Euler-Lagrange models with complex currents of three-phase electrical machines and observability issues

Euler-Lagrange models with complex currents of three-phase electrical machines and observability issues

Pierre Rouchon, in collaboration with

Duro Basic,François Malraitfrom 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

1Preprint 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 dt

Jθ˙

=np=

φe¯ ^npθ∗

ıs

−τ_L

d dt

λıs+ ¯φe^npθ

=us−Rsıs

where

I ∗ stands for complex-conjugation,=√

−1 andnpis the number of pairs of poles.

I θis the rotor mechanical angle,J andτ_Lare the inertia and load torque, respectively.

I ıs=ısα+ısβ (respu_s=u_sα+u_sβ) is the stator current (resp. voltage): complex quantities.

I λ= (L_d+Lq)/2 with inductancesL_d =Lq>0 (no saliency here).

I The stator flux isφs=λıs+ ¯φe^npθ with the constantφ >¯ 0 representing to the rotor flux due to permanent magnets.

2See, 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 LagrangianL_c+L_m:

L_c = J

2θ˙², L_m= λ 2

ıs+ ¯ıe^npθ

2

where¯ı= ¯φ/λ >0 is the permanent magnetizing current.

Euler-Lagrange setting:with additional variableq_s∈Cdefined by _dt^dq_s=ıs, take the LagrangianL=L_c+L_mas a real function ofq= (θ,qsα,q_sβ)andq˙ = ( ˙θ, ısα, ı_sβ):

L(q,q) =˙ J

2θ˙²+λ 2

(ısα+ ¯ıcosnpθ)²+ (ı_sβ+ ¯ısinnpθ)² Then the dynamics(3 real ODE)read:

d dt

∂L

∂θ˙

−^∂L_∂θ =−τ_L

d dt

∂L

∂q˙sα

−_∂q^∂L

sα =u_sα−R_sısα, _dt^d ∂L

∂q˙sβ

−_∂q^∂L

sβ =u_sβ−R_sısβ

3See, 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 coordinatesq₁andq₂correspond to a point q=q₁+q₂in the complex plane(=√

−1). The Lagrangian L(q₁,q₂,q˙₁,q˙₂)is a real function and the Euler-Lagrange equations are

d dt

∂L

∂q˙₁

− ∂L

∂q₁ =0, d dt

∂L

∂q˙₂

− ∂L

∂q₂ =0.

Using thecomplex notationq L(q,˜ q^∗,q,˙ q˙^∗)≡ L

q+q^∗

2 ,q−q^∗

2 ,q˙ + ˙q^∗

2 ,q˙ −q˙^∗ 2

.

Since 2^∂_∂q^L˜ = _∂q^∂L

1 −_∂q^∂L

2, 2_∂q^∂L˜∗ = _∂q^∂L

1 +_∂q^∂L

2 we get d

dt

∂L

∂q˙1

+∂L

∂q˙2

= ∂L

∂q1

+∂L

∂q2

that reads d

dt 2∂L˜

∂q˙^∗

!

−2∂L˜

∂q^∗ =0.

4A 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 _dt^dq_s=ıs(q_scomplex cyclic variables)and the Lagrangian

L(θ,θ, ı˙ s, ı^∗_s) = J 2θ˙²+λ

2

ıs+ ¯ıe^npθ ı^∗_s+ ¯ıe^−npθ

the usual equations

d dt

Jθ˙

=np=

λ¯ıe^npθ∗

ıs

−τ_L, d dt

λ(ıs+ ¯ıe^npθ)

=us−Rsıs

read

d dt

∂L

∂θ˙

= ∂L

∂θ −τ_L, 2d dt

∂L

∂ı^∗_s

=us−Rsıs

since _∂q^∂L∗

s =0 and _∂^∂L_q˙∗ s = _∂ı^∂L∗

s.

