Microsoft Word - A_40_Neukirchner_R.doc HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 39(1) pp. 147-152 (2011) MODELING AND PARAMETER SENSITIVITY ANALYSIS OF A SYNCHRONOUS MOTOR L. NEUKIRCHNER, A. FODOR , A. MAGYAR University of Pannonia, Department of Electrical Engineering and Information Systems, Veszprém, HUNGARY E-mail: foa@almos.uni-pannon.hu A simple dynamic model of a synchronous motor is developed in this paper based on first engineering principles that describe the mechanical phenomena together with the electrical model. The constructed state space model consists of nonlinear state equations and bi-linear output equations. The developed model has been verified under the usual regulated operating conditions when the speed and the torque are controlled, the manipulated input is the network voltage and the exciter voltage. The effect of load on the controlled synchronous motor has been analyzed by simulation using PI controllers. Model parameter sensitivity analysis has been applied to determine the model parameters to be estimated. Keywords: synchronous machine, dynamic state space model, sensitivity analysis, parameter estimation Introduction Classical synchronous motors are widely used machines when constant speed is necessary. The speed control of synchronous machines is a difficult problem since the motor speed is a linear function of the network frequency. For the control of the synchronous motor (SM) we have to use an inverter which generates the three phase sinusoidal electrical network and a DC power supply which provides the exciter voltage. The final aim of our study is to design a controller that regulates the speed and the torque of the synchronous motor. Because of the specialties and great practical importance of synchronous machines in industry, their modeling for control purposes is well investigated in the literature. Besides of the basic textbooks (see e.g. [1, 2]), there are papers that describe the modelling and use the developed models for the design of various controllers. The aim of this paper is to perform the parameter sensitivity analysis of a simple dynamic model of a synchronous motor. The result of this analysis will be the next step of the parameter estimation. The model of the synchronous motor Modelling assumptions For constructing the synchronous motor model, let us make the following assumptions: ● a symmetrical tri-phase stator winding system is assumed, ● one field winding is considered to be in the machine, ● all the windings are magnetically coupled, ● the flux linkage of the windings is a function of rotor position, ● the copper losses and the slots in the machine are neglected, ● the spatial distribution of stator fluxes and apertures wave are considered to be sinusoidal, ● the stator and rotor permeability are assumed to be infinite. ● It is also assumed that all the losses due to wiring, saturation and slots can be neglected. The four windings (three stators and one rotor) are magnetically coupled. Since the magnetic coupling between the windings is a function of the rotor position, the flux linkage of the windings is also a function of the rotor position. The actual terminal voltage v of the windings can be written in the form )()ir(=v j J 1=j jj J 1=j λ±⋅± ∑∑ & (1) where ij are the currents, rj are the winding resistances, and λj are the flux linkages. The positive directions of the stator currents point in the synchronous motor terminals. Thereafter, the two stator electromagnetic fields, both travelling at rotor speed, were identified by decomposing each stator phase current under steady state into two components, one in phase with the electromagnetic field and another phase shifted by 90°. With the above, one can construct an air-gap field with its maximal aligned to the rotor poles (d axis), while the other is aligned to the q axis (between poles). This method is called the Park's transformation [3, 4]. 148 Figure 1: The equivalent circuit of the SM As a result, the vector voltage equation is: dFqdDqRSdFq iLiR=v &+ω (2) with [ ]TqFddFq iiii = [ ]TqFddFq vvvv = where vd and vq are the direct and the quadratic components of the stator voltage of the synchronous motor, id and iq are the direct and the quadratic components of the stator current, while vF and iF are the exciter voltage and current. Flux linkage equations The synchronous motor consists of six coupled coils referred to with indices a, b and c are the stator phases coils, F is the filed coil. The linkage equations can be written in the following form: ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ λ λ λ λ FF i i i i LLLL LLLL LLLL LLLL c b a FFFcFbFa cFcccbca bFbcbbba aFacabaa c b a (3) where Lxx are the strator and rotor mutual inductances. After applying Park’s transformation, the following linkage equations are obtained: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ λ λ λ λ F q d 0 FF q Fd 