Using Hamiltonians to Model Saturation in Space Vector Representations of AC Electrical Machines

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Using Hamiltonians to Model Saturation in Space Vector Representations of AC Electrical

Machines

Duro Basic¹, Al Kassem Jebai², Fran¸cois Malrait¹, Philippe Martin², and Pierre Rouchon²

1 Schneider Electric, STIE, 33, rue Andr´e Blanchet, 27120 Pacy-sur Eure, France.

2 Mines ParisTech, Centre Automatique et Systèmes, Mathématiques et Systèmes, 60 Bd Saint-Michel, 75272 Paris cedex 06, France.

Abstract. An Hamiltonian formulation with complex fluxes and currents is proposed. This formulation is derived from a recent Lagrangian formulation with complex electrical quantities. The complexification pro- cess avoids the usual separation into real and imaginary parts and no- tably simplifies modeling issues. Simple modifications of the magnetic energy underlying standard (α, β) models yield new (α, β) models de- scribing machines with magnetic saturation and saliency. We prove that the usual expression of the electro-mechanical torque (wedge product of fluxes and currents) is related to a rotational invariance characterizing sinusoidal machines.

1 Introduction

In [1] a Lagrangian formulation with complex currents and fluxes is proposed.

In this paper we develop the Hamiltonian counterpart only sketched in [1]. For three-phase electrical machines we recall the usual model linear in fluxes, currents and voltages, and give its Hamiltonian formulation based on magnetic energies depending quadratically on fluxes. We then propose a modification of the usual magnetic energies in order to take into account magnetic saturation. We prove that if these additional terms preserve the rotational invariance of the usual magnetic energies, then the resulting electro-magnetic torque always admits the usual form and is thus still proportional to the imaginary part of the product of complex conjugate of fluxes with stator currents.

Section 2 is devoted to permanent-magnet machines. In subsection 2.1 we present the Hamiltonian formulation of the usual model with saliency effects.

In subsection 2.2 we introduce a class of saturation models and prove that, if we just replace in the usual model the constant inductances by inductances depending on the flux level, the resulting model does not admit in general a magnetic energy and thus is not correct from a physical ground. For sinusoidal machines where the magnetic energy is invariant with respect to the choice of angle origin, we prove in subsection 2.3 that the usual formula giving the electro- magnetic torque as a wedge product between the flux and current remains valid

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even in the presence of saliency and magnetic saturation. Section 3 is devoted to induction machines. In subsection 3.1 we present the Hamiltonian formulation of the usual model. In subsection 3.2 we introduce a class of saturation models. For machines with sinusoidally wound phases where the magnetic energy is invariant with respect to the choice of angle origin, we prove in subsection 3.3 that the usual formula giving the electro-magnetic torque as a wedge product between the flux and current remains valid even in the presence of magnetic saturation.

In section 4 we suggest some further developments.

The authors acknowledge John Chiasson for interesting discussions and pre- cious comments.

2 Permanent-magnet machines

2.1 Hamiltonian modeling

In the (α, β) frame (total power invariant transformation), the usual dynamic equations read (see, e.g., [2, 4]):





 d dt

Jθ˙

=np= (λı^∗_s+ ¯φe^−n^p^θ−µıse^−2n^p^θ)ıs

−τL

d

dt λıs+ ¯φe^n^p^θ−µı^∗_se^2n^p^θ

=us−Rsıs

(1)

where

– ^∗stands for complex-conjugation,=means imaginary part,=√

−1 andnp

is the number of pairs of poles.

– θ is the rotor mechanical angle, J and τL are the inertia and load torque, respectively.

– ı_s∈Cis the stator current,u_s∈Cthe stator voltage.

– λ= (L_d+L_q)/2 and µ = (L_q −L_d)/2 (inductances L_d > 0 and L_q >0, saliency whenL_d6=L_q).

– The constant ¯φ >0 represents the rotor flux due to the permanent magnets.

