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9
th International Conference on


“Technical and Physical Problems of
Electrical

Engineering”


9
-
11

September

201
3


Isik University & Istanbul Technical University

Istanbul
,
Turkey



ICTPE

Conference


www.iotpe.com


i
c
tpe@
iotpe
.com

ictpe0@gmail.com

IC
TPE
-
201
3

N
umber

1

Code

0
1POW
0
7

Pages
1
-
6


1

PAPER TITLE


N.M.
Author

1



H.A.
Author

2
,3

B
.
Author

3

S.
Author

4

R
.
T.

Author

4


1
.

Electrical Engineering Department,


University of

,
City
,
Country
,
e
-
mail.account
.1
@
---
.
---

2
.

Center of

,
Faculty of
Engineering,


University of

,
City, Country
,
e
-
mail.account
.2
@
---
.
---

3
.

Department

of …
,
Organization

of

,
City, Country, e
-
mail.account
.3
@
---
.
---

4
.



Co
mpany
,
City, Country, e
-
mail.account@
---
.
---
, e
-
mail.account
.4
@
---
.
---



Abstract
-

Permanent magnet synchronous motor
(PMSM) have a wide range of applications, such as
electric drives and machine ………………………………

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…………………………………………………………….

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……………… to ensure stability and tracking.
Simulations is carried out to verify the theoretical results.



Keywords:

PMSM, Modeling, Saturation, ……………,
…………., ……………., Lyapunov Stabi
lity.


I. INTRODUCTION


A broad spectrum of electric machines is widely used
in electromechanical systems. In addition to the required
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primary issues are studied in this paper. In particular, we
perform nonlinear modeling and analysis, controllers
design, and validate the theoretical results [1].


II. NONLINEAR MOTOR DYNAMICS

A mathematical

model of three
-
phase ,two
-
pole
permanent
-
magnet synchronous motors should be
developed. Three
-
phase, two
-
pole permanent
-
magnet
synchronous motor is illustrated in Figure 1.


A. Motor
Modeling

For the magnetically coupled abc stator windings, we
apply the
Kirchhoff voltage law to find a set of the
following differential equations:


(1)


(2)


(3)

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…………………………………………………………….


(5)


(6)

where the flux linkages are:

9
th International Conference on “Technical and Physical Problems of
Electrical

Engineering” (ICTPE
-
201
3
)

Istanbul
,
Turkey
,
9
-
11

September

201
3


2


(7)

where

is the stator resistance,

and

are the
leakage and magnetizing inductances

and

is the amplitude of the flux linkages established
by the permanent magnet.




Figure 1. Two
-
pole permanent
-
magnet synchronous motor


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B. Park Transformation

Applying the
Park transformation, we have
the
following expression for the electromagnetic torque:


(14)

Using
Equation
(11) and the park transformation, one
obtains the following differential
equation to model
permanent
-
magnet synchronous motors in the rotor
reference frame:


(15)


(16)


(17)


(18)

where
,
,
,

and
,
,
are the
quadrature,
direct
, and zero
-
axis current and voltage components.

The analysis of permanent
-
magnet synchronous
motors in the arbitrary
reference

frame u
sing the
quadrature
, direct
,

and zero
-
quantities is simple. The
electromagnetic torque is a function of the quadrature
current

and differential equation for the zero current
can be omitted from the analysis. We
have:

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That is, the total derivative of a positive
-
definite
quadratic function

is negative. Hence, an
open
-
loop system is

uniformly asymptotically stable [3].


9
th International Conference on “Technical and Physical Problems of
Electrical

Engineering” (ICTPE
-
201
3
)

Istanbul
,
Turkey
,
9
-
11

September

201
3


3

III. FEEDBACK LINEARIZATION CONTROL


As a first step toward the design, we mathematically
set up the design problem. It is easy to verify that the
linearizability
condition

is guaranteed. Let:

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Remark.

In pole
-
placement design, the specification
of optimum (desired) trans
ient responses in terms of
system models and feedback coefficients is equivalent to
the specification imposed on desired transfer functions of
closed
-
loop systems. Clearly, the desired eigenvalues can
be specified by the designer, and these eigenvalues are

used to find the corresponding feedback gains. However,
the pole
-
placement concept, while guaranteeing the
desired location of the characteristic eigenvalues can lead
to positive feedback coefficients and control constraints.

