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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@---.--- 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 ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. 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: ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. 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 ……………………………………………………………. degraded. ……………………………………………………………. 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: ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. 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 . ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. 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 ……………………………………………………………. load torque : ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. …………………………………………………………… ……………………………………………………………. ……………………………………………………………. ……………………………………………………………. 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 ……………………………………………………………. ……………………………………………………………. Voltage Maximum Loading Reactance Resistance ……………………………………………………………. Level (MVA) (p.u/Km) (p.u/Km) ……………………………………………………………. 230 397 3.85e-4 1.22e-4 ……………………………………………………………. 400 750 1.24e-4 3.5e-5 ……………………………………………………………. ……………………………………………………………. 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 postgraduate student in Uabcd, Eabcd, where he received 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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