PM machines: structure of any dynamical models

More generally, the magnetic LagrangianL_mis a real value function ofθ,ıs andı^∗_sthat is ^2π_n

p periodic versusθ. Thus any LagrangianL_PM representing a 3-phases permanent magnet machine admits the following form

LPM = J

2θ˙²+L_m(θ, ıs, ı^∗_s)

Consequently, any model (with saliency, magnetic-saturation, space-harmonics, ...) of permanent magnet machines admits the following structure (J independent ofθhere):

d dt

Jθ˙

= ∂L_m

∂θ −τ_L, d dt

2∂L_m

∂ı^∗_s

=us−Rsıs

withφ_s =2^∂L_∂ı∗^m

s asstator flux.

PM machines: saliency effects

With apositive magnetic Lagrangianof the form

L_m= λ 2

ıs+ ¯ıe^npθ ı^∗_s+ ¯ıe^−npθ

−µ 4

ı^∗_se^npθ2

+

ıse^−npθ2

whereλ= (L_d +L_q)/2 andµ= (L_q−L_d)/2 (inductances L_d >0 andL_q>0), we recover the usual model with saliency:





 d dt

Jθ˙

=n_p=

(λı^∗_s+λ¯ıe^−npθ−µı_se^−2npθ)ı_s

−τ_L d

dt

λı_s+λ¯ıe^npθ−µı^∗_se^2npθ

=u_s−R_sı_s.

PM machines: magnetic-saturation and saliency effects Inductances depend on the currentsas, e.g.,

λ=λ(|ı_s+ ¯ıe^npθ|) =λ q

(ıs+ ¯ıe^npθ)(ı^∗_s+ ¯ıe^−npθ)

whereı_s+ ¯ıe^npθ stands fortotal magnetizing current. With magnetic Lagrangien

L_m= λ

ı_s+ ¯ıe^npθ

2

ı_s+ ¯ıe^npθ

2

−µ 4

ı^∗_se^npθ2

+

ı_se^−npθ2

the dynamics read (Λ=λ+|ıs+¯ıe^npθ|

2 λ⁰):





 d dt

Jθ˙

=n_p= Λ

ı^∗_s+ ¯ıe^−npθ

−µıse^−2npθ ıs

−τL

d dt

Λ

ıs+ ¯ıe^npθ

−µı^∗_se^2npθ

=us−Rsıs

Similarlyµcould also depend on|ı_s+ ¯ıe^npθ|.

Induction machines: usual models

Dynamics withcomplex stator and rotor currents:













 d dt

Jθ˙

=n_p=

L_mı^∗_re^−npθı_s

−τ_L d

dt

L_rır +L_mıse^−npθ

=−R_rır

d dt

L_sıs+L_mıre^npθ

=u_s−R_sıs

where

I ı_r ∈C(resp. ı_s ∈C) is the rotor (resp. stator) current;

us ∈Cis the stator voltage

I R_s >0 andR_r >0 are stator and rotor resistances.

I L_s >0,L_r >0 andL_mare the inductances satisfying LsLr >L²_mfor physical reasons(positive magnetic Lagrangien). They are constant here.

I the stator (resp. rotor)flux isφs =Lsıs+Lmıre^npθ(resp.

φ_r =L_rı_r +L_mı_se^−npθ).

Induction machines: Lagrangian with complex currents The Lagrangian of the usual model is

L_m = J

2θ˙²+L_m 2

ı_s+ı_re^npθ

2

+L_fr

2 |ı_r|²+L_fs 2 |ı_s|² whereLs=Lm+L_fs andLr =Lm+L_fr withLm >0 and

0<L_fr,L_fs L_m. More generally, a physically consistent model should be obtained with a Lagrangian of the form

L_IM= J

2θ˙²+L_m(θ, ı_r, ı^∗_r, ı_s, ı^∗_s)

whereL_mis the magnetic Lagrangien expressed with the rotor angle and currents.It is ^2π_n

p periodic versusθ.Any physically admissible model reads (J independent ofθ)

d dt

Jθ˙

= ∂L_m

∂θ −τ_L, d

dtφr =−R_rır, d

dtφs =us−Rsıs, where therotor and stator fluxesare given by

φr =2∂L_m

∂ı^∗_r , φs =2∂L_m

∂ı^∗_s .