Hence, the stability,
robustness to parameter variations,
and system performance are significantly degraded.


Mathematically, feedback linearization reduces the
complexity of the corresponding analysis and design.
However, even from mathematical standpoints, the
simplific
ation and
"optimum"

performance would be
achieved in expense of large control efforts required
because of linearizing feedback (25). This leads to
saturation. It must be emphasized that the need to
linearize (19,

20,

21)
does

not exist because the open
-
loo
p system is uniformly asympotically stable.


The most critical problem is that the linearizing
feedback:

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Hence, the feedback linearizing controllers cannot be
implemented to control synchronous machines. It is
desirable, therefore, to develop other methods to solve the
motion control p
roblem, methods that do not entail the
applied voltages to the saturation limits to cancel
beneficial nonlinearities


,
and
,and

methods that do not lead to unbalanced motor operation
.


IV. THE

LYAPUNOV
-
BASED APPROACH

In this section, the design is approached using a
nonlinear model. Using
Equations
(19
)
,

(
20
) and (
21), we
have the following matrix form
:

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The feedback coefficients

and
can be
found by solving nonlinear matrix inequalities. Applying
the Lyapunov

stability theory and generalizing the results
above, the stability of the resulting closed
-
loop system
can be examined studying the criteria imposed on the
Lyapunov function. For the bounded reference signal,
using the positive
-
definite quadratic function

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he given tracking controllers extend the applicability
of
the stabilizing algorithms, and allows one to solve the
motion control problem for electromechanical systems
driven by permanent
-
magnet synchronous motors. Using
the inverse Park transformation, one derives the control
laws in the machine

variables. In particular, the
bounded PID controller is given as:

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V. SIMULATION RESULTS


In this section, we design a tracking controller for a
electromechanical system. We use a Kollmorgen

four
-
pole permanent
-
magnet synchronous motors H
-
232 with
the following rated data and parameters: 135 W, 434
rad/sec, 40 V, 0.42 N
.
m, 6.9 A,

,
,
or

and

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9
th International Conference on “Technical and Physical Problems of
Electrical

Engineering” (ICTPE
-
201
3
)

Istanbul
,
Turkey
,
9
-
11

September

201
3


4

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This controller is bounded. The sufficient criteria for
stability are
satisfied.
To study the transient
behavior,

a
controller is verified through comprehensive simulations.
Different reference velocity , loads , and initial conditi
ons


The applied phase voltages and the resulting phase
currents in the as
b
s

and
c
s

windings are illustrated in
Figure 2. Figure 3 documents the motor mech
a
nical
a
ngular velocity
. The setting time for the motor angular
velocity as motor starts from stall is 0.0025 sec. The
disturbance
attenuation

features are evident. In particular,
the assigned angular velocity with zero steady
-
state error
has been guaranteed when the rated load
torque was
applied.

Figures 2 and 3 illustrate the dynamics of the closed


loop drive for the following reference speed and load
torque:

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Figure 2. Radial
-
velocity profiles for different

rates


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9
th International Conference on “Technical and Physical Problems of
Electrical

Engineering” (ICTPE
-
201
3
)

Istanbul
,
Turkey
,
9
-
11

September

201
3


5



Figure 9. Source side voltage and current of phase (2)


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VI. CONCLUSIONS

Permanent
-
magnet synchronous motors are used in a
wide range of electromechanical systems because they
are simple
and can be easily controlled. The steady
-
state
torque
-
speed characteristics fulfil the controllability
criteria over an entire envelope of operation. In this paper
a bounded controller is designed and sufficient criteria for
stability are satisfied. Differ
ent reference velocity, loads,
and initial conditions are studied to analyze the tracking
performance of the resulting system.

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APPENDICES

A
ppendix 1
.

Construction Cost and Characteristics of
230 and 400 kV Lines

Tables 8 and 9 show the construction costs of 230 and
400 kV lines. Also, characteristics of these lines are listed
in Table 10.


