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# Differences in perfromance between SF6 and Vacuum

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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@---.---

2 Center of …, Faculty of Engineering, … University of …, City, Country, e-mail.account@---.---

3 Department of …, Organization of …, City, Country, e-mail.account@---.---

4 … Company, City, Country, e-mail.account@---.---, e-mail.account@---.---

Abstract- Permanent magnet synchronous motor                    primary issues are studied in this paper. In particular, we
(PMSM) have a wide range of applications, such as               perform nonlinear modeling and analysis, controllers
electric drives and machine ………………………………                        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
……………… to ensure stability and tracking.                        synchronous motor is illustrated in Figure 1.
Simulations is carried out to verify the theoretical results.
A. Motor Modelling
Keywords: PMSM, Modeling, Saturation, ……………,                        For the magnetically coupled abc stator windings, we
…………., ……………., …………….                                           apply the Kirchhoff voltage law to find a set of the
following differential equations:
I. INTRODUCTION
A broad spectrum of electric machines is widely used                          d as
in electromechanical systems. In addition to the required       u as  rs i as                                      (1)
dt
……………………………………………………………
…………………………………………………………….                                                          d bs
u bs    rs ibs                                      (2)
…………………………………………………………….                                                           dt
…………………………………………………………….                                                          d cs
…………………………………………………………….                                        u cs    rs ics                                      (3)
dt
…………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
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…………………………………………………………….
…………………………………………………………….                                        d bs
…………………………………………………………….                                               rs ibs  u bs                                (5)
…………………………………………………………….                                         dt
…………………………………………………………….                                        d cs
 rs ics  u cs                                (6)
…………………………………………………………….                                         dt
…………………………………………………………….
…………………………………………………………….                                               where the flux linkages are:
              1          1                            B. Park Transformation
 L1s  L m  2 L m        Lm      i                      Applying the Park transformation, we have the
 as                              2
  as 
     1 Lm L  L                1
 Lm       ibs              following expression for the electromagnetic torque:
 bs   2              1s    m
2         
 cs   1
                                             i cs 
 
1                                             P m                                                 2 
  2 Lm      Lm        L1s  L m                                                            2
Te         iab cos  r  ibs cos( r   )  ics cos( r   )
              2                                             2                              3                   3 
                                                              3P m r
 sin  r                                                           i qs .                                             (14)
                                                                 4
2 
  m sin( r   )                                        (7)
           3                                                 Using (11) and the park transformation, one obtains
sin(  2  )                                             the following differential equation to model permanent-


r
3                                             magnet synchronous motors in the rotor reference frame:

r
rs is the stator resistance, L1s and L m are the               diqs         rs                      m
             i r qs                   wr  i ds  r
r
3                dt            3                        3
leakage and magnetizing inductances Lss  L1s  L m ,                     L1s  L m               L1s  L m
2                              2                        2                       (15)
and  m is the amplitude of the flux linkages established                 1
                 r
u qs
by the permanent magnet.                                                   3
L1s  L m
2
r
dids            rs                              1
                  i ds  i qs  r 
r      r                         r
u ds   (16)
dt               3                              3
L1s  L m                        L1s  L m
2                              2
r
di 0 s     r r      1 r
  s i0s      u 0s                                     (17)
dt       L1s      L1s
dwr 3 p 2 m r Bm         P
       i qs    r     TL                                 (18)
dt    8J          J      2J

