Compensating admittances: T
12
, T
23
, T
31
Equating the real part to zero:
0
2
3
2
3
)
Re(
12
31
=
+
-
-
T
T
A
)
Re(
3
2
31
12
A
T
T
=
-
Equation
4
Equating the imaginary part to zero:
0
2
1
2
1
)
Im(
12
31
23
=
-
-
+
-
T
T
T
A
)
Im(
2
2
12
31
23
A
T
T
T
=
-
-
Equation 5
There are only two necessary equations but three unknowns. Therefore an infinite number
of solutions exist if three compensating susceptances are used. Any of the three
susceptances can be chosen arbitrarily (e.g. one susceptance is set to nil).
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
8
2.2.2 Compensator with Reactive Susceptances
Unity power factor is a third condition that can also be imposed. Here, the sum of all the
susceptances must be zero in the case of a balanced voltage supply. Therefore the total sum
of all the susceptances seen by the supply must be zero:
If
)
Im(
31
23
12
Y
Y
Y
Be
+
+
=
Then
Be
T
T
T
-
=
+
+
31
23
12
Equation 6
Combining Equation 4, Equation 5 and Equation 6 gives:
(
)
(
)
(
)
3
)
Im(
)
Re(
3
3
)
Im(
2
3
)
Im(
)
Re(
3
31
23
12
Be
A
A
T
Be
A
T
Be
A
A
T
+
+
-
=
-
=
-
-
=
Equation 7
These are as given by Czarnecki [7]. What is of particular interest is that the three
compensating elements are purely reactive. No real power is therefore required in order to
balance any unequal loads. The three load admittances Y
12
, Y
23
and Y
31
can be of any
complex value. A single-phase load can be considered as an extreme case of an unbalanced
load, where two of the load admittances are nil.
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
9
2.3 Single-Phase Load as an Extreme Case of Imbalance
The above theory is now applied to an unbalanced three-phase load consisting of only one
load admittance with the other two equal to zero as shown in Figure 7.
IP1
IP2
IN1
IN2
IN3
V3
1
IP3
V2
3
V1
2
I3
I2
T12
T23
T31
Y2
3
Y12
Y3
1
Figure 7: Single-phase load Y
12
=0, Y
31
=0
The load is a single-phase load. Admittances Y
12
and Y
31
are considered as open circuit,
hence, Y
12
=0 mS and Y
31
=0 mS. Substituting
23
23
23
23
23
23
23
B
Be
jB
G
Y
A
jB
G
Y
e
Y
=
-
-
=
-
=
+
=
=
where G
23
is the conductance and B
23
the susceptance of the single-phase load
into Equation 7 leads to:
(
)
(
)
(
)
3
)
Im(
)
Re(
3
3
)
Im(
2
3
)
Im(
)
Re(
3
23
23
23
23
23
31
23
23
23
23
23
23
23
23
23
12
B
jB
G
jB
G
T
B
jB
G
T
B
jB
G
jB
G
T
+
-
-
+
-
-
-
=
-
-
-
=
-
-
-
-
-
-
=
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
10
Finally:
23
31
23
23
23
12
3
1
3
1
G
T
B
T
G
T
=
-
=
-
=
Equation 8
Therefore the three susceptances above would correct the imbalance and reactive power
seen by the three-phase supply. If one adds an equal susceptance +B
23
to T
12
, T
23
and T
31
respectively, one would still have a balanced load from the supply point of view. The
reason is that A, the imbalance admittance, is invariable. The following results are
obtained:
23
23
31
23
23
23
23
23
12
3
1
0
3
1
B
G
T
B
B
T
B
G
T
+
=
=
+
-
=
+
-
=
Equation
9
The compensator is reduced to two elements only, as seen in Figure 8.
