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 firstname.lastname@example.org 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 + φenp θ − µı∗ e2np θ ¯ s ıs − τL dt dt d λıs + φenp θ − µı∗ e2np 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 + φenp θ − µı∗ e2np 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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