r   r     r          r     r     r
Where iqs , ids , i0 s and u ds , u qs , u0 s are the
quadrature-,direct-, and zero-axis current and voltage
components.
Figure 1. Two-pole permanent-magnet synchronous motor             The analysis of permanent-magnet synchronous
motors in the arbitrary refrence frame using the
…………………………………………………………….                                           quarature-, direct-,and zero-quantities is simple. The
…………………………………………………………….                                           electromagnetic torque is a function of the quadrature
…………………………………………………………….                                                     r
current iqs ,and differential equation for the zero
…………………………………………………………….                                                    r
…………………………………………………………….                                           current i0 s can be omitted from the analysis. We have:
…………………………………………………………….
…………………………………………………………….                                           …………………………………………………………….
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…………………………………………………………….                                           …………………………………………………………….
…………………………………………………………….                                           …………………………………………………………….
…………………………………………………………….                                           …………………………………………………………….
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That is, the total derivative of a positive-definite               IV. THE LYAPUNOV-BASED APPROACH
quadratic function V (iqs , ids , r ) is negative. Hence, an
r r                                          In this section, the design is approached using a
nonlinear model. Using (19,20,21), we have the
open-loop system is uniformly asymptotically stable [3].
following matrix form
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 codition is guaranteed. Let:                    …………………………………………………………….
…………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….
…………………………………………………………….                                           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
Remark. In pole-placement design, the specification        above, the stability of the resulting closed-loop system
of optimum (desired) transient responses in terms of            can be examined studying the criteria imposed on the
system models and feedback coefficients is equivalent to        Lyapunov function. For the bounded reference signal,
the specification imposed on desired transfer functions of      using the positive-definite quadratic function
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            …………………………………………………………….
Mathematically, feedback linearization reduces the
complexity of the corresponding analysis and design.            he given tracking controllers extend the applicability of
However, even from mathematical standpoints, the                the stabilizing algorithms, and allows one to solve the
simplification and "optimum" performance would be               motion control problem for electromechanical systems
achieved in expense of large control efforts required           driven by permanent-magnet synchronous motors. Using
because of linearizing feedback (25). This leads to             the inverse Park transformation, one derives the control
saturation. It must be emphasized that the need to              laws in the machine (abc) variables. In particular, the
linearize (19,20,21) dose not exist because the open-loop       bounded PID controller is given as:
system is uniformly asympotically stable.
The most critical problem is that the linearizing          …………………………………………………………….
feedback:                                                       …………………………………………………………….
…………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
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…………………………………………………………….                                                      V. SIMULATION RESULTS
…………………………………………………………….                                             In this section, we design a tracking controller for a
electromechanical system. We use a Kollmorgen four-
Hence, the feedback linearizing controllers cannot be       pole permanent-magnet synchronous motors H-232 with
implemented to control synchronous machines. It is              the following rated data and parameters: 135 W, 434
desirable, therefore, to develop other methods to solve the     rad/sec, 40 V, 0.42 N-m, 6.9 A, rs  0.5 ,
motion control problem, methods that do not entail the
applied voltages to the saturation limits to cancel             Lss  0.001H , L1s  0.0001H , L m  0.0006H ,
ir         ir                      m  0.069V  sec/ rad or m  0.069N  m / A,
beneficial nonlinearities ds r and qs r ,and methods
Bm  0.0000115N  m  sec/ rad , and
that do not lead to unbalanced motor operation
J  0.000017kg  m 2 .
…………………………………………………………….
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…………………………………………………………….
…………………………………………………………….
…………………………………………………………….
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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     Figure 2. Radial-velocity profiles for different      rates
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 mechnical            …………………………………………………………….
angular velocity . The setting time for the motor angular       …………………………………………………………….
velocity as motor starts from stall is 0.0025 sec . The         …………………………………………………………….
disturbance attentuation 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        …………………………………………………………….
…………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
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…………………………………………………………….                                        …………………………………………………………….
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APPENDICES
A. 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

Fix Cost of Line       Variable Cost of Line
Number of Line
Construction              Construction
Circuits
(×103 dollars)            (×103 dollars)
1                     546.5                   45.9
2                     546.5                   63.4

