A Novel Two-Step Flux Linkage Identification for PMSMs Considering Magnetic Saturation and Spatial Harmonics

Abstract

I. INTRODUCTIONIn order to achieve reliable and high-performance control of permanent magnet synchronous machines (PMSMs), the accurate model of electromagnetic characteristics of PMSMs under different load conditions is critical [1]. The conventional method is to use the finite element analysis (FEA) to derive the magnetic model of PMSM [2]. However, this method requires accurate geometric parameters and material information of the machine and the results must be validated by experiments. Therefore, it is desirable to obtain accurate machine model from measurable terminal signals.To date, considerable research work has been conducted to deal with the flux linkage identification problem for PMSMs [3]-[5]. However, either magnetic saturation or spatial harmonics effects have been ignored during the flux linkage identification in the existing methods. Therefore, this paper proposes a novel two-step flux linkage identification method for PMSMs considering both magnetic saturation and spatial harmonics. Step one is an offline estimation of the flux linkages considering magnetic saturation based on genetic algorithm (GA) , while the spatial harmonics effect has been mitigated compared with the existing method. In the second step, a state-space observer is employed to estimate the spatial harmonics online with the estimated offline data in step one. In addition, Kalman filter is applied in dq-axis inductances harmonics estimation to reduce the influence of noise and high order harmonics in the motor drive system. Therefore, the proposed method achieves to consider both magnetic saturation and spatial harmonics for PMSM flux linkage identification. The proposed approach is validated on both the surface-mounted and interior PMSMs.II. Magnetic Saturation Modeling and Parameter Estimation using GATo consider magnetic saturation in the machine model, the dq-axis flux linkages are modeled using the quadratic equation as in (1).Ψd(id, iq)= α0+α1id+α2iq+α3id2+α4iq2+α5idiq;Ψq(id, iq)= β0+β1id+β2iq+β3id2+β4iq2+β5idiq. (1)Where id, iq, Ψd, Ψq are dq-axis currents and flux linkages; α0 denotes the fundamental PM flux linkage; β0 is adopted to generalize the quadratic form of the model and compensate the offset for Ψq; α1 and β1 denote self-inductances in the dq-axis; α2,…, α5 and β2,…, β5 denote the cross-coupling effect and the influence of magnetic saturation. There are 12 parameters to be estimated, while the rank number is two. Therefore, the coefficients of flux linkages equations cannot be solved directly as it is rank-deficient. To solve this issue, a GA based estimation approach is proposed.Suppose that, for a given pair of (id, iq), the (ud, uq) under a certain speed ωm during the period T seconds (T=2πn/ωm, where n=1,2, …,N represents the number of rotations) are recorded in order to mitigate the influence of spatial harmonics. To estimate the coefficients of flux linkages, two fitness functions are designed as follows:Fd = ∑i=1..I∑j=1..J[α0 + α1id(i) + α2iq(j) + α3id2(i) +α4iq2(j) + α5id(i)iq(j)-Ψd(id(i), iq(j))];Fq = ∑i=1..I∑j=1..J[β0 + β1id(i) + β2iq(j) + β3id2(i) +β4iq2(j) + β5id(i)iq(j)-Ψq(id(i), iq(j))], (2)where i=1,…, I and j=1,…, J are directories of (id, iq). Based on (2), the coefficients of dq-axis flux linkages at each pair of (id, iq) can be obtained by using GA to achieve the minimization of Fd and Fq.III. Online Parameter Estimation Considering Spatial HarmonicsConsidering spatial harmonics, dq-axis voltage equations of a PMSM can be written as (3) under the steady state.ud = Rid-ωeLqq,0iq+ωeΨd6sin(6pθ)+sin(6pθ)(-6pωeLdd,6id-ωeLdq,6id)+cos(6pθ)(6pωeLdq,6iq-ωeLqq,6iq)+εd ; uq = Riq+ωeLdd,0id+ωeΨd1+ωeΨq6cos(6pθ)+sin(6pθ)(-6pωeLqq,6iq-ωeLdq,6iq)+cos(6pθ)(6pωeLdq,6id-ωeLdd,6id)+εq, (3)where ud, uq are dq-axis voltages; θ is the electrical rotor position; p is the number of pole pairs; R is the resistance; ωe is the rotor speed; Ψd6 and Ψq6 are the magnitudes of the 6th permanent magnet (PM) flux-linkage harmonics in dq-axis. Ldd,0,Lqq,0,Ldd,6,Lqq,6 are the magnitudes of fundamental and the 6th harmonic of self-inductances in the dq-axis; Ldq,6 is the magnitude of the 6th harmonic of mutual-inductance; εd and εq are high order harmonics and high frequency noise components in the dq-axis. The harmonic magnitudes of the PM flux linkage can be easily obtained by using FFT analysis of the back-electromotive force. To achieve an online identification of the dq-axis inductances harmonics, a state observer is employed.The state observer model is firstly built based on (3) to get the magnitude of the 6th harmonic of the self-inductance and mutual-inductance in the dq-axis. The state vector is built as:x=[ Ls6cos(6pθ); Lm6sin(6pθ); -6pωeLs6sin(6pθ); 6pωeLm6cos(6pθ)], (4)with Ls6=Ldd,6 and Lm6=Ldq,6. The estimated dq-axis flux linkage maps are shown in Fig.1 and Fig.2. Detail analysis and experimental results will be given in the full paper.IV.CONCLUSIONThis paper proposes a novel two-step method to identify flux linkages in PMSMs considering both the effects of magnetic saturation and spatial harmonics during the estimation process. In step one, GA is used to estimate the magnetic saturation coefficients of flux linkage. With the results obtained from GA, the Kalman filter based online method is proposed to estimate the spatial harmonics to achieve an accurate estimation of the flux linkage map. The co-simulation results show that the proposed approach is effective for flux linkage estimation. **