Flux observer for induction machine control. Part II – Robust synthesis and experimental implementation

This paper proposes an approach to determine the gain of closed loop flux observer for an induction machine with respect to robustness and stability criteria. In the companion paper, a sensitivity analysis of flux observer has pointed out the influence of the sampling rate and the parameters mismatch. From this study, an accurate observer can be deduced. The analytical results are checked experimentally, in steady state and during transients. For that aim, an experimental rig has been installed. This paper deals also with a new description of the induction machine drive thanks to Causal Informational Graph. This description is useful for the simulation and for the implementation of the experimental rig. PACS. 84.50.+d Electric motors – 84.60.Bk Performance characteristics of energy conversion systems; figure of merit – 84.70.+p High-current and high-voltage technology: power systems; power transmission lines and cables (including superconducting cables) – 89.20.+a Industrial and technological research and development Nomenclature (φd, φq)R (Wb) Two-phase-equivalent rotor magnetic fluxes σ = 1− M 2 C LCRLCS Leakage coefficient ωR (rad/s) Speed of rotor phase ωS (rad/s) Speed of stator phase Ω (rad/s) Mechanical speed f (N m s) Coefficient of viscous friction (Id, Iq)S (A) Two-phase-equivalent stator currents J (J s) Inertia moment LCR (H) Cyclic rotor inductance LCS (H) Cyclic stator inductance MC (H) Mutual cyclic inductance between stator and rotor p Pole pairs number RR (Ω) Rotor resistance RS (Ω) Stator resistance Te (s) Sampling time Teq = 1 1 σTS + 1− σ σTR (s) Equivalent stator current time constant TR = LCR RR (s) Rotor time constant (Vd, Vq)S (V) Two-phase-equivalent stator voltage a e-mail: [email protected] Introduction In the previous paper [1], it was proved that both bad knowledge of the parameters and sampling effects lead to errors in flux estimation. Moreover, this analysis allows observer gain values capable of minimising parameters and sampling rate influences to be distinguished. This is what is proposed in this paper. In the first part, the flux reconstitution errors are studied as a function of observer gain for the two sampling methods in case of parameter inaccuracy or not. An optimal gain value is deduced. As the flux modulus and orientation errors are computed assuming a steady state condition, the observer transient behaviour has to be verified. The simulated results will next be compared to experimental trials in order to prove the validity of the closed-loop observer gain computation. 1 Observer error analysis in steady state In this part we search the observer gain K which reduces the parameter value uncertainties and the sampling rate influence [6,8]. The analysis follows several steps: (1) With a null gain, the most sensitive operating point at steady state is determined, i.e. the pair of speed and torque for an observed flux which give the greater modulus or orientation errors. Thus the results of the previous study are used for each discrepancy source; 26 The European Physical Journal Applied Physics &DVH IXOO RUGHU VDPSOLQJ PHWKRG IRU D VSHHG HTXDO WR USP DQG D WRUTXH HTXDO WR 1P &DVH UHGXFHG RUGHU VDPSOLQJ PHWKRG IRU D VSHHG HTXDO WR USP DQG D WRUTXH HTXDO WR 1P Fig. 1. Modulus and orientation error (nominal flux, Te = 0.8 ms). (2) For these sensitive points, the gain K which minimises the modulus and orientation errors is deduced from the steady state analysis; the observer stability is verified, then the result of the synthesis is carried out; (3) The gain is tested for all operating points at steady state and the stability is verified; (4) For one or more sensitive points, speed and flux transients are simulated in order to test the observer dynamic behaviour; (5) Finally, experimental and simulated results are compared. This method is illustrated with some examples. 1.1 Sample rate influence The consequences of the sampling method on flux estimation have already been studied in the previous paper [1]. With the assumption that the parameters of the model are accurate, the error analysis has brought to light the following results [6]. In the case of the full order sampling method, the modulus and orientation errors are large at high speed because of the limited development and the orientation error depends on the torque. The most sensitive point is found at high speed (here equal to 2000 rpm) and at no load. In the case of the reduced order sampling method, the modulus error is insignificant (lower than 0.6%), the orientation error increases with the speed and does not depend on the torque. The most sensitive point, which is studied, is at high speed: 2000 rpm for a torque equal to 20 N m. The gain K that minimises the modulus and orientation error is now sought for these sensitive points. The results are presented in Figure 1. The gain matrix is still defined as: K = σo MCo 1− σo [ k1 k2sign(Ω) −k2sign(Ω) k1 ] · (1) The errors are computed for different values of the pair (k1, k2). In the case of the full order sampling method, the modulus and orientation errors are reduced nearly to zero for the following gain: K = σoMCo 1− σo [ 0 −0.1sign(Ω) 0.1sign(Ω) 0 ] · (2) In the case of the reduced order sampling method, the orientation error decreases whatever the value of K different to zero. Therefore another criterion is introduced in order to choose a gain K: the observer eigenvalues are imposed to be close to null in order to