0 F q d 0 i i i i L0kM0 0L00 kM0L0 000L (4) Voltage equations We can write Kirchoff’s voltage laws in the following form: nvriv ++= λ& (5) abcnmabcnn iLiRv &−−= (6) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 00 0 n abc abc F abc F abc F abc v i i R R v v λ λ & & (7) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = c b a abc r r r R 00 00 00 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = nnn nnn nnn nm LLL LLL LLL L ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = nnn nnn nnn n rrr rrr rrr R where vn is the neutral voltage and RF = rF. Using Park’s transformation is replaced with the equations for the d-q voltage components: ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −− = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ q F d q FF Fd q F d qFd F qd q F d i i i L LkM kML i i i rkML r Lr v v v & & & 00 0 0 00 0 ωω ω (8) The voltage equation in matrix form (8) is: dFqdFqRSdFq iLiRv &+= ω (9) The state space model for the currents is obtained by expressing dFqi& from (11), i.e. dFq 1 dFqRS 1 dFq vLiRLi ⋅−⋅⋅−= − ω −& (10) Power and torque equations The electrical energy of the SM is a sum of the following mechanical equations. dWElectr = dWMech + dWField + dWΩ (11) Time derivate of the energy equation is the power equation, which represents the energy change: PMech = PElectr – PField – PΩ (12) 149 Torque is obtained from dividing power by angular velocity. TMech = λdiq – λqid (13) dt dθ =ω (14) The torque from field enegy is given by the torque equation: ) d i d i d i(T qq d d 0 0Field θ λ + θ λ + θ λ = (15) Afterwards the torque is: TMech = TElectr – TField – TDump (16) From Newton’s law of motion we can write the speed and torque equation: ω−−=ω DTTH2 mechelectr& (17) where D is a damping constant. We can write the input power in the following equation: 2 0n 2 q 2 d 2 0 q q d d 0 0dqqdelectr ir3)iii(r ) d i d i d i()ii(P −++− θ λ + θ λ + θ λ −λ−λω= (18) (In balanced condition i0 = 0.) After we can compute the accelerating torque: DumpMechElectr DumpMech 3Electr Acc TTT TT 3 T T −− =−−= To compute eletrical torque we sould write the direct and quadratic part of the stator flux from equation (12): qqq FFddd iL ikMiL = += λ λ (19) After it the write eletrical torque can be expressed: [ ] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −= q F d dqqFqd3eletr i i i iLikMiLT (20) Using dt TAcc=ω& it is possible compute the speed of the synchronous machine. The motion equation is as follows: [ ] j MechT qFd jj dq j qF j qd T iii D 3 iL 3 ikM 3 iL = τ −ω ⋅ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ τ − τ − ττ ω& (21) The loading angle (δ) of the synchronous motor is dtR t t )(= 0 0 ωωδδ −+ ∫ (22) that can be differentiated to obtain the time derivative of δ Rωωδ −= & (23) Altogether, there are five state variables: id , iF , iq , ω and δ. The input variables (i.e. manipulable inputs and disturbances) are Tmech, vF, vd and vq. Observe that the state equations are bilinear in the state variables. The outputs of the model are the speed (ω) of the motor and the loading angle (δ) of the SM. Model analysis The state space model (10, 21, 23) has been verified by simulation against engineering intuition using parameter values of a similar machine [5]. After the basic dynamical analysis, eleven parameters have been selected for sensitivity analysis. Motor parameters The parameters are described only for phase a since the machine is assumed to have symmetrical tri-phase stator windings system. The stator mutual inductances for phase a are: )) 6 5 (2(LMLL )) 2 (2(LMLL )) 6 (2(LMLL msacca mscbbc msbaab π +Θ−−== π −Θ−−== π −Θ−−== (24) where Ms is a given constant. The phase a stator to rotor mutual inductances are given by (from phase windings to the field windings): ) 3 2 cos(MLL ) 3 2 cos(MLL )cos(MLL FFccF FFbbF FFaaF π +Θ== π −Θ== Θ== (25) where MF is a given constant. Parameters Ld, Ld, L0 and MF used by the state space model (10, 21, 23) are defined as: 3 2 2 2 3 2 3 0 = = −= −+= ++= k k L M MLL LMLL LMLL AF F ss mssq mssd (26) 150 Using the initial assumption of symmetrical tri-phase stator windings (i.e. ra = rb = rc = r) the resistance of stator windings of the machine we denoted by r. Resistance of the rotor exciter is represented by rF. Parameters Ld and Lq are the direct and quadratic stator inductances, LF is the stator exciter inductance. Dconst presents the damping constant, P is the proportional, I is the integrator parameter of the torque PI controller. The parameter values were obtained from the literature [1]: H00849.0l H0126.0l H0126.0l H138.0L H171.0L H176.0L F q d F q d = = = = = = 05.0P 1.0I 004.2D 0195.0r 5577.0r H1302.0L F AF = = = Ω= Ω= = (27) Stability analysis Eleven parameters of the synchronous motor have been selected for sensitivity analysis, and the sensitivity of the state variables has been investigated by Matlab dynamical simulation. The equilibrium point of the