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September 2014
International Journal on
ISSN 2077-3528
“Technical and Physical Problems of Engineering”
IJTPE Journal
(IJTPE)
www.iotpe.com
Published by International Organization of IOTPE
[email protected]
Issue 20
Volume 6
Number 3
Pages 1-6
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, [email protected]. Center of …, Faculty of Engineering, … University of …, City, Country, [email protected]. Department of …, Organization of …, City, Country, [email protected]. … Company, City, Country, [email protected], [email protected]
Abstract- Permanent magnet synchronous motor
(PMSM) have a wide range of applications, such as
electric drives and machine ………………………………
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……………… to ensure stability and tracking.
Simulations is carried out to verify the theoretical results.
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:
d as
(1)
u as  rs i as 
dt
d bs
(2)
u bs  rs ibs 
dt
d cs
(3)
u cs  rs i cs 
dt
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d bs
(5)
  rs i bs  u bs
dt
d cs
(6)
 rs i cs  u cs
dt
where the flux linkages are:
Keywords: PMSM, Modeling, Saturation, ……………,
…………., ……………., Lyapunov Stability.
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].
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International Journal on “Technical and Physical Problems of Engineering” (IJTPE), Iss. 12, Vol. 4, No. 3, Sep. 2012
1

 L1s  L m  2 L m
 as  
     1 Lm L  L
m
1s
 bs   2
 cs  
1
  1 Lm
 Lm
 2
2


 sin  r



2
 m sin( r   ) 

3 


sin( r  2  ) 
3 



 ias 
 i 
  bs 
  ics 
L1s  L m 

B.2. Different Sample Materials
Depending upon different materials used in the
insulation model the apparent charge varies. Parameters
in Table 1 are simulated using MATLAB to observe the
variation of Partial Discharge with different materials
used in the sample model. Each material has different
dielectric constants.
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1
 Lm
2
1
 Lm
2
(7)
C. Park Transformation
Applying the Park transformation, we have the
following expression for the electromagnetic torque:
P m 
2
Te 
iab cos  r  ibs cos( r   ) 

2 
3
(14)
2  3P m r
ics cos( r   )  
iqs
3 
4
Using Equation (11) and the park transformation, one
obtains the following differential equation to model
permanent-magnet synchronous motors in the rotor
reference frame:
r
diqs
rs
m
r

i r qs 
wr  i ds
r
3
3
dt
L1s  L m
L1s  L m
2
2
(15)
1
r

u qs
3
L1s  L m
2
r
dids
rs
1
r
r
r
(16)

i ds
 i qs
r 
u ds
3
3
dt
L1s  L m
L1s  L m
2
2
where rs is the stator resistance, L1s and L m are the
3
Lm
2
and  m is the amplitude of the flux linkages established
by the permanent magnet.
leakage and magnetizing inductances Lss  L1s 
Figure 1. Two-pole permanent-magnet synchronous motor
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di0r s
r
1 r
  s i0r s 
u 0s
dt
L1s
L1s
(17)
dwr 3 p 2 m r Bm
P

