Euler-Lagrange models with complex currents of three-phase ...

Euler-Lagrange models with complex currents of three-phase electrical machines and observability issues1 Pierre Rouchon, in collaboration with Duro Basic, François Malrait from Schneider Electric, STIE. Mines ParisTech Centre Automatique et Systèmes Mathématiques et Systèmes [email protected] http://cas.ensmp.fr/~rouchon/index.html

Electrical and Mechatronical Systems Workshop Bernoulli Center, Lausanne 18-20 February 2009 1

Preprint arXiv:0806.0387v3 [math.OC]

Outline Permanent magnet 3-phases machines Usual models Euler Lagrange setting with complex electrical variables Saliency effects Both saliency and magnetic-saturation effects Induction 3-phases machines Usual models Magnetic-saturation effects Both space-harmonics and magnetic-saturation effects Observability issues at zero stator frequency Concluding remarks

PM machines: usual models In the (α, β) frame the dynamic equations read2 :     ∗  d  ¯ np θ ıs − τL  J θ˙ = np = φe dt     d λı + φe ¯ np θ = us − Rs ıs s dt where √ I ∗ stands for complex-conjugation,  = −1 and np is the number of pairs of poles. I θ is the rotor mechanical angle, J and τL are the inertia and load torque, respectively. I ıs = ısα + ısβ (resp us = usα + usβ ) is the stator current (resp. voltage): complex quantities. I λ = (Ld + Lq )/2 with inductances Ld = Lq > 0 (no saliency here). ¯ np θ with the constant φ¯ > 0 I The stator flux is φs = λıs + φe representing to the rotor flux due to permanent magnets. 2

See, e.g., J. Chiasson: Modeling and High Performance Control of Electric Machines, Wiley-IEEE Press, 2005.

PM machines: Euler-Lagrange setting3 Lagrangian: sum of kinetic and magnetic Lagrangian Lc + Lm : 2 J λ Lc = θ˙2 , Lm = ıs + ¯ıenp θ 2 2 ¯ > 0 is the permanent magnetizing current. where ¯ı = φ/λ Euler-Lagrange setting: with additional variable qs ∈ C defined d by dt qs = ıs , take the Lagrangian L = Lc + Lm as a real ˙ ısα , ısβ ): function of q = (θ, qsα , qsβ ) and q˙ = (θ,  J λ ˙ = θ˙2 + (ısα + ¯ı cos np θ)2 + (ısβ + ¯ı sin np θ)2 L(q, q) 2 2 Then the dynamics (3 real ODE) read:   d ∂L − ∂L dt ∂θ = −τL ∂ θ˙     ∂L d ∂L ∂L d ∂L − = u − R ı , − ∂q = usβ − Rs ısβ sα s sα dt ∂ q˙ sα ∂qsα dt ∂ q˙ sβ sβ

3

See, e.g., R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez. Passivity–Based Control of Euler–Lagrange Systems. Communications and Control Engineering. Springer-Verlag, Berlin, 1998.

Euler-Lagrange equation with complex variables4 Two generalized coordinates q1 and q2 √ correspond to a point q = q1 + q2 in the complex plane ( = −1). The Lagrangian L(q1 , q2 , q˙ 1 , q˙ 2 ) is a real function and the Euler-Lagrange equations are     d ∂L ∂L d ∂L ∂L − = 0, − = 0. dt ∂ q˙ 1 ∂q1 dt ∂ q˙ 2 ∂q2 Using the complex notation q   q + q ∗ q − q ∗ q˙ + q˙ ∗ q˙ − q˙ ∗ ˜ q ∗ , q, ˙ q˙ ∗ ) ≡ L L(q, , , , . 2 2 2 2 ˜

Since 2 ∂∂qL = d dt



4

∂L ∂q1

∂L ∂L + ˙ ∂ q1 ∂ q˙ 2



˜

∂L ∂L −  ∂q , 2 ∂q ∗ = 2

∂L ∂L = + ∂q1 ∂q2

∂L ∂q1

∂L +  ∂q we get 2

that reads

d dt

∂ L˜ 2 ∗ ∂ q˙

! −2

∂ L˜ = 0. ∂q ∗

A usual method borrowed from quantum physics: see, e.g., C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg. Photons and Atoms: Introduction to Quantum Electrodynamics. Wiley, 1989.

PM machines: Lagrangian with complex stator currents With

d dt qs

= ıs (qs complex cyclic variables) and the Lagrangian

   ˙ ıs , ı∗s ) = J θ˙2 + λ ıs + ¯ıenp θ ı∗s + ¯ıe−np θ L(θ, θ, 2 2 the usual equations  ∗  d  ˙ J θ = np = λ¯ıenp θ ıs −τL , dt

 d  λ(ıs + ¯ıenp θ ) = us − Rs ıs dt

read d dt since

∂L ∂qs∗



∂L ∂ θ˙



= 0 and

∂L = − τL , ∂θ ∂L ∂ q˙ s∗

=

∂L ∂ı∗s .

d 2 dt



∂L ∂ı∗s

 = us − Rs ıs

PM machines: structure of any dynamical models More generally, the magnetic Lagrangian Lm is a real value function of θ, ıs and ı∗s that is 2π np periodic versus θ. Thus any Lagrangian LPM representing a 3-phases permanent magnet machine admits the following form LPM =

J ˙2 θ + Lm (θ, ıs , ı∗s ) 2

Consequently, any model (with saliency, magnetic-saturation, space-harmonics, ...) of permanent magnet machines admits the following structure (J independent of θ here):   d  ˙ ∂Lm d ∂Lm Jθ = − τL , 2 ∗ = us − Rs ıs dt ∂θ dt ∂ıs m with φs = 2 ∂L ∂ı∗ as stator flux. s

PM machines: saliency effects

With a positive magnetic Lagrangian of the form  2  2   µ  λ np θ ∗ np θ −np θ ∗ −np θ Lm = ıs + ¯ıe ıs e + ıs e ıs + ¯ıe − 2 4 where λ = (Ld + Lq )/2 and µ = (Lq − Ld )/2 (inductances Ld > 0 and Lq > 0), we recover the usual model with saliency:  d   dt   d dt

   J θ˙ = np = (λı∗s + λ¯ıe−np θ −µıs e−2np θ )ıs − τL   λıs + λ¯ıenp θ −µı∗s e2np θ = us − Rs ıs . 

