Modeling magnetic saturation and saliency effects via Euler-Lagrange by brm24619

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									    Modeling magnetic saturation and
   saliency effects via Euler-Lagrange
    models with complex currents for
three-phase permanent magnet machines

          Pierre Rouchon, in collaboration with
Duro Basic, François Malrait from Schneider Electric, STIE.

                           Mines ParisTech
                  Centre Automatique et Systèmes
                    Mathématiques et Systèmes
                 pierre.rouchon@mines-paristech.fr
          http://cas.ensmp.fr/~rouchon/index.html


              IFAC Workshop E-COSM’09
 Engine and Powertrain Control, Simulation and Modeling
             IFP, Rueil-Malmaison, France
           30 November – 2 December 2009
Outline


   Motivations


   Usual (α, β) model with saliency


   The (α, β) model with magnetic saturation


   A first experimental validation


   Concluding remarks
Modeling and control of electrical machines in future cars

        Specific machines with unusual geometry (e.g. wheel-mounted
        electrical machine), operating close to magnetic saturation,
        exploring large speed domains.
        Control of the electro-magnetic torque:
             precision, bandwidth and robustness
             minimizing energy losses
             deceleration versus braking, coupling with ABS.
             ESP and more generally advanced traction control system
             by monitoring each wheel separately.

    Need for dynamic modelling compatible with real-time applications
    but including effects that are not usually taken into account in
    standard (α, β) models.
    Goal: how to base such modeling on variational principles underlying
    Maxwell/Lorentz equations with magnetic energies directly expressed
    with usual complex currents, fluxes and voltage variables.
    Focus: permanent magnet machines with saliency and magnetic
    saturation.
Permanent-magnet machines: usual (α, β) models
   In the (α, β) frame the dynamic equations read1 :
       
        d      d                                               ∗
             J θ = np          λıs + φenp θ − µı∗ e2np θ
                                         ¯        s                 ıs − τL
         dt     dt
       
        d
              λıs + φenp θ − µı∗ e2np θ = us − Rs ıs .
                    ¯
       
                                s
       
         dt
    where
       ∗ (resp. ) stands for complex-conjugation (resp. imaginary
                  √
       part),  = −1 and np is the number of pairs of poles.
       θ is the rotor mechanical angle, J and τL are the inertia and
       load torque, respectively.
       ıs = ısα + ısβ (resp us = usα + usβ ) is the stator current
       (resp. voltage): complex quantities.
             L +L       L −L
       λ = d 2 q , µ = q 2 d with inductances Ld , Lq > 0.
                                     ¯                         ¯
       The stator flux is φs = λıs + φenp θ − µı∗ e2np θ with φ > 0
                                                 s
       the constant flux due to permanent magnets.
       1
        See, e.g., J. Chiasson: Modeling and High Performance Control of
    Electric Machines, Wiley-IEEE Press, 2005.
Hamiltonian dynamical models with complex variables
   The magnetic energy of the usual (α, β) model reads:
                          1                         2        1                    2
    Hm (φs , φ∗ , θ) =
              s                 (φs e−np θ ) − φ
                                                ¯       +         (φs e−np θ )
                         2Ld                                2Lq
   where φs is the stator flux. Magnetic saturations just mean that
   Ld and Lq are not constant but depend also on φs .
   Any three-phase permanent magnet machine with saliency and
   magnetic saturation can be described by a dynamic model with
   the following structure:2
     d        d           ∂Hm           d                                  ∂Hm
          J      θ   =−       − τL ,       φs = us − Rs ıs ,      ıs = 2
     dt       dt           ∂θ           dt                                 ∂φ∗
                                                                             s

   where the magnetic energy Hm is considered as a function of
   the rotor angle θ, the stator flux φs and φ∗ .
                                             s
   When inductances Ld and Lq are constant, we recover the
   usual (α, β) model (with saliency when Ld = Lq ).
      2
       D. Basic, F. Malrait, P. Rouchon: Euler-Lagrange models with complex
   currents of three-phase electrical machines and observability issues. To
   appear in IEEE Transactions on Automatic Control.
Experimental validations based on high frequency voltage injection
    Rotor blocked at θ = 0 and high frequency voltage injection:

                    r    a           d        r    a               ∂Hm
          us (t) = us + us f (Ωt),      φs = us + us f (Ωt) − Rs 2
                                     dt                            ∂φ∗
                                                                     s
                                                                    ıs

                                       r      a
    where the complex quantities us and us are constant, the
    function γ → f (γ) is 2π periodic with a zero mean .
                                                        a
    For Ω large enough(typically ∼ 1kHz), φs = φr + us F (Ωt) where
                                                    s  Ω
    F is the primitive of f , dF = f with a zero mean,
                              dγ
     2π
     0  F (γ) dγ = 0.
    Then, ıs = ır + δıs where current ripples δıs result from flux
                s
                    a
    ripples δφs = us F (Ωt):
                   Ω

            r
           us    ∂Hm r                               ∂ 2 Hm        ∂ 2 Hm
    ır =
     s        =2     (φ , (φr )∗ , 0),    δıs = 2            δφs +2 2 ∗ δφ∗ .
           Rs    ∂φ∗ s s
                   s                                ∂φs ∂φ∗s       ∂ φs s
Magnet saturation: current ripples depend on current offsets

    When Ld and Lq depend on φs , the magnetic energy

                          1                        2        1                          2
    Hm (φs , φ∗ , θ) =
              s                (φs e−np θ ) − φ
                                               ¯       +            (φs e−np θ )
                         2Ld                               2Lq

    is no longer quadratic in (φs , φ∗ ).
                                     s
                    a    r
    Voltages us = us + us F (Ωt) generate current ripples δıs
                                                                           ∗
                  ∂ 2 Hm        a
                               us F (Ωt)      ∂ 2 Hm            a
                                                               us F (Ωt)
         δıs = 2                            +2 2 ∗                             .
                 ∂φs ∂φ∗s          Ω          ∂ φs                 Ω

    depending on flux level φr given by ır = 2 ∂H∗ (φr , (φr )∗ , 0).
                            s           s     ∂φ
                                                 m
                                                    s     ss
                                                               u                   r
    Thus amplitudes of δıs are related to current offsets ır = Rs .
                                                           s     s
    Magnetic saturation implies that Ld , Lq are decreasing versus
                   2            ∂ 2 Hm
    |φs |. Thus ∂φsHm∗ and
                 ∂
                   ∂φs          ∂ 2 φ∗
                                         are increasing and amplitudes of
                                     s
    δıs increase too.
Experimental data on a 1.2 kW permanent magnet machine




   The blue curve corresponds to the amplitudes of the
   current-ripples (obtained via a simple PLL-filter), the red curve
   to the current measures ıs (d-axis aligned with the rotor θ = 0).
   Increasing dependance of the ripples versus the current offset
   ır : inductances are decreasing functions of |φs |.
    s
Concluding remarks

       The saturation models for permanent-magnet machine are
       based on variational principles. Experimental data provide
       a first validation of such modelling procedures that
       preserve the physical insight while maintaining a synthetic
       view without describing all the technological and material
       details.
       More complete validations could be done: at non zero rotor
       velocity, the proposed computations of the current ripples
       are still possible since they are obtained via usual
       perturbations/averaging techniques. Such saturation
       models can be used for control purposes: adaptation of
       usual control schemes to take into account saturation
       effects are under study.
       Modeling of interconnected machines and generators can
       also be developed with similar variational principles relying
       on complex currents, fluxes and voltages.

								
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