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22 Modeling and Parameter Identification of Electric Machines.pdf

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22 MODELING AND PARAMETER IDENTIFICATION OF ELECTRIC MACHINES
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22 Modeling and Parameter Identi?cation of Electric MachinesAli Keyhani, Wenzhe Lu, and Bogdan ProcaOhio State University,Columbus, OhioNOMENCLATURE(·): ? Estimate of (·) [ ]T: Transpose of [ ]E[·]: The operation of taking the expected value of [·]w(·): Process noise sequencev(·): Measurement noise sequenceX(·): State vectorY(·): Measured output vector in the presence of noiseQ: Covariance of the process noise sequenceR0: Covariance of the measurement noise sequenceR(·): Covariance of the state vectore(·): Estimation error, e(k) = Y(k) – ?Y(k)exp: The exponential operatordet: DeterminantY(kk– 1): ? The estimated value of Y(k) at time instant kgiven the data up to k-1U(·): Input vectorθ(·): Parameter vectorDK3023_book.fm Page 449 Thursday, March 3, 2005 6:47 PM ? 2005 by Taylor & Francis Group, LLC 22.1 INTRODUCTIONModeling the dynamical properties of a system is an important step in analysis and design of control systems. Modeling often results in a parametric model of the system that contains several unknown parameters. Experimental data are needed to estimate the unknown parameters. Electric machines are now widely used in electric/hybrid vehicles. Identifying appro- priate model structures of these machines and estimating the parameters of the models has become an important part of the automotive control design. Generally, the parameter estimation from test data can be done in frequency-domain or time-domain. Since noise, which may cause problems to parameter estimation, is an inherent part of the test data we will ?rst study the effects of noise on frequency-domain parameter estimation. To examine this issue, we will study the identi?cation of synchro- nous machine parameters from noise-corrupted measurements. Then, we will show how the time-domain maximum likelihood technique can be used to remove the effect of noise from estimated parameters. The models and the procedures to identify the parameters of synchronous, induction, and switched reluctance machines using experimental data will be presented.22.2 CASE STUDY: THE EFFECTS OF NOISE ON FREQUENCY-DOMAIN PARAMETER ESTIMATION OF SYNCHRONOUS MACHINE22.2.1 PROBLEMDESCRIPTIONA solid-rotor machine consists essentially of an in?nite number of rotor circuits. However, in practice, only a three-rotor-winding or a two-rotor-winding model is used in estimating machine parameters from test data. Experience gained in modeling of many machines shows that neither the second nor the third order model structure can be an exact mathe- matical representation of a machine. In estimating the parameters of a system, one question needs to be answered: If the assumed model structure is correct, then can one obtain a unique estimate of the parameters from noise-corrupted frequency response data? The answer to this question cannot be found from measurements, since the measurements are made on a machine with a complex, high order rotor circuit, with unknown structure and unknown parameters. If one assumes a model structure and then proceeds with estimating its parameters from actual measurements, then the structural error and the effect of noise in the mea- surements will result in inaccurate parameters. Therefore, it will not be clear whether the discrepancy between the simulated model response and the measured response is due to the effect of noise on the parameters, inadequacy of the assumed model structure, or both. Therefore, the structural identi?cation problem and the parameter estimation problem should be studied separately. There is a need to show that the measurements noise will not corrupt the estimated parameters when the parameters of an assumed structure are estimated from the frequency response measurements. In this section, a third order machine model with known parameters is simulated, and then the data are noise-corrupted using a known noise distribution. The objective is to estimate the parameters of this model from the noise-corrupted data and evaluate the estimated parameters by comparison with the known parameters.DK3023_book.fm Page 450 Thursday, March 3, 2005 6:47 PM ? 