Induction machines: magnetic-saturation With positive inductances of the form

Lm=Lm

ıs+ıre^npθ

, Ls =Lm+L_fs, Lr =Lm+L_fr

themagnetic Lagrangien remains positive L_m = Lm

ıs+ıre^npθ

2

ıs+ıre^npθ

2

+L_fr

2 ırı^∗_r +L_fs 2 ısı^∗_s and the saturation model reads













 d dt

Jθ˙

=n_p=

Λmı^∗_re^−npθıs

−τL

d dt

Λm

ır +ıse^−npθ +L_frır

=−R_rır

d dt

Λm

ıs+ıre^npθ

+L_fsıs

=us−Rsıs

withΛm =L_m+|^ıs+ıre^npθ|

2 L⁰_mfunction of

ıs+ıre^npθ .

Induction machines: space-harmonics and magnetic-saturation.

Add contribution of space harmonics to magnetic Lagrangien:

L_m

ıs+ıre^npθ 2

ıs+ıre^npθ

2

+L_fr

2 ırı^∗_r + L_fs 2 ısı^∗_s +L_ν

2

ısı^∗_re^{−σννnpθ}+ı^∗_sıre^{σννnpθ}

withL_ν >0 a small parameter (|L_ν| L_m) andσν =±1 depending on arithmetic conditions⁵. The dynamical model is changed as follows:

d dt

Jθ˙

=n_p= Λ_me^−npθ+Lνσννe^{−σννnpθ} ı^∗_rı_s

−τ_L d

dt Λ_m ı_r +ı_se^−npθ

+Lfrı_r+Lνı_se^{−σννnpθ}

=−R_rı_r d

dt Λ_m ı_s+ı_re^npθ

+L_fsı_s+Lνı_re^{σννnpθ}

=us−Rsı_s

5See 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 torqueτ_Lconstant but unknown, control inputsus and measured outputsıs6

Physical models including saliency and magnetic saturation associated to LagrangianLPM = ^J₂θ˙²+L_m(θ, ı_s, ı^∗_s),

d dt

Jθ˙

= ∂L_m

∂θ −τL, d dt

2∂L_m

∂ı^∗_s

=u_s−R_sıs

can be always written in state-space form

d

dtX =f(X,U), Y =h(X)

whereX = (τ_L, θ,θ,˙ <(ı_s),=(ı_s))withU = (<(u_s),=(u_s)), Y = (<(ı_s),=(ı_s))and _dt^dτ_L=0.

6For a nice exposure see J. Holtz: Sensorless control of induction motor drives.Proc. of the IEEE, 90(8):1359–1394, 2002.

Sensorless control around zero stator frequency

A stationary regime atzero stator frequencycorresponds then to a steady state( ¯X,U,¯ Y¯)satisfyingf( ¯X,U) =¯ 0,Y¯ =h( ¯X).

For a PM machines we get

∂L_m

∂θ (θ, ı_s, ı^∗_s)−τ_L=0, ı_s = ¯ı_s

to recover(θ, ıs, τ_L)from the stationary valuesu¯s andı¯s. This impliessevere observability difficulties:

I to any constant input and outputu¯sand¯ıs satisfying u¯_s =R_s¯ı_s correspond aone dimensional family of steady statesparameterized by the scalar variableξwith

τ_L= ^∂L_∂θ^m(ξ,¯ıs,¯ı^∗_s), θ=ξ, ıs = ¯ıs.

I the linear tangent systems around such steady-states are not observable;

The situation is similar for induction machines: including space-harmonic and magnetic-saturation does not canceled such lack of observability.

Concluding remarks

I Extensions tonetwork of machines and generators

connected via long linescan also be developed with similar variational principles and Euler-Lagrange equations with complex currents and voltages (ODE or PDE).

I Observability issues at zero stator frequency: a strong motivation for theoretical works on the followingspecific stabilizationproblem involving an unknown constant parameterp: take _dt^dx =f(x,u,p),y =h(x)a nonlinear system where{(x,p)|f(x,u,¯ p) =0,h(x) = ¯y}is a smooth curve; take any(¯x,p)¯ on this equilibrium curve;

under which conditionsis it possible to construct (without knowingp) a (dynamic)output feedbackstabilizingx aroundx¯in arobustway.