Microsoft Word - A_05_R.doc HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 38(1). pp. 21-26 (2010) PARAMETER SENSITIVITY ANALYSIS OF A SYNCHRONOUS GENERATOR A. FODOR1 , A. MAGYAR1, K. M. HANGOS1, 2 1University of Pannonia, Department of Electrical Engineering and Information Systems, Veszprém, HUNGARY E-mail: foa@almos.uni-pannon.hu 2Computer and Automation Research Institute HAS, Process Control Research Group, Budapest, HUNGARY A previously developed simple dynamic model of an industrial size synchronous generator is analyzed in this paper. The constructed state-space model consists of a nonlinear state equation and a bilinear output equation. It has been shown that the model is locally asymptotically stable with parameters obtained from the literature for a similar generator. The effect of load disturbances on the partially controlled generator has been analyzed by simulation using a PI controller. It has been found that the controlled system is stable and can follow the set-point changes in the effective power well. The sensitivity of the model for its parameters has also been investigated and parameter groups have been identified according to the system’s degree of sensitivity to them. This groups form the different candidates of parameters for subsequent parameter estimation. The ways of applying the developed methods to other generators used in the automotive industry are also outlined. Keywords: synchronous machine, dynamic state space model, parameter sensitivity Introduction Synchronous generators are popular and widely used electrical power sources in a wide range of applications including power plants and the automotive industry, too. Whatever size and application area, these generators share the most important dynamic properties, and their dynamic models have a similar structure. In almost all power plants, both the effective and reactive components of the generated power depend on the need of the consumers and on their own operability criteria. This consumer generated time-varying load is the major disturbance that should be taken care of by the generator controller. Therefore the final aim of our study is to design a controller that can control the reactive power such that its generation is minimized in such a way that the quality of the control of the effective power remains (nearly) unchanged. Because of the specialties and great practical importance of the synchronous generators in power plants, their modelling for control purposes is well investigated in the literature. Besides of the basic textbooks (see e.g. [1] and [2]), there are papers that describe the modelling and use the developed models for the design of various controllers [3, 4]. These papers, however, do not take the special circumstances found in nuclear power plants into account that may result in special generator models. The aim of this paper is to perform model verification and parameter sensitivity analysis of a simple dynamic model of a synchronous generator (SG) proposed in [5] and [6]. The result of this analysis will be the basis of a subsequent parameter estimation step. The model of the synchronous generator In this section the bilinear state-space model for a synchronous generator is presented based largely on [5] that will be used for local stability and parameter sensitivity analysis in later sections. Modelling assumptions For constructing the synchronous generator 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, ● there are two amortisseur or damper windings in the machine, ● all of the windings are magnetically coupled, ● the flux linkage of the windings is a function of the rotor position, ● the copper losses and the slots in the machine are neglected, ● the spatial distribution of the stator fluxes and apertures wave are considered to be sinusoidal, ● the stator and rotor permeability are assumed to be infinite. 22 It is also assumed that all the losses due to wiring, saturation and slots can be neglected. The six windings (three stators, one rotor and two dampers) are magnetically coupled. Since the magnetic coupling between the windings is a function of the rotor position, the flux linking 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 ,)()(= =1=1 j J j jj J j irv λ&∑∑ ±⋅± where ij are the currents, rj are the winding resistances, and λj are the flux linkages. The positive directions of the stator currents point out of the synchronous generator 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.