2005 by Taylor & Francis Group, LLC 22.2.2 PARAMETERSESTIMATIONTECHNIQUEIn the literature the second order model of synchronous machine is referred to as SSFR2 and the third order model as SSFR3. These notations will be used in this section. It is generally assumed that the synchronous machine d-axis and q-axis circuit structures can be represented by the SSFR3 or the SSFR2 models. The SSFR3 model is shown in Figure 22.1. The SSFR2 model structure can be obtained from the SSFR3 model by reducing the number of rotor body circuits from two to one and also assuming that Lf2d, which re?ects the leakage ?ux effect, is zero. The standard circuit model structure can be obtained by assuming that Lf12dis also negligible.22.2.2.1 Estimation of D-Axis Parameters from the Time ConstantsThe transfer functions of the d-axis SSFR3 equivalent circuits are: (22.1) (22.2) Using an assumed value of armature resistance, Ra, the Ld(s) is calculated from the operational impedance, Zd(s)= –Vd(s)/Id(s), and sG(s) is calculated from Ifd(s)/Id(s) when the ?eld is short-circuited. The equations that relate the circuit parameters to the time constants can be obtained from Equation 22.1 and Equation 22.2. These equations are described in terms of theFigure 22.1SSFR3 equivalent circuit structures. a R l L d I d f L 12 d f L 2 fd L fd R fd e fd I d V ad L d L 1 d R 1 d L 2 d R 2 1 d I 2 d I (a) d-axis circuit a R l L q I q V aq L q L 1 q R 1 q L 2 q R 2 1 q I 2 q I q L 3 q R 3 3 q I (b) q-axis circuit LsK Ts Ts Ts Ts Ts dd () () () () () () = +++ ++ 111 11 123 45 (() 1 6 +Ts sG s G Ts Ts Ts Ts Ts d () () () () () () = ++ +++ 11 111 78 456DK3023_book.fm Page 451 Thursday, March 3, 2005 6:47 PM ? 2005 by Taylor & Francis Group, LLC unknown vector (i.e., the circuit parameters) and the known vector , as de?ned in Table 22.1.The vector is estimated from the measured frequency response data of transfer functions. The time constants are estimated by using a curve-?tting technique described in References 15, 16, and 20. The functional form of the vector that relates to the circuit parameters (i.e., the vector ) can be derived using MACSYMA [21], a computer-aided symbolic processor. These relationships are complex and nonlinear, and can be written as: (22.3) where i= 1, …, 10. Details of these equations are given in Appendix A. In general, these 10 equations are nonlinear in nature and are not consistent with each other. This is due to the noise ζimbedded in vector . Because of the nonlinearity of these equations, a closed form solution for vector may not be possible, and a numerical technique such as Newton- Raphson method may have to be used to solve these equations iteratively. Moreover, these are a redundant set of equations, 10 equations with 9 unknown parameters. Because of the inconsistency of these equations, multiple solutions will be obtained depending on which equation is ignored. If the measured frequency response data are noise free (i.e., ζi= 0, i= 1, …, 10), then Equation A.1 through Equation A.10, given in Appendix A, would be consistent, and a unique solution will be obtained regardless of which equation is ignored. The set of nonlinear equations, , can be solved by updating as:K= 0, 1, 2 … (22.4) whereTable 22.1De?nitions for D-Axis Circuit Unknowns and KnownsUnknown Circuit Parameters Unknown Vector Known Constants Known VectorLadx1L1y0Lf12dx2Kdy1R1dx3T1T2T3y2L1dx4T1T2+ T1T3+ T2T3y3Lf2dx5T1+ T2+ T3y4R2dx6T41T5T6y5L2dx7T4T5+ T4T6+ T5T6y6Rfdx8T4+ T5+ T6y7Lfdx9Gdy 8 — T 7 T 8 y 9 — T 7+ T 8 y 10 x y x y y y x fxygxy iiii () (,) =+ += ζ 0 y x Fx fx f x f x ()[() ,() , () ] =…= 121 0 0 x xxx KKK + =+ 1 ? , DK3023_book.fm Page 452 Thursday, March 3, 2005 6:47 PM ? 2005 by Taylor & Francis Group, LLC
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