Table 11. Construction cost of 400 kV

Fix Cost of Line Variable Cost of Line
Figure 9. Source side voltage and current of phase (2)       Number of Line
Construction (×103 Construction (×103
Circuits
dollars)           dollars)
…………………………………………………………….
1                 1748.6                 92.9
…………………………………………………………….
2                 1748.6                 120.2
…………………………………………………………….
…………………………………………………………….
Table 12. Characteristics of 230 kV lines
…………………………………………………………….
…………………………………………………………….                                              Level          (MVA)              (p.u/Km)       (p.u/Km)
…………………………………………………………….
230                397               3.85e-4        1.22e-4
…………………………………………………………….
400                750               1.24e-4        3.5e-5
…………………………………………………………….
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B. 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%.
…………………………………………………………….
…………………………………………………………….                                                       ACKNOWLEDGEMENT
…………………………………………………………….                                           The great work of Ms 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 and T.G. Williams, “Algorithmic
…………………………………………………………….                                        Techniques for Transition-Optimised Voltage and
Reactive Power Control”, Proceedings of International
VI. CONCLUSIONS                             Conference on Power System Technology, Volume
Permanent-magnet synchronous motors are used in a           3, Pages 1660-1664, 13-17 Oct. 2002.
wide range of electromechanical systems because they            [2] J. Zhong, E. Nobile, A. Bose and K. Bhattacharya,
are simple and can be easily controlled. The steady-state       “Localized Reactive Power Markets Using the Concept of
torque-speed characteristics fulfil the controllability         Voltage Control Areas”, IEEE Transactions on Power
criteria over an entire envelope of operation. In this paper    Systems, Volume 19, Issue 3, Pages 1555-1561, Aug.
a bounded controller is designed and sufficient criteria for    2004.
stability are satisfied. Different reference velocity, loads,
and initial conditions are studied to analyze the tracking      …………………………………………………………….
performance of the resulting system.                            …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                        …………………………………………………………….
…………………………………………………………….                                      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
BIOGRAPHIES                               Sabcd, in the book “Iabcd Sabcd and Kabcd Mabcd for
Eabcd”. His current research interests include the broad
Nabcd Mabcd Author was born in             area of nonlinear systems, on both dynamics and control,
Photo       Tacd, Iabcd, 1967. He received the         and power electronics. He has authored or coauthored of
of the author   B.Sc. and the M.Sc. degrees from           several papers (over to 100) in journals (ISI/INSPEC or
University of Tabcd (Tabcd, Iabcd)         Rabcd Aabcd indexed) and international conference
Height: 3.0 cm  and the Ph.D. degree from University       proceedings. He is an Associate Editor of scientific
Width: 2.5 cm  of Sabcd (Tabcd, Iabcd), all in Power      journal of the University of Pabcd “Eabcd and Cabcd
Electrical Engineering, in 1989,           Sabcd” and program chair and proceeding editor of the
1992, and 1997, respectively.              International Conference on “Eabcd, Cabcd and Aabcd
Currently, he is a Professor of Power Electrical              Iabcd”, 2005, 2007 and 2009 editions.
Engineering at University of Eabcd (Babcd, Aabcd). He
is also an academic member of Power Electrical                                    Sabcd Author was born in Tabcd,
Engineering at University of Sabcd (Tabcd, Iabcd) and                Photo        Eabcd Aabcd, Iabcd in September
teaches Power System Analysis, Power System                      of the author    1940. He received the Dipl.-Ing.
Operation, and Reactive Power Control. He is the                                  degree on Sabcd Tabcd from the
secretary of International Conference on ABCD. His               Height: 3.0 cm   Rabcd, Aabcd, Gabcd in 1969. From
research interests are in the area of Power Quality,              Width: 2.5 cm   1970 to 1971 he worked for Aabcd,
Energy Management Systems, ICT in Power Engineering                               Fabcd, Gabcd on electric distribution
and Virtual E-learning Educational Systems. He is a                               system planning. From 1972 to 1977
member of the International Electrical and Electronic         he was a lecturer of Electrical Engineering at University
Engineers.                                                    of Tabcd, Tabcd, Iabcd. From 1977 to 1979 he was as
Habcd Aabcd Author was born in         M.Sc. degree on Power System. From 1980 to 2007 he
Photo           Zabcd, Iabcd, on January 23, 1951.     was a professor of Electrical Engineering of University of
of the author       He received the B.Sc. and M.S.E.       Tabcd. In February 2007 he was retired. During his
degrees in Electrical Engineering in   working in University of Tabcd he was from 1988 to
Height: 3.0 cm      1973 and 1979 and the Ph.D. degree     1989 in Rabcd, Aabcd, Gabcd and 1996 to 1997 in
Width: 2.5 cm       in electrical engineering from Mabcd   Electrical Engineering Department of University of
State University, Uabcd, in 1981.      Sabcd, Cabcd in Sabbatical leave. His research interest is
Currently, he is a full professor at   in electrical machines, modeling, parameter estimation
electrical engineering department of University of Tabcd      and vector control.
Tabcd, Iabcd. His research interests are in the application
of artificial intelligence to power system control design,                        Rabcd Tabcd Author was born in
dynamic load modeling, power system observability                    Photo        Sabcd, Nabcd, Abcd on 28
studies and voltage collapse. He is a member of Mabcd            of the author    September 1949. He is professor of
Association of Electrical and Electronic Engineers and                            power engineering (1993); chief
IEEE.                                                             Height: 3.0 cm  editor of scientific journal of “Pabcd
Width: 2.5 cm   Eabcd Pabcd” from 2000; director of
Babcd Author was born in Aabcd                                Institute of Pabcd from 2002 up to
Photo         Mabcd, Rabcd, in February 1961. He                            2009; academician and the first vice-
of the author     received a five-year degree in            president of Aabcd Nabcd Aabcd of Sabcd from 2007. He
electronic engineering from the           is laureate of Aabcd State Prize (1978); Honored Scientist
Height: 3.0 cm   University of Babcd, Rabcd, in 1986       of Aabcd (2005); co-chairman of International
Width: 2.5 cm    and the Ph.D. degree in Automatic         Conferences on “Tabcd and Pabcd Eabcd”. His research
Systems and Control from the same         areas are theory of non-linear electrical chains with
university, in 1996. He is currently a    distributed parameters, neutral earthing and ferroresonant
Professor with the University of Pabcd, Rabcd.                processes and alternative energy sources. His publications
Previously, he was in hardware design with Dabcd Rabcd        are more than 250 articles and patent and 5 monographs.

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