PM machines: magnetic-saturation and saliency effects Inductances depend on the currents as, e.g.,  q np θ ∗ n θ −n θ p p (ıs + ¯ıe )(ıs + ¯ıe ) λ = λ(|ıs + ¯ıe |) = λ where ıs + ¯ıenp θ stands for total magnetizing current. With magnetic Lagrangien
2 µ  2  2  λ ıs + ¯ıenp θ np θ ∗ np θ −np θ Lm = ıs e + ıs e ıs + ¯ıe − 2 4 the dynamics read (Λ = λ +  d   dt   d dt

|ıs +¯ıenp θ | 2

λ0 ):



      J θ˙ = np = Λ ı∗s + ¯ıe−np θ − µıs e−2np θ ıs − τL     Λ ıs + ¯ıenp θ − µı∗s e2np θ = us − Rs ıs

Similarly µ could also depend on |ıs + ¯ıenp θ |.

Induction machines: usual models Dynamics with complex stator and rotor currents:      d  ˙ = np = Lm ı∗r e−np θ ıs − τL J θ    dt     d Lr ır + Lm ıs e−np θ = −Rr ır  dt        d L ı + L ı enp θ = u − R ı s s m r s s s dt where I ır ∈ C (resp. ıs ∈ C) is the rotor (resp. stator) current; us ∈ C is the stator voltage I Rs > 0 and Rr > 0 are stator and rotor resistances. I Ls > 0, Lr > 0 and Lm are the inductances satisfying Ls Lr > L2m for physical reasons (positive magnetic Lagrangien). They are constant here. I the stator (resp. rotor) flux is φs = Ls ıs + Lm ır enp θ (resp. φr = Lr ır + Lm ıs e−np θ ).

Induction machines: Lagrangian with complex currents The Lagrangian of the usual model is 2 L J Lm Lfs 2 fr Lm = θ˙2 + |ıs | ıs + ır enp θ + |ır |2 + 2 2 2 2 where Ls = Lm + Lfs and Lr = Lm + Lfr with Lm > 0 and 0 < Lfr , Lfs  Lm . More generally, a physically consistent model should be obtained with a Lagrangian of the form J ˙2 θ + Lm (θ, ır , ı∗r , ıs , ı∗s ) 2 where Lm is the magnetic Lagrangien expressed with the rotor angle and currents. It is 2π np periodic versus θ. Any physically admissible model reads (J independent of θ) d  ˙ ∂Lm d d Jθ = − τL , φr = −Rr ır , φs = us − Rs ıs , dt ∂θ dt dt where the rotor and stator fluxes are given by LIM =

φr = 2

∂Lm , ∂ı∗r

φs = 2

∂Lm . ∂ı∗s

Induction machines: magnetic-saturation With positive inductances of the form   Lm = Lm ıs + ır enp θ , Ls = Lm + Lfs , Lr = Lm + Lfr the magnetic Lagrangien remains positive
2 L Lm ıs + ır enp θ L fr Lm = ır ı∗r + fs ıs ı∗s ıs + ır enp θ + 2 2 2 and the saturation model reads      d  ˙ = np = Λm ı∗r e−np θ ıs − τL J θ    dt       d Λm ır + ıs e−np θ + Lfr ır = −Rr ır  dt          d Λ ı + ı enp θ + L ı = u − R ı m s r s s s fs s dt with Λm = Lm +

|ıs +ır enp θ | 2

L0m function of ıs + ır enp θ .

Induction machines: space-harmonics and magnetic-saturation. Add contribution of space harmonics to magnetic Lagrangien:
2 L Lm ıs + ır enp θ L fr np θ ır ı∗ + fs ıs ı∗s ıs + ır e + 2 2 r 2  Lν  ∗ −σν νnp θ + ıs ır e + ı∗s ır eσν νnp θ 2 with Lν > 0 a small parameter (|Lν |  Lm ) and σν = ±1 depending on arithmetic conditions5 . The dynamical model is changed as follows: d dt d dt d dt 5



 J θ˙ = np =



Λm e−np θ + Lν σν νe−σν νnp θ ı∗r ıs − τL



Λm ır + ıs e−np θ + Lfr ır + Lν ıs e−σν νnp θ = −Rr ır

Λm ıs + ır enp θ + Lfs ıs + Lν ır eσν νnp θ = us − Rs ıs

See H.R. Fudeh and C.M. Ong: Modeling and analysis of induction machines containing space harmonics. Part-I: modeling and transformation. IEEE Transactions on Power Apparatus and Systems, 102:2608–2615, 1983.

Sensorless control of PM machines Sensorless control: a load torque τL constant but unknown, control inputs us and measured outputs ıs 6 Physical models including saliency and magnetic saturation associated to Lagrangian LPM = J2 θ˙2 + Lm (θ, ıs , ı∗s ),   d  ˙ ∂Lm d ∂Lm Jθ = − τL , 2 ∗ = us − Rs ıs dt ∂θ dt ∂ıs can be always written in state-space form d X = f (X , U), dt

Y = h(X )

˙