IP1
IP2
IN
1
IN
2
IN
3
V31
IP3
V2
3
V1
2
I3
I2
Y23
T23
T31
T12
Figure 8: Two-element compensator T
23
=0 mS
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
11
2.4 Resistive Single-Phase Load
If the load is purely resistive, then B
23
=0 and Equation 7 becomes:
23
31
23
23
12
3
1
0
3
1
G
T
T
G
T
=
=
-
=
Equation 10
Referring to Figure 8, and assuming that the three-phase supply voltages V
12
, V
23
and V
31
are balanced sinusoids and of positive sequence direction, the compensator currents I
N1,
I
N2
and I
N3
can be calculated as follows:
0
12
= V
V
120
23
-
= V
V
120
31
= V
V
if
23
G
V
I
=
and using Equation 10:
12
12
2
jT
V
I
N
=
-
=
3
0
23
jG
V
90
3
-
=
I
31
31
3
jT
V
I
N
-
=
-
=
3
120
23
jG
V
30
3
=
I
3
2
1
N
N
N
I
I
I
-
-
=
-
-
-
=
30
3
90
3
I
I
150
3
=
I
1
1
N
P
I
I
-
=
30
3
-
=
I
2
2
2
N
P
I
I
I
-
=
-
-
-
=
90
3
)
120
(
I
I
210
3
=
I
3
3
3
N
P
I
I
I
-
=
-
=
30
3
)
60
(
I
I
90
3
=
I
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
12
The three-phase supply current magnitudes are
3
1
that of the single-phase load currents
and they are balanced positive sequence currents. The negative and positive sequence
currents add up to make I
2
, 0, I
3
(=-I
2
)
.
These currents can be seen in Figure 9. The
compensating susceptances,
T
12
= -
3
1
G
23
and
T
31
=
3
1
G
23
"inject" the negative sequence current necessary to balance the unbalanced load consisting
of a single resistive element. The supply delivers a balanced three-phase current.
I2
I3
V12
V23
V31
IN3
IN2
IN1
IP3
IP2
IP1
postive sequence
negative sequence
Figure 9: Phasor Diagram of balancing, balanced and load currents
A relevant observation is that I
N2
=-I
P3
and I
N3
=-I
P2
. The negative sequence components are
the conjugates of the positive sequence.
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
13
2.5 Resistive and Reactive Load
The calculations for the simplest case mentioned in 2.3 will be extended to a resistive and
reactive load. The voltages are still assumed as before (balanced / symmetric). The value
for the susceptance has to be added to Equation 8.
+
=
=
-
-
=
23
23
31
23
23
23
12
3
1
0
3
1
B
G
T
T
B
G
T
Equation 11
23
2
Y
V
I
=
(
)
23
23
23
23
23
B
jI
I
jVB
VG
jB
G
V
+
=
+
=
+
=
(
)
90
0
3
1
3
1
0
23
23
23
23
12
23
2
-
-
=
-
-
=
=
B
G
V
B
G
j
V
jT
V
I
N
(
)
90
90
3
23
23
2
-
-
-
=
B
N
V
G
V
I
(
)
90
90
3
23
2
+
-
=
B
N
I
I
I
(
)
90
300
3
1
3
1
120
23
23
23
23
31
31
3
+
-
=
-
-
=
-
=
B
G
V
B
G
j
V
jT
V
I
N
(
)
30
30
3
23
23
3
-
=
B
N
V
G
V
I
(
)
30
30
3
23
3
+
=
B
N
I
I
I
(
)
(
)
+
-
+
-
-
=
-
-
=
30
30
3
90
90
3
23
23
3
2
1
B
B
N
N
N
I
I
I
I
I
I
I
(
)
240
3
150
3
23
1
+
=
B
N
I
I
I
It must be noted that the compensating reactance's currents are not balanced.
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
14
Supply side currents:
(
)
60
3
30
3
23
1
1
+
-
=
-
=
B
N
P
I
I
I
I
(
) (
)
(
)
(
)
+
-
-
-
+
-
=
-
=
90
90
3
120
120
23
23
2
2
2
B
B
N
P
I
I
jI
I
I
I
I
(
)
60
3
210
3
23
2
-
+
=
B
P
I
I
I
(
) (
)
(
)
(
)
+
-
+
=
-
=
30
30
3
60
60
23
23
3
3
3
B
B
N
P
I
I
jI
I
I
I
I
(
)
180
3
90
3
23
3
+
=
B
P
I
I
I
I2
I3
V12
V23
V31
IN3
IN2
postive sequence
negative sequence
IN1
IP1
IP
2
IP
3
IB
IB
IB
IB
IB
IB
Figure 10: Phasor Diagram of balancing, balanced and load currents
23
3
B
I
IB
=
Figure 10 shows that the supply currents are balanced.