It is proved in [1] that (1) admits the following Hamiltonian formulation d

dt

Jθ˙

=−∂Hm

∂θ −τL, d

dtφs=us−Rsıs, ıs= 2∂Hm

∂φ^∗_s (2) where the magnetic energyHm

H_m(φ_s, φ^∗_s, θ) = 1 2Ld

<(φ_se^−n^p^θ)−φ¯² + 1

2Lq

=(φ_se^−n^p^θ)²

(3)

= 1 2Ld

φse^−n^p^θ+φ^∗_se^n^p^θ

2 −φ¯

² + 1

2Lq

φse^−n^p^θ−φ^∗_se^n^p^θ 2

²

= 1 8Ld

φse^−n^p^θ+φ^∗_se^n^p^θ−2 ¯φ²

− 1 8Lq

φse^−n^p^θ−φ^∗_se^n^p^θ²

(3)

where the rotor angle θ, the stator flux φ_s and its complex conjugate φ^∗_s are considered independent variables when computing the partial derivatives ofH_m. In particular,ı_s= 2^∂H_∂φ^m∗

s

reads ı_s= ^e_2L^npθ

d φ_se^−n^p^θ+φ^∗_se^n^p^θ−2 ¯φ +^e_2L^npθ

q φ_se^−n^p^θ−φ^∗_se^n^p^θ

=

1 2Ld+_2L¹

q

φs−_L¹

d

φe¯ ^n^p^θ+

1 2Ld−_2L¹

q

φ^∗_se^2n^p^θ. Inverting this relation we recover the usual relation between φs and the stator current

φs=λıs+ ¯φe^n^p^θ−µı^∗_se^2n^p^θ. 2.2 Magnetic saturation

To take into account magnetic saturation, we keep the structure equations (2) and modify the magnetic energy H_m given in (3). For obvious physical rea- sons,Ld andLq should be decreasing functions of|φs|². The simplest magnetic saturation model will be given by setting

H_m(φ_s, φ^∗_s, θ) = Sd(|φs|²) L¯d

<(φ_se^−n^p^θ)−φ¯²

+Sq(|φs|²) L¯q

=(φ_se^−n^p^θ)² where the saturation functions Sd andSq are increasing function of |φs|² with Sd(0) =Sq(0) = 1 and where ¯Ld and ¯Lq are the unsaturated values of Ld and Lq (low stator currents). The saturation model we propose is then given by (2) with this modified Hamiltonian.

Using this Hamiltonian formulation to define the relationships betweenıs,φs

andτemas in (2) automatically maintains energy conservation. This conservation results from the fact that mixed partial derivatives are independent of order,

∂²Hm

∂θ∂φ^∗_s = ∂²Hm

∂φ^∗_s∂θ. This implies

−2∂τem

∂φ^∗_s = ∂ıs

∂θ

whereτemandısare considered as function of the independent variablesφs,φ^∗_s andθ.

On the other hand, an incorrect but seemingly ”natural” way to include saturation in the usual (α, β) model

d dt

J d

dtθ

=np=(φ^∗_sıs)−τL

d

dtφs=us−Rsıs

ı_s= 1

2Ld

+ 1 2Lq

φ_s+

1 2Ld

− 1 2Lq

φ^∗_se^2n^p^θ− φ¯ Ld

e^n^p^θ

(4)

consists in taking L_d and L_q as function of ρ² = φ_sφ^∗_s, without changing the formula for the electro-magnetic torque. If we proceed like this we get

τem=np

2

1

2Ld

− 1 2Lq

(φ^∗s)²e^2n^p^θ−(φs)²e^−2n^p^θ

− φ¯

Ld

φ^∗se^n^p^θ−φse^−n^p^θ whereL_dand L_q depend onρ²=|φ_s|² Then some computations give

−2∂τem

∂φ^∗_s −∂ıs

∂θ =np



 d

1 2Ld−_2L¹

q

dρ² (φ^∗_s)²e^2n^p^θ−(φs)²e^−2n^p^θ

− d _¯

φ Ld

dρ² φ^∗_se^n^p^θ−φse^−n^p^θ



φs.