[4, 5] As a result of the derivation in [5] the vector voltage equation is as follows: vdFDqQ = –RRSω·idFDqQ – Li ˙ dFDqQ (1) with idFDqQ = [id iF iD iq iQ] T and vdFDqQ = [vd –vF vD = 0 vq vQ = 0] T, where vd and vq are the direct and the quadratic components of the stator voltage of the SG, vD and vQ are the direct and the quadratic components of the rotor voltage of the SG, id and iq are the direct and the quadratic components of the stator current, iD and iQ are the direct and the quadratic components of the rotor current, while vF and iF are the exciter voltage and current. Furthermore, RRSω and L are the following matrices ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ +ω−ω−ω− ωω+ ω Q eDFd D F QQe RS r0000 0RrkMkML 00r00 000r0 kML00Rr =R ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + + = QQ Qeq DRD RFF DFed LkM000 kMLL000 00LMkM 00MLkM 00kMkMLL L where r is the stator resistance of the SG, rF is the exciter resistance, rD and rQ are the direct and the quadratic part of the rotor resistance of the SG, Ld, Lq, LD and LQ are the direct and the quadratic part of the stator and rotor inductance, ω is the angular velocity, and MF, MD and MR are linkage inductances (see later). The resistance Re and inductance Le represent the output transformer of the synchronous generator and the transmission-line. The state-space model for the currents is obtained by expressing i ˙ dFDqQ from (1), i.e. i ˙ dFDqQ = –L –1·RRSω·idFDqQ – L –1·vdFDqQ (2) The motion equation is the following jj dQ j dq j qD j qF j qd D 3 ikM 3 iL 3 ikM 3 ikM 3 iL = • ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ τ − τττ − τ − τ −ω& • (3) • [id iF iD iq iQ ω] Altogether, there are 6 state variables: id, iF, iD, iq, iQ and ω. The input variables (i.e. manipulatable inputs and disturbances) are: Tmech, vF, vd and vq. Observe, that the state equations are bilinear in the state variables because matrix RRSω depends linearly on ω. Note, that (3) can be used as an additional state equation for state space model (2). The loading angle (δ) of the synchronous generator is dt)(= r t 0t 0 ω−ω+δδ ∫ that we can differentiate to obtain the time derivative of the δ in per unit notation δ ˙ = ω – 1 (4) The output active power equation can be written in the following form: pout = vdid + vqiq, (5) and the reactive power is qout = vdiq – vqid. (6) Note, that output equations are bi-linear in the state and input variables. Model analysis The above model has been verified by simulation against engineering intuition using parameter values of a similar generator taken from the literature [1]. After the basic dynamical analysis, the set of model parameters is partitioned based on the model's sensitivity on them. Generator parameters Because the above developed model uses pseudo- parameters that are composed from the original physical ones, mathematical expressions are needed to describe how these parameters depend on the physical ones. The parameters are described only for a single phase “a” since the machine is assumed to have symmetrical tri-phase stator windings system. The stator mutual inductances for phase a are )) 6 (2(cosLM=L=L msbaab π −Θ−− 23 where MS is a given constant. The rotor mutual inductances are LFD = LDF = MR, LFQ = LQF = 0 and LDQ = LQD = 0. The phase a stator to rotor mutual inductances (from phase windings to the field windings) are given by: LaF = LFa = MF cos(Θ) where the parameter MF is also a given constant. The stator to rotor mutual inductance for phase a (from phase windings to the direct axis of the damper windings) is LaD = LDa = MD cos(Θ) with a given parameter MD. The phase a stator to rotor mutual inductances (from phase windings to the damper quadratic direct axis) are given by: LaQ = LQa = MQ cos(Θ) The parameters Ld , Lq , MF , MD and MR used by the state space model (2 3, 5, 6) and by the above inductance equations are defined as AQQADR ADFADD mSSdmSSd LMLM LMLM LMLLLMLL 2 3 2 3 2 3 2 3 2 3 == == ++=++= Using the initial assumption of symmetrical tri-phase stator windings we get the resistance of stator windings of the generator, where rF denotes the resistance of the rotor windings, and rD and rQ represent the resistance of the d and q axis circuit. In order to avoid working with numerical values that are in order of magnitude different, the equations have been normalized to a base value. We choose the base for rotor quantities and normalize the voltage and the torque accordingly. The variables in the normalized equations are then in „per units”. The