state space model can be obtained from the steady-state version of state equations (10, 21, 23) using the above parameter values. The equilibrium point of the system is: 2.912249P 0.375484i -2.046594i 2.978999i 779712.1 in q d F = = = = =ω (28) The state matrix of the state space model (28) has the following numerical value in this equilibrium: ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅−⋅−− ⋅⋅⋅ ⋅⋅−⋅ −−⋅⋅− −− −−− −−− −− 12 133 132 42 103330.5105201.36031.42364.6 104654.21685.1102970.1107901.6 101782.93503.4104983.2105277.2 2242.18023.5108667.8103714.3 (29) Figure 2: The Matlab Simulink model of the synchronous motor The eigenvalues of the state matrix are: 3 4 3 3 2 2,1 103064115.1 100820501.4 068256.4j10399932.3 − − − ⋅−=λ ⋅−=λ ±⋅−=λ (30) The real parts of the eigenvalues are negative but their magnitudes are small, thus the system is on the boundary of the stability domain. Parameter sensitivity analysis PI controller The applied control method of the synchronous machine is a classical PI controller (Fig. 2) that ensures stability of the equilibrium point under small perturbations. The controlled output is the mechanical torque, the manipulated 151 input is the voltage. The proportional parameter of the PI controller of the torque is 0.05 and the integrator time is 0.1 in per units. Model validation The dynamical properties of the motor have been investigated. The response of the torque controlled motor has been tested under step-like changes of the exciter voltage. The simulation results are shown in Figs. 3, 4, 5 and 6 where the quadratic linkage inductance lq the damping constant D, the stator exciter resistance rF and the stator resistance r are shown. Figure 3: Model responses for the +90% changing of parameter lq Sensitivity analysis The aim of this section is to define parameter groups according to the system’s sensitivity on them. Linkage inductances ld, lq, lF are not used by the current model, only by the flux model. As it was expected, the model is insensitive for these parameters. Note, that the linkage inductance parameters are only used for calculating the fluxes of the machine (Fig. 3). Response to the exciter voltage step change of the controlled motor (means the deviation form the steady-state value) Sensitivity of the model to the controller parameters P (proportional) and I (integrator) and the damping constant D has also been investigated. This is why the output and the steady state value of the system variables do not change even for a considerably large change of D (Fig. 4). Figure 4: Model responses for the +90% changing of parameter D Figure 5: Model responses for the +90% changing of parameter r stator resistance Not sensitive: These are the linkage inductances ld, lq, lF and damping constant D. The state space model is insensitive for them, the parameter values cannot be determined from measurement data using any parameter estimation method. Sensitive: The stator resistance r, the proportional controller parameter P, the integrator controller parameter I and the stator inductances Ld and Lq. Critically sensitive: The rotor exciter resistance rF and the rotor exciter inductance LF. 152 Figure 6: Model responses for the +10% changing of parameter rF rotor exciter resistance Conclusions and future works Based on the results presented here, it is possible to select the candidate parameters for model parameter estimation based on real data that is a further aim of the authors. The four parameters to be estimated in a later work are P (proportional parameter of the controller), LF (rotor exciter inductance), rF (rotor exciter resistance), and r (stator resistance). The final aim of is to develop a simple yet detailed state space model of the induction motor for control purposes which gives us the possibility to develop and analyze different control strategies for the synchronous motor. ACKNOWLEDGEMENT We acknowledge the financial support of this work for the Hungarian State and the European Union under the TAMOP-4.2.1/B-09/1/KONV-2010-0003 project. REFERENCES 1. P. VAS: Artifical-intelligence-Based Electrical Machines and Drives, Oxford University Press, (1999) 2. T. W. MON, M. M. AUNG: Simulation of Synchronous Machine in Stability Study for Power System, International Journal of Electrical Systems Science and Engineering, (2008), 49–54 3. P. M. ANDERSON, A. A. FOUAD: Power-Systems- Control and Stability, Iowa State University Press, Ames Iowa, (1977) 4. A. FODOR, A. MAGYAR, K . M . HANGOS: Parameter Sensitivity Analysis of a Synchronous Generator, Hungarian Journal of Industrial Chemistry, (2010) 5. A. FODOR, A. MAGYAR, K . M . HANGOS: Dynamic modelling and model analysis of a large industrial synchronous generator, Proc. of Applied Electronics 2010, Pilsen, Czech Republic, (2010), 91–96 << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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