Thus such modeling does not in general respect the commutation condition

−2^∂τ_∂φ^em∗ s

= ^∂ı_∂θ^s: no magnetic energy exists for such non-physical models. The correct current relationships include additional terms with derivatives of the functions S_d andS_q:

ıs= 1

2Ld

+ 1 2Lq

φs+

1 2Ld

− 1 2Lq

φ^∗_se^2n^p^θ− φ¯ Ld

e^n^p^θ

+S_d⁰(|φs|²) 2 ¯Ld

φs <(φse^−n^p^θ)−φ¯²

+S_q⁰(|φs|²) 2 ¯Lq

φs =(φse^−n^p^θ)² .

2.3 Sinusoidal models

Assume that the magnetic energyHmadmits the following rotational invariance associated to sinusoidal back electro-magnetic force (bemf):

∀φs∈C, ∀θ, ξ∈S¹, Hm(e^n^p^ξφ_s, e^−n^p^ξφ^∗_s, ξ+θ) =Hm(φ_s, φ^∗_s, θ).

Then withH(ψ, ψ^∗) =Hm(ψ, ψ^∗,0),Hmadmits the following form Hm(φs, φ^∗_s, θ)≡ H(φse^−n^p^θ, φ^∗_se^n^p^θ).

In this case

τem=−∂Hm

∂θ =−np

∂H

∂ψ^∗φ^∗_se^n^p^θ−∂H

∂ψφse^−n^p^θ

ı_s= 2∂Hm

∂φ^∗_s = 2e^n^p^θ∂H

∂ψ^∗.

Since H is a real quantity ı^∗_s =e^−n^p^{θ ∂H}_∂ψ. Thus we recover the usual formula relating the electro-magnetic torque to the fluxφs and currentıs:

τ_em=n_pφ^∗_sı_s−φ_sı^∗_s

2 =n_p=(φ^∗_sı_s). (4)

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When H_m does not admit such rotational invariance, τ_em is different from n_p=(φ^∗_sı_s). Thus (4) is a direct consequence of rotational invariance. The saturation models considered in the previous subsection admit this rotational invariance and yield electro-magnetic torques satisfying (4).

A simple example of a non sinusoidal model is a machine with a trapezoidal bemf F(n_pθ) (a sinusoidal model corresponds to F(n_pθ) = cos(n_pθ)). In this case we change the Hamiltonian in (2) by

Hm= 1

2L_d <(φs(F(npθ) +F(npθ+^π₂)))−φ¯² + 1

2L_q =(φ_s(F(n_pθ) +F(n_pθ+^π₂)))² where e^−n^p^θ in (3) is replaced byF(n_pθ) +F(n_pθ+^π₂). This Hamiltonian is not rotationally invariant.

3 Induction machines

3.1 Hamiltonian modeling

We will now proceed as for permanent-magnet machines. The standardT-model of an induction machine admit the following form:









 d dt

Jθ˙

=n_p= L_mı^∗_re^−n^p^θı_s

−τ_L d

dt L_m ı_r+ı_se^−n^p^θ

+L_frı_r

=−R_rı_r d

dt Lm ıs+ıre^n^p^θ

+Lfsıs

=us−Rsıs

(5)

where

– n_pis the number of pairs of poles,θis the rotor mechanical angle,J andτ_L are the inertia and load torque, respectively.

– ı_r ∈ C is the rotor current (in the rotor frame, different from the (d, q) frame) ,ı_s∈Cthe stator current (in the stator frame, i.e. the (α, β) frame) and u_s ∈ C the stator voltage (in the stator frame). The stator and rotor resistances areRs>0 andRr>0.

– The inductancesL_m,L_frandL_fsare positive parameters withL_fr, L_fsL_m. – The stator (resp. rotor) flux is φ_s =L_m ı_s+ı_re^n^p^θ

+L_fsı_s (resp. φ_r = Lm ır+ıse^−n^p^θ

+Lfrır).