parameters of the synchronous generator were obtained from the literature [1]. The stator base quantities, the rated power, output voltage, output current and the angular frequency base values are: s/rad f2= A 6158=I kV 8.66=3/kV 15=V MVA 53.333=/3MVA 160=S e B B B πω The physical parameters of the synchronous machine and the external network in dimensionless values are: Ld = 1.700 ld = 0.150 LMD =0.02838 Lq = 1.640 lq =0.150 LMQ = 0.2836 LD = 1.605 lF = 0.101 r = 0.001096 LQ = 1.526 lD = 0.055 rF = 0.00074 LAD = 1.550 lQ = 0.036 rD = 0.0131 LAQ = 1.490 rQ = 0.054 Re = 0.2 V∞ = 0.828 Le = 1.640 D = 2.004 Local stability analysis The steady-state values of the state variables can be obtained from the steady-state version of state equations (2, 3) using the above parameters. Equation (1) implies that the expected value to iD and iQ are 0, that coincide with the engineering intuition. The equilibrium point of the system is: 10 Q 9 D F q d 105.3334899=i 108.6242856=i 2.97899982=i 0.66750001=i 1.9132609=i 0.9990691= − − ⋅− ⋅− − ω The state matrix A of the locally linearized state-space model x˙ = Ax + Bu has the following numerical value in this equilibrium: ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −−−−−⋅− −−−−− − − − −−−− − 0.00110.00050.00080.00020.0002108 1.00050.12340.03522.58392.58393.5009 1.02470.09010.03612.64642.64643.5855 1.47371.61102.20570.09640.00440.0228 0.80250.87731.20110.07720.00490.0124 2.32852.54553.48510.01420.00040.0361 6 The eigenvalues of the state matrix are: 0.123426=104.724291= 101.67235=0.100024= 0.997704103.619088= 6 4 5 3 43 2 1,2 −⋅− ⋅−− ±⋅− − − − λλ λλ λ e j It is apparent that the real parts of the eigenvalues are negative but their magnitudes are small, thus the system is on the boundary of the stability domain. PI controller The control scheme of the synchronous machine is a classical PI controller that ensures stability of the equilibrium point under small perturbations [4]. The controlled output is the speed (ω), the manipulated input is the mechanical torque Tmech. The proportional parameter of the PI controller of the speed is 0.05 and the integrator time is 0.1 in per units. Model validation The dynamic properties of the generator have been investigated in such a way that a single synchronous machine was connected to an infinite bus that models the electrical network. The response of the speed controlled generator has been tested under step-like changes of the exciter voltage. The simulation results are shown in Fig. 1, where the exciter voltage vF and the torque angle δ are shown. 24 When the exciter voltage is increased the loading angle must be decreased as it can be seen in the Fig. 1. Figure 1: Response to the exciter voltage step change of the controlled generator (Δ means the deviation form the steady-state value) Sensitivity analysis The aim of this sub-section is to define parameter groups according to the system's sensitivity on them. Linkage inductances ld, lq, lD, lQ, LMD and LMQ are not used by the current model, only by the flux model [5]. It is not expected that the output and the state variables of system change when these parameters are perturbed, see Fig. 2. As it was expected, the model is not sensitive to these parameters. Note, that the linkage inductance parameters are only used for determining the fluxes of the generator. Sensitivity of the model to the controller parameters P and I and the dumping constant D has also been investigated. Since the PI controller controls ω by modifying the value of Tmech , the controller keeps ω at synchronous speed. This is why the output and the steady state value of the system variables do not change (as it is apparent in Fig. 3) even for a considerably large change of D. Sensitivity analysis of the resistance of the stator and the resistance of the transmission line led to the same result. A ±20% perturbation in them resulted in a small change in currents id, iq and iF. This causes the change of the effective and the reactive power of the generator, as shown in Fig. 4. The analysis of the effect of the rotor resistance rF showed, that the ±20% perturbation of rF kept the quadratic component of the stator current (iq) constant, but currents id and iF were changed. The output of the generator also changed, as it is shown in Fig. 5. Figure 2: The model states and outputs for a ±20% change of ld Figure 3: The model states and outputs for a ±90% change of D 25 Figure 4: The model states and outputs for a ±20% change of rresist Figure 