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
15
2.6 Extension from an One-Phase to a Two-Phase Load
Previous explanations show that it is enough to know the single-phase load admittance
23
23
23
jB
G
Y
+
=
to derive the compensating susceptances to balance the load. (Repeating
Equation 8)
+
=
-
-
=
23
23
23
23
3
1
31
3
1
12
B
G
T
B
G
T
This implies that it is necessary to work out the values G
23
and B
23
in order to get T
12
and
T
31
.
Together with the previously introduced fact that every three-phase load can be
transformed into an equivalent two-phase load, it is now possible to extend the single-
phase definition of A to the two-phase case below:
13
23
Y
a
Y
A
-
-
=
)
Im(
13
23
Y
Y
Be
+
=
This leads to following components of A:
13
13
23
2
3
2
1
)
Re(
B
G
G
A
+
+
-
=
Equation 12
13
13
23
2
1
2
3
)
Im(
B
G
B
A
+
-
-
=
Equation 13
and Be:
13
23
B
B
Be
+
=
Equation 14
Re(A), Im(A) and Be are the values that have to be substituted into Equation 7 (presented
again below) to obtain T
12
, T
23
and T
31
for the general case.
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
16
(
)
(
)
(
)
3
)
Im(
)
Re(
3
3
)
Im(
2
3
)
Im(
)
Re(
3
31
23
12
Be
A
A
T
Be
A
T
Be
A
A
T
+
+
-
=
-
=
-
-
=
The author has refrained from doing the substitution so as not to spoil the overview. The
phasor diagrams and calculations for this general case are abandoned as well because of
the same reason. The principle was shown for the case of a single-phase load. The general
case could be shown with a similar example.
2.7 Further Equations
Some more equations have to be introduced, as they are necessary to derive the load
admittances.
=
=
n
i
i
p
n
P
1
1
Equation 15
2
rms
V
P
G
=
Equation 16
=
=
n
i
i
rms
v
n
V
1
2
2
1
Equation 17
2
2
2
rms
rms
V
I
Y
=
Equation 18
=
=
n
i
i
rms
i
n
I
1
2
2
1
Equation 19
2
2
2
G
Y
B
-
=
Equation 20
P = Real Power
V = Voltage
I = Current
Y = Admittance
G = Conductance
B = Susceptance
n = Number of Instantaneous Values
Lower case letters indicate instantaneous values. All values are absolute values.
THESIS: Beuschel Andreas
Chapter 2
FH REGENSBURG
17
2.8 Conclusion
It is possible to derive the compensating susceptances from the values of the load
admittances. Equation 12, Equation 13 and Equation 14 shown above therefore have to be
implemented in the DSP. The compensating susceptances can be calculated by
implementing Equation 7 into the DSP. In order to obtain the load admittance the
equations from Chapter 2.7 have to be considered.
THESIS: Beuschel Andreas
Chapter 3
FH REGENSBURG
18
3 Computer
Simulations
Before starting with the programming or building of anything, it was necessary to prove
the facts shown in Chapter 1. This part of the thesis was important but should not have
taken too much time. A combination of three programs was therefore used to obtain the
desired results: Orcad, Matlab and Simulink.
The transformation from a three-phase to an equivalent two-phase load was compatible
with Orcad. In this simulation program based on Spice, it is easy to add waveforms from
different circuit diagrams into one output window. Matlab was used to do the calculations
to retrieve all the different values of A, Be and T
xy
as well as the device parameters
(capacity, inductance) of each part in the circuit. Finally, the compensation was set up with
Simulink, which is an extension of Matlab but can be considered as a self-contained
program. For Simulink, different libraries are obtainable. Power Toolbox was found to be
the most suitable as it contained all mandatory parts for establishing a working simulation.
3.1 Orcad
Two 3-phase circuits were established. The only difference was that one contained the
original three-phase load. In the other circuit only two impedances between lines 2-3 and
3-1 (but not 2-1) represented the load (as shown in Chapter 2.1). If the derived equations 1
and 2 were correct, the resulting currents must be the same. The equivalent values were
already calculated with a Matlab m-file.