The Hamiltonian formulation proposed in [1] reads:

d dt

Jθ˙

=−∂Hm

∂θ −τL, d

dtφr=−2Rr

∂Hm

∂φ^∗_r , d

dtφs=us−2Rs

∂Hm

∂φ^∗_s (6)

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where the magnetic energy H_m now depends on θ, the rotor flux φ_r and its complex conjugate φ^∗_r, the stator flux φ_s and its complex conjugate φ^∗_s. The rotor (resp. stator) current is then given by 2^∂H_∂φ^m∗

r

(resp. 2^∂H_∂φ^m∗ s

). For the standard model (5), we have

Hm= _2L¹

f(φs−e^n^p^θφr)(φ^∗_s−e^−n^p^θφ^∗_r) +_2L¹

sφsφ^∗_s+_2L¹

rφrφ^∗_r (7) withLf =^L^fs_L^L^fr

m +Lfs+Lfr,Ls=Lfs+^L^fs_L^+L^fr

fr LmandLr=Lfr+^L^fs_L^+L^fr

fs Lm. Such Hamiltonian formulations based on fluxes are also named π-models whereasT- models based on currents correspond to Lagrangian formulations (see, e.g., [5]).

3.2 Magnetic saturation

As in section 2.2, we will take into account magnetic saturation, we assume that in (7), L_s, L_r andL_f are decreasing function of |φ_s|². Then the magnetic saturation model will be given by :

1 Ls

=S_s(|φ_s|²) L¯s

, 1 Lr

=S_r(|φ_s|²) L¯r

, 1 Lf

= S_f(|φ_s|²) L¯f

where the saturation functions Ss, Sr and Sf are increasing function of |φs|² with Ss(0) =Sr(0) =Sf(0) = 1 and where ¯Ls , ¯Lr and ¯Lf are the unsaturated values ofL_s,L_r andL_f. The saturated Hamiltonian is then

Hm=^S^f^(|φ_{2 ¯}_L^s^|²⁾

f |φs−e^n^p^θφr|²+^S^s^(|φ_{2 ¯}_L^s^|²⁾

s |φs|²+^S^r^(|φ_{2 ¯}_L^s^|²⁾

r |φr|²

With the dynamic equations then given by (6). This saturation model is the Hamiltonian counter-part of the saturation model proposed in [5].

3.3 Sinusoidal models

The HamiltonianHm here above admits the following rotational invariance associated to a sinusoidal bemf:

∀φs∈C, ∀θ, ξ∈S¹,

H_m(e^n^p^ξφ_s, φ_r, e^−n^p^ξφ^∗_s, φ^∗_r, ξ+θ) =H_m(φ_s, φ_r, φ^∗_s, φ^∗_r, θ).

Then with H(ψs, ψr, ψ^∗_s, ψ^∗_r) =Hm(ψs, ψr, ψ^∗_s, ψ_r^∗,0), Hm admits the following form

Hm(φs, φr, φ^∗_s, φ^∗_r, θ)≡ H(e^−n^p^θφs, φr, e^n^p^θφ^∗_s, φ^∗_r).

In this case

τ_em=−∂H_m

∂θ =−np

∂H

∂ψ_s^∗φ^∗_se^n^p^θ− ∂H

∂ψs

φ_se^−n^p^θ

Since ı_s = 2^∂H_∂φ^m∗ s

= 2e^n^p^{θ ∂H}_∂ψ∗ s

and ı^∗_s = 2^∂H_∂φ^m

s = 2e^−n^p^{θ ∂H}_∂ψ

s we recover the usual formula relating the electro-magnetic torque to stator flux and current:

τem=−np

φ^∗_sıs−φsı^∗_s

2 =np=(φ^∗_sıs).

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4 Concluding remarks

It remains also to validate experimentally such magnetic-saturation models. Sub- stantial modifications to such Hamiltonian formulation are needed to include, in parallel to magnetic-saturation, magnetic hysteresis and the associated energy losses [3].

References

1. D. Basic, F. Malrait, and P. Rouchon. Euler-Lagrange models with complex currents of three-phase electrical machines and observability issues. IEEE Trans. Automatic Control, 55(1):212–217, 2010.

2. J. Chiasson. Modeling and High Performance Control of Electric Machines. IEEE Press Series on Power Engineering. Wiley-IEEE Press, 2005.

3. E. Della Torre. Magnetic Hysteresis. IEEE Press, 1999.

4. W. Leonhard. Control of Electrical Drives. Elsevier, 1985.

5. C.R. Sullivan and S.R. Sanders. Models for induction machines with magnetic saturation of the main flux path.IEEE Trans. on Industry Applications, 31(4):907–

914, 1995.