5: The model states and outputs for a ±20% change of rF The sensitivity of the model states and outputs to the inductance of the rotor (LF) and the inductance of the direct axis (LD) has also been analyzed. The results show only a moderate reaction in id and iF to the parameter perturbations, and the equilibrium state of the system kept unchanged. However, decreasing the value of the parameters to the 90 percent of their nominal value destabilized the system. The results of a ±9% perturbation in LF are shown in Fig. 6. A small perturbation of the outputs is noticeable. Figure 6: The model states and outputs for a ±9% change of LF Finally, the sensitivity of the model (2, 3, 5, 6) to the linkage inductance LAD has been examined. When the parameter has been changed ±5%, currents iD and iF changed only a little. On the other hand, the steady-state of the system has shifted as it can be seen in Fig. 7. A parameter variation of more than 5% destabilized the system. As a result of the sensitivity analysis, it is possible to define the following groups of parameters: ● Not sensitive inductances ld, lq, lD, lQ, LMD, LMQ, LAQ, LQ damping constant D and the controller parameters P and I. Since the state space model of interest is insensitive for them, the values of these parameters cannot be determined from measurement data using any parameter estimation method. ● Less sensitive: resistances of the stator r and the transmission-line Re. ● More sensitive: resistance rF of the rotor and the inductance of transmission-line Le. These parameters are candidates for parameter estimation. 26 ● Critically sensitive: linkage inductance LAD, inductances LD and LF. These parameters can be estimated very well. Figure 7: The model states and outputs for a ±4% change of LAD Conclusion and further work The simple bilinear dynamic model of an industrial size synchronous generator described in [5] and [6] has been investigated in this paper. It has been shown that the model is locally asymptotically stable around a physically meaningful equilibrium state with parameters obtained from the literature for a similar generator. The effect of load disturbances on the partially controlled generator has been analyzed by simulation by using a traditional PI controller. It has been found that the controlled system is stable and can follow the setpoint changes in the effective power well. Eighteen parameters of the system have been selected for sensitivity analysis, and the sensitivity of the state variables and outputs has been investigated for all of them. As a result, the parameters have been partitioned to four groups. Based on the results presented here, the further aim of the authors is to estimate the parameters of the model for a real system from measurements. The sensitivity analysis enables us to select the candidates for estimation that are rF, Le, LAD, LD and LF. It is important to emphasize that this model can be and will be used as a starting point for the model development of a permanent magnet synchronous motor (PMSM) which board spectrum, that is widely used in the automotive industry. This becomes possible by changing the exciter coil of the classical synchronous machine to a permanent magnet: this way the model of the PMSM is obtained, which is one variant of the brushless direct current motors (BLDC 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. This work was also supported in part by the Hungarian Research Fund through grant K67625. REFERENCES 1. P. M. ANDERSON, A. A. FOUAD: Power-Systems- Control and Stability, The IOWA State University Press, Ames Iowa, 1977. 2. P. M. ANDERSON, B. L. AGRAWAL, J. E. VAN NESS: Subsynchronous Resonance in Power Systems, IEEE Press, New York, 1990. 3. A. LOUKIANOV, J. M. CANEDO, V. I. UTKIN, J. CABRERA-VAZQUEZ: Discontinuous controller for power systems: sliding-mode bock control approach, IEEE Trans. on Industrial Electronics, 51, 2004, 340–353. 4. F. P. DE MELLO, C. CONCORDIA: Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. Power Application Systems, PAS-88:316–329, 1969. 5. A. FODOR, A. MAGYAR, K. M. HANGOS: Dynamic Modelling and Model Analysis of a Large Industrial Synchronous Generator Proc. of Applied Electronics 2010, Plzen, Czech Republic. 6. A. FODOR, A. MAGYAR, K. M. HANGOS: Parameter Sensitivity Analysis of an Industrial Synchronous Generator Proc. of PhD Workshop 2010, Veszprém, Hungary. << /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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