THESIS: Beuschel Andreas
Chapter 3
FH REGENSBURG
19
V2
V3
0
I
I
R12
100
R23
100
R31
100
V1
I
V2b
V1b
R31res
50
R23res
50
I
0
L23res
91.9m
C31res
110.3u
I
V3b
I
Figure 11: (left) Balanced resistive load
Figure 12: (right) Equivalent two-phase Load to Figure 11
Figure 13: Resulting line currents of previous figures
THESIS: Beuschel Andreas
Chapter 3
FH REGENSBURG
20
I
V3
V1
C23
90.9u
L12
318.3m
I
0
R23
150
R31
33
R12
70
V2
C31
70.7u
I
C31res
122.1u
I
V2b
I
V3b
R23res
65.63
0
V1b
L23res
435m
I
R31res
32.24
Figure 14: (left) Unbalanced load with reactances
Figure 15: (right) Equivalent two-phase load to Figure 14
Figure 16: Resulting line currents of previous two figures
Figure 11 to Figure 16 show the circuitries and the results in 2 different cases. First, a
balanced resistive load (Figure 11) and then an unbalanced reactive load (Figure 14) is
replaced by the equivalent two-phase load. Figure 13 and Figure 16 show the line currents
measured from the current probes in the respective figures. The colours from the probes
and the waveform match. Initially, these currents differ slightly but after two cycles (first
case ½ cycle) they are identical. The slight differences are a simulation result from Orcad.
It assumes different initial start values for different reactances.
THESIS: Beuschel Andreas
Chapter 3
FH REGENSBURG
21
3.2 Matlab m-file
Matlab m-files are programmed with the basic Matlab language. They can contain simple
instructions as well as complicated structures including a GUI (Graphical User Interface).
It was not the author's intention to create a GUI because the output in the regular Matlab
Window was enough. Hence, only basic instructions were used to calculate the results.
Once programmed, the same calculations can be repeated with different starting values as
often as is required. This is the simple advantage of m-files and was the fundamental
reason for implementing m-files. During the project two m-files were programmed because
of alternating starting values. "Calculations_Z_serial.m" starts with values of serially
connected impedances. Whereas parallel admittances are the starting values in
"Calculations_Y_parallel.m" These files are shown in Appendix B
.
Single steps in the
program are:
- Definition of starting values
- Get inverse of starting value
- Define phasors a and a
2
- Get values for Capacitors and/or Inductors
- Calculation of equivalent two-impedance and two-admittance load
- Derive data according to Czarnecki (A, Be, T
12
, T
23
, T
31
)
- Get parameters for compensating Capacitors and Inductors
Example of an output in the Matlab window:
Z12 = 7.0000e+001 +1.0000e+002i
Z23 = 1.5000e+002 -3.5000e+001i
Z31 = 33.0000 -45.0000i
Y12 = 0.0047 - 0.0067i
Y23 = 0.0063 + 0.0015i
Y31 = 0.0106 + 0.0145i
L12 = 0.3183
C23 = 9.0946e-005
C31 = 7.0736e-005
Z23b = 6.5629e+001 +1.3655e+002i
L23b = 0.4347
Z31b = 32.2410 -26.0625i
C31b = 1.2213e-004
A = 0.0197 - 0.0027i
Be = 0.0092
T12 = 9.1795
T23 = -4.8812
T31 = -13.5130
CT12 = 2.9219e-005
LT23 = 0.6521
LT31 = 0.2356
END = 0
Details
- Seiten
- Erscheinungsform
- Originalausgabe
- Jahr
- 2002
- ISBN (eBook)
- 9783832468361
- ISBN (Paperback)
- 9783838668369
- DOI
- 10.3239/9783832468361
- Dateigröße
- 2.6 MB
- Sprache
- Englisch
- Institution / Hochschule
- Fachhochschule Regensburg – Elektrotechnik
- Erscheinungsdatum
- 2003 (Mai)
- Note
- 1,0
- Schlagworte
- digital signal processor remote monitoring tms320f243 texas instruments czarnecki