Table 10. Construction cost of 230 kV


Variable Cost of Line
Construction

(×10
3

dollars)

Fix Cost of Line
Construction


(×10
3

dollars)

Number of Line
Circuits

45.9

546.5

1

63.4

546.5

2

Table 11. Construction cost of 400 kV


Variable Cost of Line
Construction

(×10
3

dollars)

Fix Cost of Line
Construction

(×10
3

dollars)

Number of Line
Circuits

92.9

1748.6

1

120.2

1748.6

2


Table 12. Characteristics of 230 kV lines


Resistance
(p.u/Km)

Reactance
(p.u/Km)

Maximum Loading
(MVA)

Voltage
Level

1.22e
-
4

3.85e
-
4

397

230

3.5e
-
5

1.24e
-
4

750

400


Appendix 2
. GA and Other Required Data

Load growth
coefficient = 1.08;
Inflation
coefficient
for loss = 1.15; Loss cost in now = 36.1 (
$/MWh
);
Number of initial population = 5;
End condition: 3500
iteration after obtaining best fitness (
N
=3500);
LL
max

=
30%.


NOMENCLATURES

: The number of gas and heat units

: The number of water units

: The whole production expense

: The fuel expense of its unit

:
The load of net in
t

moment

:
The cycling reserve load

:
The production power of
i

heat unit

:
The production power of
j

water unit

:
The reserve power of
i

water unit

:
The reserve power of
j

water unit

:
The starting expense of
i

heat unit

:
The starting expense of
j

water unit

: The subtitle
related to interval

: The time of a complete period under consideration

…………………………………………………………….


ACKNOWLEDGEMENT
S

The great work of Mr.

Eabcd Nancd that was a
doctoral thesis and other parts for power research at the
University of
Cabcd, Iabcd, was a great help for
developing this paper. With the cooperation of my Ph.D.
thesis’s supervisor Prof. Sabcd Kabcd that spent a
valuable part of his time for the paper.


REFERENCES

[1]
G.A. Taylor, M. Rashidinejad, Y.H. Song, M.R.
Irving, M.
E. Bradley
,

T.G. Williams,


Algori
thmic
Techniques for Transition
Optimized

Voltage and
Reactive Power Control”,
International Conference on
Power System Technology,

Vol
.

3,

No. 2, pp.

1660
-
1664,
13
-
17 Oct. 2002.

[2]

J. Zhong, E. Nobile, A. Bose
,

K. Bhattacharya,

Localized Reactive Power Markets Using the Concept of
Voltage Control Areas”,

IEEE Transactions on Power
Systems,

Vol
.

19,

Issue 3, pp. 1555
-
1561,

Aug. 2004.

9
th International Conference on “Technical and Physical Problems of
Electrical

Engineering” (ICTPE
-
201
3
)

Istanbul
,
Turkey
,
9
-
11

September

201
3


6

[3]

A.B. Author1, “Title of the Book”, Ghijk Press, City,
Country, Month Year.

[4] A. Author1, C.D Author2, “Title of the Conference
Paper”, 8th International Conference on Technical and
Physical Problems of Electrical Engineering (ICTPE
-
2013), No. 10, Code
01CPE05
, pp. 42
-
48, City, Country,
9
-
11 September 2013.

[5] A.B. Author1, C.
Author2, D.E.F. Author3, “Title of
the Journal Paper”, International Journal on Technical and
Physical Problems of Engineering (IJTPE), Issue 12, Vol.
4, No. 3,
pp. 79
-
86, September 2012.

[6] http://www.abcd.efgh_ijk.LMN/.../opqr.pdf .

……………………………………………………
……….


BIOGRAPHIES


Nabcd Mabcd
was born in
Tacd,
Iabcd,
1967. He received the B.Sc.
and the M.Sc. degrees from
University of Tabcd (Tabcd, Iabcd)
and the Ph.D. degree from
University of Sabcd (Tabcd, Iabcd),
all in Power Electrical Engineering,
in 1989, 1
992, and 1997,
respectively. Currently, he is a Professor of Power
Electrical Engineering at University of Eabcd (Babcd,
Aabcd). He is also an academic member of Power
Electrical Engineering at University of Sabcd (Tabcd,
Iabcd) and teaches Power System An
alysis, Power
System Operation, and Reactive Power Control. He is the
secretary of International Conference on ABCD. His
research interests are in the area of Power Quality,
Energy Management Systems, ICT in Power Engineering
and Virtual E
-
learning Educati
onal Systems. He is a
member of the International Electrical and Electronic
Engineers.