increase the response time of the observer: K = σoMCo 1− σo [ −1 −3.5sign(Ω) 3.5sign(Ω) −1 ] · (3) This study was carried out in the steady state for two particular operating points. Now it must be verified that E. Delmotte et al.: Flux observer for induction machine control 27 &DVH IXOO RUGHU VDPSOLQJ PHWKRG » » 1⁄4 o « « ¬ a : : V V VLJQ VLJQ 0 . R &R R &DVH UHGXFHG RUGHU VDPSOLQJ PHWKRG » » 1⁄4 o « « ¬ a : : V V VLJQ VLJQ 0 . R &R R Fig. 2. Modulus and orientation error (nominal flux, Te = 0.8 ms). the observer behaviour is satisfactory over a large range of operating points. The results are given in Figure 2. In the case of the full order sampling method, the modulus error is decreased and the orientation error is lower than 2◦. In the case of the reduced order sampling method, the orientation error is reduced but the modulus error is increased. Nevertheless this error remains insignificant (lower than 2%). Thought to these results, the observer decreases the modulus and orientation errors due to the sample rate and the sampling method. Naturally the observer stability was verified by computing the eigenvalues the modulus of the eigenvalues varies from 0 to 0.998 according to the speed. 1.2 Influence of rotor resistance This study concerns the influence of parameters uncertainties in flux reconstitution [9]. With a sampling rate equal to 0.8 ms, the modulus and orientation errors are analysed for the two sampling methods. The parameters are assumed to be the same for the observer and the machine, except the rotor resistance. As previously in the first paper, an error is artificially introduced in the rotor resistance machine for the simulations and the calculations as follows: rRr = 1− RR RRo = −0.33 . (4) The effects of both sampling rate and rotor resistance variation are studied. The previous analysis has highlighted the discrepancies in flux reconstitution due to these effects (Fig. 7 of the previous paper). In the case of the full order sampling method, the influence of the sampling rate is greatest at high speed while at low speed the rotor resistance variation is prominent. The modulus and orientation errors depend on the torque. Two sensitive points therefore appear: at high speed (here equal to 2000 rpm) and without load; at standstill and at high load (here the torque is equal to 20 N m). In the case of the reduced order sampling method, the rotor resistance variation is prominent: its effects add to the sampling rate effects. The most sensitive point is at high torque and at high speed (here respectively equal to 2000 rpm and 20 N m). As before the gain K, which minimises the modulus and orientation error for the sensitive points, is sought (Fig. 3). In the case of the full order sampling method, two gains are obtained for the two sensitive points: At 2000 rpm: K = σoMCo 1− σo [ 0 −0.075 0.075 0 ] (5) at 0 rpm: K = σoMCo 1− σo [−1 0 0 −1 ] · (6) 28 The European Physical Journal Applied Physics &DVH IXOO RUGHU VDPSOLQJ PHWKRG IRU D VSHHG HTXDO WR USP DQG D WRUTXH HTXDO WR 1P &DVH UHGXFHG RUGHU VDPSOLQJ PHWKRG IRU D VSHHG HTXDO WR USP DQG D WRUTXH HTXDO WR 1P Fig. 3. Modulus and orientation error (nominal flux, Te = 0.8 ms). Speed varying gain therefore appears to be the right solution. For example the following law is possible: K = σo MCo 1− σo [ k1 k2sign(Ω) −k2sign(Ω) k1 ] (7) with  k1 = − exp ( −|Ω| Ω1 ) k2 = 0.075 ( 1− exp ( − |Ω| Ω2 )) and { Ω1 = 200 tr/mn Ω2 = 500 tr/mn. In the case of the reduced order sampling method, from Figure 3, the appropriate gain K seems to be: K = σo MCo 1− σo [−1 0 0 −1 ] = Kt. (8) The two gains are now tested for all the operating points. The results of modulus and orientation errors for these gains are given in Figure 4. In the case of the full order sampling method, the modulus error is decreased in comparison to the computation with a null gain; the orientation error is lower than 6◦. In the case of the reduced order sampling method, the orientation and modulus errors are reduced. So in conclusion, the observer decreases the modulus and orientation errors due to the sample rate and the rotor resistance variation. Once again, the observer stability was verified. A similar approach may be used to analyse observer robustness together with other electrical machine parameters. For example a variation of the mutual cyclic inductance simulates a change of the magnetic state. It is possible to take into account the saturation with inductances that depend on the magnetizing current [10,12]. In the case of the observer of the rotor flux, the influence of the stator resistance is neglected with regard to the rotor resistance [6]. These studies enable choice of an appropriate observer gain, able to cope with parameter uncertainty. However, the analysis was carried out for the steady state assumption. It is now also necessary to verify the gain choice validity for transient state by simulation and experimental trials. We now propose a new way to describe the experimental and simulated system in order to precisely define the experimental and simulated conditions. 2 Description of the experimental and simulated system 2.1 Electrotechnical system functional analysis In order to control the induction machine speed, the system is built along two axes: the power axis and the control axis [2] (Fig. 5). The power axis contains the power source (a three phase rectifier), the static converter (a three phase inverter E. Delmotte et al.: Flux observer for induction machine control 29 &DVH IXOO RUGHU VDPSOLQJ PHWKRG » » 1⁄4 o « « ¬ a : : V V