i qs 
r 
TL
dt
8J
J
2J
(18)
r
r
r
where, iqs
, ids
, i0rs and uds
, uqsr , u0r s are the quadrature,
direct, and zero-axis current and voltage components.
The analysis of permanent-magnet synchronous
motors in the arbitrary reference frame using the
quadrature, direct, and zero-quantities is simple. The
electromagnetic torque is a function of the quadrature
r
current iqs
and differential equation for the zero current
B. Factors Affecting Partial Discharge
Partial discharge activity in a solid dielectric material
depends upon applying voltage, dielectric constant of the
material and the size of the void. These are considered as
the important factors affecting Partial Discharge in solid
dielectrics.
i0rs can be omitted from the analysis. We have:
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That is, the total derivative of a positive-definite
r r
, ids , r ) is negative. Hence, an
quadratic function V (iqs
B.1. Applied Voltage
When the applied high voltage is increased, the
electric field is enhanced and the liberation electron rate
is increased. As a result more Partial Discharges will
occur. High voltage ranging from 4 KV to 20 KV is
applied to the simulation model to observe the various
activities due to the presence of void, which is done with
the help of a laboratory experiment setup.
open-loop system is uniformly asymptotically stable [3].
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International Journal on “Technical and Physical Problems of Engineering” (IJTPE), Iss. 12, Vol. 4, No. 3, Sep. 2012
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) transient 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
simplification 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 openloop 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 problem, methods that do not entail the
applied voltages to the saturation limits to cancel
r
beneficial nonlinearities ids
r , and iqsr r ,and
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 k p , k i , and k d 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 (abc) 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 fourpole 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, rs  0.5  ,
Lss  0.001 [H] , L1s  0.0001 [H], L m  0.0006[H],
 m  0.069 [V.sec/rad ] or  m  0.069 [N.m/A] ,
Bm  0.0000115 [N.m.sec/r ad], and
J  0.000017 [Kg.m2 ]
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methods that do not lead to unbalanced motor operation .
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International Journal on “Technical and Physical Problems of Engineering” (IJTPE), Iss. 12, Vol. 4, No. 3, Sep. 2012
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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 conditions
The applied phase voltages and the resulting phase
currents in the as bs and cs windings are illustrated in
Figure 2. Figure 3 documents the motor mechanical
angular 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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International Journal on “Technical and Physical Problems of Engineering” (IJTPE), Iss. 12, Vol. 4, No. 3, Sep. 2012
Table 11. Construction cost of 400 kV
Number of Line
Circuits
Fix Cost of Line
Construction
(×103 dollars)
Variable Cost of Line
Construction
(×103 dollars)
1
2
1748.6
1748.6
92.9
120.2
Table 12. Characteristics of 230 kV lines
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546.5
546.5
45.9
63.4
230
400
397
750
3.85e-4
1.24e-4
1.22e-4
3.5e-5
B. Symbols / Parameters
I : The number of gas and heat units
J : The number of water units
C : The whole production expense
Cti ( Pti (t )) : The fuel expense of its unit
Pd (t ) : The load of net in t moment
Pr (t ) : The cycling reserve load
Pti (t ) : The production power of i heat unit
Phj (t ) : The production power of j water unit
rti ( Pti (t )) : The reserve power of i water unit
rhj ( Phj (t )) : The reserve power of j water unit
S ti (t ) : The starting expense of i heat unit
S hj (t ) : The starting expense of j water unit
t : The subtitle related to interval
T : The time of a complete period under consideration
ACKNOWLEDGEMENTS
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.
Table 10. Construction cost of 230 kV
1
2
Resistance
(p.u/Km)
A. Acronyms
CHCP
Combined Heat, Cooling and Power
CHP
Combined Heat and Power
COP
Coefficient of Performance
DG
Distributed Generation
APPENDICES
Appendix 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.
Variable Cost of Line
Construction
(×103 dollars)
Reactance
(p.u/Km)
NOMENCLATURES
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. Different reference velocity, loads,
and initial conditions are studied to analyze the tracking
performance of the resulting system.
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Fix Cost of Line
Construction
(×103 dollars)
Maximum Loading
(MVA)
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); LLmax =
30%.
Figure 9. Source side voltage and current of phase (2)
Number of Line
Circuits
Voltage
Level
REFERENCES
[1] G.A. Taylor, M. Rashidinejad, Y.H. Song, M.R.
Irving, M.E. Bradley, T.G. Williams, “Algorithmic
Techniques for Transition Optimized Voltage and
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International Journal on “Technical and Physical Problems of Engineering” (IJTPE), Iss. 12, Vol. 4, No. 3, Sep. 2012
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.
[3] A.B. Author1, “Title of the Book”, Ghijk Press, City,
Country, Year.
[4] A. Author1, C.D Author2, “Title of the Conference
Paper”, 8th International Conference on Technical and
Physical Problems of Electrical Engineering (ICTPE2013), 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.
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, in the book “Iabcd Sabcd and Kabcd Mabcd for
Eabcd”. His current research interests include the broad
area of nonlinear systems, on both dynamics and control,
and power electronics. He has authored or coauthored of
several papers (over to 100) in journals (ISI/INSPEC or
Rabcd Aabcd indexed) and international conference
proceedings. He is an Associate Editor of scientific
journal of the University of Pabcd “Eabcd and Cabcd
Sabcd” and program chair and proceeding editor of the
International Conference on “Eabcd, Cabcd and Aabcd
Iabcd”, 2005, 2007 and 2009 editions.
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,
1992,
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 Analysis, 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 Educational Systems. He is a
member of the International Electrical and Electronic
Engineers.
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.
Habcd Sabcd 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.
Rabcd Tabcd was born in Sabcd,
Nabcd, Abcd on 28 September 1949.
He is professor of power engineering
(1993); chief editor of scientific
journal of “Pabcd Eabcd Pabcd”
from 2000; director of Institute of
Pabcd from 2002 up to 2009;
academician and the first vicepresident of Aabcd Nabcd Aabcd of Sabcd from 2007. He
is laureate of Aabcd State Prize (1978); Honored Scientist
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International Journal on “Technical and Physical Problems of Engineering” (IJTPE), Iss. 12, Vol. 4, No. 3, Sep. 2012
of Aabcd (2005); co-chairman of International
Conferences on “Tabcd and Pabcd Eabcd”. His research
areas are theory of non-linear electrical chains with
distributed parameters, neutral earthing and ferroresonant
processes and alternative energy sources. His publications
are more than 250 articles and patent and 5 monographs.
7