Habcd
S
abcd
was born in Zabcd,
Iabcd, on January 23, 1951. He
received the B.Sc. and M.S.E
.
degrees in Electrical Engineering in
1973 and 1979 and the Ph.D. degree
in electrical engineering from Mabcd
State University, Uabcd, in 1981.
Currently, he is a full professor at
electrical engineering department of University of Tabcd
Tabcd, Iabcd. His

research interests are in the application
of artificial intelligence to power system control design,
dynamic load modeling, power system observability
studies and voltage collapse. He is a member of Mabcd
Association of Electrical and Electronic Engineers

and
IEEE.


Babcd
Kabcd

was born in Aabcd
Mabcd, Rabcd, in February 1961. He
received a five
-
year degree in
electronic engineering from the
University of Babcd, Rabcd, in 1986
and the Ph.D. degree in Automatic
Systems and Control from the same
university
, in 1996. He is currently a Professor with the
University of Pabcd, Rabcd. Previously, he was in
hardware design with Dabcd Rabcd SA, Rabcd. He has
authored of six books in Power Converter area, one in
Theory and Control Systems, one in Fuzzy Control, one

in Hardware topologies for PC and Devices, and one in
Medical Electronics and Informatics. Also, he is co
-
authored of the book Fundamentals of Electromagnetic
Compatibility, Theory and Practice and of a book chapter
-

“Iabcd Cabcd of the Eabcd Gabcd Sabcd
I 楮 瑨攠book
“Iabcd Sabcd

慮d
䭡h捤 䵡b捤

for Eabcd”. His current
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sys瑥msI on bo瑨 dynam楣i 慮d 捯n瑲o氬l 慮d pow敲
敬散瑲on楣献i 䡥 has author敤 or 捯慵瑨or敤 of sev敲慬a
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pro捥敤楮gsK 䡥e 楳 慮 䅳Ao捩慴c bd楴ir of s捩敮瑩f楣
journal of the University of Pabcd “Eabcd and Cabcd
Sabcd” and program chair and proceeding editor of the
International Conference on “E
慢捤I C慢捤 慮d A慢捤
Iabcd”, 2005, 2007 and 2009 editions.


Sabcd
Eabcd

was born in Tabcd,
Eabcd Aabcd, Iabcd in September
1940. He received

the Dipl.
-
Ing.
degree on Sabcd Tabcd from the
Rabcd, Aabcd, Gabcd in 1969. From
1970 to 1971 he worked for Aabcd
,
Fabcd, Gabcd on electric distribution
system planning. From 1972 to 1977
he was a lecturer

of

Electrical Engineering at University
of Tabcd, Tabcd, Iabcd. From 1977 to 1979 he was as
postgraduate student

in Uabcd, Eabcd, where he received
M.Sc. degree on

Power System. From 1980 to 2007 he
was a professor of Electrical Engineering

of University of
Tabcd. In February 2007 he was retired. During his
working in University of Tabcd he was from 1988 to
1989 in Rabcd, Aabcd, Gabcd

and 1996 to 1997 in
Electrical
Engineering Department of University of
Sabcd,

Cabcd

in Sabbatical leave. His research interest is
in electrical machines, modeling, parameter estimation
and vector control.


Rabcd Tabcd

was born in Sabcd,
Nabcd, Abcd on 28 September 1949.
He is professo
r of power engineering
(1993); chief editor of scientific
journal of “Pabcd Eabcd Pabcd”
from 2MMM㬠 d楲散瑯r of fns瑩瑵瑥t of
m慢捤 from 2MM2 up 瑯 2MMV;
慣慤敭楣楡i and 瑨攠 f楲st v楣i
J
pr敳楤敮琠tf A慢捤 乡k捤 Aab捤 of 卡p捤 from 2MMTK 䡥
楳 污ur敡瑥tof A慢
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of A慢捤 E2MMRF㬠 捯
J
捨慩am慮 of fn瑥tn慴楯n慬
Conferences on “Tabcd and Pabcd Eabcd”. His research
慲敡s 慲攠 瑨敯ry of non
J
l楮敡r 敬散瑲楣慬i 捨慩ns w楴h
d楳瑲楢u瑥t p慲am整敲sI n敵瑲a氠敡r瑨ing and f敲ror敳en慮t
pro
捥ss敳⁡nd 慬瑥an慴av攠en敲gy sour捥sK 䡩s pub汩捡瑩lns
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